Subgroups — #11736820

Order 896.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90213	C₂	1	1	yes	maximal, characteristic
2	C₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#89772	C₂	1	1	yes	maximal, characteristic, complemented
4	#89612	C₂	1	1	yes	maximal, characteristic, complemented
5	#89612	C₂	1	1	yes	maximal, characteristic, complemented
6	#89609	C₂	1	1	yes	maximal, characteristic, complemented
7	D₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#89499	C₂	1	1	yes	maximal, characteristic
9	#89499	C₂	1	1	yes	maximal, characteristic
10	#89491	C₂	1	1	yes	maximal, characteristic, complemented
11	#89490	C₂	1	1	yes	maximal, characteristic, complemented
12	#89480	C₂	1	1	yes	maximal, characteristic, complemented
13	#89477	C₂	1	1	yes	maximal, characteristic, complemented
14	#89461	C₂	1	1	yes	maximal, characteristic, complemented
15	#89457	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#1157	C₂²	1	1	yes	characteristic
17	C₇×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×#1131	C₂²	1	1	yes	characteristic
20	C₂×#1131	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
23	#6456	C₄	1	1	yes	characteristic
24	#6456	C₄	1	1	yes	characteristic
25	#6450	C₄	1	1	yes	characteristic
26	#6450	C₄	1	1	yes	characteristic
27	(C₂²⋊D₇)⋊C₄	C₄	1	1	yes	characteristic
28	(C₂²⋊D₇)⋊C₄	C₄	1	1	yes	characteristic
29	(C₂²⋊D₇)⋊C₄	C₄	1	1	yes	characteristic
30	(C₂²⋊D₇)⋊C₄	C₄	1	1	yes	characteristic
31	#6435	C₂²	1	1	yes	characteristic
32	#6435	C₂²	1	1	yes	characteristic
33	C₄²×D₇	C₂²	1	1	yes	characteristic
34	C₄²×D₇	C₂²	1	1	yes	characteristic
35	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₇×#97	C₂²	1	1	yes	characteristic
40	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
42	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
43	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
47	#6389	C₂²	1	1	yes	characteristic
48	#6389	C₂²	1	1	yes	characteristic
49	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
53	C₈×Dic₇	C₂²	1	1	yes	characteristic
54	C₈×Dic₇	C₂²	1	1	yes	characteristic
55	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
57	#6377	C₂²	1	1	yes	characteristic
58	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
59	#2925	—	7	1	no	maximal, Sylow
60	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
65	#1157	—	2	2	no	—
66	#1157	—	2	2	no	—
67	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₄²×C₇	C₂³	1	1	yes	characteristic
85	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
89	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
90	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
92	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
99	C₄×Dic₇	C₂³	1	1	yes	characteristic
100	C₄×Dic₇	C₂³	1	1	yes	characteristic
101	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
105	#1131	—	2	2	no	—
106	#1131	—	2	2	no	—
107	C₈×D₇	—	2	2	no	—
108	C₈×D₇	—	2	2	no	—
109	C₄×#41	—	7	1	no	—
110	#433	—	7	1	no	—
111	#433	—	7	1	no	—
112	#433	—	7	1	no	—
113	#433	D₇	1	1	yes	characteristic, complemented
114	#432	—	7	1	no	—
115	#432	—	7	1	no	—
116	#431	—	7	1	no	—
117	#431	—	7	1	no	—
118	#423	—	7	1	no	—
119	C₂×C₄⋊C₈	—	7	1	no	—
120	#408	—	7	1	no	—
121	#408	—	7	1	no	—
122	#405	—	7	1	no	—
123	#405	—	7	1	no	—
124	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	1	no	—
129	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₇	—	2	2	no	—
136	C₂²⋊D₇	—	2	2	no	—
137	C₂×Dic₇	—	2	2	no	—
138	C₂×Dic₇	—	4	1	no	—
139	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
142	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
144	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
145	D₂₈	—	2	2	no	—
146	C₄×D₇	—	2	1	no	—
147	C₄×D₇	—	2	1	no	—
148	C₄×D₇	—	2	2	no	—
149	C₄×D₇	—	2	2	no	—
150	C₄×D₇	—	4	1	no	—
151	C₄×D₇	—	4	1	no	—
152	C₄×D₇	#41	1	2	yes	—
153	C₄×D₇	#41	1	2	yes	—
154	Dic₁₄	—	2	2	no	—
155	C₇×C₈	—	2	1	no	—
156	C₇×C₈	—	2	1	no	—
157	C₇×C₈	—	2	1	no	—
158	C₇×C₈	—	2	1	no	—
159	C₇⋊C₈	—	2	1	no	—
160	C₇⋊C₈	—	2	1	no	—
161	C₇⋊C₈	—	2	1	no	—
162	C₇⋊C₈	—	2	1	no	—
163	C₂×#41	—	7	1	no	—
164	C₂×M₁₆	—	7	1	no	—
165	C₂×M₁₆	—	7	1	no	—
166	C₂²×C₈	—	7	1	no	—
167	C₂²×C₈	—	7	1	no	—
168	C₄×Q₈	—	7	1	no	—
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	1	no	—
172	C₄×D₄	—	7	1	no	—
173	C₄×D₄	—	7	1	no	—
174	C₄×D₄	—	7	1	no	—
175	C₄×D₄	—	7	1	no	—
176	#117	—	7	1	no	—
177	#117	—	7	1	no	—
178	#117	C₂×D₇	1	1	yes	characteristic
179	C₂×C₄²	—	7	1	no	—
180	C₂×C₄²	—	7	1	no	—
181	C₂×C₄²	—	7	1	no	—
182	C₄⋊C₈	—	7	1	no	—
183	C₄⋊C₈	—	7	1	no	—
184	C₄⋊C₈	—	7	1	no	—
185	C₄⋊C₈	—	7	1	no	—
186	C₄⋊C₈	—	7	1	no	—
187	C₄⋊C₈	—	7	1	no	—
188	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
189	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
190	C₂²⋊C₈	—	7	1	no	—
191	C₂²⋊C₈	—	7	1	no	—
192	C₂²⋊C₈	—	7	1	no	—
193	C₂²⋊C₈	—	7	1	no	—
194	C₂²⋊C₈	—	7	1	no	—
195	C₂²⋊C₈	—	7	1	no	—
196	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
197	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
198	#97	—	7	1	no	—
199	#97	—	7	1	no	—
200	#97	—	7	1	no	—
201	#97	C₂×D₇	1	1	yes	characteristic
202	C₄×C₈	—	7	1	no	—
203	C₄×C₈	—	7	1	no	—
204	C₄×C₈	—	7	1	no	—
205	C₄×C₈	C₂×D₇	1	1	yes	characteristic
206	C₂²×C₇	—	2	1	no	—
207	C₂²×C₇	—	2	1	no	—
208	C₂²×C₇	—	2	1	no	—
209	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
210	C₂×D₇	—	2	1	no	—
211	C₂×D₇	—	2	1	no	—
212	C₂×D₇	—	2	1	no	—
213	C₂×D₇	—	2	1	no	—
214	C₂×D₇	—	2	1	no	—
215	C₂×D₇	#117	1	2	yes	—
216	C₄×C₇	—	2	1	no	—
217	C₄×C₇	—	2	1	no	—
218	C₄×C₇	—	2	1	no	—
219	C₄×C₇	—	2	1	no	—
220	C₄×C₇	—	2	1	no	—
221	C₄×C₇	C₂×#41	1	1	yes	characteristic
222	C₄×C₇	C₂×#41	1	1	yes	characteristic
223	Dic₇	—	2	1	no	—
224	Dic₇	—	2	1	no	—
225	Dic₇	—	2	1	no	—
226	Dic₇	—	2	1	no	—
227	Dic₇	#117	1	2	yes	—
228	Dic₇	#131	1	2	yes	—
229	#41	—	14	2	no	—
230	#41	—	14	2	no	—
231	C₂×Q₈	—	7	1	no	—
232	C₂×D₄	—	7	1	no	—
233	C₂×D₄	—	7	1	no	—
234	C₂×D₄	—	7	1	no	—
235	C₂²×C₄	—	7	1	no	—
236	C₂²×C₄	—	7	1	no	—
237	C₂²×C₄	—	7	1	no	—
238	C₂²×C₄	—	7	1	no	—
239	C₂²×C₄	—	7	1	no	—
240	C₂²×C₄	—	7	1	no	—
241	C₂²×C₄	—	7	1	no	—
242	C₂²×C₄	—	7	1	no	—
243	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
244	M₁₆	—	14	2	no	—
245	M₁₆	—	14	2	no	—
246	C₂×C₈	—	7	1	no	—
247	C₂×C₈	—	7	1	no	—
248	C₂×C₈	—	7	1	no	—
249	C₂×C₈	—	7	1	no	—
250	C₂×C₈	—	14	2	no	—
251	C₂×C₈	—	14	2	no	—
252	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
256	C₄⋊C₄	—	7	1	no	—
257	C₄⋊C₄	—	7	1	no	—
258	C₄⋊C₄	—	7	1	no	—
259	C₄⋊C₄	—	7	1	no	—
260	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
261	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
262	C₂²⋊C₄	—	7	1	no	—
263	C₂²⋊C₄	—	7	1	no	—
264	C₂²⋊C₄	—	7	1	no	—
265	C₂²⋊C₄	—	7	1	no	—
266	C₂²⋊C₄	C₄×D₇	1	1	yes	characteristic
267	C₂²⋊C₄	C₄×D₇	1	1	yes	characteristic
268	C₄²	—	7	1	no	—
269	C₄²	—	7	1	no	—
270	C₄²	—	7	1	no	—
271	C₄²	—	7	1	no	—
272	C₄²	—	7	1	no	—
273	C₄²	—	7	1	no	—
274	C₄²	—	7	1	no	—
275	C₄²	—	7	1	no	—
276	C₄²	C₂²×D₇	1	1	yes	characteristic
277	C₄²	C₂²×D₇	1	1	yes	characteristic
278	C₂×C₇	—	4	1	no	—
279	C₂×C₇	C₂×#117	1	1	yes	characteristic
280	C₂×C₇	C₂×#131	1	1	yes	characteristic
281	C₂×C₇	#568	1	1	yes	characteristic
282	D₇	—	2	2	no	—
283	D₇	—	4	1	no	—
284	C₂³	—	7	1	no	—
285	C₂³	—	7	1	no	—
286	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
287	Q₈	—	14	2	no	—
288	D₄	—	14	2	no	—
289	D₄	—	14	2	no	—
290	D₄	—	14	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	7	1	no	—
294	C₂×C₄	—	7	1	no	—
295	C₂×C₄	—	7	2	no	—
296	C₂×C₄	—	7	2	no	—
297	C₂×C₄	—	7	1	no	—
298	C₂×C₄	—	7	1	no	—
299	C₂×C₄	—	7	1	no	—
300	C₂×C₄	—	7	1	no	—
301	C₂×C₄	—	14	2	no	—
302	C₂×C₄	—	14	2	no	—
303	C₂×C₄	—	14	2	no	—
304	C₂×C₄	—	14	1	no	—
305	C₂×C₄	—	14	1	no	—
306	C₂×C₄	—	28	1	no	—
307	C₂×C₄	—	28	1	no	—
308	C₂×C₄	—	28	1	no	—
309	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	14	1	no	—
320	C₈	—	14	1	no	—
321	C₈	—	14	1	no	—
322	C₈	—	14	1	no	—
323	C₇	#2925	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	7	2	no	—
328	C₂²	—	14	1	no	—
329	C₂²	—	14	1	no	—
330	C₂²	—	14	1	no	—
331	C₂²	—	14	1	no	—
332	C₂²	—	14	1	no	—
333	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	7	2	no	—
340	C₄	—	7	2	no	—
341	C₄	—	14	1	no	—
342	C₄	—	14	1	no	—
343	C₄	—	14	1	no	—
344	C₄	—	14	1	no	—
345	C₄	D₇×#41	1	1	yes	characteristic
346	C₄	D₇×#41	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	14	2	no	—
349	C₂	—	28	1	no	—
350	C₂	D₇×#117	1	1	yes	characteristic
351	C₂	D₇×#131	1	1	yes	characteristic
352	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#433	C₂
prime radical	C₂×#1157	C₂²
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#433	D₇
hypocenter	C₇	#2925
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₇×#433	C₂	semidirect
3. #89772	C₂	semidirect
4. #89612	C₂	semidirect
5. #89612	C₂	semidirect
6. #89609	C₂	semidirect
7. D₇×C₄⋊C₈	C₂	semidirect
10. #89491	C₂	semidirect
11. #89490	C₂	semidirect
12. #89480	C₂	semidirect
13. #89477	C₂	semidirect
14. #89461	C₂	semidirect
15. #89457	C₂	semidirect
113. #433	D₇	semidirect
323. C₇	#2925	semidirect

Maximal subgroups

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

subgroup	index
#90213	2
C₇×#433	2
#89772	2
#89612	2
#89609	2
D₇×C₄⋊C₈	2
#89499	2
#89491	2
#89490	2
#89480	2
#89477	2
#89461	2
#89457	2
#2925	7