Subgroups — C₇×#18963 #13196179

Order 1792.

844 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3247	C₂	1	1	yes	maximal, characteristic
4	C₇×#3210	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1638	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1636	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1633	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1632	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1623	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1550	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1495	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#563	—	2	2	no	—
17	C₇×Q₈²	C₂²	1	1	yes	characteristic
18	C₇×#557	—	2	1	no	—
19	C₇×#557	C₂²	1	1	yes	characteristic
20	C₇×#556	—	2	1	no	—
21	C₇×#556	—	2	1	no	—
22	C₇×#556	C₂²	1	1	yes	characteristic
23	C₇×#554	C₂²	1	1	yes	characteristic
24	C₇×#552	—	2	1	no	—
25	C₇×#544	—	2	1	no	—
26	C₇×#542	—	2	1	no	—
27	C₇×#541	—	2	1	no	—
28	C₇×#541	—	2	1	no	—
29	C₇×#541	—	2	2	no	—
30	C₇×#533	C₂²	1	1	yes	characteristic
31	C₇×#533	C₂²	1	1	yes	characteristic
32	C₇×#532	C₂²	1	1	yes	characteristic
33	C₇×#529	C₂²	1	1	yes	characteristic
34	C₇×#494	C₂²	1	1	yes	characteristic
35	C₇×#492	C₂²	1	1	yes	characteristic
36	C₇×#487	C₂²	1	1	yes	characteristic
37	C₇×#486	C₂²	1	1	yes	characteristic
38	C₇×#478	—	2	1	no	—
39	C₇×#477	C₂²	1	1	yes	characteristic
40	C₇×#477	C₂²	1	1	yes	characteristic
41	C₇×#477	C₂²	1	1	yes	characteristic
42	C₇×#476	C₂²	1	1	yes	characteristic
43	C₇×#474	—	2	1	no	—
44	C₇×#468	—	2	1	no	—
45	C₇×#468	—	2	1	no	—
46	C₇×#466	—	2	1	no	—
47	C₇×#465	—	2	1	no	—
48	C₇×#464	—	2	2	no	—
49	C₇×#463	—	2	1	no	—
50	C₇×#461	—	2	1	no	—
51	C₇×#460	C₂²	1	1	yes	characteristic
52	C₇×#460	C₂²	1	1	yes	characteristic
53	C₇×#460	C₂²	1	1	yes	characteristic
54	C₇×#459	C₂²	1	1	yes	characteristic
55	C₇×#452	—	2	2	no	—
56	C₇×#431	C₂²	1	1	yes	characteristic
57	C₇×#423	C₂²	1	1	yes	characteristic
58	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₇×#333	C₂²	1	1	yes	characteristic
60	C₇×#333	C₂²	1	1	yes	characteristic
61	C₇×#332	C₂²	1	1	yes	characteristic
62	C₇×#332	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	#18963	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×Q₁₆	—	2	2	no	—
74	C₂×C₇×SD₁₆	—	2	1	no	—
75	C₂×C₇×SD₁₆	—	2	1	no	—
76	C₂×C₇×SD₁₆	—	2	2	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂×C₇×D₈	—	2	1	no	—
79	C₂×C₇×M₁₆	—	2	1	no	—
80	C₂²×C₇×C₈	—	2	1	no	—
81	C₇×C₄⋊Q₈	—	2	2	no	—
82	C₇×C₄⋊Q₈	—	2	1	no	—
83	C₇×C₄⋊Q₈	—	2	2	no	—
84	C₇×C₄⋊Q₈	—	2	1	no	—
85	C₇×C₄⋊Q₈	—	2	1	no	—
86	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
87	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
88	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₇×#127	C₂³	1	1	yes	characteristic
92	C₇×#126	—	2	2	no	—
93	C₇×#126	—	2	2	no	—
94	C₇×#126	—	2	2	no	—
95	C₇×#126	—	2	2	no	—
96	C₇×#126	—	2	2	no	—
97	C₇×#126	—	2	2	no	—
98	C₇×#126	—	2	1	no	—
99	C₇×#125	—	2	2	no	—
100	C₇×#125	—	2	2	no	—
101	C₇×#125	—	2	2	no	—
102	C₇×#125	—	2	2	no	—
103	C₇×#125	—	2	1	no	—
104	C₇×#125	—	2	2	no	—
105	C₇×#125	—	2	1	no	—
106	C₇×#125	—	2	1	no	—
107	C₇×#125	—	2	1	no	—
108	C₇×#125	D₄	1	1	yes	characteristic
109	C₇×#124	—	2	2	no	—
110	C₇×#124	—	2	1	no	—
111	C₇×#124	—	2	1	no	—
112	C₇×#124	D₄	1	1	yes	characteristic
113	C₇×#123	—	2	2	no	—
114	C₇×#123	—	2	1	no	—
115	C₇×C₂²⋊Q₈	—	2	2	no	—
116	C₇×C₂²⋊Q₈	—	2	2	no	—
117	C₇×C₂²⋊Q₈	—	2	1	no	—
118	C₇×C₂²⋊Q₈	—	2	1	no	—
119	C₇×C₂²⋊Q₈	—	2	2	no	—
120	C₇×C₂²⋊Q₈	—	2	1	no	—
121	C₇×C₂²⋊Q₈	—	2	1	no	—
122	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
123	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
124	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
125	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
126	C₇×#121	—	2	1	no	—
127	C₇×#121	—	2	1	no	—
128	C₇×#121	D₄	1	1	yes	characteristic
129	C₄×C₇×Q₈	—	2	2	no	—
130	C₄×C₇×Q₈	—	2	1	no	—
131	C₄×C₇×Q₈	—	2	1	no	—
132	C₄×C₇×Q₈	—	2	1	no	—
133	C₄×C₇×Q₈	—	2	1	no	—
134	C₄×C₇×Q₈	—	2	1	no	—
135	C₄×C₇×Q₈	—	2	1	no	—
136	C₄×C₇×Q₈	—	2	1	no	—
137	C₄×C₇×Q₈	D₄	1	2	yes	—
138	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
139	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
140	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
141	C₄×C₇×D₄	—	2	1	no	—
142	C₄×C₇×D₄	—	2	1	no	—
143	C₄×C₇×D₄	—	2	1	no	—
144	C₄×C₇×D₄	—	2	1	no	—
145	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
146	C₇×#117	—	2	2	no	—
147	C₇×#117	—	2	1	no	—
148	C₇×#117	—	2	1	no	—
149	C₇×#117	—	2	1	no	—
150	C₇×#117	—	2	1	no	—
151	C₂×C₇×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
153	C₇×#107	—	2	1	no	—
154	C₇×#107	—	2	1	no	—
155	C₇×#107	—	2	1	no	—
156	C₇×#106	—	2	1	no	—
157	C₇×C₄⋊C₈	—	2	1	no	—
158	C₇×C₄⋊C₈	—	2	1	no	—
159	C₇×C₄⋊C₈	—	2	1	no	—
160	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₇×#103	—	2	1	no	—
165	C₇×#103	—	2	1	no	—
166	C₇×#103	—	2	1	no	—
167	C₇×#103	—	2	2	no	—
168	C₇×#102	—	2	1	no	—
169	C₇×#102	—	2	1	no	—
170	C₇×#102	—	2	1	no	—
171	C₇×#102	—	2	1	no	—
172	C₇×#102	—	2	1	no	—
173	C₇×#102	—	2	2	no	—
174	C₇×C₂²⋊C₈	—	2	1	no	—
175	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
176	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
177	#3555	C₂×C₇	1	1	yes	characteristic, complemented
178	#3254	C₂×C₇	1	1	yes	characteristic, complemented
179	#3247	C₂×C₇	1	1	yes	characteristic
180	#3210	C₂×C₇	1	1	yes	characteristic, complemented
181	#1638	C₂×C₇	1	1	yes	characteristic, complemented
182	#1636	C₂×C₇	1	1	yes	characteristic, complemented
183	#1633	C₂×C₇	1	1	yes	characteristic, complemented
184	#1632	C₂×C₇	1	1	yes	characteristic, complemented
185	#1628	C₂×C₇	1	1	yes	characteristic, complemented
186	#1626	C₂×C₇	1	1	yes	characteristic, complemented
187	#1623	C₂×C₇	1	1	yes	characteristic, complemented
188	#1622	C₂×C₇	1	1	yes	characteristic, complemented
189	#1550	C₂×C₇	1	1	yes	characteristic, complemented
190	#1495	C₂×C₇	1	1	yes	characteristic, complemented
191	#1481	C₂×C₇	1	1	yes	characteristic, complemented
192	C₇×#41	—	8	2	no	—
193	C₂×C₇×Q₈	—	2	2	no	—
194	C₂×C₇×Q₈	—	2	1	no	—
195	C₂×C₇×Q₈	—	2	1	no	—
196	C₂×C₇×Q₈	—	2	1	no	—
197	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
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×D₄	1	2	yes	—
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	—	4	1	no	—
206	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₇	—	2	2	no	—
208	C₂²×C₄×C₇	—	2	1	no	—
209	C₂²×C₄×C₇	—	4	1	no	—
210	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
213	C₇×Q₁₆	—	8	2	no	—
214	C₇×SD₁₆	—	8	1	no	—
215	C₇×SD₁₆	—	8	1	no	—
216	C₇×SD₁₆	—	8	1	no	—
217	C₇×SD₁₆	—	8	2	no	—
218	C₇×D₈	—	8	1	no	—
219	C₇×M₁₆	—	8	1	no	—
220	C₂×C₇×C₈	—	2	1	no	—
221	C₂×C₇×C₈	—	2	1	no	—
222	C₂×C₇×C₈	—	2	1	no	—
223	C₂×C₇×C₈	—	2	1	no	—
224	C₂×C₇×C₈	—	2	1	no	—
225	C₂×C₇×C₈	—	2	1	no	—
226	C₂×C₇×C₈	—	8	1	no	—
227	C₇×C₄⋊C₄	—	2	2	no	—
228	C₇×C₄⋊C₄	—	2	2	no	—
229	C₇×C₄⋊C₄	—	2	2	no	—
230	C₇×C₄⋊C₄	—	2	2	no	—
231	C₇×C₄⋊C₄	—	2	2	no	—
232	C₇×C₄⋊C₄	—	2	2	no	—
233	C₇×C₄⋊C₄	—	2	2	no	—
234	C₇×C₄⋊C₄	—	2	2	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	1	no	—
237	C₇×C₄⋊C₄	—	2	1	no	—
238	C₇×C₄⋊C₄	—	2	1	no	—
239	C₇×C₄⋊C₄	—	2	1	no	—
240	C₇×C₄⋊C₄	—	2	1	no	—
241	C₇×C₄⋊C₄	—	2	1	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	1	no	—
244	C₇×C₄⋊C₄	—	2	2	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	—	2	1	no	—
253	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
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×C₂²⋊C₄	—	2	2	no	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	2	no	—
261	C₇×C₂²⋊C₄	—	2	1	no	—
262	C₇×C₂²⋊C₄	—	2	2	no	—
263	C₇×C₂²⋊C₄	—	2	1	no	—
264	C₇×C₂²⋊C₄	—	2	1	no	—
265	C₇×C₂²⋊C₄	—	2	1	no	—
266	C₇×C₂²⋊C₄	—	2	1	no	—
267	C₇×C₂²⋊C₄	—	2	1	no	—
268	C₇×C₂²⋊C₄	—	2	1	no	—
269	C₇×C₂²⋊C₄	—	4	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	2	no	—
274	C₄²×C₇	—	2	2	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	—	2	1	no	—
277	C₄²×C₇	—	2	1	no	—
278	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
281	C₄²×C₇	C₂⁴	1	1	yes	characteristic
282	#563	—	2	2	no	—
283	Q₈²	C₂²×C₇	1	1	yes	characteristic
284	#557	—	2	1	no	—
285	#557	C₂²×C₇	1	1	yes	characteristic
286	#556	—	2	1	no	—
287	#556	—	2	1	no	—
288	#556	C₂²×C₇	1	1	yes	characteristic
289	#554	C₂²×C₇	1	1	yes	characteristic
290	#552	—	2	1	no	—
291	#544	—	2	1	no	—
292	#542	—	2	1	no	—
293	#541	—	2	1	no	—
294	#541	—	2	1	no	—
295	#541	—	2	2	no	—
296	#533	C₂²×C₇	1	1	yes	characteristic
297	#533	C₂²×C₇	1	1	yes	characteristic
298	#532	C₂²×C₇	1	1	yes	characteristic
299	#529	C₂²×C₇	1	1	yes	characteristic
300	#494	C₂²×C₇	1	1	yes	characteristic
301	#492	C₂²×C₇	1	1	yes	characteristic
302	#487	C₂²×C₇	1	1	yes	characteristic
303	#486	C₂²×C₇	1	1	yes	characteristic
304	#478	—	2	1	no	—
305	#477	C₂²×C₇	1	1	yes	characteristic
306	#477	C₂²×C₇	1	1	yes	characteristic
307	#477	C₂²×C₇	1	1	yes	characteristic
308	#476	C₂²×C₇	1	1	yes	characteristic
309	#474	—	2	1	no	—
310	#468	—	2	1	no	—
311	#468	—	2	1	no	—
312	#466	—	2	1	no	—
313	#465	—	2	1	no	—
314	#464	—	2	2	no	—
315	#463	—	2	1	no	—
316	#461	—	2	1	no	—
317	#460	C₂²×C₇	1	1	yes	characteristic
318	#460	C₂²×C₇	1	1	yes	characteristic
319	#460	C₂²×C₇	1	1	yes	characteristic
320	#459	C₂²×C₇	1	1	yes	characteristic
321	#452	—	2	2	no	—
322	#431	C₂²×C₇	1	1	yes	characteristic
323	#423	C₂²×C₇	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
325	#333	C₂²×C₇	1	1	yes	characteristic
326	#333	C₂²×C₇	1	1	yes	characteristic
327	#332	C₂²×C₇	1	1	yes	characteristic
328	#332	C₂²×C₇	1	1	yes	characteristic
329	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
330	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
331	Q₈⋊C₈	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	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
336	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
337	C₂³×C₇	—	2	1	no	—
338	C₂³×C₇	—	4	1	no	—
339	C₂³×C₇	#120	1	1	yes	characteristic
340	C₇×Q₈	—	2	1	no	—
341	C₇×Q₈	—	2	1	no	—
342	C₇×Q₈	—	2	1	no	—
343	C₇×Q₈	—	2	1	no	—
344	C₇×Q₈	—	2	1	no	—
345	C₇×Q₈	—	2	1	no	—
346	C₇×Q₈	—	4	2	no	—
347	C₇×Q₈	—	4	2	no	—
348	C₇×Q₈	—	4	1	no	—
349	C₇×Q₈	—	4	1	no	—
350	C₇×Q₈	—	8	1	no	—
351	C₇×Q₈	—	8	2	no	—
352	C₇×D₄	—	2	1	no	—
353	C₇×D₄	—	2	1	no	—
354	C₇×D₄	—	4	2	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	—	8	1	no	—
360	C₇×D₄	—	8	2	no	—
361	C₂×C₄×C₇	—	2	1	no	—
362	C₂×C₄×C₇	—	2	1	no	—
363	C₂×C₄×C₇	—	2	1	no	—
364	C₂×C₄×C₇	—	2	1	no	—
365	C₂×C₄×C₇	—	2	1	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	2	2	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	2	no	—
373	C₂×C₄×C₇	—	2	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	—	8	2	no	—
379	C₂×C₄×C₇	—	8	2	no	—
380	C₂×C₄×C₇	#120	1	1	yes	characteristic
381	C₂×C₄×C₇	#120	1	1	yes	characteristic
382	C₂×C₄×C₇	#120	1	1	yes	characteristic
383	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
384	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
385	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
386	C₇×C₈	—	4	1	no	—
387	C₇×C₈	—	4	1	no	—
388	C₇×C₈	—	4	1	no	—
389	C₇×C₈	—	4	1	no	—
390	C₇×C₈	—	4	1	no	—
391	C₇×C₈	—	4	1	no	—
392	C₂×#41	—	2	1	no	—
393	C₂×Q₁₆	—	2	2	no	—
394	C₂×SD₁₆	—	2	2	no	—
395	C₂×SD₁₆	—	2	1	no	—
396	C₂×SD₁₆	—	2	1	no	—
397	C₂×SD₁₆	—	2	1	no	—
398	C₂×D₈	—	2	1	no	—
399	C₂×M₁₆	—	2	1	no	—
400	C₂²×C₈	—	2	1	no	—
401	C₄⋊Q₈	—	2	2	no	—
402	C₄⋊Q₈	—	2	1	no	—
403	C₄⋊Q₈	—	2	2	no	—
404	C₄⋊Q₈	—	2	1	no	—
405	C₄⋊Q₈	—	2	1	no	—
406	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
407	C₄⋊Q₈	C₇×D₄	1	1	yes	characteristic
408	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
409	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
410	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
411	#127	C₂³×C₇	1	1	yes	characteristic
412	#126	—	2	2	no	—
413	#126	—	2	2	no	—
414	#126	—	2	2	no	—
415	#126	—	2	2	no	—
416	#126	—	2	2	no	—
417	#126	—	2	2	no	—
418	#126	—	2	1	no	—
419	#125	—	2	2	no	—
420	#125	—	2	2	no	—
421	#125	—	2	2	no	—
422	#125	—	2	2	no	—
423	#125	—	2	1	no	—
424	#125	—	2	2	no	—
425	#125	—	2	1	no	—
426	#125	—	2	1	no	—
427	#125	—	2	1	no	—
428	#125	C₇×D₄	1	1	yes	characteristic
429	#124	—	2	2	no	—
430	#124	—	2	1	no	—
431	#124	—	2	1	no	—
432	#124	C₇×D₄	1	1	yes	characteristic
433	#123	—	2	2	no	—
434	#123	—	2	1	no	—
435	C₂²⋊Q₈	—	2	2	no	—
436	C₂²⋊Q₈	—	2	2	no	—
437	C₂²⋊Q₈	—	2	1	no	—
438	C₂²⋊Q₈	—	2	1	no	—
439	C₂²⋊Q₈	—	2	2	no	—
440	C₂²⋊Q₈	—	2	1	no	—
441	C₂²⋊Q₈	—	2	1	no	—
442	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
443	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
444	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
445	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
446	#121	—	2	1	no	—
447	#121	—	2	1	no	—
448	#121	C₇×D₄	1	1	yes	characteristic
449	C₄×Q₈	—	2	2	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	—	2	1	no	—
453	C₄×Q₈	—	2	1	no	—
454	C₄×Q₈	—	2	1	no	—
455	C₄×Q₈	—	2	1	no	—
456	C₄×Q₈	—	2	1	no	—
457	C₄×Q₈	C₇×D₄	1	2	yes	—
458	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
459	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
460	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
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	C₂³×C₇	1	1	yes	characteristic
466	#117	—	2	2	no	—
467	#117	—	2	1	no	—
468	#117	—	2	1	no	—
469	#117	—	2	1	no	—
470	#117	—	2	1	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
473	#107	—	2	1	no	—
474	#107	—	2	1	no	—
475	#107	—	2	1	no	—
476	#106	—	2	1	no	—
477	C₄⋊C₈	—	2	1	no	—
478	C₄⋊C₈	—	2	1	no	—
479	C₄⋊C₈	—	2	1	no	—
480	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
481	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
482	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
483	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
484	#103	—	2	1	no	—
485	#103	—	2	1	no	—
486	#103	—	2	1	no	—
487	#103	—	2	2	no	—
488	#102	—	2	1	no	—
489	#102	—	2	2	no	—
490	#102	—	2	1	no	—
491	#102	—	2	1	no	—
492	#102	—	2	1	no	—
493	#102	—	2	1	no	—
494	C₂²⋊C₈	—	2	1	no	—
495	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
496	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
497	C₂²×C₇	—	2	1	no	—
498	C₂²×C₇	—	2	1	no	—
499	C₂²×C₇	—	2	1	no	—
500	C₂²×C₇	—	4	1	no	—
501	C₂²×C₇	—	4	1	no	—
502	C₂²×C₇	—	4	1	no	—
503	C₂²×C₇	—	8	1	no	—
504	C₂²×C₇	—	8	1	no	—
505	C₂²×C₇	—	8	1	no	—
506	C₂²×C₇	C₂×#120	1	1	yes	characteristic
507	C₄×C₇	—	2	1	no	—
508	C₄×C₇	—	2	1	no	—
509	C₄×C₇	—	2	1	no	—
510	C₄×C₇	—	2	1	no	—
511	C₄×C₇	—	4	1	no	—
512	C₄×C₇	—	4	1	no	—
513	C₄×C₇	—	4	1	no	—
514	C₄×C₇	—	4	1	no	—
515	C₄×C₇	—	4	2	no	—
516	C₄×C₇	—	4	1	no	—
517	C₄×C₇	—	4	2	no	—
518	C₄×C₇	—	4	1	no	—
519	C₄×C₇	—	4	1	no	—
520	C₄×C₇	—	8	1	no	—
521	C₄×C₇	#577	1	1	yes	characteristic
522	C₄×C₇	#577	1	1	yes	characteristic
523	C₄×C₇	#578	1	1	yes	characteristic
524	C₄×C₇	#578	1	1	yes	characteristic
525	#41	—	8	2	no	—
526	C₂×Q₈	—	2	2	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
531	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
532	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
533	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	—	4	1	no	—
539	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	4	1	no	—
543	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
544	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
545	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
546	Q₁₆	—	8	2	no	—
547	SD₁₆	—	8	1	no	—
548	SD₁₆	—	8	1	no	—
549	SD₁₆	—	8	1	no	—
550	SD₁₆	—	8	2	no	—
551	D₈	—	8	1	no	—
552	M₁₆	—	8	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	—	8	1	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
587	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
588	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
589	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
590	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
591	C₂²⋊C₄	—	2	2	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	1	no	—
595	C₂²⋊C₄	—	2	2	no	—
596	C₂²⋊C₄	—	2	1	no	—
597	C₂²⋊C₄	—	2	1	no	—
598	C₂²⋊C₄	—	2	1	no	—
599	C₂²⋊C₄	—	2	1	no	—
600	C₂²⋊C₄	—	2	1	no	—
601	C₂²⋊C₄	—	2	1	no	—
602	C₂²⋊C₄	—	4	1	no	—
603	C₂²⋊C₄	—	4	1	no	—
604	C₄²	—	2	2	no	—
605	C₄²	—	2	2	no	—
606	C₄²	—	2	2	no	—
607	C₄²	—	2	2	no	—
608	C₄²	—	2	1	no	—
609	C₄²	—	2	1	no	—
610	C₄²	—	2	1	no	—
611	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
612	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
613	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
614	C₄²	C₂⁴×C₇	1	1	yes	characteristic
615	C₂×C₇	—	4	1	no	—
616	C₂×C₇	—	8	1	no	—
617	C₂×C₇	—	16	1	no	—
618	C₂×C₇	#3013	1	1	yes	characteristic
619	C₂×C₇	#3013	1	1	yes	characteristic
620	C₂×C₇	#3021	1	1	yes	characteristic
621	C₂³	—	2	1	no	—
622	C₂³	—	4	1	no	—
623	C₂³	C₇×#120	1	1	yes	characteristic
624	Q₈	—	2	1	no	—
625	Q₈	—	2	1	no	—
626	Q₈	—	2	1	no	—
627	Q₈	—	2	1	no	—
628	Q₈	—	2	1	no	—
629	Q₈	—	2	1	no	—
630	Q₈	—	4	2	no	—
631	Q₈	—	4	2	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	8	1	no	—
635	Q₈	—	8	2	no	—
636	D₄	—	2	1	no	—
637	D₄	—	2	1	no	—
638	D₄	—	4	2	no	—
639	D₄	—	4	1	no	—
640	D₄	—	4	1	no	—
641	D₄	—	4	1	no	—
642	D₄	—	4	1	no	—
643	D₄	—	8	1	no	—
644	D₄	—	8	2	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	8	2	no	—
663	C₂×C₄	—	8	2	no	—
664	C₂×C₄	C₇×#120	1	1	yes	characteristic
665	C₂×C₄	C₇×#120	1	1	yes	characteristic
666	C₂×C₄	C₇×#120	1	1	yes	characteristic
667	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
668	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
669	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
670	C₈	—	4	1	no	—
671	C₈	—	4	1	no	—
672	C₈	—	4	1	no	—
673	C₈	—	4	1	no	—
674	C₈	—	4	1	no	—
675	C₈	—	4	1	no	—
676	C₇	#18963	1	1	yes	characteristic, direct product factor, Sylow
677	C₂²	—	2	1	no	—
678	C₂²	—	2	1	no	—
679	C₂²	—	2	1	no	—
680	C₂²	—	4	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	1	no	—
683	C₂²	—	8	1	no	—
684	C₂²	—	8	1	no	—
685	C₂²	—	8	1	no	—
686	C₂²	C₂×C₇×#120	1	1	yes	characteristic
687	C₄	—	2	1	no	—
688	C₄	—	2	1	no	—
689	C₄	—	2	1	no	—
690	C₄	—	2	1	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	1	no	—
694	C₄	—	4	1	no	—
695	C₄	—	4	2	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	1	no	—
700	C₄	—	8	1	no	—
701	C₄	C₇×#577	1	1	yes	characteristic
702	C₄	C₇×#577	1	1	yes	characteristic
703	C₄	C₇×#578	1	1	yes	characteristic
704	C₄	C₇×#578	1	1	yes	characteristic
705	C₂	—	4	1	no	—
706	C₂	—	8	1	no	—
707	C₂	—	16	1	no	—
708	C₂	C₇×#3013	1	1	yes	characteristic
709	C₂	C₇×#3013	1	1	yes	characteristic
710	C₂	C₇×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3247	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
1. C₇×#3555	C₂	semidirect
2. C₇×#3254	C₂	semidirect
4. C₇×#3210	C₂	semidirect
5. C₇×#1638	C₂	semidirect
6. C₇×#1636	C₂	semidirect
7. C₇×#1633	C₂	semidirect
8. C₇×#1632	C₂	semidirect
9. C₇×#1628	C₂	semidirect
10. C₇×#1626	C₂	semidirect
11. C₇×#1623	C₂	semidirect
12. C₇×#1622	C₂	semidirect
13. C₇×#1550	C₂	semidirect
14. C₇×#1495	C₂	semidirect
15. C₇×#1481	C₂	semidirect
71. #18963	C₇	direct (N × H)
177. #3555	C₂×C₇	semidirect
178. #3254	C₂×C₇	semidirect
180. #3210	C₂×C₇	semidirect
181. #1638	C₂×C₇	semidirect
182. #1636	C₂×C₇	semidirect
183. #1633	C₂×C₇	semidirect
184. #1632	C₂×C₇	semidirect
185. #1628	C₂×C₇	semidirect
186. #1626	C₂×C₇	semidirect
187. #1623	C₂×C₇	semidirect
188. #1622	C₂×C₇	semidirect
189. #1550	C₂×C₇	semidirect
190. #1495	C₂×C₇	semidirect
191. #1481	C₂×C₇	semidirect
676. C₇	#18963	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×#3555	2
C₇×#3254	2
C₇×#3247	2
C₇×#3210	2
C₇×#1638	2
C₇×#1636	2
C₇×#1633	2
C₇×#1632	2
C₇×#1628	2
C₇×#1626	2
C₇×#1623	2
C₇×#1622	2
C₇×#1550	2
C₇×#1495	2
C₇×#1481	2
#18963	7

Subgroups — C7×#18963 #13196179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18963 #13196179