Subgroups — C₇×#25985 #13203201

Order 1792.

842 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3084	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3083	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2898	C₂	1	1	yes	maximal, characteristic
4	C₇×#2898	C₂	1	1	yes	maximal, characteristic
5	C₇×#2097	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2064	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#2064	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2028	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1866	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1866	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1860	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1860	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1815	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1793	C₂	1	1	yes	maximal, characteristic
16	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂×C₇×#131	C₂²	1	1	yes	characteristic
18	C₂×C₇×#131	C₂²	1	1	yes	characteristic
19	C₇×#544	C₂²	1	1	yes	characteristic
20	C₇×#520	C₂²	1	1	yes	characteristic
21	C₇×#520	C₂²	1	1	yes	characteristic
22	C₇×#479	—	2	1	no	—
23	C₇×#479	—	2	1	no	—
24	C₇×#478	—	2	1	no	—
25	C₇×#478	—	2	1	no	—
26	C₇×#477	—	2	1	no	—
27	C₇×#476	—	2	1	no	—
28	C₇×#475	—	2	1	no	—
29	C₇×#474	—	2	1	no	—
30	C₂×C₇×#108	C₂²	1	1	yes	characteristic
31	C₂×C₇×#108	C₂²	1	1	yes	characteristic
32	C₇×#428	C₂²	1	1	yes	characteristic
33	C₇×#428	C₂²	1	1	yes	characteristic
34	C₂×C₇×#107	C₂²	1	1	yes	characteristic
35	C₂×C₇×#106	C₂²	1	1	yes	characteristic
36	C₇×#424	C₂²	1	1	yes	characteristic
37	C₇×#424	C₂²	1	1	yes	characteristic
38	C₂×C₇×#104	C₂²	1	1	yes	characteristic
39	C₂×C₇×#104	C₂²	1	1	yes	characteristic
40	C₇×#419	—	2	1	no	—
41	C₇×#419	—	2	1	no	—
42	C₇×#419	C₂²	1	1	yes	characteristic
43	C₇×#419	C₂²	1	1	yes	characteristic
44	C₇×#417	—	2	1	no	—
45	C₇×#417	—	2	1	no	—
46	C₇×#417	C₂²	1	1	yes	characteristic
47	C₇×#417	C₂²	1	1	yes	characteristic
48	C₇×#417	C₂²	1	1	yes	characteristic
49	C₇×#417	C₂²	1	1	yes	characteristic
50	C₇×#416	—	2	1	no	—
51	C₇×#416	—	2	1	no	—
52	C₇×#416	C₂²	1	1	yes	characteristic
53	C₇×#416	C₂²	1	1	yes	characteristic
54	C₂×C₇×#103	—	2	1	no	—
55	C₂×C₇×#103	—	2	1	no	—
56	C₂×C₇×#103	C₂²	1	1	yes	characteristic
57	C₂×C₇×#103	C₂²	1	1	yes	characteristic
58	C₂×C₇×#101	C₂²	1	1	yes	characteristic
59	C₇×#391	C₂²	1	1	yes	characteristic
60	C₇×#382	C₂²	1	1	yes	characteristic
61	C₇×#343	C₂²	1	1	yes	characteristic
62	C₇×#343	C₂²	1	1	yes	characteristic
63	C₇×#343	C₂²	1	1	yes	characteristic
64	C₇×#339	C₂²	1	1	yes	characteristic
65	C₇×#339	C₂²	1	1	yes	characteristic
66	C₇×#336	C₂²	1	1	yes	characteristic
67	#25985	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₇×2₋¹⁺⁴	—	2	2	no	—
69	C₇×2₋¹⁺⁴	—	2	2	no	—
70	C₇×2₋¹⁺⁴	—	4	1	no	—
71	C₇×2₋¹⁺⁴	Q₈	1	4	yes	complemented
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×#41	—	2	1	no	—
74	C₂×C₇×#41	—	4	1	no	—
75	C₂×C₇×#41	C₂³	1	1	yes	characteristic
76	C₂×C₇×#41	C₂³	1	1	yes	characteristic
77	C₂²×C₇×Q₈	—	2	1	no	—
78	C₂²×C₇×Q₈	—	2	1	no	—
79	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
80	C₇×#131	—	2	2	no	—
81	C₇×#131	—	2	2	no	—
82	C₇×#131	—	4	1	no	—
83	C₇×#131	—	4	1	no	—
84	C₂×C₇×M₁₆	—	2	1	no	—
85	C₂×C₇×M₁₆	—	2	1	no	—
86	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
90	C₂²×C₇×C₈	—	2	1	no	—
91	C₂²×C₇×C₈	—	2	1	no	—
92	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
93	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
94	C₇×C₄⋊Q₈	—	2	1	no	—
95	C₇×C₄⋊Q₈	—	2	1	no	—
96	C₇×#125	—	2	1	no	—
97	C₇×#125	—	2	1	no	—
98	C₇×C₂²⋊Q₈	—	4	1	no	—
99	C₄×C₇×Q₈	—	2	1	no	—
100	C₄×C₇×Q₈	—	2	1	no	—
101	C₄×C₇×D₄	—	2	1	no	—
102	C₄×C₇×D₄	—	2	1	no	—
103	C₄×C₇×D₄	—	4	1	no	—
104	C₄×C₇×D₄	—	4	1	no	—
105	C₇×#117	—	2	1	no	—
106	C₇×#117	—	2	1	no	—
107	C₇×#117	C₂³	1	1	yes	characteristic
108	C₇×#117	C₂³	1	1	yes	characteristic
109	C₂×C₇×C₄⋊C₄	—	2	1	no	—
110	C₂×C₇×C₄⋊C₄	—	2	1	no	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
112	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
115	C₇×#108	—	4	1	no	—
116	C₇×#108	—	4	1	no	—
117	C₇×#107	—	2	1	no	—
118	C₇×#107	—	2	1	no	—
119	C₇×#107	—	2	1	no	—
120	C₇×#107	—	2	1	no	—
121	C₇×#106	—	2	1	no	—
122	C₇×#106	—	2	1	no	—
123	C₇×#106	—	2	1	no	—
124	C₇×#106	—	2	1	no	—
125	C₇×C₄⋊C₈	—	2	1	no	—
126	C₇×C₄⋊C₈	—	2	1	no	—
127	C₇×C₄⋊C₈	—	2	1	no	—
128	C₇×C₄⋊C₈	—	2	1	no	—
129	C₇×#104	—	2	2	no	—
130	C₇×#104	—	2	2	no	—
131	C₇×#103	—	2	1	no	—
132	C₇×#103	—	2	1	no	—
133	C₇×#103	—	2	1	no	—
134	C₇×#103	—	2	1	no	—
135	C₇×#103	—	2	1	no	—
136	C₇×#103	—	2	1	no	—
137	C₇×#103	—	2	1	no	—
138	C₇×#103	—	2	1	no	—
139	C₇×#103	—	2	1	no	—
140	C₇×#103	—	2	1	no	—
141	C₇×#103	—	2	1	no	—
142	C₇×#103	—	2	1	no	—
143	C₇×#103	D₄	1	1	yes	characteristic
144	C₇×#103	D₄	1	1	yes	characteristic
145	C₇×#103	D₄	1	1	yes	characteristic
146	C₇×#103	D₄	1	1	yes	characteristic
147	C₇×#102	—	2	1	no	—
148	C₇×#102	—	2	1	no	—
149	C₇×#102	—	2	1	no	—
150	C₇×#102	—	2	1	no	—
151	C₇×#102	D₄	1	1	yes	characteristic
152	C₇×#102	D₄	1	1	yes	characteristic
153	C₇×#102	D₄	1	1	yes	characteristic
154	C₇×#102	D₄	1	1	yes	characteristic
155	C₇×#101	—	4	1	no	—
156	C₇×#95	—	4	1	no	—
157	C₇×#95	—	4	1	no	—
158	#3084	C₂×C₇	1	1	yes	characteristic, complemented
159	#3083	C₂×C₇	1	1	yes	characteristic, complemented
160	#2898	C₂×C₇	1	1	yes	characteristic
161	#2898	C₂×C₇	1	1	yes	characteristic
162	#2097	C₂×C₇	1	1	yes	characteristic, complemented
163	#2064	C₂×C₇	1	1	yes	characteristic, complemented
164	#2064	C₂×C₇	1	1	yes	characteristic, complemented
165	#2028	C₂×C₇	1	1	yes	characteristic, complemented
166	#1866	C₂×C₇	1	1	yes	characteristic, complemented
167	#1866	C₂×C₇	1	1	yes	characteristic, complemented
168	#1860	C₂×C₇	1	1	yes	characteristic, complemented
169	#1860	C₂×C₇	1	1	yes	characteristic, complemented
170	#1816	C₂×C₇	1	1	yes	characteristic, complemented
171	#1815	C₂×C₇	1	1	yes	characteristic, complemented
172	#1793	C₂×C₇	1	1	yes	characteristic
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	4	no	—
178	C₇×#41	—	2	4	no	—
179	C₇×#41	—	4	4	no	—
180	C₇×#41	—	4	2	no	—
181	C₇×#41	—	4	2	no	—
182	C₇×#41	—	4	4	no	—
183	C₇×#41	C₂×Q₈	1	2	yes	—
184	C₇×#41	C₂×Q₈	1	2	yes	—
185	C₇×#41	#41	1	2	yes	—
186	C₇×#41	#41	1	2	yes	—
187	C₂×C₇×Q₈	—	2	1	no	—
188	C₂×C₇×Q₈	—	2	4	no	—
189	C₂×C₇×Q₈	—	2	4	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	—	2	1	no	—
194	C₂×C₇×Q₈	—	4	1	no	—
195	C₂×C₇×Q₈	—	4	1	no	—
196	C₂×C₇×Q₈	—	4	2	no	—
197	C₂×C₇×Q₈	—	4	2	no	—
198	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
199	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
200	C₂×C₇×Q₈	C₂×Q₈	1	2	yes	—
201	C₂×C₇×D₄	—	2	1	no	—
202	C₂×C₇×D₄	—	2	1	no	—
203	C₂×C₇×D₄	—	4	1	no	—
204	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₇	—	2	1	no	—
207	C₂²×C₄×C₇	—	2	1	no	—
208	C₂²×C₄×C₇	—	2	1	no	—
209	C₂²×C₄×C₇	—	2	1	no	—
210	C₂²×C₄×C₇	—	2	1	no	—
211	C₂²×C₄×C₇	—	2	1	no	—
212	C₂²×C₄×C₇	—	2	1	no	—
213	C₂²×C₄×C₇	—	4	1	no	—
214	C₂²×C₄×C₇	—	4	1	no	—
215	C₂²×C₄×C₇	—	4	1	no	—
216	C₂²×C₄×C₇	—	4	1	no	—
217	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
218	C₇×M₁₆	—	2	1	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	—	4	2	no	—
227	C₇×M₁₆	—	4	2	no	—
228	C₂×C₇×C₈	—	2	1	no	—
229	C₂×C₇×C₈	—	2	1	no	—
230	C₂×C₇×C₈	—	2	1	no	—
231	C₂×C₇×C₈	—	2	1	no	—
232	C₂×C₇×C₈	—	2	1	no	—
233	C₂×C₇×C₈	—	2	1	no	—
234	C₂×C₇×C₈	—	2	1	no	—
235	C₂×C₇×C₈	—	2	1	no	—
236	C₂×C₇×C₈	—	4	2	no	—
237	C₂×C₇×C₈	—	4	2	no	—
238	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
239	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
240	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
241	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
242	C₇×C₄⋊C₄	—	2	1	no	—
243	C₇×C₄⋊C₄	—	2	1	no	—
244	C₇×C₄⋊C₄	—	2	1	no	—
245	C₇×C₄⋊C₄	—	2	1	no	—
246	C₇×C₄⋊C₄	—	2	1	no	—
247	C₇×C₄⋊C₄	—	2	1	no	—
248	C₇×C₄⋊C₄	—	4	1	no	—
249	C₇×C₄⋊C₄	—	4	1	no	—
250	C₇×C₄⋊C₄	—	4	1	no	—
251	C₇×C₄⋊C₄	—	4	1	no	—
252	C₇×C₄⋊C₄	—	4	1	no	—
253	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
254	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
255	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
257	C₇×C₂²⋊C₄	—	2	1	no	—
258	C₇×C₂²⋊C₄	—	2	1	no	—
259	C₇×C₂²⋊C₄	—	4	1	no	—
260	C₇×C₂²⋊C₄	—	4	1	no	—
261	C₄²×C₇	—	2	1	no	—
262	C₄²×C₇	—	2	1	no	—
263	C₄²×C₇	—	2	1	no	—
264	C₄²×C₇	—	2	1	no	—
265	C₄²×C₇	—	4	1	no	—
266	C₄²×C₇	—	4	1	no	—
267	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
268	C₂×#131	C₂²×C₇	1	1	yes	characteristic
269	C₂×#131	C₂²×C₇	1	1	yes	characteristic
270	#544	C₂²×C₇	1	1	yes	characteristic
271	#520	C₂²×C₇	1	1	yes	characteristic
272	#520	C₂²×C₇	1	1	yes	characteristic
273	#479	—	2	1	no	—
274	#479	—	2	1	no	—
275	#478	—	2	1	no	—
276	#478	—	2	1	no	—
277	#477	—	2	1	no	—
278	#476	—	2	1	no	—
279	#475	—	2	1	no	—
280	#474	—	2	1	no	—
281	C₂×#108	C₂²×C₇	1	1	yes	characteristic
282	C₂×#108	C₂²×C₇	1	1	yes	characteristic
283	#428	C₂²×C₇	1	1	yes	characteristic
284	#428	C₂²×C₇	1	1	yes	characteristic
285	C₂×#107	C₂²×C₇	1	1	yes	characteristic
286	C₂×#106	C₂²×C₇	1	1	yes	characteristic
287	#424	C₂²×C₇	1	1	yes	characteristic
288	#424	C₂²×C₇	1	1	yes	characteristic
289	C₂×#104	C₂²×C₇	1	1	yes	characteristic
290	C₂×#104	C₂²×C₇	1	1	yes	characteristic
291	#419	—	2	1	no	—
292	#419	—	2	1	no	—
293	#419	C₂²×C₇	1	1	yes	characteristic
294	#419	C₂²×C₇	1	1	yes	characteristic
295	#417	—	2	1	no	—
296	#417	—	2	1	no	—
297	#417	C₂²×C₇	1	1	yes	characteristic
298	#417	C₂²×C₇	1	1	yes	characteristic
299	#417	C₂²×C₇	1	1	yes	characteristic
300	#417	C₂²×C₇	1	1	yes	characteristic
301	#416	—	2	1	no	—
302	#416	—	2	1	no	—
303	#416	C₂²×C₇	1	1	yes	characteristic
304	#416	C₂²×C₇	1	1	yes	characteristic
305	C₂×#103	—	2	1	no	—
306	C₂×#103	—	2	1	no	—
307	C₂×#103	C₂²×C₇	1	1	yes	characteristic
308	C₂×#103	C₂²×C₇	1	1	yes	characteristic
309	C₂×#101	C₂²×C₇	1	1	yes	characteristic
310	#391	C₂²×C₇	1	1	yes	characteristic
311	#382	C₂²×C₇	1	1	yes	characteristic
312	#343	C₂²×C₇	1	1	yes	characteristic
313	#343	C₂²×C₇	1	1	yes	characteristic
314	#343	C₂²×C₇	1	1	yes	characteristic
315	#339	C₂²×C₇	1	1	yes	characteristic
316	#339	C₂²×C₇	1	1	yes	characteristic
317	#336	C₂²×C₇	1	1	yes	characteristic
318	C₂³×C₇	—	2	1	no	—
319	C₂³×C₇	—	2	1	no	—
320	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
321	C₇×Q₈	—	2	1	no	—
322	C₇×Q₈	—	2	1	no	—
323	C₇×Q₈	—	2	2	no	—
324	C₇×Q₈	—	2	2	no	—
325	C₇×Q₈	—	4	4	no	—
326	C₇×Q₈	—	4	1	no	—
327	C₇×Q₈	—	4	1	no	—
328	C₇×Q₈	—	16	1	no	—
329	C₇×Q₈	C₂²⋊Q₈	1	2	yes	—
330	C₇×Q₈	C₂²⋊Q₈	1	2	yes	—
331	C₇×D₄	—	2	2	no	—
332	C₇×D₄	—	2	2	no	—
333	C₇×D₄	—	2	2	no	—
334	C₇×D₄	—	2	2	no	—
335	C₇×D₄	—	2	1	no	—
336	C₇×D₄	—	2	1	no	—
337	C₇×D₄	—	4	4	no	—
338	C₇×D₄	C₂²⋊Q₈	1	2	yes	—
339	C₇×D₄	C₂²⋊Q₈	1	2	yes	—
340	C₂×C₄×C₇	—	2	1	no	—
341	C₂×C₄×C₇	—	2	1	no	—
342	C₂×C₄×C₇	—	2	2	no	—
343	C₂×C₄×C₇	—	2	2	no	—
344	C₂×C₄×C₇	—	2	2	no	—
345	C₂×C₄×C₇	—	2	2	no	—
346	C₂×C₄×C₇	—	2	2	no	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	2	1	no	—
349	C₂×C₄×C₇	—	2	1	no	—
350	C₂×C₄×C₇	—	2	1	no	—
351	C₂×C₄×C₇	—	4	4	no	—
352	C₂×C₄×C₇	—	4	4	no	—
353	C₂×C₄×C₇	—	4	1	no	—
354	C₂×C₄×C₇	—	4	1	no	—
355	C₂×C₄×C₇	—	4	1	no	—
356	C₂×C₄×C₇	—	4	1	no	—
357	C₂×C₄×C₇	—	4	1	no	—
358	C₂×C₄×C₇	—	4	1	no	—
359	C₂×C₄×C₇	—	4	1	no	—
360	C₂×C₄×C₇	—	4	1	no	—
361	C₂×C₄×C₇	—	8	1	no	—
362	C₂×C₄×C₇	—	8	1	no	—
363	C₂×C₄×C₇	—	8	1	no	—
364	C₂×C₄×C₇	—	16	1	no	—
365	C₂×C₄×C₇	—	16	1	no	—
366	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
367	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
368	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
369	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
370	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
371	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
372	C₇×C₈	—	2	1	no	—
373	C₇×C₈	—	2	1	no	—
374	C₇×C₈	—	2	1	no	—
375	C₇×C₈	—	2	1	no	—
376	C₇×C₈	—	4	1	no	—
377	C₇×C₈	—	4	1	no	—
378	C₇×C₈	—	4	1	no	—
379	C₇×C₈	—	4	1	no	—
380	2₋¹⁺⁴	—	2	2	no	—
381	2₋¹⁺⁴	—	2	2	no	—
382	2₋¹⁺⁴	—	4	1	no	—
383	2₋¹⁺⁴	C₇×Q₈	1	4	yes	complemented
384	C₂×#41	—	2	1	no	—
385	C₂×#41	—	2	1	no	—
386	C₂×#41	—	4	1	no	—
387	C₂×#41	C₂³×C₇	1	1	yes	characteristic
388	C₂×#41	C₂³×C₇	1	1	yes	characteristic
389	C₂²×Q₈	—	2	1	no	—
390	C₂²×Q₈	—	2	1	no	—
391	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
392	#131	—	2	2	no	—
393	#131	—	2	2	no	—
394	#131	—	4	1	no	—
395	#131	—	4	1	no	—
396	C₂×M₁₆	—	2	1	no	—
397	C₂×M₁₆	—	2	1	no	—
398	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
399	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
400	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
401	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
402	C₂²×C₈	—	2	1	no	—
403	C₂²×C₈	—	2	1	no	—
404	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
405	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
406	C₄⋊Q₈	—	2	1	no	—
407	C₄⋊Q₈	—	2	1	no	—
408	#125	—	2	1	no	—
409	#125	—	2	1	no	—
410	C₂²⋊Q₈	—	4	1	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×Q₈	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	4	1	no	—
416	C₄×D₄	—	4	1	no	—
417	#117	—	2	1	no	—
418	#117	—	2	1	no	—
419	#117	C₂³×C₇	1	1	yes	characteristic
420	#117	C₂³×C₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
427	#108	—	4	1	no	—
428	#108	—	4	1	no	—
429	#107	—	2	1	no	—
430	#107	—	2	1	no	—
431	#107	—	2	1	no	—
432	#107	—	2	1	no	—
433	#106	—	2	1	no	—
434	#106	—	2	1	no	—
435	#106	—	2	1	no	—
436	#106	—	2	1	no	—
437	C₄⋊C₈	—	2	1	no	—
438	C₄⋊C₈	—	2	1	no	—
439	C₄⋊C₈	—	2	1	no	—
440	C₄⋊C₈	—	2	1	no	—
441	#104	—	2	2	no	—
442	#104	—	2	2	no	—
443	#103	—	2	1	no	—
444	#103	—	2	1	no	—
445	#103	—	2	1	no	—
446	#103	—	2	1	no	—
447	#103	—	2	1	no	—
448	#103	—	2	1	no	—
449	#103	—	2	1	no	—
450	#103	—	2	1	no	—
451	#103	—	2	1	no	—
452	#103	—	2	1	no	—
453	#103	—	2	1	no	—
454	#103	—	2	1	no	—
455	#103	C₇×D₄	1	1	yes	characteristic
456	#103	C₇×D₄	1	1	yes	characteristic
457	#103	C₇×D₄	1	1	yes	characteristic
458	#103	C₇×D₄	1	1	yes	characteristic
459	#102	—	2	1	no	—
460	#102	—	2	1	no	—
461	#102	—	2	1	no	—
462	#102	—	2	1	no	—
463	#102	C₇×D₄	1	1	yes	characteristic
464	#102	C₇×D₄	1	1	yes	characteristic
465	#102	C₇×D₄	1	1	yes	characteristic
466	#102	C₇×D₄	1	1	yes	characteristic
467	#101	—	4	1	no	—
468	#95	—	4	1	no	—
469	#95	—	4	1	no	—
470	C₂²×C₇	—	2	1	no	—
471	C₂²×C₇	—	2	1	no	—
472	C₂²×C₇	—	2	2	no	—
473	C₂²×C₇	—	2	2	no	—
474	C₂²×C₇	—	4	1	no	—
475	C₂²×C₇	—	4	1	no	—
476	C₂²×C₇	—	4	1	no	—
477	C₂²×C₇	—	4	1	no	—
478	C₂²×C₇	#543	1	1	yes	characteristic
479	C₂²×C₇	D₄²	1	1	yes	characteristic
480	C₂²×C₇	#551	1	1	yes	characteristic
481	C₄×C₇	—	2	2	no	—
482	C₄×C₇	—	2	2	no	—
483	C₄×C₇	—	4	2	no	—
484	C₄×C₇	—	4	1	no	—
485	C₄×C₇	—	4	1	no	—
486	C₄×C₇	—	4	1	no	—
487	C₄×C₇	—	4	1	no	—
488	C₄×C₇	—	8	1	no	—
489	C₄×C₇	—	8	1	no	—
490	C₄×C₇	—	8	1	no	—
491	C₄×C₇	—	8	1	no	—
492	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
493	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
494	C₄×C₇	#546	1	1	yes	characteristic
495	C₄×C₇	#546	1	1	yes	characteristic
496	#41	—	2	1	no	—
497	#41	—	2	1	no	—
498	#41	—	2	1	no	—
499	#41	—	2	1	no	—
500	#41	—	2	4	no	—
501	#41	—	2	4	no	—
502	#41	—	4	4	no	—
503	#41	—	4	2	no	—
504	#41	—	4	2	no	—
505	#41	—	4	4	no	—
506	#41	C₂×C₇×Q₈	1	2	yes	—
507	#41	C₂×C₇×Q₈	1	2	yes	—
508	#41	C₇×#41	1	2	yes	—
509	#41	C₇×#41	1	2	yes	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	4	no	—
512	C₂×Q₈	—	2	4	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	—	2	1	no	—
517	C₂×Q₈	—	4	1	no	—
518	C₂×Q₈	—	4	1	no	—
519	C₂×Q₈	—	4	2	no	—
520	C₂×Q₈	—	4	2	no	—
521	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
522	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
523	C₂×Q₈	C₂×C₇×Q₈	1	2	yes	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	4	1	no	—
527	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
528	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	1	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	1	no	—
536	C₂²×C₄	—	4	1	no	—
537	C₂²×C₄	—	4	1	no	—
538	C₂²×C₄	—	4	1	no	—
539	C₂²×C₄	—	4	1	no	—
540	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
541	M₁₆	—	2	1	no	—
542	M₁₆	—	2	1	no	—
543	M₁₆	—	2	1	no	—
544	M₁₆	—	2	1	no	—
545	M₁₆	—	2	1	no	—
546	M₁₆	—	2	1	no	—
547	M₁₆	—	2	1	no	—
548	M₁₆	—	2	1	no	—
549	M₁₆	—	4	2	no	—
550	M₁₆	—	4	2	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	4	2	no	—
560	C₂×C₈	—	4	2	no	—
561	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
562	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
563	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
564	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	4	1	no	—
572	C₄⋊C₄	—	4	1	no	—
573	C₄⋊C₄	—	4	1	no	—
574	C₄⋊C₄	—	4	1	no	—
575	C₄⋊C₄	—	4	1	no	—
576	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
577	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
578	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
579	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
580	C₂²⋊C₄	—	2	1	no	—
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	4	1	no	—
583	C₂²⋊C₄	—	4	1	no	—
584	C₄²	—	2	1	no	—
585	C₄²	—	2	1	no	—
586	C₄²	—	2	1	no	—
587	C₄²	—	2	1	no	—
588	C₄²	—	4	1	no	—
589	C₄²	—	4	1	no	—
590	C₂×C₇	—	2	1	no	—
591	C₂×C₇	—	2	1	no	—
592	C₂×C₇	—	4	2	no	—
593	C₂×C₇	—	4	2	no	—
594	C₂×C₇	#2437	1	1	yes	characteristic
595	C₂×C₇	#3289	1	1	yes	characteristic
596	C₂×C₇	#3290	1	1	yes	characteristic
597	C₂³	—	2	1	no	—
598	C₂³	—	2	1	no	—
599	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
600	Q₈	—	2	1	no	—
601	Q₈	—	2	1	no	—
602	Q₈	—	2	2	no	—
603	Q₈	—	2	2	no	—
604	Q₈	—	4	4	no	—
605	Q₈	—	4	1	no	—
606	Q₈	—	4	1	no	—
607	Q₈	—	16	1	no	—
608	Q₈	C₇×C₂²⋊Q₈	1	2	yes	—
609	Q₈	C₇×C₂²⋊Q₈	1	2	yes	—
610	D₄	—	2	2	no	—
611	D₄	—	2	2	no	—
612	D₄	—	2	2	no	—
613	D₄	—	2	2	no	—
614	D₄	—	2	1	no	—
615	D₄	—	2	1	no	—
616	D₄	—	4	4	no	—
617	D₄	C₇×C₂²⋊Q₈	1	2	yes	—
618	D₄	C₇×C₂²⋊Q₈	1	2	yes	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	4	4	no	—
631	C₂×C₄	—	4	4	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	8	1	no	—
641	C₂×C₄	—	8	1	no	—
642	C₂×C₄	—	8	1	no	—
643	C₂×C₄	—	16	1	no	—
644	C₂×C₄	—	16	1	no	—
645	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
646	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
647	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
648	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
649	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
650	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
651	C₈	—	2	1	no	—
652	C₈	—	2	1	no	—
653	C₈	—	2	1	no	—
654	C₈	—	2	1	no	—
655	C₈	—	4	1	no	—
656	C₈	—	4	1	no	—
657	C₈	—	4	1	no	—
658	C₈	—	4	1	no	—
659	C₇	#25985	1	1	yes	characteristic, direct product factor, Sylow
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	4	1	no	—
665	C₂²	—	4	1	no	—
666	C₂²	—	4	1	no	—
667	C₂²	—	4	1	no	—
668	C₂²	C₇×#543	1	1	yes	characteristic
669	C₂²	C₇×D₄²	1	1	yes	characteristic
670	C₂²	C₇×#551	1	1	yes	characteristic
671	C₄	—	2	2	no	—
672	C₄	—	2	2	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	1	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	1	no	—
678	C₄	—	8	1	no	—
679	C₄	—	8	1	no	—
680	C₄	—	8	1	no	—
681	C₄	—	8	1	no	—
682	C₄	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
683	C₄	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
684	C₄	C₇×#546	1	1	yes	characteristic
685	C₄	C₇×#546	1	1	yes	characteristic
686	C₂	—	2	1	no	—
687	C₂	—	2	1	no	—
688	C₂	—	4	2	no	—
689	C₂	—	4	2	no	—
690	C₂	C₇×#2437	1	1	yes	characteristic
691	C₂	C₇×#3289	1	1	yes	characteristic
692	C₂	C₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3084	C₂	semidirect
2. C₇×#3083	C₂	semidirect
5. C₇×#2097	C₂	semidirect
6. C₇×#2064	C₂	semidirect
7. C₇×#2064	C₂	semidirect
8. C₇×#2028	C₂	semidirect
9. C₇×#1866	C₂	semidirect
10. C₇×#1866	C₂	semidirect
11. C₇×#1860	C₂	semidirect
12. C₇×#1860	C₂	semidirect
13. C₇×#1816	C₂	semidirect
14. C₇×#1815	C₂	semidirect
67. #25985	C₇	direct (N × H)
71. C₇×2₋¹⁺⁴	Q₈	semidirect
158. #3084	C₂×C₇	semidirect
159. #3083	C₂×C₇	semidirect
162. #2097	C₂×C₇	semidirect
163. #2064	C₂×C₇	semidirect
164. #2064	C₂×C₇	semidirect
165. #2028	C₂×C₇	semidirect
166. #1866	C₂×C₇	semidirect
167. #1866	C₂×C₇	semidirect
168. #1860	C₂×C₇	semidirect
169. #1860	C₂×C₇	semidirect
170. #1816	C₂×C₇	semidirect
171. #1815	C₂×C₇	semidirect
383. 2₋¹⁺⁴	C₇×Q₈	semidirect
659. C₇	#25985	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×#3084	2
C₇×#3083	2
C₇×#2898	2
C₇×#2097	2
C₇×#2064	2
C₇×#2028	2
C₇×#1866	2
C₇×#1860	2
C₇×#1816	2
C₇×#1815	2
C₇×#1793	2
#25985	7

Subgroups — C7×#25985 #13203201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#25985 #13203201