Subgroups — #11739419

Order 896.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90420	C₂	1	1	yes	maximal, characteristic
2	#90380	C₂	1	1	yes	maximal, characteristic
3	C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#89851	C₂	1	1	yes	maximal, characteristic, complemented
5	#89836	C₂	1	1	yes	maximal, characteristic, complemented
6	#89653	C₂	1	1	yes	maximal, characteristic
7	#89652	C₂	1	1	yes	maximal, characteristic
8	#89650	C₂	1	1	yes	maximal, characteristic, complemented
9	#89648	C₂	1	1	yes	maximal, characteristic, complemented
10	#89633	C₂	1	1	yes	maximal, characteristic
11	D₇×#106	C₂	1	1	yes	maximal, characteristic
12	#89628	C₂	1	1	yes	maximal, characteristic, complemented
13	#89627	C₂	1	1	yes	maximal, characteristic, complemented
14	#89480	C₂	1	1	yes	maximal, characteristic
15	#89474	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₇×#125	C₂²	1	1	yes	characteristic
18	#6509	—	2	1	no	—
19	#6507	—	2	1	no	—
20	C₂×#1131	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
22	#6460	—	2	1	no	—
23	#6459	—	2	1	no	—
24	#6459	—	2	1	no	—
25	#6455	C₂²	1	1	yes	characteristic
26	#6455	C₂²	1	1	yes	characteristic
27	D₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	D₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	#6453	C₂²	1	1	yes	characteristic
30	#6453	C₂²	1	1	yes	characteristic
31	#6452	—	2	1	no	—
32	#6452	—	2	1	no	—
33	#6451	—	2	1	no	—
34	#6451	C₂²	1	1	yes	characteristic
35	#6451	C₂²	1	1	yes	characteristic
36	#6435	C₂²	1	1	yes	characteristic
37	#6433	C₂²	1	1	yes	characteristic
38	#6432	C₂²	1	1	yes	characteristic
39	C₇×#107	C₂²	1	1	yes	characteristic
40	C₇×#107	C₂²	1	1	yes	characteristic
41	C₇×#106	C₂²	1	1	yes	characteristic
42	C₇×#106	C₂²	1	1	yes	characteristic
43	C₇×#97	C₂²	1	1	yes	characteristic
44	#6392	C₂²	1	1	yes	characteristic
45	#6392	C₂²	1	1	yes	characteristic
46	#6391	C₂²	1	1	yes	characteristic
47	#6391	C₂²	1	1	yes	characteristic
48	#6389	C₂²	1	1	yes	characteristic
49	C₈×Dic₇	C₂²	1	1	yes	characteristic
50	#6382	C₂²	1	1	yes	characteristic
51	#6382	C₂²	1	1	yes	characteristic
52	#6382	C₂²	1	1	yes	characteristic
53	#6382	C₂²	1	1	yes	characteristic
54	#6382	C₂²	1	1	yes	characteristic
55	#6382	C₂²	1	1	yes	characteristic
56	#6381	C₂²	1	1	yes	characteristic
57	#6381	C₂²	1	1	yes	characteristic
58	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
59	#3163	—	7	1	no	maximal, Sylow
60	C₂×C₇×Q₈	—	2	1	no	—
61	C₂×C₄×D₇	—	2	1	no	—
62	C₂×C₄×D₇	—	2	1	no	—
63	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₄	—	2	1	no	—
65	C₂×Dic₁₄	—	2	1	no	—
66	C₂×Dic₁₄	—	2	1	no	—
67	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	2	1	no	—
70	C₇×C₄⋊C₄	—	2	1	no	—
71	C₇×C₄⋊C₄	—	2	1	no	—
72	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₄²×C₇	C₂³	1	1	yes	characteristic
77	(C₂×D₇)⋊C₄	—	2	1	no	—
78	(C₂×D₇)⋊C₄	—	2	1	no	—
79	(C₂×D₇)⋊C₄	D₄	1	1	yes	characteristic
80	(C₂×D₇)⋊C₄	D₄	1	1	yes	characteristic
81	C₄⋊Dic₇	—	2	1	no	—
82	C₄⋊Dic₇	—	2	1	no	—
83	C₄⋊Dic₇	—	2	1	no	—
84	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
88	Dic₇⋊C₄	—	2	1	no	—
89	Dic₇⋊C₄	—	2	1	no	—
90	Dic₇⋊C₄	—	2	1	no	—
91	Dic₇⋊C₄	—	2	1	no	—
92	Dic₇⋊C₄	—	2	1	no	—
93	Dic₇⋊C₄	—	2	1	no	—
94	Dic₇⋊C₄	—	2	1	no	—
95	Dic₇⋊C₄	—	2	1	no	—
96	Dic₇⋊C₄	D₄	1	1	yes	characteristic
97	Dic₇⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₇	—	2	1	no	—
99	C₄×Dic₇	—	2	1	no	—
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	#1131	Q₈	1	4	yes	complemented
104	C₈×D₇	Q₈	1	4	yes	complemented
105	#544	—	7	1	no	—
106	#544	—	7	1	no	—
107	#501	—	7	1	no	—
108	#501	—	7	1	no	—
109	#501	—	7	1	no	—
110	#501	D₇	1	1	yes	characteristic, complemented
111	#499	—	7	1	no	—
112	#499	—	7	1	no	—
113	#498	—	7	1	no	—
114	#498	—	7	1	no	—
115	#428	—	7	1	no	—
116	#428	—	7	1	no	—
117	#427	—	7	1	no	—
118	C₂×#106	—	7	1	no	—
119	#405	—	7	1	no	—
120	C₂²×D₇	C₂×D₄	1	1	yes	characteristic
121	C₇×Q₈	—	8	1	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
129	C₂×Dic₇	—	2	1	no	—
130	C₂×Dic₇	—	2	1	no	—
131	C₂×Dic₇	—	2	1	no	—
132	C₂×Dic₇	—	2	1	no	—
133	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
136	C₄×D₇	—	8	1	no	—
137	C₄×D₇	—	8	1	no	—
138	C₄×D₇	C₂×Q₈	1	2	yes	—
139	C₄×D₇	C₂×Q₈	1	2	yes	—
140	Dic₁₄	—	8	1	no	—
141	Dic₁₄	—	8	1	no	—
142	Dic₁₄	—	8	1	no	—
143	C₇×C₈	C₂×Q₈	1	2	yes	—
144	C₇×C₈	C₂×Q₈	1	2	yes	—
145	C₇⋊C₈	C₂×Q₈	1	2	yes	—
146	C₇⋊C₈	C₂×Q₈	1	2	yes	—
147	C₂×M₁₆	—	7	1	no	—
148	C₂²×C₈	—	7	1	no	—
149	C₄⋊Q₈	—	7	1	no	—
150	C₄⋊Q₈	—	7	1	no	—
151	C₄⋊Q₈	—	7	1	no	—
152	C₄⋊Q₈	—	14	1	no	—
153	C₄⋊Q₈	—	14	1	no	—
154	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
155	#125	—	7	1	no	—
156	#125	—	7	1	no	—
157	#125	—	7	1	no	—
158	#125	—	14	1	no	—
159	#125	—	14	1	no	—
160	#125	C₂×D₇	1	1	yes	characteristic
161	C₂²⋊Q₈	—	14	1	no	—
162	C₂²⋊Q₈	—	14	1	no	—
163	C₂²⋊Q₈	—	14	1	no	—
164	C₂²⋊Q₈	—	14	1	no	—
165	#117	—	7	1	no	—
166	#117	—	7	1	no	—
167	#117	—	7	1	no	—
168	C₂×C₄⋊C₄	—	7	1	no	—
169	C₂×C₄⋊C₄	—	7	1	no	—
170	#107	—	7	1	no	—
171	#107	—	7	1	no	—
172	#107	—	7	1	no	—
173	#107	—	7	1	no	—
174	#107	C₂×D₇	1	1	yes	characteristic
175	#107	C₂×D₇	1	1	yes	characteristic
176	#106	—	7	1	no	—
177	#106	—	7	1	no	—
178	#106	—	7	1	no	—
179	#106	—	7	1	no	—
180	#106	—	7	1	no	—
181	#106	—	7	1	no	—
182	#106	—	7	1	no	—
183	#106	—	7	1	no	—
184	#106	C₂×D₇	1	1	yes	characteristic
185	#106	C₂×D₇	1	1	yes	characteristic
186	#97	—	7	1	no	—
187	#97	C₂×D₇	1	1	yes	characteristic
188	C₄×C₈	—	7	1	no	—
189	C₄×C₈	—	7	1	no	—
190	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×D₇	—	2	1	no	—
192	C₂×D₇	—	2	1	no	—
193	C₂×D₇	C₄⋊Q₈	1	2	yes	—
194	C₄×C₇	—	2	1	no	—
195	C₄×C₇	—	2	1	no	—
196	C₄×C₇	—	4	1	no	—
197	C₄×C₇	—	4	1	no	—
198	C₄×C₇	—	4	1	no	—
199	C₄×C₇	—	4	1	no	—
200	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
201	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
202	Dic₇	—	2	1	no	—
203	Dic₇	—	2	1	no	—
204	Dic₇	—	4	1	no	—
205	Dic₇	—	4	1	no	—
206	Dic₇	—	4	1	no	—
207	Dic₇	—	4	1	no	—
208	Dic₇	C₄⋊Q₈	1	2	yes	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	14	1	no	—
211	C₂×Q₈	—	14	1	no	—
212	C₂×Q₈	—	14	1	no	—
213	C₂²×C₄	—	7	1	no	—
214	C₂²×C₄	—	14	1	no	—
215	C₂²×C₄	—	14	1	no	—
216	M₁₆	—	7	4	no	—
217	C₂×C₈	—	7	1	no	—
218	C₂×C₈	—	7	1	no	—
219	C₂×C₈	—	7	4	no	—
220	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	7	1	no	—
226	C₄⋊C₄	—	7	1	no	—
227	C₄⋊C₄	—	7	1	no	—
228	C₄⋊C₄	—	7	1	no	—
229	C₄⋊C₄	—	7	1	no	—
230	C₄⋊C₄	—	7	1	no	—
231	C₄⋊C₄	—	14	1	no	—
232	C₄⋊C₄	—	14	1	no	—
233	C₄⋊C₄	—	14	1	no	—
234	C₄⋊C₄	—	14	1	no	—
235	C₄⋊C₄	—	14	1	no	—
236	C₄⋊C₄	—	14	1	no	—
237	C₄⋊C₄	—	14	1	no	—
238	C₄⋊C₄	—	14	1	no	—
239	C₄⋊C₄	—	14	1	no	—
240	C₄⋊C₄	—	14	1	no	—
241	C₄⋊C₄	—	14	1	no	—
242	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
243	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
246	C₂²⋊C₄	—	7	1	no	—
247	C₂²⋊C₄	—	7	1	no	—
248	C₂²⋊C₄	—	14	1	no	—
249	C₂²⋊C₄	—	14	1	no	—
250	C₄²	—	7	1	no	—
251	C₄²	—	14	1	no	—
252	C₄²	—	14	1	no	—
253	C₄²	C₂²×D₇	1	1	yes	characteristic
254	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₇	#577	1	1	yes	characteristic
256	C₂×C₇	#577	1	1	yes	characteristic
257	D₇	—	2	2	no	—
258	C₂³	—	7	1	no	—
259	Q₈	—	8	1	no	—
260	Q₈	—	56	1	no	—
261	Q₈	—	56	1	no	—
262	Q₈	—	56	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	7	1	no	—
268	C₂×C₄	—	7	2	no	—
269	C₂×C₄	—	7	2	no	—
270	C₂×C₄	—	7	1	no	—
271	C₂×C₄	—	7	1	no	—
272	C₂×C₄	—	14	1	no	—
273	C₂×C₄	—	14	1	no	—
274	C₂×C₄	—	14	1	no	—
275	C₂×C₄	—	14	1	no	—
276	C₂×C₄	—	56	1	no	—
277	C₂×C₄	—	56	1	no	—
278	C₂×C₄	D₄×D₇	1	1	yes	characteristic
279	C₂×C₄	D₄×D₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
281	C₈	—	7	2	no	—
282	C₈	—	7	2	no	—
283	C₈	Q₈×D₇	1	2	yes	—
284	C₈	Q₈×D₇	1	2	yes	—
285	C₇	#3163	1	1	yes	characteristic, complemented, Sylow
286	C₂²	—	7	2	no	—
287	C₂²	—	14	1	no	—
288	C₂²	—	14	1	no	—
289	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	7	2	no	—
297	C₄	—	14	1	no	—
298	C₄	—	14	1	no	—
299	C₄	—	28	1	no	—
300	C₄	—	28	1	no	—
301	C₄	—	28	1	no	—
302	C₄	—	28	1	no	—
303	C₄	C₂×Q₈×D₇	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₇	1	1	yes	characteristic
305	C₂	—	14	2	no	—
306	C₂	D₇×C₄⋊Q₈	1	1	yes	characteristic
307	C₂	#90466	1	1	yes	characteristic
308	C₂	#90469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#501	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂²×D₇	C₂×D₄
hypercenter	#501	D₇
hypocenter	C₇	#3163
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. C₇×#501	C₂	semidirect
4. #89851	C₂	semidirect
5. #89836	C₂	semidirect
8. #89650	C₂	semidirect
9. #89648	C₂	semidirect
12. #89628	C₂	semidirect
13. #89627	C₂	semidirect
15. #89474	C₂	semidirect
103. #1131	Q₈	semidirect
104. C₈×D₇	Q₈	semidirect
110. #501	D₇	semidirect
285. C₇	#3163	semidirect

Maximal subgroups

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

subgroup	index
#90420	2
#90380	2
C₇×#501	2
#89851	2
#89836	2
#89653	2
#89652	2
#89650	2
#89648	2
#89633	2
D₇×#106	2
#89628	2
#89627	2
#89480	2
#89474	2
#3163	7