Subgroups — C₇×#25199 #13202415

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3069	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3069	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3019	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#455	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2069	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2050	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2049	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#2004	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1967	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1932	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1914	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1902	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1861	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₇×#575	C₂²	1	1	yes	characteristic
18	C₇×#575	C₂²	1	1	yes	characteristic
19	C₂×C₇×#137	C₂²	1	1	yes	characteristic
20	C₂×C₇×#137	C₂²	1	1	yes	characteristic
21	C₂×C₇×#137	C₂²	1	1	yes	characteristic, complemented
22	C₇×#568	C₂²	1	1	yes	characteristic
23	C₇×#568	C₂²	1	1	yes	characteristic
24	C₇×#568	C₂²	1	1	yes	characteristic, complemented
25	C₇×#535	C₂²	1	1	yes	characteristic
26	C₂×C₇×#124	C₂²	1	1	yes	characteristic
27	C₇×#473	—	2	2	no	—
28	C₇×#473	—	2	2	no	—
29	C₇×#471	—	2	2	no	—
30	C₇×#471	—	2	2	no	—
31	C₇×#455	—	2	2	no	—
32	C₇×#455	—	2	2	no	—
33	C₇×#455	—	2	2	no	—
34	C₇×#453	—	2	2	no	—
35	C₇×#444	—	2	2	no	—
36	C₇×#443	—	2	2	no	—
37	C₇×#430	C₂²	1	1	yes	characteristic, complemented
38	C₇×#430	C₂²	1	1	yes	characteristic, complemented
39	C₇×#430	C₂²	1	1	yes	characteristic, complemented
40	C₇×#423	C₂²	1	1	yes	characteristic, complemented
41	C₇×#421	C₂²	1	1	yes	characteristic
42	C₇×#421	C₂²	1	1	yes	characteristic
43	C₂×C₇×#104	C₂²	1	1	yes	characteristic
44	C₂×C₇×#104	C₂²	1	1	yes	characteristic
45	C₂×C₇×#104	C₂²	1	1	yes	characteristic, complemented
46	C₇×#419	C₂²	1	1	yes	characteristic, complemented
47	C₇×#413	—	2	2	no	—
48	C₇×#413	C₂²	1	1	yes	characteristic
49	C₇×#413	C₂²	1	1	yes	characteristic
50	C₇×#413	C₂²	1	1	yes	characteristic, complemented
51	C₂×C₇×#101	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₇×#101	C₂²	1	1	yes	characteristic, complemented
53	C₂×C₇×#100	—	2	2	no	—
54	C₂×C₇×#100	C₂²	1	1	yes	characteristic
55	C₂×C₇×#100	C₂²	1	1	yes	characteristic
56	C₂×C₇×#100	C₂²	1	1	yes	characteristic
57	C₂×C₇×#100	C₂²	1	1	yes	characteristic
58	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
59	C₇×#344	C₂²	1	1	yes	characteristic, complemented
60	C₇×#344	C₂²	1	1	yes	characteristic, complemented
61	C₇×#339	C₂²	1	1	yes	characteristic, complemented
62	C₇×#338	C₂²	1	1	yes	characteristic, complemented
63	#25199	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₇×H(5,𝔽₂)	—	2	2	no	—
65	C₇×H(5,𝔽₂)	—	2	2	no	—
66	C₇×H(5,𝔽₂)	—	4	1	no	—
67	C₇×H(5,𝔽₂)	—	4	1	no	—
68	C₂×C₇×#41	—	2	1	no	—
69	C₂×C₇×#41	—	2	1	no	—
70	C₂×C₇×#41	C₂³	1	1	yes	characteristic
71	C₂×C₇×#41	C₂³	1	1	yes	characteristic
72	C₂×C₇×#41	C₂³	1	1	yes	characteristic
73	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
74	C₂²×C₇×D₄	—	2	2	no	—
75	C₂²×C₇×D₄	—	2	1	no	—
76	C₂²×C₇×D₄	—	2	1	no	—
77	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
78	C₇×#137	—	2	1	no	—
79	C₇×#137	—	2	2	no	—
80	C₇×#137	—	2	2	no	—
81	C₇×#137	—	2	1	no	—
82	C₇×#137	—	2	1	no	—
83	C₇×#137	—	2	2	no	—
84	C₇×#137	—	2	2	no	—
85	C₇×#137	—	4	1	no	—
86	C₇×#137	—	4	1	no	—
87	C₇×#137	D₄	1	2	yes	—
88	C₇×AGL(1,ℤ/8)	—	2	1	no	—
89	C₇×AGL(1,ℤ/8)	—	2	1	no	—
90	C₇×AGL(1,ℤ/8)	—	2	1	no	—
91	C₇×AGL(1,ℤ/8)	D₄	1	2	yes	—
92	C₇×#135	—	2	2	no	—
93	C₇×#135	—	4	1	no	—
94	C₂×C₇×Q₁₆	—	2	1	no	—
95	C₂×C₇×Q₁₆	—	2	1	no	—
96	C₂×C₇×Q₁₆	—	2	1	no	—
97	C₂×C₇×SD₁₆	—	2	1	no	—
98	C₂×C₇×SD₁₆	—	2	1	no	—
99	C₂×C₇×SD₁₆	—	2	1	no	—
100	C₇×#131	—	2	2	no	—
101	C₇×#131	—	2	2	no	—
102	C₇×#131	—	2	2	no	—
103	C₇×#131	—	2	2	no	—
104	C₇×#131	—	4	1	no	—
105	C₇×#131	—	4	1	no	—
106	C₂×C₇×M₁₆	—	2	2	no	—
107	C₂×C₇×M₁₆	—	2	2	no	—
108	C₂×C₇×M₁₆	—	4	1	no	—
109	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
110	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
111	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
112	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
113	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
114	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
115	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
116	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
117	C₇×#127	—	2	1	no	—
118	C₇×#124	—	2	1	no	—
119	C₇×#124	—	2	1	no	—
120	C₇×#124	—	2	1	no	—
121	C₇×#124	—	4	1	no	—
122	C₇×#121	—	4	1	no	—
123	C₇×#120	—	2	2	no	—
124	C₇×#117	C₂³	1	1	yes	characteristic
125	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
127	C₇×#108	—	2	2	no	—
128	C₇×#108	—	2	2	no	—
129	C₇×#108	—	2	2	no	—
130	C₇×C₄⋊C₈	—	4	1	no	—
131	C₇×#104	—	2	2	no	—
132	C₇×#104	—	2	2	no	—
133	C₇×#104	—	2	1	no	—
134	C₇×#104	—	2	1	no	—
135	C₇×#104	—	2	1	no	—
136	C₇×#104	—	2	1	no	—
137	C₇×#104	D₄	1	2	yes	complemented
138	C₇×#104	D₄	1	2	yes	complemented
139	C₇×#103	—	4	1	no	—
140	C₇×#101	—	2	1	no	—
141	C₇×#101	—	2	1	no	—
142	C₇×#101	—	2	2	no	—
143	C₇×#101	—	2	2	no	—
144	C₇×#101	D₄	1	2	yes	complemented
145	C₇×#100	—	2	1	no	—
146	C₇×#100	—	2	1	no	—
147	C₇×#100	—	2	2	no	—
148	C₇×#100	—	2	2	no	—
149	C₇×#100	—	2	2	no	—
150	C₇×#100	—	2	2	no	—
151	C₇×#100	D₄	1	2	yes	complemented
152	C₇×#97	—	2	1	no	—
153	C₄×C₇×C₈	—	2	1	no	—
154	#3069	C₂×C₇	1	1	yes	characteristic, complemented
155	#3069	C₂×C₇	1	1	yes	characteristic, complemented
156	#3019	C₂×C₇	1	1	yes	characteristic
157	C₂×#455	C₂×C₇	1	1	yes	characteristic, complemented
158	#2956	C₂×C₇	1	1	yes	characteristic, complemented
159	#2888	C₂×C₇	1	1	yes	characteristic, complemented
160	#2069	C₂×C₇	1	1	yes	characteristic, complemented
161	#2050	C₂×C₇	1	1	yes	characteristic, complemented
162	#2049	C₂×C₇	1	1	yes	characteristic, complemented
163	#2004	C₂×C₇	1	1	yes	characteristic, complemented
164	#1967	C₂×C₇	1	1	yes	characteristic, complemented
165	#1932	C₂×C₇	1	1	yes	characteristic, complemented
166	#1914	C₂×C₇	1	1	yes	characteristic, complemented
167	#1902	C₂×C₇	1	1	yes	characteristic, complemented
168	#1861	C₂×C₇	1	1	yes	characteristic, complemented
169	C₂⁴×C₇	—	2	2	no	—
170	C₂⁴×C₇	#41	1	2	yes	—
171	C₇×#41	—	2	1	no	—
172	C₇×#41	—	2	1	no	—
173	C₇×#41	—	2	1	no	—
174	C₇×#41	—	2	1	no	—
175	C₇×#41	—	2	1	no	—
176	C₇×#41	—	2	1	no	—
177	C₇×#41	—	2	1	no	—
178	C₇×#41	—	2	1	no	—
179	C₇×#41	—	2	1	no	—
180	C₇×#41	—	2	1	no	—
181	C₇×#41	—	4	2	no	—
182	C₇×#41	—	4	2	no	—
183	C₇×#41	—	4	2	no	—
184	C₇×#41	—	4	2	no	—
185	C₇×#41	C₂×D₄	1	2	yes	—
186	C₇×#41	C₂×D₄	1	2	yes	—
187	C₂×C₇×Q₈	—	2	1	no	—
188	C₂×C₇×Q₈	—	2	1	no	—
189	C₂×C₇×Q₈	—	2	1	no	—
190	C₂×C₇×Q₈	—	2	1	no	—
191	C₂×C₇×Q₈	—	2	1	no	—
192	C₂×C₇×Q₈	—	2	1	no	—
193	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₇×Q₈	#41	1	1	yes	characteristic
196	C₂×C₇×D₄	—	2	1	no	—
197	C₂×C₇×D₄	—	2	2	no	—
198	C₂×C₇×D₄	—	2	2	no	—
199	C₂×C₇×D₄	—	2	1	no	—
200	C₂×C₇×D₄	—	2	1	no	—
201	C₂×C₇×D₄	—	2	1	no	—
202	C₂×C₇×D₄	—	2	1	no	—
203	C₂×C₇×D₄	—	2	1	no	—
204	C₂×C₇×D₄	—	2	1	no	—
205	C₂×C₇×D₄	—	2	1	no	—
206	C₂×C₇×D₄	—	2	1	no	—
207	C₂×C₇×D₄	—	4	2	no	—
208	C₂×C₇×D₄	—	4	2	no	—
209	C₂×C₇×D₄	—	4	4	no	—
210	C₂×C₇×D₄	—	4	4	no	—
211	C₂×C₇×D₄	—	4	2	no	—
212	C₂×C₇×D₄	—	4	2	no	—
213	C₂×C₇×D₄	—	4	2	no	—
214	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
215	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
216	C₂×C₇×D₄	#41	1	1	yes	characteristic
217	C₂²×C₄×C₇	—	2	1	no	—
218	C₂²×C₄×C₇	—	2	2	no	—
219	C₂²×C₄×C₇	—	2	1	no	—
220	C₂²×C₄×C₇	—	2	1	no	—
221	C₂²×C₄×C₇	—	2	1	no	—
222	C₂²×C₄×C₇	—	2	1	no	—
223	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
224	C₇×Q₁₆	—	2	2	no	—
225	C₇×Q₁₆	—	4	2	no	—
226	C₇×Q₁₆	—	4	2	no	—
227	C₇×Q₁₆	—	4	1	no	—
228	C₇×Q₁₆	—	4	2	no	—
229	C₇×SD₁₆	—	2	2	no	—
230	C₇×SD₁₆	—	2	2	no	—
231	C₇×SD₁₆	—	4	2	no	—
232	C₇×SD₁₆	—	4	2	no	—
233	C₇×SD₁₆	—	4	1	no	—
234	C₇×SD₁₆	—	4	1	no	—
235	C₇×SD₁₆	—	4	2	no	—
236	C₇×D₈	—	2	2	no	—
237	C₇×D₈	—	4	1	no	—
238	C₇×M₁₆	—	2	1	no	—
239	C₇×M₁₆	—	2	1	no	—
240	C₇×M₁₆	—	2	1	no	—
241	C₇×M₁₆	—	2	1	no	—
242	C₇×M₁₆	—	2	1	no	—
243	C₇×M₁₆	—	2	1	no	—
244	C₇×M₁₆	—	2	1	no	—
245	C₇×M₁₆	—	2	1	no	—
246	C₇×M₁₆	—	2	1	no	—
247	C₇×M₁₆	—	2	1	no	—
248	C₇×M₁₆	—	2	1	no	—
249	C₇×M₁₆	—	2	1	no	—
250	C₇×M₁₆	—	4	2	no	—
251	C₇×M₁₆	—	4	2	no	—
252	C₇×M₁₆	—	4	2	no	—
253	C₇×M₁₆	C₂×D₄	1	2	yes	—
254	C₇×M₁₆	C₂×D₄	1	2	yes	—
255	C₇×M₁₆	C₂×D₄	1	2	yes	—
256	C₇×M₁₆	C₂×D₄	1	2	yes	—
257	C₂×C₇×C₈	—	2	1	no	—
258	C₂×C₇×C₈	—	2	1	no	—
259	C₂×C₇×C₈	—	2	1	no	—
260	C₂×C₇×C₈	—	2	1	no	—
261	C₂×C₇×C₈	—	2	1	no	—
262	C₂×C₇×C₈	—	2	1	no	—
263	C₂×C₇×C₈	—	2	1	no	—
264	C₂×C₇×C₈	—	2	1	no	—
265	C₂×C₇×C₈	—	4	2	no	—
266	C₂×C₇×C₈	—	4	2	no	—
267	C₂×C₇×C₈	—	4	2	no	—
268	C₇×C₄⋊C₄	—	2	1	no	—
269	C₇×C₂²⋊C₄	—	2	1	no	—
270	C₇×C₂²⋊C₄	—	4	2	no	—
271	C₇×C₂²⋊C₄	—	4	2	no	—
272	C₇×C₂²⋊C₄	—	4	2	no	—
273	C₄²×C₇	—	2	1	no	—
274	C₄²×C₇	—	2	1	no	—
275	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
276	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
277	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
278	#575	C₂²×C₇	1	1	yes	characteristic
279	#575	C₂²×C₇	1	1	yes	characteristic
280	C₂×#137	C₂²×C₇	1	1	yes	characteristic
281	C₂×#137	C₂²×C₇	1	1	yes	characteristic
282	C₂×#137	C₂²×C₇	1	1	yes	characteristic, complemented
283	#568	C₂²×C₇	1	1	yes	characteristic
284	#568	C₂²×C₇	1	1	yes	characteristic
285	#568	C₂²×C₇	1	1	yes	characteristic, complemented
286	#535	C₂²×C₇	1	1	yes	characteristic
287	C₂×#124	C₂²×C₇	1	1	yes	characteristic
288	#473	—	2	2	no	—
289	#473	—	2	2	no	—
290	#471	—	2	2	no	—
291	#471	—	2	2	no	—
292	#455	—	2	2	no	—
293	#455	—	2	2	no	—
294	#455	—	2	2	no	—
295	#453	—	2	2	no	—
296	#444	—	2	2	no	—
297	#443	—	2	2	no	—
298	#430	C₂²×C₇	1	1	yes	characteristic, complemented
299	#430	C₂²×C₇	1	1	yes	characteristic, complemented
300	#430	C₂²×C₇	1	1	yes	characteristic, complemented
301	#423	C₂²×C₇	1	1	yes	characteristic, complemented
302	#421	C₂²×C₇	1	1	yes	characteristic
303	#421	C₂²×C₇	1	1	yes	characteristic
304	C₂×#104	C₂²×C₇	1	1	yes	characteristic
305	C₂×#104	C₂²×C₇	1	1	yes	characteristic
306	C₂×#104	C₂²×C₇	1	1	yes	characteristic, complemented
307	#419	C₂²×C₇	1	1	yes	characteristic, complemented
308	#413	—	2	2	no	—
309	#413	C₂²×C₇	1	1	yes	characteristic
310	#413	C₂²×C₇	1	1	yes	characteristic
311	#413	C₂²×C₇	1	1	yes	characteristic, complemented
312	C₂×#101	C₂²×C₇	1	1	yes	characteristic, complemented
313	C₂×#101	C₂²×C₇	1	1	yes	characteristic, complemented
314	C₂×#100	—	2	2	no	—
315	C₂×#100	C₂²×C₇	1	1	yes	characteristic
316	C₂×#100	C₂²×C₇	1	1	yes	characteristic
317	C₂×#100	C₂²×C₇	1	1	yes	characteristic
318	C₂×#100	C₂²×C₇	1	1	yes	characteristic
319	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
320	#344	C₂²×C₇	1	1	yes	characteristic, complemented
321	#344	C₂²×C₇	1	1	yes	characteristic, complemented
322	#339	C₂²×C₇	1	1	yes	characteristic, complemented
323	#338	C₂²×C₇	1	1	yes	characteristic, complemented
324	C₂³×C₇	—	2	1	no	—
325	C₂³×C₇	—	2	2	no	—
326	C₂³×C₇	—	2	1	no	—
327	C₂³×C₇	—	2	2	no	—
328	C₂³×C₇	—	2	2	no	—
329	C₂³×C₇	—	2	1	no	—
330	C₂³×C₇	—	4	4	no	—
331	C₂³×C₇	—	8	2	no	—
332	C₂³×C₇	—	16	2	no	—
333	C₂³×C₇	C₂×#41	1	1	yes	characteristic
334	C₇×Q₈	—	2	1	no	—
335	C₇×Q₈	—	2	1	no	—
336	C₇×Q₈	—	2	1	no	—
337	C₇×Q₈	—	2	1	no	—
338	C₇×Q₈	—	2	1	no	—
339	C₇×Q₈	—	4	1	no	—
340	C₇×Q₈	—	4	1	no	—
341	C₇×Q₈	—	4	1	no	—
342	C₇×Q₈	—	4	1	no	—
343	C₇×Q₈	#121	1	2	yes	—
344	C₇×D₄	—	2	2	no	—
345	C₇×D₄	—	2	2	no	—
346	C₇×D₄	—	2	1	no	—
347	C₇×D₄	—	2	1	no	—
348	C₇×D₄	—	2	1	no	—
349	C₇×D₄	—	2	1	no	—
350	C₇×D₄	—	2	1	no	—
351	C₇×D₄	—	4	4	no	—
352	C₇×D₄	—	4	4	no	—
353	C₇×D₄	—	4	1	no	—
354	C₇×D₄	—	4	1	no	—
355	C₇×D₄	—	4	1	no	—
356	C₇×D₄	—	4	1	no	—
357	C₇×D₄	—	4	1	no	—
358	C₇×D₄	—	4	1	no	—
359	C₇×D₄	—	4	1	no	—
360	C₇×D₄	—	4	1	no	—
361	C₇×D₄	—	16	2	no	—
362	C₇×D₄	#121	1	2	yes	—
363	C₂×C₄×C₇	—	2	1	no	—
364	C₂×C₄×C₇	—	2	2	no	—
365	C₂×C₄×C₇	—	2	2	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	2	1	no	—
369	C₂×C₄×C₇	—	2	1	no	—
370	C₂×C₄×C₇	—	4	4	no	—
371	C₂×C₄×C₇	—	4	1	no	—
372	C₂×C₄×C₇	—	4	1	no	—
373	C₂×C₄×C₇	—	4	1	no	—
374	C₂×C₄×C₇	—	4	1	no	—
375	C₂×C₄×C₇	—	4	1	no	—
376	C₂×C₄×C₇	—	4	1	no	—
377	C₂×C₄×C₇	—	4	1	no	—
378	C₂×C₄×C₇	—	4	1	no	—
379	C₂×C₄×C₇	—	8	1	no	—
380	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
381	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
382	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
383	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
384	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
385	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
386	C₇×C₈	—	2	2	no	—
387	C₇×C₈	—	2	2	no	—
388	C₇×C₈	—	2	2	no	—
389	C₇×C₈	—	2	2	no	—
390	C₇×C₈	—	4	1	no	—
391	C₇×C₈	—	4	1	no	—
392	C₇×C₈	—	4	1	no	—
393	C₇×C₈	—	4	1	no	—
394	H(5,𝔽₂)	—	2	2	no	—
395	H(5,𝔽₂)	—	2	2	no	—
396	H(5,𝔽₂)	—	4	1	no	—
397	H(5,𝔽₂)	—	4	1	no	—
398	C₂×#41	—	2	1	no	—
399	C₂×#41	—	2	1	no	—
400	C₂×#41	C₂³×C₇	1	1	yes	characteristic
401	C₂×#41	C₂³×C₇	1	1	yes	characteristic
402	C₂×#41	C₂³×C₇	1	1	yes	characteristic
403	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
404	C₂²×D₄	—	2	2	no	—
405	C₂²×D₄	—	2	1	no	—
406	C₂²×D₄	—	2	1	no	—
407	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
408	#137	—	2	1	no	—
409	#137	—	2	2	no	—
410	#137	—	2	2	no	—
411	#137	—	2	1	no	—
412	#137	—	2	1	no	—
413	#137	—	2	2	no	—
414	#137	—	2	2	no	—
415	#137	—	4	1	no	—
416	#137	—	4	1	no	—
417	#137	C₇×D₄	1	2	yes	—
418	AGL(1,ℤ/8)	—	2	1	no	—
419	AGL(1,ℤ/8)	—	2	1	no	—
420	AGL(1,ℤ/8)	—	2	1	no	—
421	AGL(1,ℤ/8)	C₇×D₄	1	2	yes	—
422	#135	—	2	2	no	—
423	#135	—	4	1	no	—
424	C₂×Q₁₆	—	2	1	no	—
425	C₂×Q₁₆	—	2	1	no	—
426	C₂×Q₁₆	—	2	1	no	—
427	C₂×SD₁₆	—	2	1	no	—
428	C₂×SD₁₆	—	2	1	no	—
429	C₂×SD₁₆	—	2	1	no	—
430	#131	—	2	2	no	—
431	#131	—	2	2	no	—
432	#131	—	2	2	no	—
433	#131	—	2	2	no	—
434	#131	—	4	1	no	—
435	#131	—	4	1	no	—
436	C₂×M₁₆	—	2	2	no	—
437	C₂×M₁₆	—	2	2	no	—
438	C₂×M₁₆	—	4	1	no	—
439	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
441	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
442	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
443	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
444	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
445	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
446	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
447	#127	—	2	1	no	—
448	#124	—	2	1	no	—
449	#124	—	2	1	no	—
450	#124	—	2	1	no	—
451	#124	—	4	1	no	—
452	#121	—	4	1	no	—
453	#120	—	2	2	no	—
454	#117	C₂³×C₇	1	1	yes	characteristic
455	C₂×C₂²⋊C₄	—	2	2	no	—
456	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
457	#108	—	2	2	no	—
458	#108	—	2	2	no	—
459	#108	—	2	2	no	—
460	C₄⋊C₈	—	4	1	no	—
461	#104	—	2	2	no	—
462	#104	—	2	2	no	—
463	#104	—	2	1	no	—
464	#104	—	2	1	no	—
465	#104	—	2	1	no	—
466	#104	—	2	1	no	—
467	#104	C₇×D₄	1	2	yes	complemented
468	#104	C₇×D₄	1	2	yes	complemented
469	#103	—	4	1	no	—
470	#101	—	2	1	no	—
471	#101	—	2	1	no	—
472	#101	—	2	2	no	—
473	#101	—	2	2	no	—
474	#101	C₇×D₄	1	2	yes	complemented
475	#100	—	2	1	no	—
476	#100	—	2	1	no	—
477	#100	—	2	2	no	—
478	#100	—	2	2	no	—
479	#100	—	2	2	no	—
480	#100	—	2	2	no	—
481	#100	C₇×D₄	1	2	yes	complemented
482	#97	—	2	1	no	—
483	C₄×C₈	—	2	1	no	—
484	C₂²×C₇	—	2	2	no	—
485	C₂²×C₇	—	4	1	no	—
486	C₂²×C₇	—	4	1	no	—
487	C₂²×C₇	—	4	1	no	—
488	C₂²×C₇	—	4	2	no	—
489	C₂²×C₇	—	8	1	no	—
490	C₂²×C₇	—	8	1	no	—
491	C₂²×C₇	—	8	2	no	—
492	C₂²×C₇	—	8	2	no	—
493	C₂²×C₇	—	8	2	no	—
494	C₂²×C₇	—	8	1	no	—
495	C₂²×C₇	—	16	4	no	—
496	C₂²×C₇	#525	1	1	yes	characteristic
497	C₂²×C₇	#546	1	1	yes	characteristic
498	C₂²×C₇	#546	1	1	yes	characteristic
499	C₄×C₇	—	2	2	no	—
500	C₄×C₇	—	4	1	no	—
501	C₄×C₇	—	4	1	no	—
502	C₄×C₇	—	4	1	no	—
503	C₄×C₇	—	4	1	no	—
504	C₄×C₇	—	8	1	no	—
505	C₄×C₇	—	8	1	no	—
506	C₄×C₇	C₂×#121	1	1	yes	characteristic
507	C₄×C₇	#538	1	1	yes	characteristic
508	C₄×C₇	D₄²	1	1	yes	characteristic
509	C₄×C₇	D₄²	1	1	yes	characteristic
510	C₂⁴	—	2	2	no	—
511	C₂⁴	C₇×#41	1	2	yes	—
512	#41	—	2	1	no	—
513	#41	—	2	1	no	—
514	#41	—	2	1	no	—
515	#41	—	2	1	no	—
516	#41	—	2	1	no	—
517	#41	—	2	1	no	—
518	#41	—	2	1	no	—
519	#41	—	2	1	no	—
520	#41	—	2	1	no	—
521	#41	—	2	1	no	—
522	#41	—	4	2	no	—
523	#41	—	4	2	no	—
524	#41	—	4	2	no	—
525	#41	—	4	2	no	—
526	#41	C₂×C₇×D₄	1	2	yes	—
527	#41	C₂×C₇×D₄	1	2	yes	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
535	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
536	C₂×Q₈	C₇×#41	1	1	yes	characteristic
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	—	2	1	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	—	2	1	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	1	no	—
548	C₂×D₄	—	4	2	no	—
549	C₂×D₄	—	4	2	no	—
550	C₂×D₄	—	4	4	no	—
551	C₂×D₄	—	4	4	no	—
552	C₂×D₄	—	4	2	no	—
553	C₂×D₄	—	4	2	no	—
554	C₂×D₄	—	4	2	no	—
555	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
556	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
557	C₂×D₄	C₇×#41	1	1	yes	characteristic
558	C₂²×C₄	—	2	1	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
565	Q₁₆	—	2	2	no	—
566	Q₁₆	—	4	2	no	—
567	Q₁₆	—	4	2	no	—
568	Q₁₆	—	4	1	no	—
569	Q₁₆	—	4	2	no	—
570	SD₁₆	—	2	2	no	—
571	SD₁₆	—	2	2	no	—
572	SD₁₆	—	4	2	no	—
573	SD₁₆	—	4	2	no	—
574	SD₁₆	—	4	1	no	—
575	SD₁₆	—	4	1	no	—
576	SD₁₆	—	4	2	no	—
577	D₈	—	2	2	no	—
578	D₈	—	4	1	no	—
579	M₁₆	—	2	1	no	—
580	M₁₆	—	2	1	no	—
581	M₁₆	—	2	1	no	—
582	M₁₆	—	2	1	no	—
583	M₁₆	—	2	1	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	—	2	1	no	—
586	M₁₆	—	2	1	no	—
587	M₁₆	—	2	1	no	—
588	M₁₆	—	2	1	no	—
589	M₁₆	—	2	1	no	—
590	M₁₆	—	2	1	no	—
591	M₁₆	—	4	2	no	—
592	M₁₆	—	4	2	no	—
593	M₁₆	—	4	2	no	—
594	M₁₆	C₂×C₇×D₄	1	2	yes	—
595	M₁₆	C₂×C₇×D₄	1	2	yes	—
596	M₁₆	C₂×C₇×D₄	1	2	yes	—
597	M₁₆	C₂×C₇×D₄	1	2	yes	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	2	1	no	—
605	C₂×C₈	—	2	1	no	—
606	C₂×C₈	—	4	2	no	—
607	C₂×C₈	—	4	2	no	—
608	C₂×C₈	—	4	2	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₂²⋊C₄	—	2	1	no	—
611	C₂²⋊C₄	—	4	2	no	—
612	C₂²⋊C₄	—	4	2	no	—
613	C₂²⋊C₄	—	4	2	no	—
614	C₄²	—	2	1	no	—
615	C₄²	—	2	1	no	—
616	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
617	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
618	C₂×C₇	—	2	1	no	—
619	C₂×C₇	—	2	1	no	—
620	C₂×C₇	—	2	1	no	—
621	C₂×C₇	—	4	2	no	—
622	C₂×C₇	—	8	1	no	—
623	C₂×C₇	—	8	2	no	—
624	C₂×C₇	—	8	1	no	—
625	C₂×C₇	#2424	1	1	yes	characteristic
626	C₂³	—	2	1	no	—
627	C₂³	—	2	2	no	—
628	C₂³	—	2	1	no	—
629	C₂³	—	2	2	no	—
630	C₂³	—	2	2	no	—
631	C₂³	—	2	1	no	—
632	C₂³	—	4	4	no	—
633	C₂³	—	8	2	no	—
634	C₂³	—	16	2	no	—
635	C₂³	C₂×C₇×#41	1	1	yes	characteristic
636	Q₈	—	2	1	no	—
637	Q₈	—	2	1	no	—
638	Q₈	—	2	1	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	4	1	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	—	4	1	no	—
645	Q₈	C₇×#121	1	2	yes	—
646	D₄	—	2	2	no	—
647	D₄	—	2	2	no	—
648	D₄	—	2	1	no	—
649	D₄	—	2	1	no	—
650	D₄	—	2	1	no	—
651	D₄	—	2	1	no	—
652	D₄	—	2	1	no	—
653	D₄	—	4	4	no	—
654	D₄	—	4	4	no	—
655	D₄	—	4	1	no	—
656	D₄	—	4	1	no	—
657	D₄	—	4	1	no	—
658	D₄	—	4	1	no	—
659	D₄	—	4	1	no	—
660	D₄	—	4	1	no	—
661	D₄	—	4	1	no	—
662	D₄	—	4	1	no	—
663	D₄	—	16	2	no	—
664	D₄	C₇×#121	1	2	yes	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	4	4	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	8	1	no	—
682	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
683	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
684	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
685	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
686	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
687	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
688	C₈	—	2	2	no	—
689	C₈	—	2	2	no	—
690	C₈	—	2	2	no	—
691	C₈	—	2	2	no	—
692	C₈	—	4	1	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₇	#25199	1	1	yes	characteristic, direct product factor, Sylow
697	C₂²	—	2	2	no	—
698	C₂²	—	4	1	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	2	no	—
702	C₂²	—	8	1	no	—
703	C₂²	—	8	1	no	—
704	C₂²	—	8	2	no	—
705	C₂²	—	8	2	no	—
706	C₂²	—	8	2	no	—
707	C₂²	—	8	1	no	—
708	C₂²	—	16	4	no	—
709	C₂²	C₇×#525	1	1	yes	characteristic
710	C₂²	C₇×#546	1	1	yes	characteristic
711	C₂²	C₇×#546	1	1	yes	characteristic
712	C₄	—	2	2	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	1	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	8	1	no	—
718	C₄	—	8	1	no	—
719	C₄	C₂×C₇×#121	1	1	yes	characteristic
720	C₄	C₇×#538	1	1	yes	characteristic
721	C₄	C₇×D₄²	1	1	yes	characteristic
722	C₄	C₇×D₄²	1	1	yes	characteristic
723	C₂	—	2	1	no	—
724	C₂	—	2	1	no	—
725	C₂	—	2	1	no	—
726	C₂	—	4	2	no	—
727	C₂	—	8	2	no	—
728	C₂	—	8	1	no	—
729	C₂	—	8	1	no	—
730	C₂	C₇×#2424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₇×#3019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3069	C₂	semidirect
2. C₇×#3069	C₂	semidirect
4. C₂×C₇×#455	C₂	semidirect
5. C₇×#2956	C₂	semidirect
6. C₇×#2888	C₂	semidirect
7. C₇×#2069	C₂	semidirect
8. C₇×#2050	C₂	semidirect
9. C₇×#2049	C₂	semidirect
10. C₇×#2004	C₂	semidirect
11. C₇×#1967	C₂	semidirect
12. C₇×#1932	C₂	semidirect
13. C₇×#1914	C₂	semidirect
14. C₇×#1902	C₂	semidirect
15. C₇×#1861	C₂	semidirect
21. C₂×C₇×#137	C₂²	semidirect
24. C₇×#568	C₂²	semidirect
37. C₇×#430	C₂²	semidirect
38. C₇×#430	C₂²	semidirect
39. C₇×#430	C₂²	semidirect
40. C₇×#423	C₂²	semidirect
45. C₂×C₇×#104	C₂²	semidirect
46. C₇×#419	C₂²	semidirect
50. C₇×#413	C₂²	semidirect
51. C₂×C₇×#101	C₂²	semidirect
52. C₂×C₇×#101	C₂²	semidirect
58. C₄×C₇×M₁₆	C₂²	semidirect
59. C₇×#344	C₂²	semidirect
60. C₇×#344	C₂²	semidirect
61. C₇×#339	C₂²	semidirect
62. C₇×#338	C₂²	semidirect
63. #25199	C₇	direct (N × H)
137. C₇×#104	D₄	semidirect
138. C₇×#104	D₄	semidirect
144. C₇×#101	D₄	semidirect
151. C₇×#100	D₄	semidirect
154. #3069	C₂×C₇	semidirect
155. #3069	C₂×C₇	semidirect
157. C₂×#455	C₂×C₇	semidirect
158. #2956	C₂×C₇	semidirect
159. #2888	C₂×C₇	semidirect
160. #2069	C₂×C₇	semidirect
161. #2050	C₂×C₇	semidirect
162. #2049	C₂×C₇	semidirect
163. #2004	C₂×C₇	semidirect
164. #1967	C₂×C₇	semidirect
165. #1932	C₂×C₇	semidirect
166. #1914	C₂×C₇	semidirect
167. #1902	C₂×C₇	semidirect
168. #1861	C₂×C₇	semidirect
282. C₂×#137	C₂²×C₇	semidirect
285. #568	C₂²×C₇	semidirect
298. #430	C₂²×C₇	semidirect
299. #430	C₂²×C₇	semidirect
300. #430	C₂²×C₇	semidirect
301. #423	C₂²×C₇	semidirect
306. C₂×#104	C₂²×C₇	semidirect
307. #419	C₂²×C₇	semidirect
311. #413	C₂²×C₇	semidirect
312. C₂×#101	C₂²×C₇	semidirect
313. C₂×#101	C₂²×C₇	semidirect
319. C₄×M₁₆	C₂²×C₇	semidirect
320. #344	C₂²×C₇	semidirect
321. #344	C₂²×C₇	semidirect
322. #339	C₂²×C₇	semidirect
323. #338	C₂²×C₇	semidirect
467. #104	C₇×D₄	semidirect
468. #104	C₇×D₄	semidirect
474. #101	C₇×D₄	semidirect
481. #100	C₇×D₄	semidirect
696. C₇	#25199	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3069	2
C₇×#3019	2
C₂×C₇×#455	2
C₇×#2956	2
C₇×#2888	2
C₇×#2069	2
C₇×#2050	2
C₇×#2049	2
C₇×#2004	2
C₇×#1967	2
C₇×#1932	2
C₇×#1914	2
C₇×#1902	2
C₇×#1861	2
#25199	7

Subgroups — C7×#25199 #13202415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#25199 #13202415