Subgroups — #11737367

Order 896.

322 conjugacy classes of subgroups.

The subgroup lattice has 228 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#89796	C₂	1	1	yes	maximal, characteristic, complemented
4	#89612	C₂	1	2	yes	maximal
5	#89607	C₂	1	1	yes	maximal, characteristic
6	#89603	C₂	1	1	yes	maximal, characteristic, complemented
7	#89600	C₂	1	2	yes	maximal, complemented
8	#89462	C₂	1	2	yes	maximal
9	#89457	C₂	1	1	yes	maximal, characteristic
10	#89451	C₂	1	2	yes	maximal, complemented
11	C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
13	#6509	C₄	1	1	yes	characteristic
14	#6507	C₄	1	1	yes	characteristic
15	#6460	C₄	1	1	yes	characteristic
16	#6459	C₄	1	2	yes	—
17	#6452	C₄	1	2	yes	—
18	#6451	C₄	1	1	yes	characteristic
19	#6435	C₂²	1	2	yes	—
20	C₄²×D₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₄	C₂²	1	2	yes	—
22	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
23	C₇×C₄⋊C₈	C₂²	1	2	yes	—
24	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂²	1	2	yes	—
27	(C₂×D₇)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
30	#6389	C₂²	1	2	yes	—
31	#6389	C₂²	1	2	yes	—
32	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
33	Dic₇⋊C₈	C₂²	1	2	yes	—
34	Dic₇⋊C₈	C₂²	1	2	yes	—
35	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×Dic₇	C₂²	1	2	yes	—
37	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
39	#6377	C₂²	1	2	yes	—
40	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
41	#2985	—	7	1	no	maximal, Sylow
42	C₂×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₁₄	C₂×C₄	1	2	yes	—
48	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂³	1	2	yes	—
51	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₄²×C₇	C₂³	1	2	yes	—
55	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₇	C₂×C₄	1	2	yes	—
59	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
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	C₂³	1	1	yes	characteristic
67	C₄×Dic₇	C₂³	1	2	yes	—
68	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₈	C₂³	1	2	yes	—
71	#533	—	7	1	no	—
72	#446	—	7	1	no	—
73	#446	—	7	2	no	—
74	#446	—	7	1	no	—
75	#446	—	7	2	no	—
76	C₈×Q₈	—	7	1	no	—
77	C₈×Q₈	D₇	1	1	yes	characteristic, complemented
78	#433	—	7	2	no	—
79	#433	—	7	2	no	—
80	#432	—	7	1	no	—
81	#431	—	7	1	no	—
82	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
83	C₇×Q₈	—	2	2	no	—
84	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₇	C₂²×C₄	1	2	yes	—
93	C₂×Dic₇	C₂²×C₄	1	2	yes	—
94	C₄×D₇	—	2	2	no	—
95	C₄×D₇	—	2	1	no	—
96	C₄×D₇	—	2	1	no	—
97	C₄×D₇	—	4	1	no	—
98	Dic₁₄	—	2	2	no	—
99	Dic₁₄	—	4	2	no	—
100	C₇×C₈	—	2	1	no	—
101	C₇×C₈	—	2	1	no	—
102	C₇×C₈	—	2	2	no	—
103	C₇⋊C₈	—	2	1	no	—
104	C₇⋊C₈	—	2	2	no	—
105	C₇⋊C₈	—	2	1	no	—
106	C₄⋊Q₈	—	7	1	no	—
107	C₄⋊Q₈	—	7	1	no	—
108	#125	—	7	2	no	—
109	C₂²⋊Q₈	—	7	1	no	—
110	C₂²⋊Q₈	—	7	2	no	—
111	C₂²⋊Q₈	—	7	1	no	—
112	C₄×Q₈	—	7	1	no	—
113	C₄×Q₈	—	7	2	no	—
114	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
115	#117	—	7	2	no	—
116	C₂×C₄²	—	7	1	no	—
117	C₄⋊C₈	—	7	1	no	—
118	C₄⋊C₈	—	7	1	no	—
119	C₄⋊C₈	—	7	2	no	—
120	C₄⋊C₈	—	7	2	no	—
121	C₄⋊C₈	—	7	1	no	—
122	C₄⋊C₈	—	7	2	no	—
123	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
124	C₄⋊C₈	C₂×D₇	1	2	yes	—
125	C₂²⋊C₈	—	7	1	no	—
126	C₂²⋊C₈	—	7	1	no	—
127	C₂²⋊C₈	—	7	2	no	—
128	#97	—	7	2	no	—
129	#97	—	7	2	no	—
130	#97	—	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	C₄×C₈	C₂×D₇	1	2	yes	—
135	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₇	—	2	1	no	—
137	C₂×D₇	—	2	1	no	—
138	C₂×D₇	—	2	1	no	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	—	2	2	no	—
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	#131	1	2	yes	—
143	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₇	—	2	1	no	—
146	Dic₇	—	2	2	no	—
147	Dic₇	—	2	1	no	—
148	Dic₇	—	2	1	no	—
149	Dic₇	—	2	2	no	—
150	C₂×Q₈	—	7	1	no	—
151	C₂×Q₈	—	7	2	no	—
152	C₂×Q₈	C₄×D₇	1	1	yes	characteristic
153	C₂²×C₄	—	7	1	no	—
154	C₂²×C₄	—	7	1	no	—
155	C₂²×C₄	—	7	1	no	—
156	C₂×C₈	—	7	2	no	—
157	C₂×C₈	—	7	1	no	—
158	C₂×C₈	—	7	1	no	—
159	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
160	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
161	C₂×C₈	C₂²×D₇	1	2	yes	—
162	C₄⋊C₄	—	7	2	no	—
163	C₄⋊C₄	—	7	2	no	—
164	C₄⋊C₄	—	7	2	no	—
165	C₄⋊C₄	—	7	2	no	—
166	C₄⋊C₄	—	7	2	no	—
167	C₄⋊C₄	—	7	2	no	—
168	C₄⋊C₄	—	7	1	no	—
169	C₄⋊C₄	C₄×D₇	1	2	yes	—
170	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
171	C₂²⋊C₄	—	7	2	no	—
172	C₂²⋊C₄	—	7	2	no	—
173	C₄²	—	7	1	no	—
174	C₄²	—	7	1	no	—
175	C₄²	—	7	2	no	—
176	C₄²	—	7	1	no	—
177	C₄²	C₂²×D₇	1	1	yes	characteristic
178	C₄²	C₂²×D₇	1	2	yes	—
179	C₂×C₇	#519	1	1	yes	characteristic
180	C₂×C₇	C₂×#131	1	1	yes	characteristic
181	C₂×C₇	#568	1	1	yes	characteristic
182	D₇	—	4	1	no	—
183	C₂³	—	7	1	no	—
184	Q₈	—	2	2	no	—
185	Q₈	—	14	2	no	—
186	Q₈	—	28	2	no	—
187	C₂×C₄	—	7	1	no	—
188	C₂×C₄	—	7	1	no	—
189	C₂×C₄	—	7	1	no	—
190	C₂×C₄	—	7	2	no	—
191	C₂×C₄	—	7	2	no	—
192	C₂×C₄	—	14	2	no	—
193	C₂×C₄	—	14	1	no	—
194	C₂×C₄	—	14	1	no	—
195	C₂×C₄	—	28	1	no	—
196	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	14	1	no	—
205	C₈	—	14	2	no	—
206	C₈	—	14	1	no	—
207	C₇	#2985	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	14	1	no	—
209	C₂²	—	14	1	no	—
210	C₂²	—	14	1	no	—
211	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	14	1	no	—
216	C₄	—	14	2	no	—
217	C₄	—	14	1	no	—
218	C₄	—	14	2	no	—
219	C₄	—	14	1	no	—
220	C₄	#6464	1	2	yes	—
221	C₄	#6551	1	1	yes	characteristic
222	C₄	#6554	1	1	yes	characteristic
223	C₂	—	28	1	no	—
224	C₂	#90263	1	1	yes	characteristic
225	C₂	C₂×#6464	1	1	yes	characteristic
226	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×C₈×Q₈	C₂
prime radical	C₂²×D₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₈×Q₈	D₇
hypocenter	C₇	#2985
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₇×C₈×Q₈	C₂	semidirect
3. #89796	C₂	semidirect
6. #89603	C₂	semidirect
7. #89600	C₂	semidirect
10. #89451	C₂	semidirect
11. C₈×Dic₁₄	C₂	semidirect
77. C₈×Q₈	D₇	semidirect
207. C₇	#2985	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×C₈×Q₈	2
#89796	2
#89612	2
#89607	2
#89603	2
#89600	2
#89462	2
#89457	2
#89451	2
C₈×Dic₁₄	2
#2985	7