Subgroups — #11736849

Order 896.

406 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1159	C₂	1	1	yes	maximal, characteristic
2	C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#89883	C₂	1	2	yes	maximal, complemented
4	C₂×#6389	C₂	1	2	yes	maximal, complemented
5	D₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#89607	C₂	1	1	yes	maximal, characteristic, complemented
7	#89603	C₂	1	1	yes	maximal, characteristic, complemented
8	#89502	C₂	1	2	yes	maximal, complemented
9	#89491	C₂	1	2	yes	maximal, complemented
10	C₄×#1131	C₂	1	1	yes	maximal, characteristic, complemented
11	#89451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#1159	C₄	1	1	yes	characteristic
13	C₂²×C₇×C₈	C₂²	1	2	yes	—
14	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	D₄×Dic₇	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₇	C₂²	1	2	yes	—
18	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
19	C₂×#1131	C₂²	1	1	yes	characteristic
20	C₂×#1131	C₂²	1	1	yes	characteristic
21	#6455	C₄	1	1	yes	characteristic
22	#6450	C₄	1	1	yes	characteristic
23	(C₂²⋊D₇)⋊C₄	C₄	1	2	yes	—
24	#6440	C₄	1	2	yes	—
25	C₄²×D₇	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
32	#6389	C₂²	1	2	yes	—
33	#6389	C₂²	1	2	yes	—
34	#6389	C₂²	1	1	yes	characteristic
35	#6389	C₂²	1	2	yes	—
36	#6389	C₂²	1	1	yes	characteristic
37	Dic₇⋊C₈	C₂²	1	2	yes	—
38	Dic₇⋊C₈	C₂²	1	2	yes	—
39	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
41	#2970	—	7	1	no	maximal, Sylow
42	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₇	C₂×C₄	1	2	yes	—
46	C₂²×Dic₇	C₂×C₄	1	2	yes	—
47	#1159	—	2	4	no	—
48	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	—	2	4	no	—
53	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂³	1	2	yes	—
55	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₄²×C₇	C₂³	1	1	yes	characteristic
59	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
60	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
62	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₇	C₂×C₄	1	2	yes	—
67	C₄×Dic₇	C₂×C₄	1	2	yes	—
68	C₄×Dic₇	C₂³	1	2	yes	—
69	C₄×Dic₇	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₈	—	2	4	no	—
71	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇⋊C₈	C₂³	1	2	yes	—
73	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
74	#1131	—	2	2	no	—
75	#1131	—	2	2	no	—
76	C₄×#41	—	7	1	no	—
77	#446	—	7	1	no	—
78	#446	—	7	1	no	—
79	#435	—	7	2	no	—
80	#435	—	7	2	no	—
81	C₈×D₄	—	7	1	no	—
82	C₈×D₄	D₇	1	1	yes	characteristic, complemented
83	#432	—	7	1	no	—
84	#432	—	7	2	no	—
85	C₄×M₁₆	—	7	1	no	—
86	C₂×#97	—	7	2	no	—
87	C₂³×C₇	C₂²×C₄	1	2	yes	—
88	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
89	C₇×D₄	M₁₆	1	4	yes	complemented
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₇	—	2	4	no	—
97	C₂×Dic₇	—	2	4	no	—
98	C₂×Dic₇	—	4	2	no	—
99	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
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	C₄×D₇	—	2	2	no	—
105	C₄×D₇	—	2	1	no	—
106	C₄×D₇	—	2	1	no	—
107	C₄×D₇	—	4	1	no	—
108	Dic₁₄	—	2	2	no	—
109	C₇×C₈	—	2	1	no	—
110	C₇×C₈	—	2	2	no	—
111	C₇×C₈	#41	1	2	yes	complemented
112	C₇⋊C₈	—	2	1	no	—
113	C₇⋊C₈	—	2	2	no	—
114	C₇⋊C₈	#41	1	2	yes	complemented
115	C₂×#41	—	7	1	no	—
116	C₂×M₁₆	—	7	1	no	—
117	C₂×M₁₆	—	7	1	no	—
118	C₂²×C₈	—	7	2	no	—
119	C₂²×C₈	C₂×D₇	1	2	yes	—
120	C₄×Q₈	—	7	1	no	—
121	C₄×Q₈	—	7	1	no	—
122	C₄×D₄	—	7	1	no	—
123	C₄×D₄	—	7	2	no	—
124	C₄×D₄	—	7	2	no	—
125	C₄×D₄	C₂×D₇	1	1	yes	characteristic
126	#117	—	7	1	no	—
127	#117	—	7	2	no	—
128	C₂×C₄²	—	7	2	no	—
129	C₂×C₄²	—	7	1	no	—
130	C₄⋊C₈	—	7	2	no	—
131	C₄⋊C₈	—	7	2	no	—
132	C₄⋊C₈	—	7	1	no	—
133	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
134	C₂²⋊C₈	—	7	2	no	—
135	C₂²⋊C₈	—	7	2	no	—
136	C₂²⋊C₈	C₂×D₇	1	2	yes	—
137	#97	—	7	1	no	—
138	#97	—	7	2	no	—
139	#97	—	7	2	no	—
140	#97	—	7	2	no	—
141	#97	—	7	1	no	—
142	C₄×C₈	—	7	1	no	—
143	C₄×C₈	C₂×D₇	1	1	yes	characteristic
144	C₂²×C₇	—	2	2	no	—
145	C₂²×C₇	—	2	2	no	—
146	C₂²×C₇	C₂×M₁₆	1	4	yes	—
147	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×D₇	—	2	1	no	—
149	C₂×D₇	—	2	1	no	—
150	C₂×D₇	—	2	1	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	1	no	—
153	C₄×C₇	C₂×M₁₆	1	2	yes	—
154	C₄×C₇	C₂×#41	1	1	yes	characteristic
155	C₄×C₇	C₂×#41	1	1	yes	characteristic
156	Dic₇	—	2	1	no	—
157	Dic₇	—	2	2	no	—
158	Dic₇	—	2	1	no	—
159	Dic₇	—	2	1	no	—
160	Dic₇	—	2	2	no	—
161	#41	—	14	4	no	—
162	C₂×Q₈	—	7	1	no	—
163	C₂×D₄	—	7	2	no	—
164	C₂×D₄	C₄×D₇	1	1	yes	characteristic
165	C₂²×C₄	—	7	2	no	—
166	C₂²×C₄	—	7	2	no	—
167	C₂²×C₄	—	7	1	no	—
168	C₂²×C₄	—	7	1	no	—
169	C₂²×C₄	—	7	1	no	—
170	C₂²×C₄	C₂²×D₇	1	2	yes	—
171	M₁₆	—	14	2	no	—
172	M₁₆	—	14	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	7	1	no	—
175	C₂×C₈	—	7	1	no	—
176	C₂×C₈	—	7	2	no	—
177	C₂×C₈	—	14	4	no	—
178	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₇	1	2	yes	—
180	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
181	C₄⋊C₄	—	7	2	no	—
182	C₄⋊C₄	—	7	2	no	—
183	C₄⋊C₄	—	7	1	no	—
184	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
185	C₂²⋊C₄	—	7	2	no	—
186	C₂²⋊C₄	—	7	2	no	—
187	C₂²⋊C₄	C₄×D₇	1	2	yes	—
188	C₄²	—	7	2	no	—
189	C₄²	—	7	1	no	—
190	C₄²	—	7	1	no	—
191	C₄²	—	7	2	no	—
192	C₄²	—	7	1	no	—
193	C₄²	—	7	2	no	—
194	C₄²	C₂²×D₇	1	1	yes	characteristic
195	C₂×C₇	—	2	4	no	—
196	C₂×C₇	C₄×#41	1	1	yes	characteristic
197	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
198	C₂×C₇	#568	1	1	yes	characteristic
199	D₇	—	4	1	no	—
200	C₂³	—	7	1	no	—
201	C₂³	C₂×C₄×D₇	1	2	yes	—
202	Q₈	—	14	2	no	—
203	D₄	—	14	4	no	—
204	D₄	#1131	1	4	yes	complemented
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	7	1	no	—
208	C₂×C₄	—	7	1	no	—
209	C₂×C₄	—	7	2	no	—
210	C₂×C₄	—	7	1	no	—
211	C₂×C₄	—	7	2	no	—
212	C₂×C₄	—	14	4	no	—
213	C₂×C₄	—	14	2	no	—
214	C₂×C₄	—	14	1	no	—
215	C₂×C₄	—	14	1	no	—
216	C₂×C₄	—	28	1	no	—
217	C₂×C₄	—	28	2	no	—
218	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
221	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	7	2	no	—
225	C₈	—	14	1	no	—
226	C₈	—	14	2	no	—
227	C₈	#1159	1	2	yes	complemented
228	C₇	#2970	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	14	1	no	—
232	C₂²	—	14	1	no	—
233	C₂²	—	14	1	no	—
234	C₂²	C₂×#1131	1	4	yes	—
235	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	14	1	no	—
239	C₄	—	14	2	no	—
240	C₄	—	14	1	no	—
241	C₄	—	14	2	no	—
242	C₄	—	14	1	no	—
243	C₄	C₂×#1131	1	2	yes	—
244	C₄	C₂×#1159	1	1	yes	characteristic
245	C₄	D₇×#41	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	28	1	no	—
248	C₂	C₄×#1159	1	1	yes	characteristic
249	C₂	C₂²×#1131	1	1	yes	characteristic
250	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×C₈×D₄	C₂
prime radical	C₂×#1159	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₈×D₄	D₇
hypocenter	C₇	#2970
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₇×C₈×D₄	C₂	semidirect
3. #89883	C₂	semidirect
4. C₂×#6389	C₂	semidirect
5. D₄×C₇⋊C₈	C₂	semidirect
6. #89607	C₂	semidirect
7. #89603	C₂	semidirect
8. #89502	C₂	semidirect
9. #89491	C₂	semidirect
10. C₄×#1131	C₂	semidirect
11. #89451	C₂	semidirect
82. C₈×D₄	D₇	semidirect
89. C₇×D₄	M₁₆	semidirect
111. C₇×C₈	#41	semidirect
114. C₇⋊C₈	#41	semidirect
204. D₄	#1131	semidirect
227. C₈	#1159	semidirect
228. C₇	#2970	semidirect

Maximal subgroups

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

subgroup	index
C₄×#1159	2
C₇×C₈×D₄	2
#89883	2
C₂×#6389	2
D₄×C₇⋊C₈	2
#89607	2
#89603	2
#89502	2
#89491	2
C₄×#1131	2
#89451	2
#2970	7