Subgroups — #11737413

Order 896.

314 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90269	C₂	1	1	yes	maximal, characteristic
2	C₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#89796	C₂	1	1	yes	maximal, characteristic, complemented
4	#89612	C₂	1	1	yes	maximal, characteristic
5	#89612	C₂	1	1	yes	maximal, characteristic
6	#89607	C₂	1	1	yes	maximal, characteristic
7	#89603	C₂	1	1	yes	maximal, characteristic, complemented
8	#89600	C₂	1	1	yes	maximal, characteristic, complemented
9	#89600	C₂	1	1	yes	maximal, characteristic, complemented
10	#89481	C₂	1	1	yes	maximal, characteristic
11	#89477	C₂	1	1	yes	maximal, characteristic
12	#89473	C₂	1	1	yes	maximal, characteristic, complemented
13	#89473	C₂	1	1	yes	maximal, characteristic, complemented
14	#89462	C₂	1	1	yes	maximal, characteristic
15	#89451	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
17	#6509	C₄	1	1	yes	characteristic
18	#6507	C₄	1	1	yes	characteristic
19	#6460	C₄	1	1	yes	characteristic
20	#6459	C₄	1	1	yes	characteristic
21	#6459	C₄	1	1	yes	characteristic
22	#6452	C₄	1	1	yes	characteristic
23	#6452	C₄	1	1	yes	characteristic
24	#6451	C₄	1	1	yes	characteristic
25	#6435	C₂²	1	1	yes	characteristic
26	#6435	C₂²	1	1	yes	characteristic
27	C₄²×D₇	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
31	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₇×#97	C₂²	1	1	yes	characteristic
35	C₇×#97	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
37	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
38	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
39	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
40	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
41	#6389	C₂²	1	1	yes	characteristic
42	#6389	C₂²	1	1	yes	characteristic
43	#6389	C₂²	1	1	yes	characteristic
44	#6389	C₂²	1	1	yes	characteristic
45	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
46	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₇	C₂²	1	1	yes	characteristic
52	C₈×Dic₇	C₂²	1	1	yes	characteristic
53	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
54	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
55	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
56	#6377	C₂²	1	1	yes	characteristic
57	#6377	C₂²	1	1	yes	characteristic
58	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
59	#2987	—	7	1	no	maximal, Sylow
60	C₂×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
66	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₄²×C₇	C₂³	1	1	yes	characteristic
75	C₄²×C₇	C₂³	1	1	yes	characteristic
76	C₄²×C₇	C₂³	1	1	yes	characteristic
77	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
78	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
79	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
80	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
83	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
84	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
85	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
86	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
87	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
88	Dic₇⋊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	Dic₇⋊C₄	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	C₂³	1	1	yes	characteristic
97	C₄×Dic₇	C₂³	1	1	yes	characteristic
98	C₄×Dic₇	C₂³	1	1	yes	characteristic
99	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
103	#533	—	7	1	no	—
104	#446	—	7	1	no	—
105	#446	—	7	1	no	—
106	#446	—	7	1	no	—
107	#446	—	7	1	no	—
108	#446	—	7	1	no	—
109	#446	—	7	1	no	—
110	#446	—	7	1	no	—
111	#446	D₇	1	1	yes	characteristic, complemented
112	#433	—	7	1	no	—
113	#433	—	7	1	no	—
114	#433	—	7	1	no	—
115	#433	—	7	1	no	—
116	#432	—	7	1	no	—
117	#432	—	7	1	no	—
118	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
119	C₇×Q₈	—	4	1	no	—
120	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
134	C₄×D₇	—	2	1	no	—
135	C₄×D₇	—	2	1	no	—
136	C₄×D₇	—	4	1	no	—
137	C₄×D₇	—	4	1	no	—
138	Dic₁₄	—	4	1	no	—
139	Dic₁₄	—	4	1	no	—
140	Dic₁₄	—	4	1	no	—
141	C₇×C₈	—	2	1	no	—
142	C₇×C₈	—	2	1	no	—
143	C₇×C₈	—	2	1	no	—
144	C₇×C₈	—	2	1	no	—
145	C₇⋊C₈	—	2	1	no	—
146	C₇⋊C₈	—	2	1	no	—
147	C₇⋊C₈	—	2	1	no	—
148	C₇⋊C₈	—	2	1	no	—
149	C₄⋊Q₈	—	7	1	no	—
150	C₄⋊Q₈	—	7	1	no	—
151	#125	—	7	1	no	—
152	#125	—	7	1	no	—
153	C₂²⋊Q₈	—	7	1	no	—
154	C₂²⋊Q₈	—	7	1	no	—
155	C₂²⋊Q₈	—	7	1	no	—
156	C₂²⋊Q₈	—	7	1	no	—
157	C₄×Q₈	—	7	1	no	—
158	C₄×Q₈	—	7	1	no	—
159	C₄×Q₈	—	7	1	no	—
160	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
161	#117	—	7	1	no	—
162	#117	—	7	1	no	—
163	C₂×C₄²	—	7	1	no	—
164	C₄⋊C₈	—	7	1	no	—
165	C₄⋊C₈	—	7	1	no	—
166	C₄⋊C₈	—	7	1	no	—
167	C₄⋊C₈	—	7	1	no	—
168	C₄⋊C₈	—	7	1	no	—
169	C₄⋊C₈	—	7	1	no	—
170	C₄⋊C₈	—	7	1	no	—
171	C₄⋊C₈	—	7	1	no	—
172	C₄⋊C₈	—	7	1	no	—
173	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
174	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
175	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
176	C₂²⋊C₈	—	7	1	no	—
177	C₂²⋊C₈	—	7	1	no	—
178	C₂²⋊C₈	—	7	1	no	—
179	C₂²⋊C₈	—	7	1	no	—
180	#97	—	7	1	no	—
181	#97	—	7	1	no	—
182	#97	—	7	1	no	—
183	#97	—	7	1	no	—
184	#97	—	7	1	no	—
185	#97	—	7	1	no	—
186	#97	C₂×D₇	1	1	yes	characteristic
187	#97	C₂×D₇	1	1	yes	characteristic
188	C₄×C₈	—	7	1	no	—
189	C₄×C₈	—	7	1	no	—
190	C₄×C₈	—	7	1	no	—
191	C₄×C₈	C₂×D₇	1	1	yes	characteristic
192	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×D₇	—	2	1	no	—
194	C₂×D₇	—	2	1	no	—
195	C₂×D₇	—	2	1	no	—
196	C₄×C₇	—	2	1	no	—
197	C₄×C₇	—	2	1	no	—
198	C₄×C₇	—	2	1	no	—
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	2	1	no	—
201	C₄×C₇	—	2	1	no	—
202	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
203	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
204	Dic₇	—	2	1	no	—
205	Dic₇	—	2	1	no	—
206	Dic₇	—	2	1	no	—
207	Dic₇	—	2	1	no	—
208	Dic₇	—	2	1	no	—
209	Dic₇	—	2	1	no	—
210	Dic₇	—	2	1	no	—
211	C₂×Q₈	—	7	1	no	—
212	C₂×Q₈	—	7	1	no	—
213	C₂×Q₈	—	7	1	no	—
214	C₂×Q₈	C₄×D₇	1	1	yes	characteristic
215	C₂²×C₄	—	7	1	no	—
216	C₂²×C₄	—	7	1	no	—
217	C₂²×C₄	—	7	1	no	—
218	C₂×C₈	—	7	1	no	—
219	C₂×C₈	—	7	1	no	—
220	C₂×C₈	—	7	1	no	—
221	C₂×C₈	—	7	1	no	—
222	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
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	—	7	1	no	—
232	C₄⋊C₄	—	7	1	no	—
233	C₄⋊C₄	—	7	1	no	—
234	C₄⋊C₄	—	7	1	no	—
235	C₄⋊C₄	—	7	1	no	—
236	C₄⋊C₄	—	7	1	no	—
237	C₄⋊C₄	—	7	1	no	—
238	C₄⋊C₄	—	7	1	no	—
239	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
240	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
241	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
242	C₂²⋊C₄	—	7	1	no	—
243	C₂²⋊C₄	—	7	1	no	—
244	C₂²⋊C₄	—	7	1	no	—
245	C₂²⋊C₄	—	7	1	no	—
246	C₄²	—	7	1	no	—
247	C₄²	—	7	1	no	—
248	C₄²	—	7	1	no	—
249	C₄²	—	7	1	no	—
250	C₄²	—	7	1	no	—
251	C₄²	C₂²×D₇	1	1	yes	characteristic
252	C₄²	C₂²×D₇	1	1	yes	characteristic
253	C₄²	C₂²×D₇	1	1	yes	characteristic
254	C₂×C₇	#519	1	1	yes	characteristic
255	C₂×C₇	#568	1	1	yes	characteristic
256	C₂×C₇	#568	1	1	yes	characteristic
257	D₇	—	4	1	no	—
258	C₂³	—	7	1	no	—
259	Q₈	—	4	1	no	—
260	Q₈	—	28	1	no	—
261	Q₈	—	28	1	no	—
262	Q₈	—	28	1	no	—
263	C₂×C₄	—	7	1	no	—
264	C₂×C₄	—	7	1	no	—
265	C₂×C₄	—	7	1	no	—
266	C₂×C₄	—	7	1	no	—
267	C₂×C₄	—	7	1	no	—
268	C₂×C₄	—	7	1	no	—
269	C₂×C₄	—	7	1	no	—
270	C₂×C₄	—	14	1	no	—
271	C₂×C₄	—	14	1	no	—
272	C₂×C₄	—	28	1	no	—
273	C₂×C₄	—	28	1	no	—
274	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	14	1	no	—
286	C₈	—	14	1	no	—
287	C₈	—	14	1	no	—
288	C₈	—	14	1	no	—
289	C₇	#2987	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	14	1	no	—
291	C₂²	—	14	1	no	—
292	C₂²	—	14	1	no	—
293	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	14	1	no	—
301	C₄	—	14	1	no	—
302	C₄	—	14	1	no	—
303	C₄	—	14	1	no	—
304	C₄	—	14	1	no	—
305	C₄	—	14	1	no	—
306	C₄	—	14	1	no	—
307	C₄	#6551	1	1	yes	characteristic
308	C₄	#6554	1	1	yes	characteristic
309	C₂	—	28	1	no	—
310	C₂	#90263	1	1	yes	characteristic
311	C₂	#90436	1	1	yes	characteristic
312	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#446	C₂
prime radical	C₂²×D₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#446	D₇
hypocenter	C₇	#2987
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₇×#446	C₂	semidirect
3. #89796	C₂	semidirect
7. #89603	C₂	semidirect
8. #89600	C₂	semidirect
9. #89600	C₂	semidirect
12. #89473	C₂	semidirect
13. #89473	C₂	semidirect
15. #89451	C₂	semidirect
111. #446	D₇	semidirect
289. C₇	#2987	semidirect

Maximal subgroups

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

subgroup	index
#90269	2
C₇×#446	2
#89796	2
#89612	2
#89607	2
#89603	2
#89600	2
#89481	2
#89477	2
#89473	2
#89462	2
#89451	2
#2987	7