Subgroups — #11736517

Order 896.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	#89898	C₂	1	1	yes	maximal, characteristic
4	#89894	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#89873	C₂	1	1	yes	maximal, characteristic, complemented
7	#89771	C₂	1	1	yes	maximal, characteristic, complemented
8	#89612	C₂	1	2	yes	maximal, complemented
9	#89612	C₂	1	2	yes	maximal, complemented
10	#89607	C₂	1	2	yes	maximal, complemented
11	#89605	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×D₇	C₂²	1	1	yes	characteristic
13	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#117	C₂²	1	1	yes	characteristic
16	#6489	C₂²	1	1	yes	characteristic
17	C₂×#1136	C₂²	1	1	yes	characteristic
18	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#6455	C₄	1	1	yes	characteristic
20	#6455	C₄	1	1	yes	characteristic
21	D₇×C₄⋊C₄	C₄	1	1	yes	characteristic
22	D₇×C₄⋊C₄	C₄	1	1	yes	characteristic
23	D₇×C₂²⋊C₄	C₄	1	2	yes	—
24	#6440	C₄	1	2	yes	—
25	#6435	C₂²	1	2	yes	—
26	C₄²×D₇	C₂²	1	2	yes	—
27	C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₇×C₄⋊C₈	C₂²	1	2	yes	complemented
29	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
34	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
35	#6389	C₂²	1	2	yes	complemented
36	#6389	C₂²	1	2	yes	complemented
37	C₈×Dic₇	C₂²	1	2	yes	complemented
38	C₈×Dic₇	C₂²	1	2	yes	complemented
39	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
40	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
41	#2924	—	7	1	no	maximal, Sylow
42	C₂³×D₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₇	—	2	2	no	—
46	C₂×C₄×D₇	—	2	2	no	—
47	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
53	C₇×M₁₆	—	2	2	no	—
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂³	1	2	yes	—
58	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₄²×C₇	C₂³	1	2	yes	—
62	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
63	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
64	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
67	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
68	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₇	C₂×C₄	1	2	yes	—
70	C₄×Dic₇	C₂×C₄	1	2	yes	—
71	C₄×Dic₇	C₂³	1	2	yes	—
72	#1136	—	2	2	no	—
73	C₂×C₇⋊C₈	—	2	2	no	—
74	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₇⋊C₈	C₂³	1	2	yes	—
77	C₂×#117	—	7	1	no	—
78	#433	—	7	2	no	—
79	#433	—	7	2	no	—
80	#432	—	7	2	no	—
81	#431	—	7	2	no	—
82	#424	—	7	1	no	—
83	#424	D₇	1	1	yes	characteristic, complemented
84	#407	—	7	1	no	—
85	C₂×C₂²⋊C₈	—	7	1	no	—
86	#405	—	7	1	no	—
87	#405	—	7	1	no	—
88	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₇	—	2	2	no	—
90	C₂²×D₇	—	2	2	no	—
91	C₂²×D₇	—	4	1	no	—
92	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	#41	1	2	yes	—
98	C₂×C₄×C₇	#41	1	2	yes	—
99	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₇	—	2	2	no	—
101	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₇	C₂²×C₄	1	2	yes	—
103	C₂×Dic₇	C₂²×C₄	1	2	yes	—
104	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₇	—	2	4	no	—
106	C₄×D₇	—	2	2	no	—
107	C₄×D₇	—	2	2	no	—
108	C₄×D₇	—	2	1	no	—
109	C₄×D₇	—	2	1	no	—
110	C₄×D₇	—	2	1	no	—
111	C₄×D₇	—	2	1	no	—
112	C₄×D₇	—	4	2	no	—
113	C₇×C₈	—	2	1	no	—
114	C₇×C₈	—	2	1	no	—
115	C₇×C₈	—	2	2	no	—
116	C₇⋊C₈	—	2	1	no	—
117	C₇⋊C₈	—	2	1	no	—
118	C₇⋊C₈	—	2	2	no	—
119	C₂³×C₄	—	7	1	no	—
120	C₂×M₁₆	—	7	1	no	—
121	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
122	C₂²×C₈	—	7	1	no	—
123	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
124	#117	—	7	1	no	—
125	#117	—	7	1	no	—
126	#117	—	7	1	no	—
127	#117	—	7	2	no	—
128	#117	—	7	2	no	—
129	#117	C₂×D₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	7	1	no	—
131	C₂×C₄⋊C₄	—	7	1	no	—
132	C₂×C₂²⋊C₄	—	7	2	no	—
133	C₂×C₄²	—	7	2	no	—
134	C₄⋊C₈	—	7	2	no	—
135	C₄⋊C₈	—	7	2	no	—
136	C₄⋊C₈	C₂×D₇	1	2	yes	complemented
137	C₄⋊C₈	C₂×D₇	1	2	yes	complemented
138	C₂²⋊C₈	—	7	1	no	—
139	C₂²⋊C₈	—	7	1	no	—
140	C₂²⋊C₈	—	7	1	no	—
141	C₂²⋊C₈	—	7	1	no	—
142	C₂²⋊C₈	—	7	2	no	—
143	C₂²⋊C₈	—	7	2	no	—
144	#97	—	7	2	no	—
145	#97	—	7	2	no	—
146	C₄×C₈	—	7	2	no	—
147	C₄×C₈	—	7	2	no	—
148	C₂²×C₇	—	2	1	no	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	#117	1	2	yes	—
151	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×D₇	—	2	1	no	—
153	C₂×D₇	—	2	1	no	—
154	C₂×D₇	—	2	1	no	—
155	C₂×D₇	—	2	1	no	—
156	C₂×D₇	—	2	1	no	—
157	C₂×D₇	—	4	4	no	—
158	C₂×D₇	#131	1	2	yes	—
159	C₄×C₇	—	2	2	no	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	#117	1	2	yes	—
162	C₄×C₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₇	C₂×#41	1	1	yes	characteristic
164	Dic₇	—	2	1	no	—
165	Dic₇	—	2	1	no	—
166	Dic₇	—	2	2	no	—
167	Dic₇	—	2	2	no	—
168	C₂⁴	—	7	1	no	—
169	C₂²×C₄	—	7	1	no	—
170	C₂²×C₄	—	7	2	no	—
171	C₂²×C₄	—	7	1	no	—
172	C₂²×C₄	—	7	1	no	—
173	C₂²×C₄	—	7	1	no	—
174	C₂²×C₄	—	7	1	no	—
175	C₂²×C₄	—	7	2	no	—
176	C₂²×C₄	—	14	2	no	—
177	C₂²×C₄	—	14	2	no	—
178	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	14	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	7	1	no	—
183	C₂×C₈	—	7	2	no	—
184	C₂×C₈	—	7	1	no	—
185	C₂×C₈	—	14	2	no	—
186	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₇	1	2	yes	—
188	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
189	C₄⋊C₄	—	7	2	no	—
190	C₄⋊C₄	—	7	2	no	—
191	C₄⋊C₄	—	7	1	no	—
192	C₄⋊C₄	—	7	1	no	—
193	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
194	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
195	C₂²⋊C₄	—	7	2	no	—
196	C₂²⋊C₄	—	7	2	no	—
197	C₂²⋊C₄	—	7	2	no	—
198	C₂²⋊C₄	C₄×D₇	1	2	yes	—
199	C₄²	—	7	2	no	—
200	C₄²	—	7	2	no	—
201	C₄²	—	7	2	no	—
202	C₄²	C₂²×D₇	1	2	yes	—
203	C₂×C₇	—	2	2	no	—
204	C₂×C₇	C₂×#117	1	1	yes	characteristic
205	C₂×C₇	C₂×#131	1	1	yes	characteristic
206	C₂×C₇	#568	1	1	yes	characteristic
207	D₇	—	2	2	no	—
208	D₇	—	4	1	no	—
209	C₂³	—	7	1	no	—
210	C₂³	—	7	1	no	—
211	C₂³	—	14	2	no	—
212	C₂³	—	14	2	no	—
213	C₂³	—	28	1	no	—
214	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
215	C₂×C₄	—	7	1	no	—
216	C₂×C₄	—	7	2	no	—
217	C₂×C₄	—	7	2	no	—
218	C₂×C₄	—	7	1	no	—
219	C₂×C₄	—	14	4	no	—
220	C₂×C₄	—	14	2	no	—
221	C₂×C₄	—	14	2	no	—
222	C₂×C₄	—	14	2	no	—
223	C₂×C₄	—	14	1	no	—
224	C₂×C₄	—	14	1	no	—
225	C₂×C₄	—	14	1	no	—
226	C₂×C₄	—	14	1	no	—
227	C₂×C₄	—	28	2	no	—
228	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
231	C₂×C₄	#1159	1	2	yes	—
232	C₂×C₄	#1161	1	2	yes	—
233	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	2	no	—
237	C₈	—	14	1	no	—
238	C₈	—	14	1	no	—
239	C₈	—	14	2	no	—
240	C₇	#2924	1	1	yes	characteristic, complemented, Sylow
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	7	2	no	—
244	C₂²	—	14	1	no	—
245	C₂²	—	14	1	no	—
246	C₂²	—	14	1	no	—
247	C₂²	—	14	1	no	—
248	C₂²	—	14	1	no	—
249	C₂²	—	28	4	no	—
250	C₂²	#6455	1	2	yes	—
251	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	14	2	no	—
255	C₄	—	14	2	no	—
256	C₄	—	14	1	no	—
257	C₄	—	14	1	no	—
258	C₄	#6455	1	2	yes	—
259	C₄	C₂×#1159	1	1	yes	characteristic
260	C₄	C₂×#1161	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	14	2	no	—
263	C₂	—	28	1	no	—
264	C₂	C₂×#6455	1	1	yes	characteristic
265	C₂	D₇×#131	1	1	yes	characteristic
266	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#424	C₂
prime radical	C₂³×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#424	D₇
hypocenter	C₇	#2924
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. C₇×#424	C₂	semidirect
4. #89894	C₂	semidirect
6. #89873	C₂	semidirect
7. #89771	C₂	semidirect
8. #89612	C₂	semidirect
9. #89612	C₂	semidirect
10. #89607	C₂	semidirect
11. #89605	C₂	semidirect
27. C₇×C₄⋊C₈	C₂²	semidirect
28. C₇×C₄⋊C₈	C₂²	semidirect
35. #6389	C₂²	semidirect
36. #6389	C₂²	semidirect
37. C₈×Dic₇	C₂²	semidirect
38. C₈×Dic₇	C₂²	semidirect
39. C₄⋊(C₇⋊C₈)	C₂²	semidirect
40. C₄⋊(C₇⋊C₈)	C₂²	semidirect
83. #424	D₇	semidirect
136. C₄⋊C₈	C₂×D₇	semidirect
137. C₄⋊C₈	C₂×D₇	semidirect
240. C₇	#2924	semidirect

Maximal subgroups

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

subgroup	index
D₇×#117	2
C₇×#424	2
#89898	2
#89894	2
C₂×(C₂×D₇)⋊C₈	2
#89873	2
#89771	2
#89612	2
#89607	2
#89605	2
#2924	7