Subgroups — C₇×#18808 #13196024

Order 1792.

912 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3480	C₂	1	1	yes	maximal, characteristic
2	C₇×#3228	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3113	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1653	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1652	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1650	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1648	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1642	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1641	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1640	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1640	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1553	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#559	—	2	1	no	—
17	C₇×#559	C₂²	1	1	yes	characteristic
18	C₇×#557	C₂²	1	1	yes	characteristic
19	C₇×#556	—	2	1	no	—
20	C₇×#556	C₂²	1	1	yes	characteristic
21	C₇×#553	—	2	2	no	—
22	C₇×#553	—	2	1	no	—
23	C₇×#550	C₂²	1	1	yes	characteristic
24	C₇×#548	—	2	1	no	—
25	C₇×#533	C₂²	1	1	yes	characteristic
26	C₇×#532	—	2	1	no	—
27	C₇×#532	C₂²	1	1	yes	characteristic
28	C₇×#529	—	2	1	no	—
29	C₇×#529	—	2	2	no	—
30	C₄×C₇×#41	—	2	1	no	—
31	C₄×C₇×#41	—	2	1	no	—
32	C₄×C₇×#41	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
34	C₇×#500	C₂²	1	1	yes	characteristic
35	C₇×#498	C₂²	1	1	yes	characteristic
36	C₇×#486	C₂²	1	1	yes	characteristic
37	C₇×#486	C₂²	1	1	yes	characteristic
38	C₇×#485	—	2	1	no	—
39	C₇×#483	—	2	1	no	—
40	C₇×#482	—	2	2	no	—
41	C₇×#478	—	2	2	no	—
42	C₇×#475	C₂²	1	1	yes	characteristic
43	C₇×#475	C₂²	1	1	yes	characteristic
44	C₇×#475	C₂²	1	1	yes	characteristic
45	C₇×#474	C₂²	1	1	yes	characteristic
46	C₇×#462	C₂²	1	1	yes	characteristic
47	C₇×#462	C₂²	1	1	yes	characteristic
48	C₇×#462	C₂²	1	1	yes	characteristic
49	C₇×#461	C₂²	1	1	yes	characteristic
50	C₇×#452	—	2	1	no	—
51	C₇×#451	—	2	1	no	—
52	C₇×#441	—	2	1	no	—
53	C₇×#441	—	2	1	no	—
54	C₄×C₇×Q₁₆	—	2	1	no	—
55	C₄×C₇×SD₁₆	—	2	1	no	—
56	C₇×#432	C₂²	1	1	yes	characteristic
57	C₇×#431	C₂²	1	1	yes	characteristic
58	C₇×#431	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#331	C₂²	1	1	yes	characteristic
62	C₇×#331	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	#18808	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×#41	—	2	1	no	—
74	C₂×C₇×#41	D₄	1	1	yes	characteristic
75	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
76	C₂×C₇×Q₁₆	—	2	1	no	—
77	C₂×C₇×Q₁₆	—	2	1	no	—
78	C₂×C₇×Q₁₆	—	2	1	no	—
79	C₂×C₇×SD₁₆	—	2	1	no	—
80	C₇×C₄⋊Q₈	—	2	1	no	—
81	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₇×#127	—	2	1	no	—
85	C₇×#127	C₂³	1	1	yes	characteristic
86	C₇×#126	—	2	2	no	—
87	C₇×#126	—	2	2	no	—
88	C₇×#126	—	2	2	no	—
89	C₇×#125	—	2	2	no	—
90	C₇×#125	—	2	1	no	—
91	C₇×#125	—	2	1	no	—
92	C₇×#124	—	2	2	no	—
93	C₇×#124	—	2	1	no	—
94	C₇×#124	—	2	1	no	—
95	C₇×#124	—	2	1	no	—
96	C₇×#124	—	2	1	no	—
97	C₇×#123	—	2	2	no	—
98	C₇×#123	—	2	2	no	—
99	C₇×#123	—	2	2	no	—
100	C₇×C₂²⋊Q₈	—	2	2	no	—
101	C₇×C₂²⋊Q₈	—	2	1	no	—
102	C₇×C₂²⋊Q₈	—	2	1	no	—
103	C₇×C₂²⋊Q₈	—	2	1	no	—
104	C₇×#121	—	2	2	no	—
105	C₇×#121	—	2	2	no	—
106	C₇×#121	—	2	2	no	—
107	C₇×#121	—	2	1	no	—
108	C₇×#121	—	2	1	no	—
109	C₇×#121	D₄	1	2	yes	—
110	C₄×C₇×Q₈	—	2	1	no	—
111	C₄×C₇×Q₈	—	2	1	no	—
112	C₄×C₇×Q₈	—	2	1	no	—
113	C₄×C₇×Q₈	—	2	1	no	—
114	C₄×C₇×Q₈	—	2	1	no	—
115	C₄×C₇×Q₈	—	2	1	no	—
116	C₄×C₇×Q₈	—	2	1	no	—
117	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
118	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
119	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
120	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
121	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
122	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
123	C₄×C₇×D₄	—	2	2	no	—
124	C₄×C₇×D₄	—	2	2	no	—
125	C₄×C₇×D₄	—	2	2	no	—
126	C₄×C₇×D₄	—	2	2	no	—
127	C₄×C₇×D₄	—	2	2	no	—
128	C₄×C₇×D₄	—	2	1	no	—
129	C₄×C₇×D₄	—	2	1	no	—
130	C₄×C₇×D₄	—	2	1	no	—
131	C₄×C₇×D₄	—	2	1	no	—
132	C₄×C₇×D₄	—	2	1	no	—
133	C₄×C₇×D₄	—	2	1	no	—
134	C₄×C₇×D₄	D₄	1	2	yes	—
135	C₄×C₇×D₄	D₄	1	1	yes	characteristic
136	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
137	C₇×#117	—	2	2	no	—
138	C₇×#117	—	2	2	no	—
139	C₇×#117	—	2	1	no	—
140	C₇×#117	—	2	1	no	—
141	C₇×#117	—	2	1	no	—
142	C₇×#117	D₄	1	1	yes	characteristic
143	C₂×C₇×C₄⋊C₄	—	2	1	no	—
144	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
145	C₂×C₄²×C₇	—	2	2	no	—
146	C₂×C₄²×C₇	—	2	1	no	—
147	C₂×C₄²×C₇	—	2	1	no	—
148	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
149	C₇×#107	—	2	1	no	—
150	C₇×#107	—	2	2	no	—
151	C₇×#106	—	2	1	no	—
152	C₇×#106	—	2	1	no	—
153	C₇×#106	—	2	1	no	—
154	C₇×#106	—	2	2	no	—
155	C₇×C₄⋊C₈	—	2	1	no	—
156	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
157	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₇×#103	—	2	1	no	—
161	C₇×#103	—	2	1	no	—
162	C₇×#103	—	2	1	no	—
163	C₇×#103	—	2	1	no	—
164	C₇×#103	—	2	1	no	—
165	C₇×#103	—	2	1	no	—
166	C₇×#103	—	2	1	no	—
167	C₇×#102	—	2	1	no	—
168	C₇×#102	—	2	2	no	—
169	C₇×#102	—	2	2	no	—
170	C₇×C₂²⋊C₈	—	2	1	no	—
171	C₇×C₂²⋊C₈	—	2	1	no	—
172	C₇×C₂²⋊C₈	—	2	1	no	—
173	C₇×#97	—	2	1	no	—
174	C₄×C₇×C₈	—	2	1	no	—
175	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
176	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
177	#3480	C₂×C₇	1	1	yes	characteristic
178	#3228	C₂×C₇	1	1	yes	characteristic, complemented
179	#3113	C₂×C₇	1	1	yes	characteristic, complemented
180	#3107	C₂×C₇	1	1	yes	characteristic, complemented
181	#1653	C₂×C₇	1	1	yes	characteristic, complemented
182	#1652	C₂×C₇	1	1	yes	characteristic, complemented
183	#1650	C₂×C₇	1	1	yes	characteristic, complemented
184	#1648	C₂×C₇	1	1	yes	characteristic, complemented
185	#1642	C₂×C₇	1	1	yes	characteristic, complemented
186	#1641	C₂×C₇	1	1	yes	characteristic, complemented
187	#1640	C₂×C₇	1	1	yes	characteristic, complemented
188	#1640	C₂×C₇	1	1	yes	characteristic, complemented
189	#1553	C₂×C₇	1	1	yes	characteristic, complemented
190	#1488	C₂×C₇	1	1	yes	characteristic, complemented
191	#1479	C₂×C₇	1	1	yes	characteristic, complemented
192	C₇×#41	—	4	1	no	—
193	C₇×#41	—	4	1	no	—
194	C₇×#41	—	4	2	no	—
195	C₇×#41	—	4	2	no	—
196	C₇×#41	—	4	2	no	—
197	C₇×#41	—	4	2	no	—
198	C₂×C₇×Q₈	—	2	1	no	—
199	C₂×C₇×Q₈	—	2	1	no	—
200	C₂×C₇×Q₈	—	2	2	no	—
201	C₂×C₇×Q₈	—	2	1	no	—
202	C₂×C₇×Q₈	—	2	1	no	—
203	C₂×C₇×Q₈	—	4	1	no	—
204	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
207	C₂×C₇×D₄	—	2	2	no	—
208	C₂×C₇×D₄	—	2	1	no	—
209	C₂×C₇×D₄	—	2	2	no	—
210	C₂×C₇×D₄	—	2	2	no	—
211	C₂×C₇×D₄	—	2	1	no	—
212	C₂×C₇×D₄	—	2	1	no	—
213	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
214	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₇	—	2	2	no	—
216	C₂²×C₄×C₇	—	2	1	no	—
217	C₂²×C₄×C₇	—	2	2	no	—
218	C₂²×C₄×C₇	—	2	1	no	—
219	C₂²×C₄×C₇	—	2	1	no	—
220	C₂²×C₄×C₇	—	2	2	no	—
221	C₂²×C₄×C₇	—	2	1	no	—
222	C₂²×C₄×C₇	—	2	1	no	—
223	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
226	C₇×Q₁₆	—	4	2	no	—
227	C₇×Q₁₆	—	4	2	no	—
228	C₇×Q₁₆	—	4	2	no	—
229	C₇×SD₁₆	—	4	2	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	C₇×C₄⋊C₄	—	2	2	no	—
237	C₇×C₄⋊C₄	—	2	2	no	—
238	C₇×C₄⋊C₄	—	2	2	no	—
239	C₇×C₄⋊C₄	—	2	2	no	—
240	C₇×C₄⋊C₄	—	2	1	no	—
241	C₇×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
249	C₇×C₄⋊C₄	—	2	1	no	—
250	C₇×C₄⋊C₄	—	2	1	no	—
251	C₇×C₄⋊C₄	—	2	1	no	—
252	C₇×C₄⋊C₄	—	4	1	no	—
253	C₇×C₄⋊C₄	—	4	1	no	—
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×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
258	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	2	no	—
261	C₇×C₂²⋊C₄	—	2	2	no	—
262	C₇×C₂²⋊C₄	—	2	2	no	—
263	C₇×C₂²⋊C₄	—	2	2	no	—
264	C₇×C₂²⋊C₄	—	2	2	no	—
265	C₇×C₂²⋊C₄	—	2	1	no	—
266	C₇×C₂²⋊C₄	—	2	2	no	—
267	C₇×C₂²⋊C₄	—	2	1	no	—
268	C₇×C₂²⋊C₄	—	2	2	no	—
269	C₇×C₂²⋊C₄	—	2	1	no	—
270	C₇×C₂²⋊C₄	—	4	1	no	—
271	C₄²×C₇	—	2	2	no	—
272	C₄²×C₇	—	2	2	no	—
273	C₄²×C₇	—	2	1	no	—
274	C₄²×C₇	—	2	1	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	—	2	1	no	—
277	C₄²×C₇	—	2	1	no	—
278	C₄²×C₇	—	2	1	no	—
279	C₄²×C₇	—	2	1	no	—
280	C₄²×C₇	—	4	1	no	—
281	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
284	C₄²×C₇	C₂⁴	1	1	yes	characteristic
285	#559	—	2	1	no	—
286	#559	C₂²×C₇	1	1	yes	characteristic
287	#557	C₂²×C₇	1	1	yes	characteristic
288	#556	—	2	1	no	—
289	#556	C₂²×C₇	1	1	yes	characteristic
290	#553	—	2	2	no	—
291	#553	—	2	1	no	—
292	#550	C₂²×C₇	1	1	yes	characteristic
293	#548	—	2	1	no	—
294	#533	C₂²×C₇	1	1	yes	characteristic
295	#532	—	2	1	no	—
296	#532	C₂²×C₇	1	1	yes	characteristic
297	#529	—	2	1	no	—
298	#529	—	2	2	no	—
299	C₄×#41	—	2	1	no	—
300	C₄×#41	—	2	1	no	—
301	C₄×#41	C₂²×C₇	1	1	yes	characteristic
302	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
303	#500	C₂²×C₇	1	1	yes	characteristic
304	#498	C₂²×C₇	1	1	yes	characteristic
305	#486	C₂²×C₇	1	1	yes	characteristic
306	#486	C₂²×C₇	1	1	yes	characteristic
307	#485	—	2	1	no	—
308	#483	—	2	1	no	—
309	#482	—	2	2	no	—
310	#478	—	2	2	no	—
311	#475	C₂²×C₇	1	1	yes	characteristic
312	#475	C₂²×C₇	1	1	yes	characteristic
313	#475	C₂²×C₇	1	1	yes	characteristic
314	#474	C₂²×C₇	1	1	yes	characteristic
315	#462	C₂²×C₇	1	1	yes	characteristic
316	#462	C₂²×C₇	1	1	yes	characteristic
317	#462	C₂²×C₇	1	1	yes	characteristic
318	#461	C₂²×C₇	1	1	yes	characteristic
319	#452	—	2	1	no	—
320	#451	—	2	1	no	—
321	#441	—	2	1	no	—
322	#441	—	2	1	no	—
323	C₄×Q₁₆	—	2	1	no	—
324	C₄×SD₁₆	—	2	1	no	—
325	#432	C₂²×C₇	1	1	yes	characteristic
326	#431	C₂²×C₇	1	1	yes	characteristic
327	#431	C₂²×C₇	1	1	yes	characteristic
328	#332	C₂²×C₇	1	1	yes	characteristic
329	#332	C₂²×C₇	1	1	yes	characteristic
330	#331	C₂²×C₇	1	1	yes	characteristic
331	#331	C₂²×C₇	1	1	yes	characteristic
332	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
335	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
336	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
337	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
338	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
339	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
340	C₂³×C₇	—	2	1	no	—
341	C₂³×C₇	—	2	2	no	—
342	C₂³×C₇	#120	1	1	yes	characteristic
343	C₇×Q₈	—	2	1	no	—
344	C₇×Q₈	—	2	1	no	—
345	C₇×Q₈	—	2	1	no	—
346	C₇×Q₈	—	2	1	no	—
347	C₇×Q₈	—	2	1	no	—
348	C₇×Q₈	—	4	2	no	—
349	C₇×Q₈	—	4	1	no	—
350	C₇×Q₈	—	4	1	no	—
351	C₇×Q₈	—	4	1	no	—
352	C₇×Q₈	—	4	1	no	—
353	C₇×Q₈	—	4	1	no	—
354	C₇×Q₈	—	4	1	no	—
355	C₇×Q₈	#135	1	2	yes	—
356	C₇×D₄	—	2	1	no	—
357	C₇×D₄	—	2	1	no	—
358	C₇×D₄	—	4	2	no	—
359	C₇×D₄	—	4	4	no	—
360	C₇×D₄	—	4	2	no	—
361	C₇×D₄	—	4	4	no	—
362	C₇×D₄	—	4	1	no	—
363	C₇×D₄	—	4	1	no	—
364	C₇×D₄	—	4	1	no	—
365	C₇×D₄	—	4	1	no	—
366	C₇×D₄	—	8	2	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	—	2	1	no	—
371	C₂×C₄×C₇	—	2	1	no	—
372	C₂×C₄×C₇	—	2	1	no	—
373	C₂×C₄×C₇	—	2	1	no	—
374	C₂×C₄×C₇	—	2	1	no	—
375	C₂×C₄×C₇	—	2	1	no	—
376	C₂×C₄×C₇	—	2	1	no	—
377	C₂×C₄×C₇	—	2	2	no	—
378	C₂×C₄×C₇	—	2	1	no	—
379	C₂×C₄×C₇	—	4	1	no	—
380	C₂×C₄×C₇	—	4	2	no	—
381	C₂×C₄×C₇	—	4	4	no	—
382	C₂×C₄×C₇	—	4	2	no	—
383	C₂×C₄×C₇	—	4	4	no	—
384	C₂×C₄×C₇	—	4	1	no	—
385	C₂×C₄×C₇	—	4	1	no	—
386	C₂×C₄×C₇	—	4	1	no	—
387	C₂×C₄×C₇	—	4	1	no	—
388	C₂×C₄×C₇	—	8	1	no	—
389	C₂×C₄×C₇	—	8	1	no	—
390	C₂×C₄×C₇	—	8	2	no	—
391	C₂×C₄×C₇	#120	1	1	yes	characteristic
392	C₂×C₄×C₇	#120	1	1	yes	characteristic
393	C₂×C₄×C₇	#120	1	1	yes	characteristic
394	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
395	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
396	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
397	C₇×C₈	—	4	1	no	—
398	C₇×C₈	—	4	1	no	—
399	C₇×C₈	—	4	1	no	—
400	C₇×C₈	—	4	1	no	—
401	C₇×C₈	—	4	1	no	—
402	C₇×C₈	—	4	1	no	—
403	C₂×#41	—	2	1	no	—
404	C₂×#41	—	2	1	no	—
405	C₂×#41	C₇×D₄	1	1	yes	characteristic
406	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
407	C₂×Q₁₆	—	2	1	no	—
408	C₂×Q₁₆	—	2	1	no	—
409	C₂×Q₁₆	—	2	1	no	—
410	C₂×SD₁₆	—	2	1	no	—
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
413	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
414	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
415	#127	—	2	1	no	—
416	#127	C₂³×C₇	1	1	yes	characteristic
417	#126	—	2	2	no	—
418	#126	—	2	2	no	—
419	#126	—	2	2	no	—
420	#125	—	2	2	no	—
421	#125	—	2	1	no	—
422	#125	—	2	1	no	—
423	#124	—	2	2	no	—
424	#124	—	2	1	no	—
425	#124	—	2	1	no	—
426	#124	—	2	1	no	—
427	#124	—	2	1	no	—
428	#123	—	2	2	no	—
429	#123	—	2	2	no	—
430	#123	—	2	2	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	—	2	1	no	—
433	C₂²⋊Q₈	—	2	1	no	—
434	C₂²⋊Q₈	—	2	1	no	—
435	#121	—	2	2	no	—
436	#121	—	2	2	no	—
437	#121	—	2	2	no	—
438	#121	—	2	1	no	—
439	#121	—	2	1	no	—
440	#121	C₇×D₄	1	2	yes	—
441	C₄×Q₈	—	2	1	no	—
442	C₄×Q₈	—	2	1	no	—
443	C₄×Q₈	—	2	1	no	—
444	C₄×Q₈	—	2	1	no	—
445	C₄×Q₈	—	2	1	no	—
446	C₄×Q₈	—	2	1	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
449	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
450	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
451	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
452	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
453	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
454	C₄×D₄	—	2	2	no	—
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	1	no	—
460	C₄×D₄	—	2	1	no	—
461	C₄×D₄	—	2	1	no	—
462	C₄×D₄	—	2	1	no	—
463	C₄×D₄	—	2	1	no	—
464	C₄×D₄	—	2	1	no	—
465	C₄×D₄	C₇×D₄	1	1	yes	characteristic
466	C₄×D₄	C₇×D₄	1	2	yes	—
467	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
468	#117	—	2	2	no	—
469	#117	—	2	2	no	—
470	#117	—	2	1	no	—
471	#117	—	2	1	no	—
472	#117	—	2	1	no	—
473	#117	C₇×D₄	1	1	yes	characteristic
474	C₂×C₄⋊C₄	—	2	1	no	—
475	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
476	C₂×C₄²	—	2	2	no	—
477	C₂×C₄²	—	2	1	no	—
478	C₂×C₄²	—	2	1	no	—
479	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
480	#107	—	2	1	no	—
481	#107	—	2	2	no	—
482	#106	—	2	1	no	—
483	#106	—	2	1	no	—
484	#106	—	2	1	no	—
485	#106	—	2	2	no	—
486	C₄⋊C₈	—	2	1	no	—
487	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
488	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
489	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
490	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
491	#103	—	2	1	no	—
492	#103	—	2	1	no	—
493	#103	—	2	1	no	—
494	#103	—	2	1	no	—
495	#103	—	2	1	no	—
496	#103	—	2	1	no	—
497	#103	—	2	1	no	—
498	#102	—	2	1	no	—
499	#102	—	2	2	no	—
500	#102	—	2	2	no	—
501	C₂²⋊C₈	—	2	1	no	—
502	C₂²⋊C₈	—	2	1	no	—
503	C₂²⋊C₈	—	2	1	no	—
504	#97	—	2	1	no	—
505	C₄×C₈	—	2	1	no	—
506	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
507	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
508	C₂²×C₇	—	2	1	no	—
509	C₂²×C₇	—	2	1	no	—
510	C₂²×C₇	—	2	1	no	—
511	C₂²×C₇	—	4	2	no	—
512	C₂²×C₇	—	4	1	no	—
513	C₂²×C₇	—	4	1	no	—
514	C₂²×C₇	—	4	2	no	—
515	C₂²×C₇	—	4	2	no	—
516	C₂²×C₇	—	4	1	no	—
517	C₂²×C₇	C₂×#120	1	1	yes	characteristic
518	C₄×C₇	—	2	1	no	—
519	C₄×C₇	—	2	2	no	—
520	C₄×C₇	—	2	1	no	—
521	C₄×C₇	—	2	1	no	—
522	C₄×C₇	—	2	1	no	—
523	C₄×C₇	—	4	1	no	—
524	C₄×C₇	—	4	1	no	—
525	C₄×C₇	—	4	1	no	—
526	C₄×C₇	—	4	2	no	—
527	C₄×C₇	—	4	1	no	—
528	C₄×C₇	—	4	1	no	—
529	C₄×C₇	—	4	1	no	—
530	C₄×C₇	—	8	1	no	—
531	C₄×C₇	—	8	1	no	—
532	C₄×C₇	C₂×#135	1	1	yes	characteristic
533	C₄×C₇	#575	1	1	yes	characteristic
534	C₄×C₇	#575	1	1	yes	characteristic
535	C₄×C₇	#575	1	1	yes	characteristic
536	#41	—	4	1	no	—
537	#41	—	4	1	no	—
538	#41	—	4	2	no	—
539	#41	—	4	2	no	—
540	#41	—	4	2	no	—
541	#41	—	4	2	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	—	2	1	no	—
544	C₂×Q₈	—	2	2	no	—
545	C₂×Q₈	—	2	1	no	—
546	C₂×Q₈	—	2	1	no	—
547	C₂×Q₈	—	4	1	no	—
548	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
549	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
550	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	2	2	no	—
554	C₂×D₄	—	2	2	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	1	no	—
557	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
558	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	2	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	2	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
568	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
569	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
570	Q₁₆	—	4	2	no	—
571	Q₁₆	—	4	2	no	—
572	Q₁₆	—	4	2	no	—
573	SD₁₆	—	4	2	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	2	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	2	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	4	1	no	—
597	C₄⋊C₄	—	4	1	no	—
598	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
599	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
600	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
601	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
602	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	2	no	—
608	C₂²⋊C₄	—	2	2	no	—
609	C₂²⋊C₄	—	2	1	no	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	1	no	—
612	C₂²⋊C₄	—	2	2	no	—
613	C₂²⋊C₄	—	2	1	no	—
614	C₂²⋊C₄	—	4	1	no	—
615	C₄²	—	2	2	no	—
616	C₄²	—	2	2	no	—
617	C₄²	—	2	1	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	2	1	no	—
620	C₄²	—	2	1	no	—
621	C₄²	—	2	1	no	—
622	C₄²	—	2	1	no	—
623	C₄²	—	2	1	no	—
624	C₄²	—	4	1	no	—
625	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
626	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
627	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
628	C₄²	C₂⁴×C₇	1	1	yes	characteristic
629	C₂×C₇	—	4	1	no	—
630	C₂×C₇	—	8	1	no	—
631	C₂×C₇	—	8	2	no	—
632	C₂×C₇	#3002	1	1	yes	characteristic
633	C₂×C₇	#3005	1	1	yes	characteristic
634	C₂×C₇	#3022	1	1	yes	characteristic
635	C₂³	—	2	1	no	—
636	C₂³	—	2	2	no	—
637	C₂³	C₇×#120	1	1	yes	characteristic
638	Q₈	—	2	1	no	—
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	2	1	no	—
642	Q₈	—	2	1	no	—
643	Q₈	—	4	2	no	—
644	Q₈	—	4	1	no	—
645	Q₈	—	4	1	no	—
646	Q₈	—	4	1	no	—
647	Q₈	—	4	1	no	—
648	Q₈	—	4	1	no	—
649	Q₈	—	4	1	no	—
650	Q₈	C₇×#135	1	2	yes	—
651	D₄	—	2	1	no	—
652	D₄	—	2	1	no	—
653	D₄	—	4	2	no	—
654	D₄	—	4	4	no	—
655	D₄	—	4	2	no	—
656	D₄	—	4	4	no	—
657	D₄	—	4	1	no	—
658	D₄	—	4	1	no	—
659	D₄	—	4	1	no	—
660	D₄	—	4	1	no	—
661	D₄	—	8	2	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	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	—	2	2	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	2	no	—
676	C₂×C₄	—	4	4	no	—
677	C₂×C₄	—	4	2	no	—
678	C₂×C₄	—	4	4	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	—	8	1	no	—
684	C₂×C₄	—	8	1	no	—
685	C₂×C₄	—	8	2	no	—
686	C₂×C₄	C₇×#120	1	1	yes	characteristic
687	C₂×C₄	C₇×#120	1	1	yes	characteristic
688	C₂×C₄	C₇×#120	1	1	yes	characteristic
689	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
690	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
691	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
692	C₈	—	4	1	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₈	—	4	1	no	—
697	C₈	—	4	1	no	—
698	C₇	#18808	1	1	yes	characteristic, direct product factor, Sylow
699	C₂²	—	2	1	no	—
700	C₂²	—	2	1	no	—
701	C₂²	—	2	1	no	—
702	C₂²	—	4	2	no	—
703	C₂²	—	4	1	no	—
704	C₂²	—	4	1	no	—
705	C₂²	—	4	2	no	—
706	C₂²	—	4	2	no	—
707	C₂²	—	4	1	no	—
708	C₂²	C₂×C₇×#120	1	1	yes	characteristic
709	C₄	—	2	1	no	—
710	C₄	—	2	2	no	—
711	C₄	—	2	1	no	—
712	C₄	—	2	1	no	—
713	C₄	—	2	1	no	—
714	C₄	—	4	1	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	4	1	no	—
721	C₄	—	8	1	no	—
722	C₄	—	8	1	no	—
723	C₄	C₂×C₇×#135	1	1	yes	characteristic
724	C₄	C₇×#575	1	1	yes	characteristic
725	C₄	C₇×#575	1	1	yes	characteristic
726	C₄	C₇×#575	1	1	yes	characteristic
727	C₂	—	4	1	no	—
728	C₂	—	8	1	no	—
729	C₂	—	8	2	no	—
730	C₂	C₇×#3002	1	1	yes	characteristic
731	C₂	C₇×#3005	1	1	yes	characteristic
732	C₂	C₇×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3480	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3228	C₂	semidirect
3. C₇×#3113	C₂	semidirect
4. C₇×#3107	C₂	semidirect
5. C₇×#1653	C₂	semidirect
6. C₇×#1652	C₂	semidirect
7. C₇×#1650	C₂	semidirect
8. C₇×#1648	C₂	semidirect
9. C₇×#1642	C₂	semidirect
10. C₇×#1641	C₂	semidirect
11. C₇×#1640	C₂	semidirect
12. C₇×#1640	C₂	semidirect
13. C₇×#1553	C₂	semidirect
14. C₇×#1488	C₂	semidirect
15. C₇×#1479	C₂	semidirect
71. #18808	C₇	direct (N × H)
178. #3228	C₂×C₇	semidirect
179. #3113	C₂×C₇	semidirect
180. #3107	C₂×C₇	semidirect
181. #1653	C₂×C₇	semidirect
182. #1652	C₂×C₇	semidirect
183. #1650	C₂×C₇	semidirect
184. #1648	C₂×C₇	semidirect
185. #1642	C₂×C₇	semidirect
186. #1641	C₂×C₇	semidirect
187. #1640	C₂×C₇	semidirect
188. #1640	C₂×C₇	semidirect
189. #1553	C₂×C₇	semidirect
190. #1488	C₂×C₇	semidirect
191. #1479	C₂×C₇	semidirect
698. C₇	#18808	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×#3480	2
C₇×#3228	2
C₇×#3113	2
C₇×#3107	2
C₇×#1653	2
C₇×#1652	2
C₇×#1650	2
C₇×#1648	2
C₇×#1642	2
C₇×#1641	2
C₇×#1640	2
C₇×#1553	2
C₇×#1488	2
C₇×#1479	2
#18808	7

Subgroups — C7×#18808 #13196024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18808 #13196024