Subgroups — C₇×#24808 #13202024

Order 1792.

820 conjugacy classes of subgroups.

The subgroup lattice has 658 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×#3068	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#471	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#471	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2955	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2069	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2049	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#2048	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1953	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1903	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1902	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1813	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#575	C₂²	1	1	yes	characteristic
17	C₂×C₇×#137	C₂²	1	1	yes	characteristic
18	C₂×C₇×#137	C₂²	1	1	yes	characteristic
19	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
20	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
21	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
22	C₇×#568	C₂²	1	1	yes	characteristic
23	C₇×#568	C₂²	1	1	yes	characteristic
24	C₂×C₇×#131	C₂²	1	1	yes	characteristic
25	C₂×C₇×#131	C₂²	1	1	yes	characteristic
26	C₇×#473	—	2	2	no	—
27	C₇×#472	—	2	2	no	—
28	C₇×#471	—	2	2	no	—
29	C₇×#471	—	2	2	no	—
30	C₇×#471	—	2	2	no	—
31	C₇×#471	—	2	2	no	—
32	C₇×#471	—	2	2	no	—
33	C₇×#471	—	2	2	no	—
34	C₇×#444	—	2	2	no	—
35	C₇×#443	—	2	2	no	—
36	C₇×#430	—	2	2	no	—
37	C₇×#430	C₂²	1	1	yes	characteristic
38	C₇×#430	C₂²	1	1	yes	characteristic
39	C₇×#430	C₂²	1	1	yes	characteristic
40	C₇×#430	C₂²	1	1	yes	characteristic
41	C₇×#430	C₂²	1	1	yes	characteristic
42	C₂×C₇×#108	—	2	2	no	—
43	C₂×C₇×#108	C₂²	1	1	yes	characteristic
44	C₂×C₇×#108	C₂²	1	1	yes	characteristic
45	C₂×C₇×#108	C₂²	1	1	yes	characteristic
46	C₇×#421	C₂²	1	1	yes	characteristic
47	C₇×#421	C₂²	1	1	yes	characteristic
48	C₇×#419	C₂²	1	1	yes	characteristic
49	C₇×#418	C₂²	1	1	yes	characteristic
50	C₇×#413	C₂²	1	1	yes	characteristic
51	C₇×#413	C₂²	1	1	yes	characteristic
52	C₂×C₇×#101	C₂²	1	1	yes	characteristic
53	C₂×C₇×#101	C₂²	1	1	yes	characteristic
54	C₂×C₇×#101	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	C₇×#405	C₂²	1	1	yes	characteristic
59	C₇×#344	C₂²	1	1	yes	characteristic
60	C₇×#344	C₂²	1	1	yes	characteristic
61	C₇×#338	C₂²	1	1	yes	characteristic
62	C₇×#338	C₂²	1	1	yes	characteristic
63	#24808	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₇×#41	C₂³	1	1	yes	characteristic
65	C₂×C₇×#41	C₂³	1	1	yes	characteristic
66	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
67	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
68	C₇×#137	—	2	1	no	—
69	C₇×#137	—	2	2	no	—
70	C₇×#137	—	2	2	no	—
71	C₇×#137	—	2	2	no	—
72	C₇×#137	—	4	1	no	—
73	C₇×#137	D₄	1	2	yes	—
74	C₇×AGL(1,ℤ/8)	—	2	1	no	—
75	C₇×AGL(1,ℤ/8)	—	2	2	no	—
76	C₇×AGL(1,ℤ/8)	—	2	2	no	—
77	C₇×AGL(1,ℤ/8)	—	2	2	no	—
78	C₇×AGL(1,ℤ/8)	—	4	1	no	—
79	C₇×AGL(1,ℤ/8)	D₄	1	2	yes	—
80	C₇×#135	—	2	2	no	—
81	C₂×C₇×Q₁₆	—	2	1	no	—
82	C₂×C₇×Q₁₆	—	2	1	no	—
83	C₂×C₇×SD₁₆	—	2	1	no	—
84	C₂×C₇×SD₁₆	—	2	1	no	—
85	C₂×C₇×SD₁₆	—	2	1	no	—
86	C₂×C₇×SD₁₆	—	2	1	no	—
87	C₂×C₇×SD₁₆	—	2	1	no	—
88	C₂×C₇×SD₁₆	—	2	1	no	—
89	C₂×C₇×SD₁₆	—	2	1	no	—
90	C₂×C₇×SD₁₆	—	2	1	no	—
91	C₂×C₇×SD₁₆	—	2	1	no	—
92	C₂×C₇×SD₁₆	—	2	1	no	—
93	C₂×C₇×D₈	—	2	1	no	—
94	C₂×C₇×D₈	—	2	1	no	—
95	C₇×#131	—	2	1	no	—
96	C₇×#131	—	2	1	no	—
97	C₇×#131	—	2	2	no	—
98	C₇×#131	—	2	2	no	—
99	C₇×#131	—	2	1	no	—
100	C₇×#131	—	2	1	no	—
101	C₇×#131	—	2	1	no	—
102	C₇×#131	—	2	1	no	—
103	C₇×#131	—	4	1	no	—
104	C₇×#131	—	4	1	no	—
105	C₇×#131	D₄	1	2	yes	—
106	C₇×#131	D₄	1	2	yes	—
107	C₂×C₇×M₁₆	—	2	1	no	—
108	C₂×C₇×M₁₆	—	2	1	no	—
109	C₂×C₇×M₁₆	—	2	2	no	—
110	C₂×C₇×M₁₆	—	2	2	no	—
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	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
118	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
119	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
120	C₂²×C₇×C₈	—	2	1	no	—
121	C₂²×C₇×C₈	—	2	1	no	—
122	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
123	C₇×#117	C₂³	1	1	yes	characteristic
124	C₇×#108	—	2	2	no	—
125	C₇×#108	—	2	2	no	—
126	C₇×#108	—	2	2	no	—
127	C₇×#108	—	2	2	no	—
128	C₇×#108	—	2	2	no	—
129	C₇×#108	—	2	1	no	—
130	C₇×#108	—	2	1	no	—
131	C₇×#108	—	2	1	no	—
132	C₇×#108	—	2	1	no	—
133	C₇×#108	D₄	1	2	yes	—
134	C₇×#108	D₄	1	2	yes	—
135	C₇×#104	—	2	2	no	—
136	C₇×#104	—	2	2	no	—
137	C₇×#103	—	4	1	no	—
138	C₇×#102	—	4	1	no	—
139	C₇×#101	—	2	1	no	—
140	C₇×#101	—	2	1	no	—
141	C₇×#101	—	2	2	no	—
142	C₇×#101	—	2	2	no	—
143	C₇×#101	—	2	2	no	—
144	C₇×#100	—	2	1	no	—
145	C₇×#100	—	2	1	no	—
146	C₇×#100	—	2	2	no	—
147	C₇×#100	—	2	2	no	—
148	C₇×#100	—	2	2	no	—
149	C₇×#97	—	2	1	no	—
150	C₄×C₇×C₈	—	2	1	no	—
151	#3069	C₂×C₇	1	1	yes	characteristic, complemented
152	#3068	C₂×C₇	1	1	yes	characteristic
153	C₂×#471	C₂×C₇	1	1	yes	characteristic, complemented
154	C₂×#471	C₂×C₇	1	1	yes	characteristic, complemented
155	#2955	C₂×C₇	1	1	yes	characteristic, complemented
156	#2916	C₂×C₇	1	1	yes	characteristic, complemented
157	#2070	C₂×C₇	1	1	yes	characteristic, complemented
158	#2069	C₂×C₇	1	1	yes	characteristic, complemented
159	#2049	C₂×C₇	1	1	yes	characteristic, complemented
160	#2048	C₂×C₇	1	1	yes	characteristic, complemented
161	#1977	C₂×C₇	1	1	yes	characteristic, complemented
162	#1953	C₂×C₇	1	1	yes	characteristic, complemented
163	#1903	C₂×C₇	1	1	yes	characteristic, complemented
164	#1902	C₂×C₇	1	1	yes	characteristic, complemented
165	#1813	C₂×C₇	1	1	yes	characteristic, complemented
166	C₂⁴×C₇	—	2	1	no	—
167	C₇×#41	—	2	1	no	—
168	C₇×#41	—	2	1	no	—
169	C₇×#41	—	2	1	no	—
170	C₇×#41	—	2	1	no	—
171	C₇×#41	—	2	1	no	—
172	C₇×#41	—	2	1	no	—
173	C₇×#41	C₂×D₄	1	2	yes	—
174	C₇×#41	C₂×D₄	1	2	yes	—
175	C₂×C₇×Q₈	—	2	1	no	—
176	C₂×C₇×Q₈	—	2	1	no	—
177	C₂×C₇×Q₈	—	2	1	no	—
178	C₂×C₇×Q₈	—	2	1	no	—
179	C₂×C₇×Q₈	—	2	1	no	—
180	C₂×C₇×Q₈	—	2	1	no	—
181	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
182	C₂×C₇×Q₈	#41	1	1	yes	characteristic
183	C₂×C₇×D₄	—	2	1	no	—
184	C₂×C₇×D₄	—	2	1	no	—
185	C₂×C₇×D₄	—	2	1	no	—
186	C₂×C₇×D₄	—	2	1	no	—
187	C₂×C₇×D₄	—	2	1	no	—
188	C₂×C₇×D₄	—	2	1	no	—
189	C₂×C₇×D₄	—	2	1	no	—
190	C₂×C₇×D₄	—	2	1	no	—
191	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
192	C₂×C₇×D₄	#41	1	1	yes	characteristic
193	C₂²×C₄×C₇	—	2	1	no	—
194	C₂²×C₄×C₇	—	2	1	no	—
195	C₂²×C₄×C₇	—	2	1	no	—
196	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
197	C₇×Q₁₆	—	2	2	no	—
198	C₇×Q₁₆	—	4	2	no	—
199	C₇×Q₁₆	—	4	2	no	—
200	C₇×SD₁₆	—	2	2	no	—
201	C₇×SD₁₆	—	2	2	no	—
202	C₇×SD₁₆	—	4	2	no	—
203	C₇×SD₁₆	—	4	2	no	—
204	C₇×SD₁₆	—	4	2	no	—
205	C₇×SD₁₆	—	4	2	no	—
206	C₇×SD₁₆	—	4	1	no	—
207	C₇×SD₁₆	—	4	1	no	—
208	C₇×SD₁₆	—	4	1	no	—
209	C₇×SD₁₆	—	4	1	no	—
210	C₇×D₈	—	2	2	no	—
211	C₇×D₈	—	4	2	no	—
212	C₇×D₈	—	4	2	no	—
213	C₇×M₁₆	—	2	1	no	—
214	C₇×M₁₆	—	2	1	no	—
215	C₇×M₁₆	—	2	1	no	—
216	C₇×M₁₆	—	2	1	no	—
217	C₇×M₁₆	—	2	2	no	—
218	C₇×M₁₆	—	2	2	no	—
219	C₇×M₁₆	—	2	1	no	—
220	C₇×M₁₆	—	2	1	no	—
221	C₇×M₁₆	—	2	1	no	—
222	C₇×M₁₆	—	2	1	no	—
223	C₇×M₁₆	—	2	1	no	—
224	C₇×M₁₆	—	2	1	no	—
225	C₇×M₁₆	—	2	1	no	—
226	C₇×M₁₆	—	2	1	no	—
227	C₇×M₁₆	—	2	1	no	—
228	C₇×M₁₆	—	2	1	no	—
229	C₇×M₁₆	—	4	2	no	—
230	C₇×M₁₆	—	4	2	no	—
231	C₇×M₁₆	—	4	1	no	—
232	C₇×M₁₆	—	4	1	no	—
233	C₇×M₁₆	C₂×D₄	1	2	yes	—
234	C₇×M₁₆	C₂×D₄	1	2	yes	—
235	C₇×M₁₆	C₂×D₄	1	2	yes	—
236	C₇×M₁₆	C₂×D₄	1	2	yes	—
237	C₂×C₇×C₈	—	2	1	no	—
238	C₂×C₇×C₈	—	2	1	no	—
239	C₂×C₇×C₈	—	2	1	no	—
240	C₂×C₇×C₈	—	2	1	no	—
241	C₂×C₇×C₈	—	2	2	no	—
242	C₂×C₇×C₈	—	2	2	no	—
243	C₂×C₇×C₈	—	2	1	no	—
244	C₂×C₇×C₈	—	2	1	no	—
245	C₂×C₇×C₈	—	2	1	no	—
246	C₂×C₇×C₈	—	2	1	no	—
247	C₂×C₇×C₈	—	2	1	no	—
248	C₂×C₇×C₈	—	4	2	no	—
249	C₂×C₇×C₈	—	4	2	no	—
250	C₂×C₇×C₈	—	4	1	no	—
251	C₂×C₇×C₈	—	4	1	no	—
252	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
253	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
254	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
255	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
256	C₂×C₇×C₈	#41	1	1	yes	characteristic
257	C₂×C₇×C₈	#41	1	1	yes	characteristic
258	C₇×C₄⋊C₄	—	2	1	no	—
259	C₇×C₂²⋊C₄	—	2	1	no	—
260	C₄²×C₇	—	2	1	no	—
261	#575	C₂²×C₇	1	1	yes	characteristic
262	C₂×#137	C₂²×C₇	1	1	yes	characteristic
263	C₂×#137	C₂²×C₇	1	1	yes	characteristic
264	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	1	yes	characteristic
265	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	1	yes	characteristic
266	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
267	#568	C₂²×C₇	1	1	yes	characteristic
268	#568	C₂²×C₇	1	1	yes	characteristic
269	C₂×#131	C₂²×C₇	1	1	yes	characteristic
270	C₂×#131	C₂²×C₇	1	1	yes	characteristic
271	#473	—	2	2	no	—
272	#472	—	2	2	no	—
273	#471	—	2	2	no	—
274	#471	—	2	2	no	—
275	#471	—	2	2	no	—
276	#471	—	2	2	no	—
277	#471	—	2	2	no	—
278	#471	—	2	2	no	—
279	#444	—	2	2	no	—
280	#443	—	2	2	no	—
281	#430	—	2	2	no	—
282	#430	C₂²×C₇	1	1	yes	characteristic
283	#430	C₂²×C₇	1	1	yes	characteristic
284	#430	C₂²×C₇	1	1	yes	characteristic
285	#430	C₂²×C₇	1	1	yes	characteristic
286	#430	C₂²×C₇	1	1	yes	characteristic
287	C₂×#108	—	2	2	no	—
288	C₂×#108	C₂²×C₇	1	1	yes	characteristic
289	C₂×#108	C₂²×C₇	1	1	yes	characteristic
290	C₂×#108	C₂²×C₇	1	1	yes	characteristic
291	#421	C₂²×C₇	1	1	yes	characteristic
292	#421	C₂²×C₇	1	1	yes	characteristic
293	#419	C₂²×C₇	1	1	yes	characteristic
294	#418	C₂²×C₇	1	1	yes	characteristic
295	#413	C₂²×C₇	1	1	yes	characteristic
296	#413	C₂²×C₇	1	1	yes	characteristic
297	C₂×#101	C₂²×C₇	1	1	yes	characteristic
298	C₂×#101	C₂²×C₇	1	1	yes	characteristic
299	C₂×#101	C₂²×C₇	1	1	yes	characteristic
300	C₂×#100	C₂²×C₇	1	1	yes	characteristic
301	C₂×#100	C₂²×C₇	1	1	yes	characteristic
302	C₂×#100	C₂²×C₇	1	1	yes	characteristic
303	#405	C₂²×C₇	1	1	yes	characteristic
304	#344	C₂²×C₇	1	1	yes	characteristic
305	#344	C₂²×C₇	1	1	yes	characteristic
306	#338	C₂²×C₇	1	1	yes	characteristic
307	#338	C₂²×C₇	1	1	yes	characteristic
308	C₂³×C₇	—	2	1	no	—
309	C₂³×C₇	—	2	1	no	—
310	C₂³×C₇	—	4	1	no	—
311	C₂³×C₇	—	4	1	no	—
312	C₂³×C₇	—	4	1	no	—
313	C₂³×C₇	—	16	1	no	—
314	C₂³×C₇	2₋¹⁺⁴	1	1	yes	characteristic
315	C₇×Q₈	—	2	1	no	—
316	C₇×Q₈	—	2	1	no	—
317	C₇×Q₈	—	2	1	no	—
318	C₇×Q₈	—	4	1	no	—
319	C₇×Q₈	—	4	1	no	—
320	C₇×Q₈	—	4	1	no	—
321	C₇×Q₈	—	4	1	no	—
322	C₇×Q₈	#121	1	2	yes	—
323	C₇×D₄	—	2	2	no	—
324	C₇×D₄	—	2	2	no	—
325	C₇×D₄	—	2	1	no	—
326	C₇×D₄	—	2	1	no	—
327	C₇×D₄	—	2	1	no	—
328	C₇×D₄	—	4	1	no	—
329	C₇×D₄	—	4	1	no	—
330	C₇×D₄	—	4	1	no	—
331	C₇×D₄	—	4	1	no	—
332	C₇×D₄	—	4	1	no	—
333	C₇×D₄	—	4	1	no	—
334	C₇×D₄	#121	1	2	yes	—
335	C₂×C₄×C₇	—	2	1	no	—
336	C₂×C₄×C₇	—	2	2	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	1	no	—
339	C₂×C₄×C₇	—	4	1	no	—
340	C₂×C₄×C₇	—	4	1	no	—
341	C₂×C₄×C₇	—	4	1	no	—
342	C₂×C₄×C₇	—	4	1	no	—
343	C₂×C₄×C₇	—	4	1	no	—
344	C₂×C₄×C₇	—	4	1	no	—
345	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
346	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
347	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
348	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
349	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
350	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
351	C₇×C₈	—	2	2	no	—
352	C₇×C₈	—	2	2	no	—
353	C₇×C₈	—	2	2	no	—
354	C₇×C₈	—	2	2	no	—
355	C₇×C₈	—	4	1	no	—
356	C₇×C₈	—	4	1	no	—
357	C₇×C₈	—	4	1	no	—
358	C₇×C₈	—	4	1	no	—
359	C₇×C₈	—	4	1	no	—
360	C₇×C₈	#121	1	1	yes	characteristic
361	C₇×C₈	#121	1	1	yes	characteristic
362	C₇×C₈	#121	1	1	yes	characteristic
363	C₇×C₈	#121	1	1	yes	characteristic
364	C₂×#41	C₂³×C₇	1	1	yes	characteristic
365	C₂×#41	C₂³×C₇	1	1	yes	characteristic
366	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
367	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
368	#137	—	2	1	no	—
369	#137	—	2	2	no	—
370	#137	—	2	2	no	—
371	#137	—	2	2	no	—
372	#137	—	4	1	no	—
373	#137	C₇×D₄	1	2	yes	—
374	AGL(1,ℤ/8)	—	2	1	no	—
375	AGL(1,ℤ/8)	—	2	2	no	—
376	AGL(1,ℤ/8)	—	2	2	no	—
377	AGL(1,ℤ/8)	—	2	2	no	—
378	AGL(1,ℤ/8)	—	4	1	no	—
379	AGL(1,ℤ/8)	C₇×D₄	1	2	yes	—
380	#135	—	2	2	no	—
381	C₂×Q₁₆	—	2	1	no	—
382	C₂×Q₁₆	—	2	1	no	—
383	C₂×SD₁₆	—	2	1	no	—
384	C₂×SD₁₆	—	2	1	no	—
385	C₂×SD₁₆	—	2	1	no	—
386	C₂×SD₁₆	—	2	1	no	—
387	C₂×SD₁₆	—	2	1	no	—
388	C₂×SD₁₆	—	2	1	no	—
389	C₂×SD₁₆	—	2	1	no	—
390	C₂×SD₁₆	—	2	1	no	—
391	C₂×SD₁₆	—	2	1	no	—
392	C₂×SD₁₆	—	2	1	no	—
393	C₂×D₈	—	2	1	no	—
394	C₂×D₈	—	2	1	no	—
395	#131	—	2	1	no	—
396	#131	—	2	1	no	—
397	#131	—	2	2	no	—
398	#131	—	2	2	no	—
399	#131	—	2	1	no	—
400	#131	—	2	1	no	—
401	#131	—	2	1	no	—
402	#131	—	2	1	no	—
403	#131	—	4	1	no	—
404	#131	—	4	1	no	—
405	#131	C₇×D₄	1	2	yes	—
406	#131	C₇×D₄	1	2	yes	—
407	C₂×M₁₆	—	2	1	no	—
408	C₂×M₁₆	—	2	1	no	—
409	C₂×M₁₆	—	2	2	no	—
410	C₂×M₁₆	—	2	2	no	—
411	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
412	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
413	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
414	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
415	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
416	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
417	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
418	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
419	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
420	C₂²×C₈	—	2	1	no	—
421	C₂²×C₈	—	2	1	no	—
422	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
423	#117	C₂³×C₇	1	1	yes	characteristic
424	#108	—	2	2	no	—
425	#108	—	2	2	no	—
426	#108	—	2	2	no	—
427	#108	—	2	2	no	—
428	#108	—	2	2	no	—
429	#108	—	2	1	no	—
430	#108	—	2	1	no	—
431	#108	—	2	1	no	—
432	#108	—	2	1	no	—
433	#108	C₇×D₄	1	2	yes	—
434	#108	C₇×D₄	1	2	yes	—
435	#104	—	2	2	no	—
436	#104	—	2	2	no	—
437	#103	—	4	1	no	—
438	#102	—	4	1	no	—
439	#101	—	2	1	no	—
440	#101	—	2	1	no	—
441	#101	—	2	2	no	—
442	#101	—	2	2	no	—
443	#101	—	2	2	no	—
444	#100	—	2	1	no	—
445	#100	—	2	1	no	—
446	#100	—	2	2	no	—
447	#100	—	2	2	no	—
448	#100	—	2	2	no	—
449	#97	—	2	1	no	—
450	C₄×C₈	—	2	1	no	—
451	C₂²×C₇	—	2	2	no	—
452	C₂²×C₇	—	4	1	no	—
453	C₂²×C₇	—	4	1	no	—
454	C₂²×C₇	—	8	1	no	—
455	C₂²×C₇	—	8	1	no	—
456	C₂²×C₇	—	8	1	no	—
457	C₂²×C₇	—	16	1	no	—
458	C₂²×C₇	—	16	1	no	—
459	C₂²×C₇	—	16	1	no	—
460	C₂²×C₇	#537	1	1	yes	characteristic
461	C₂²×C₇	#547	1	1	yes	characteristic
462	C₂²×C₇	#548	1	1	yes	characteristic
463	C₄×C₇	—	2	2	no	—
464	C₄×C₇	—	4	1	no	—
465	C₄×C₇	—	8	1	no	—
466	C₄×C₇	—	8	1	no	—
467	C₄×C₇	C₂×#121	1	1	yes	characteristic
468	C₄×C₇	C₂×#121	1	1	yes	characteristic
469	C₄×C₇	D₄²	1	1	yes	characteristic
470	C₄×C₇	#553	1	1	yes	characteristic
471	C₂⁴	—	2	1	no	—
472	#41	—	2	1	no	—
473	#41	—	2	1	no	—
474	#41	—	2	1	no	—
475	#41	—	2	1	no	—
476	#41	—	2	1	no	—
477	#41	—	2	1	no	—
478	#41	C₂×C₇×D₄	1	2	yes	—
479	#41	C₂×C₇×D₄	1	2	yes	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
487	C₂×Q₈	C₇×#41	1	1	yes	characteristic
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
497	C₂×D₄	C₇×#41	1	1	yes	characteristic
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
502	Q₁₆	—	2	2	no	—
503	Q₁₆	—	4	2	no	—
504	Q₁₆	—	4	2	no	—
505	SD₁₆	—	2	2	no	—
506	SD₁₆	—	2	2	no	—
507	SD₁₆	—	4	2	no	—
508	SD₁₆	—	4	2	no	—
509	SD₁₆	—	4	2	no	—
510	SD₁₆	—	4	2	no	—
511	SD₁₆	—	4	1	no	—
512	SD₁₆	—	4	1	no	—
513	SD₁₆	—	4	1	no	—
514	SD₁₆	—	4	1	no	—
515	D₈	—	2	2	no	—
516	D₈	—	4	2	no	—
517	D₈	—	4	2	no	—
518	M₁₆	—	2	1	no	—
519	M₁₆	—	2	1	no	—
520	M₁₆	—	2	1	no	—
521	M₁₆	—	2	1	no	—
522	M₁₆	—	2	2	no	—
523	M₁₆	—	2	2	no	—
524	M₁₆	—	2	1	no	—
525	M₁₆	—	2	1	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	4	2	no	—
535	M₁₆	—	4	2	no	—
536	M₁₆	—	4	1	no	—
537	M₁₆	—	4	1	no	—
538	M₁₆	C₂×C₇×D₄	1	2	yes	—
539	M₁₆	C₂×C₇×D₄	1	2	yes	—
540	M₁₆	C₂×C₇×D₄	1	2	yes	—
541	M₁₆	C₂×C₇×D₄	1	2	yes	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	2	no	—
547	C₂×C₈	—	2	2	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	4	2	no	—
554	C₂×C₈	—	4	2	no	—
555	C₂×C₈	—	4	1	no	—
556	C₂×C₈	—	4	1	no	—
557	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
558	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
559	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
560	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
561	C₂×C₈	C₇×#41	1	1	yes	characteristic
562	C₂×C₈	C₇×#41	1	1	yes	characteristic
563	C₄⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₄²	—	2	1	no	—
566	C₂×C₇	—	2	1	no	—
567	C₂×C₇	—	2	1	no	—
568	C₂×C₇	—	2	1	no	—
569	C₂×C₇	—	4	2	no	—
570	C₂×C₇	—	8	1	no	—
571	C₂×C₇	—	16	1	no	—
572	C₂×C₇	#2416	1	1	yes	characteristic
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	1	no	—
578	C₂³	—	16	1	no	—
579	C₂³	C₇×2₋¹⁺⁴	1	1	yes	characteristic
580	Q₈	—	2	1	no	—
581	Q₈	—	2	1	no	—
582	Q₈	—	2	1	no	—
583	Q₈	—	4	1	no	—
584	Q₈	—	4	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	C₇×#121	1	2	yes	—
588	D₄	—	2	2	no	—
589	D₄	—	2	2	no	—
590	D₄	—	2	1	no	—
591	D₄	—	2	1	no	—
592	D₄	—	2	1	no	—
593	D₄	—	4	1	no	—
594	D₄	—	4	1	no	—
595	D₄	—	4	1	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	1	no	—
598	D₄	—	4	1	no	—
599	D₄	C₇×#121	1	2	yes	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
611	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
612	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
613	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
614	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
615	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
616	C₈	—	2	2	no	—
617	C₈	—	2	2	no	—
618	C₈	—	2	2	no	—
619	C₈	—	2	2	no	—
620	C₈	—	4	1	no	—
621	C₈	—	4	1	no	—
622	C₈	—	4	1	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	1	no	—
625	C₈	C₇×#121	1	1	yes	characteristic
626	C₈	C₇×#121	1	1	yes	characteristic
627	C₈	C₇×#121	1	1	yes	characteristic
628	C₈	C₇×#121	1	1	yes	characteristic
629	C₇	#24808	1	1	yes	characteristic, direct product factor, Sylow
630	C₂²	—	2	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	8	1	no	—
634	C₂²	—	8	1	no	—
635	C₂²	—	8	1	no	—
636	C₂²	—	16	1	no	—
637	C₂²	—	16	1	no	—
638	C₂²	—	16	1	no	—
639	C₂²	C₇×#537	1	1	yes	characteristic
640	C₂²	C₇×#547	1	1	yes	characteristic
641	C₂²	C₇×#548	1	1	yes	characteristic
642	C₄	—	2	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	C₂×C₇×#121	1	1	yes	characteristic
647	C₄	C₂×C₇×#121	1	1	yes	characteristic
648	C₄	C₇×D₄²	1	1	yes	characteristic
649	C₄	C₇×#553	1	1	yes	characteristic
650	C₂	—	2	1	no	—
651	C₂	—	2	1	no	—
652	C₂	—	2	1	no	—
653	C₂	—	4	2	no	—
654	C₂	—	8	1	no	—
655	C₂	—	16	1	no	—
656	C₂	C₇×#2416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#2416
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₇×#3068	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
3. C₂×C₇×#471	C₂	semidirect
4. C₂×C₇×#471	C₂	semidirect
5. C₇×#2955	C₂	semidirect
6. C₇×#2916	C₂	semidirect
7. C₇×#2070	C₂	semidirect
8. C₇×#2069	C₂	semidirect
9. C₇×#2049	C₂	semidirect
10. C₇×#2048	C₂	semidirect
11. C₇×#1977	C₂	semidirect
12. C₇×#1953	C₂	semidirect
13. C₇×#1903	C₂	semidirect
14. C₇×#1902	C₂	semidirect
15. C₇×#1813	C₂	semidirect
63. #24808	C₇	direct (N × H)
151. #3069	C₂×C₇	semidirect
153. C₂×#471	C₂×C₇	semidirect
154. C₂×#471	C₂×C₇	semidirect
155. #2955	C₂×C₇	semidirect
156. #2916	C₂×C₇	semidirect
157. #2070	C₂×C₇	semidirect
158. #2069	C₂×C₇	semidirect
159. #2049	C₂×C₇	semidirect
160. #2048	C₂×C₇	semidirect
161. #1977	C₂×C₇	semidirect
162. #1953	C₂×C₇	semidirect
163. #1903	C₂×C₇	semidirect
164. #1902	C₂×C₇	semidirect
165. #1813	C₂×C₇	semidirect
629. C₇	#24808	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×#3068	2
C₂×C₇×#471	2
C₇×#2955	2
C₇×#2916	2
C₇×#2070	2
C₇×#2069	2
C₇×#2049	2
C₇×#2048	2
C₇×#1977	2
C₇×#1953	2
C₇×#1903	2
C₇×#1902	2
C₇×#1813	2
#24808	7

Subgroups — C7×#24808 #13202024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24808 #13202024