Subgroups — #11741237

Order 896.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	#90513	C₂	1	1	yes	maximal, characteristic
3	C₂×#6464	C₂	1	1	yes	maximal, characteristic, complemented
4	#90008	C₂	1	1	yes	maximal, characteristic, complemented
5	#89917	C₂	1	4	yes	maximal, complemented
6	#89916	C₂	1	4	yes	maximal, complemented
7	#89911	C₂	1	1	yes	maximal, characteristic, complemented
8	#89896	C₂	1	1	yes	maximal, characteristic, complemented
9	#89873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#41	C₂²	1	1	yes	characteristic
11	C₂×#1157	C₂²	1	1	yes	characteristic
12	C₇×#131	—	2	1	no	—
13	C₇×#131	—	2	1	no	—
14	C₇×#131	C₂²	1	2	yes	complemented
15	C₇×#131	C₂²	1	2	yes	complemented
16	C₂×C₇×M₁₆	—	2	1	no	—
17	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	1	no	—
19	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
20	#6514	C₄	1	2	yes	complemented
21	#6513	C₂²	1	4	yes	—
22	#6512	C₄	1	2	yes	complemented
23	#6505	C₄	1	2	yes	complemented
24	#6495	C₄	1	2	yes	complemented
25	#6489	C₂²	1	1	yes	characteristic
26	C₂×#1136	C₂²	1	1	yes	characteristic
27	C₂×#1136	C₂²	1	1	yes	characteristic
28	#6464	—	2	1	no	—
29	#6464	—	2	1	no	—
30	#6464	C₂²	1	2	yes	complemented
31	#6464	C₂²	1	2	yes	complemented
32	C₂×#1131	—	2	1	no	—
33	C₂×C₈×D₇	—	2	1	no	—
34	#6411	C₂²	1	4	yes	complemented
35	#6399	C₂²	1	4	yes	complemented
36	#6396	C₂²	1	4	yes	complemented
37	#6389	C₂²	1	2	yes	complemented
38	C₈×Dic₇	C₂²	1	2	yes	complemented
39	#2955	—	7	1	no	maximal, Sylow
40	C₇×#41	—	2	1	no	—
41	C₇×#41	—	2	1	no	—
42	C₇×#41	C₂×C₄	1	2	yes	—
43	C₇×#41	C₂³	1	2	yes	—
44	C₂×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇×D₄	—	2	1	no	—
46	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₇	—	2	1	no	—
50	#1157	—	2	1	no	—
51	#1157	—	2	1	no	—
52	#1157	C₂×C₄	1	2	yes	—
53	#1157	C₂³	1	2	yes	—
54	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₇	—	2	1	no	—
56	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
57	C₇×M₁₆	—	2	1	no	—
58	C₇×M₁₆	—	2	2	no	—
59	C₇×M₁₆	—	2	2	no	—
60	C₇×M₁₆	C₂³	1	2	yes	—
61	C₂×C₇×C₈	—	2	1	no	—
62	C₂×C₇×C₈	—	2	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	D₄	1	1	yes	characteristic
65	C₂×C₇×C₈	D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	D₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₇	—	2	1	no	—
71	#1144	C₂×C₄	1	4	yes	—
72	#1143	C₂×C₄	1	4	yes	—
73	#1142	C₂×C₄	1	4	yes	—
74	#1141	C₂×C₄	1	4	yes	—
75	C₄⋊Dic₇	—	2	1	no	—
76	C₄×Dic₇	C₂³	1	2	yes	—
77	#1136	—	2	1	no	—
78	#1136	C₂×C₄	1	2	yes	—
79	#1136	C₂×C₄	1	2	yes	—
80	#1136	C₂³	1	2	yes	—
81	C₂×C₇⋊C₈	—	2	1	no	—
82	C₂×C₇⋊C₈	C₂³	1	2	yes	—
83	#1131	—	2	2	no	—
84	#1131	—	2	2	no	—
85	C₈×D₇	—	2	2	no	—
86	C₈×D₇	—	2	2	no	—
87	#575	—	7	1	no	—
88	C₂×#131	—	7	1	no	—
89	C₂×#131	D₇	1	1	yes	characteristic, complemented
90	#444	—	7	4	no	—
91	#443	—	7	4	no	—
92	#430	—	7	1	no	—
93	#421	—	7	1	no	—
94	#421	—	7	1	no	—
95	#405	—	7	1	no	—
96	C₂³×C₇	—	2	1	no	—
97	C₂³×C₇	#41	1	1	yes	characteristic
98	C₂²×D₇	—	2	1	no	—
99	C₇×Q₈	—	2	1	no	—
100	C₇×Q₈	C₂²×C₄	1	2	yes	—
101	C₇×D₄	—	2	2	no	—
102	C₇×D₄	—	2	2	no	—
103	C₇×D₄	—	2	1	no	—
104	C₇×D₄	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	#41	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₇	—	2	2	no	—
115	C₂²⋊D₇	—	2	2	no	—
116	C₂×Dic₇	—	2	1	no	—
117	C₂×Dic₇	—	2	2	no	—
118	D₂₈	—	2	1	no	—
119	D₂₈	C₂²×C₄	1	2	yes	—
120	C₄×D₇	—	2	2	no	—
121	C₄×D₇	—	2	2	no	—
122	Dic₁₄	—	2	1	no	—
123	Dic₁₄	C₂²×C₄	1	2	yes	—
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	C₂×D₄	1	1	yes	characteristic
126	C₇×C₈	C₂×D₄	1	1	yes	characteristic
127	C₇×C₈	C₂×D₄	1	1	yes	characteristic
128	C₇×C₈	C₂×D₄	1	1	yes	characteristic
129	C₇⋊C₈	—	2	2	no	—
130	C₇⋊C₈	C₂²×C₄	1	4	yes	—
131	C₂×#41	—	7	1	no	—
132	C₂×#41	C₂×D₇	1	1	yes	characteristic
133	#137	—	7	2	no	—
134	#137	—	7	2	no	—
135	AGL(1,ℤ/8)	—	7	2	no	—
136	AGL(1,ℤ/8)	—	7	2	no	—
137	#135	—	7	4	no	—
138	#131	—	2	1	no	—
139	#131	—	2	1	no	—
140	#131	—	7	2	no	—
141	#131	—	7	2	no	—
142	#131	—	14	1	no	—
143	#131	—	14	1	no	—
144	#131	C₂×D₇	1	2	yes	complemented
145	#131	C₂×D₇	1	2	yes	complemented
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	—	7	1	no	—
148	C₂×M₁₆	—	7	1	no	—
149	C₂×M₁₆	—	14	1	no	—
150	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	—	14	1	no	—
153	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
154	#117	—	7	1	no	—
155	#108	—	7	4	no	—
156	#104	—	7	4	no	—
157	#104	—	7	4	no	—
158	#97	—	7	2	no	—
159	C₄×C₈	—	7	2	no	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	4	1	no	—
162	C₂²×C₇	—	4	2	no	—
163	C₂²×C₇	C₄×D₄	1	1	yes	characteristic
164	C₂²×C₇	C₄×D₄	1	1	yes	characteristic
165	C₂²×C₇	C₂×#41	1	1	yes	characteristic
166	C₂×D₇	—	2	2	no	—
167	C₂×D₇	—	4	2	no	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	C₄×D₄	1	1	yes	characteristic
170	C₄×C₇	C₄×D₄	1	1	yes	characteristic
171	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
172	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
173	Dic₇	—	2	2	no	—
174	Dic₇	—	4	2	no	—
175	#41	—	2	1	no	—
176	#41	—	2	1	no	—
177	#41	—	7	2	no	—
178	#41	—	7	2	no	—
179	#41	—	14	1	no	—
180	#41	—	14	1	no	—
181	#41	C₄×D₇	1	2	yes	—
182	#41	C₂²×D₇	1	2	yes	—
183	C₂×Q₈	—	7	1	no	—
184	C₂×Q₈	C₄×D₇	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	7	1	no	—
187	C₂×D₄	—	14	1	no	—
188	C₂×D₄	C₄×D₇	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	14	1	no	—
191	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
192	Q₁₆	—	7	4	no	—
193	SD₁₆	—	7	4	no	—
194	SD₁₆	—	7	4	no	—
195	D₈	—	7	4	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	7	2	no	—
200	M₁₆	—	7	2	no	—
201	M₁₆	—	7	2	no	—
202	M₁₆	—	14	1	no	—
203	M₁₆	—	14	2	no	—
204	M₁₆	—	14	2	no	—
205	M₁₆	C₂²×D₇	1	2	yes	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	7	2	no	—
210	C₂×C₈	—	14	1	no	—
211	C₂×C₈	—	14	2	no	—
212	C₂×C₈	—	14	2	no	—
213	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
218	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
219	C₄⋊C₄	—	14	1	no	—
220	C₂²⋊C₄	—	14	1	no	—
221	C₄²	—	7	2	no	—
222	C₂×C₇	—	2	1	no	—
223	C₂×C₇	—	2	1	no	—
224	C₂×C₇	—	2	1	no	—
225	C₂×C₇	—	4	2	no	—
226	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
227	D₇	—	4	2	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	14	1	no	—
230	C₂³	#1157	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	7	2	no	—
233	Q₈	—	14	1	no	—
234	Q₈	C₂×C₄×D₇	1	2	yes	—
235	D₄	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	2	1	no	—
238	D₄	—	7	2	no	—
239	D₄	—	14	2	no	—
240	D₄	—	14	2	no	—
241	D₄	—	14	1	no	—
242	D₄	C₂×C₄×D₇	1	2	yes	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	14	1	no	—
247	C₂×C₄	—	14	2	no	—
248	C₂×C₄	—	14	2	no	—
249	C₂×C₄	—	14	2	no	—
250	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
252	C₂×C₄	#1157	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
255	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	7	4	no	—
258	C₈	—	14	2	no	—
259	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
260	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
263	C₇	#2955	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	14	2	no	—
268	C₂²	—	28	2	no	—
269	C₂²	C₄×C₂²⋊D₇	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₇	1	1	yes	characteristic
271	C₂²	C₂×#1157	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	14	2	no	—
274	C₄	—	28	2	no	—
275	C₄	C₄×C₂²⋊D₇	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₇	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	28	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₇
commutator	C₄×C₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₂×C₇×#131	C₂
socle	C₂×C₇	C₂×C₄×D₄
prime radical	#90513	C₂
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×#131	D₇
hypocenter	C₇	#2955
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#131	C₂	semidirect
3. C₂×#6464	C₂	semidirect
4. #90008	C₂	semidirect
5. #89917	C₂	semidirect
6. #89916	C₂	semidirect
7. #89911	C₂	semidirect
8. #89896	C₂	semidirect
9. #89873	C₂	semidirect
14. C₇×#131	C₂²	semidirect
15. C₇×#131	C₂²	semidirect
20. #6514	C₄	semidirect
22. #6512	C₄	semidirect
23. #6505	C₄	semidirect
24. #6495	C₄	semidirect
30. #6464	C₂²	semidirect
31. #6464	C₂²	semidirect
34. #6411	C₂²	semidirect
35. #6399	C₂²	semidirect
36. #6396	C₂²	semidirect
37. #6389	C₂²	semidirect
38. C₈×Dic₇	C₂²	semidirect
89. C₂×#131	D₇	semidirect
144. #131	C₂×D₇	semidirect
145. #131	C₂×D₇	semidirect
263. C₇	#2955	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#131	2
#90513	2
C₂×#6464	2
#90008	2
#89917	2
#89916	2
#89911	2
#89896	2
#89873	2
#2955	7