Subgroups — #11736779

Order 896.

392 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1157	C₂	1	1	yes	maximal, characteristic
2	C₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	#89697	C₂	1	1	yes	maximal, characteristic, complemented
4	#89609	C₂	1	1	yes	maximal, characteristic, complemented
5	#89607	C₂	1	1	yes	maximal, characteristic, complemented
6	#89605	C₂	1	1	yes	maximal, characteristic, complemented
7	D₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#89499	C₂	1	2	yes	maximal
9	#89492	C₂	1	2	yes	maximal, complemented
10	#89481	C₂	1	2	yes	maximal, complemented
11	#89480	C₂	1	2	yes	maximal, complemented
12	C₂×#1157	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₇	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₇	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₇	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₇	C₄	1	1	yes	characteristic
18	#6489	C₂²	1	1	yes	characteristic
19	C₂×#1131	C₂²	1	2	yes	—
20	C₂×C₈×D₇	C₂²	1	2	yes	—
21	C₄×D₂₈	C₄	1	1	yes	characteristic
22	C₄×D₂₈	C₄	1	1	yes	characteristic
23	#6435	C₂²	1	1	yes	characteristic
24	#6435	C₂²	1	1	yes	characteristic
25	C₄²×D₇	C₂²	1	1	yes	characteristic
26	C₄²×D₇	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₄	C₄	1	1	yes	characteristic
28	C₄×Dic₁₄	C₄	1	1	yes	characteristic
29	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₇×#97	C₂²	1	2	yes	—
33	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
34	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
35	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
36	#6389	C₂²	1	2	yes	—
37	Dic₇⋊C₈	C₂²	1	2	yes	—
38	Dic₇⋊C₈	C₂²	1	2	yes	—
39	C₈×Dic₇	C₂²	1	2	yes	—
40	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄×C₇⋊C₈	C₂²	1	2	yes	—
43	#2925	—	7	1	no	maximal, Sylow
44	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
49	#1157	—	2	2	no	—
50	#1157	—	2	2	no	—
51	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂³	1	2	yes	—
60	C₂×C₇×C₈	C₂³	1	2	yes	—
61	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₇	C₂³	1	1	yes	characteristic
64	C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₂²⋊Dic₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₇	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
70	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
73	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₇	C₂³	1	1	yes	characteristic
82	C₄×Dic₇	C₂³	1	1	yes	characteristic
83	C₂×C₇⋊C₈	C₂³	1	2	yes	—
84	C₂×C₇⋊C₈	C₂³	1	2	yes	—
85	#1131	—	2	4	no	—
86	C₈×D₇	—	2	4	no	—
87	C₄×#41	—	7	1	no	—
88	#433	—	7	2	no	—
89	#433	—	7	2	no	—
90	#432	—	7	1	no	—
91	#432	D₇	1	1	yes	characteristic, complemented
92	#431	—	7	1	no	—
93	#431	—	7	1	no	—
94	#423	—	7	1	no	—
95	C₂×C₄⋊C₈	—	7	1	no	—
96	#408	—	7	2	no	—
97	#405	—	7	2	no	—
98	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	4	1	no	—
105	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
111	C₂²⋊D₇	—	2	2	no	—
112	C₂²⋊D₇	—	2	2	no	—
113	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
119	D₂₈	—	2	2	no	—
120	C₄×D₇	—	2	2	no	—
121	C₄×D₇	—	2	2	no	—
122	C₄×D₇	—	2	1	no	—
123	C₄×D₇	—	2	1	no	—
124	C₄×D₇	—	4	1	no	—
125	C₄×D₇	—	4	1	no	—
126	C₄×D₇	#41	1	2	yes	—
127	C₄×D₇	#41	1	2	yes	—
128	Dic₁₄	—	2	2	no	—
129	C₇×C₈	—	2	2	no	—
130	C₇×C₈	—	2	2	no	—
131	C₇⋊C₈	—	2	2	no	—
132	C₇⋊C₈	—	2	2	no	—
133	C₂×#41	—	7	1	no	—
134	C₂×M₁₆	—	7	2	no	—
135	C₂²×C₈	—	7	2	no	—
136	C₄×Q₈	—	7	1	no	—
137	C₄×Q₈	—	7	1	no	—
138	C₄×D₄	—	7	1	no	—
139	C₄×D₄	—	7	1	no	—
140	C₄×D₄	—	7	1	no	—
141	C₄×D₄	—	7	1	no	—
142	C₄×D₄	—	7	1	no	—
143	C₄×D₄	—	7	1	no	—
144	#117	—	7	1	no	—
145	#117	—	7	1	no	—
146	#117	—	7	1	no	—
147	C₂×C₄²	—	7	1	no	—
148	C₂×C₄²	—	7	1	no	—
149	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
150	C₄⋊C₈	—	7	2	no	—
151	C₄⋊C₈	—	7	2	no	—
152	C₄⋊C₈	—	7	1	no	—
153	C₄⋊C₈	—	7	1	no	—
154	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
155	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
156	C₂²⋊C₈	—	7	2	no	—
157	C₂²⋊C₈	—	7	2	no	—
158	C₂²⋊C₈	—	7	2	no	—
159	C₂²⋊C₈	C₂×D₇	1	2	yes	—
160	#97	—	7	2	no	—
161	#97	C₂×D₇	1	2	yes	—
162	C₄×C₈	—	7	2	no	—
163	C₄×C₈	—	7	2	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×D₇	—	2	1	no	—
169	C₂×D₇	—	2	1	no	—
170	C₂×D₇	—	2	1	no	—
171	C₂×D₇	—	2	1	no	—
172	C₂×D₇	—	2	1	no	—
173	C₂×D₇	#117	1	2	yes	—
174	C₄×C₇	—	2	1	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	—	2	1	no	—
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	#131	1	2	yes	—
179	C₄×C₇	C₂×#41	1	1	yes	characteristic
180	C₄×C₇	C₂×#41	1	1	yes	characteristic
181	Dic₇	—	2	1	no	—
182	Dic₇	—	2	1	no	—
183	Dic₇	—	2	1	no	—
184	Dic₇	—	2	1	no	—
185	Dic₇	—	2	1	no	—
186	Dic₇	#117	1	2	yes	—
187	#41	—	14	2	no	—
188	#41	—	14	2	no	—
189	C₂×Q₈	—	7	1	no	—
190	C₂×D₄	—	7	1	no	—
191	C₂×D₄	—	7	1	no	—
192	C₂×D₄	—	7	1	no	—
193	C₂²×C₄	—	7	1	no	—
194	C₂²×C₄	—	7	1	no	—
195	C₂²×C₄	—	7	1	no	—
196	C₂²×C₄	—	7	1	no	—
197	C₂²×C₄	—	7	1	no	—
198	C₂²×C₄	—	7	1	no	—
199	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
202	M₁₆	—	14	4	no	—
203	C₂×C₈	—	7	2	no	—
204	C₂×C₈	—	7	2	no	—
205	C₂×C₈	—	14	4	no	—
206	C₂×C₈	C₂²×D₇	1	2	yes	—
207	C₂×C₈	C₂²×D₇	1	2	yes	—
208	C₄⋊C₄	—	7	1	no	—
209	C₄⋊C₄	—	7	1	no	—
210	C₄⋊C₄	—	7	1	no	—
211	C₄⋊C₄	—	7	1	no	—
212	C₄⋊C₄	—	7	1	no	—
213	C₄⋊C₄	—	7	1	no	—
214	C₂²⋊C₄	—	7	1	no	—
215	C₂²⋊C₄	—	7	1	no	—
216	C₂²⋊C₄	—	7	1	no	—
217	C₂²⋊C₄	—	7	1	no	—
218	C₂²⋊C₄	—	7	1	no	—
219	C₂²⋊C₄	—	7	1	no	—
220	C₄²	—	7	1	no	—
221	C₄²	—	7	1	no	—
222	C₄²	—	7	1	no	—
223	C₄²	—	7	1	no	—
224	C₄²	—	7	1	no	—
225	C₄²	—	7	1	no	—
226	C₄²	C₄×D₇	1	1	yes	characteristic
227	C₄²	C₄×D₇	1	1	yes	characteristic
228	C₄²	C₂²×D₇	1	1	yes	characteristic
229	C₄²	C₂²×D₇	1	1	yes	characteristic
230	C₂×C₇	—	4	1	no	—
231	C₂×C₇	C₂×#117	1	1	yes	characteristic
232	C₂×C₇	C₂×#131	1	1	yes	characteristic
233	C₂×C₇	#568	1	1	yes	characteristic
234	D₇	—	2	2	no	—
235	D₇	—	4	1	no	—
236	C₂³	—	7	1	no	—
237	C₂³	—	7	1	no	—
238	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
239	Q₈	—	14	2	no	—
240	D₄	—	14	2	no	—
241	D₄	—	14	2	no	—
242	D₄	—	14	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	7	1	no	—
248	C₂×C₄	—	7	2	no	—
249	C₂×C₄	—	7	2	no	—
250	C₂×C₄	—	7	1	no	—
251	C₂×C₄	—	7	1	no	—
252	C₂×C₄	—	7	1	no	—
253	C₂×C₄	—	7	1	no	—
254	C₂×C₄	—	7	1	no	—
255	C₂×C₄	—	14	2	no	—
256	C₂×C₄	—	14	2	no	—
257	C₂×C₄	—	14	1	no	—
258	C₂×C₄	—	14	1	no	—
259	C₂×C₄	—	28	1	no	—
260	C₂×C₄	—	28	1	no	—
261	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
266	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	14	2	no	—
270	C₈	—	14	2	no	—
271	C₇	#2925	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	7	2	no	—
276	C₂²	—	14	1	no	—
277	C₂²	—	14	1	no	—
278	C₂²	—	14	1	no	—
279	C₂²	—	14	1	no	—
280	C₂²	—	14	1	no	—
281	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	7	2	no	—
287	C₄	—	14	1	no	—
288	C₄	—	14	1	no	—
289	C₄	—	14	1	no	—
290	C₄	—	14	1	no	—
291	C₄	—	14	1	no	—
292	C₄	#6470	1	2	yes	—
293	C₄	D₇×#41	1	1	yes	characteristic
294	C₄	D₇×#41	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	—	14	2	no	—
297	C₂	—	28	1	no	—
298	C₂	D₇×#117	1	1	yes	characteristic
299	C₂	C₂×#6470	1	1	yes	characteristic
300	C₂	#90436	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#432	C₂
prime radical	C₂×#1157	C₂²
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#432	D₇
hypocenter	C₇	#2925
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₇×#432	C₂	semidirect
3. #89697	C₂	semidirect
4. #89609	C₂	semidirect
5. #89607	C₂	semidirect
6. #89605	C₂	semidirect
7. D₇×C₄⋊C₈	C₂	semidirect
9. #89492	C₂	semidirect
10. #89481	C₂	semidirect
11. #89480	C₂	semidirect
91. #432	D₇	semidirect
271. C₇	#2925	semidirect

Maximal subgroups

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

subgroup	index
C₄×#1157	2
C₇×#432	2
#89697	2
#89609	2
#89607	2
#89605	2
D₇×C₄⋊C₈	2
#89499	2
#89492	2
#89481	2
#89480	2
#2925	7