Subgroups — #11736853

Order 896.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90248	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	#89874	C₂	1	2	yes	maximal, complemented
5	#89784	C₂	1	1	yes	maximal, characteristic, complemented
6	#89609	C₂	1	1	yes	maximal, characteristic, complemented
7	#89603	C₂	1	1	yes	maximal, characteristic, complemented
8	#89501	C₂	1	2	yes	maximal, complemented
9	#89492	C₂	1	2	yes	maximal, complemented
10	#89457	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×C₈	C₂²	1	2	yes	—
13	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
14	#6501	C₄	1	1	yes	characteristic
15	#6499	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₇	C₂²	1	2	yes	—
17	#6489	C₂²	1	2	yes	—
18	C₂×#1136	C₂²	1	2	yes	—
19	C₂×#1131	C₂²	1	2	yes	—
20	#6460	C₄	1	1	yes	characteristic
21	#6453	C₄	1	1	yes	characteristic
22	#6446	C₄	1	2	yes	—
23	#6442	C₄	1	2	yes	—
24	C₄²×D₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
26	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
30	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
32	#6389	C₂²	1	2	yes	—
33	Dic₇⋊C₈	C₂²	1	2	yes	—
34	Dic₇⋊C₈	C₂²	1	2	yes	—
35	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₇⋊C₈	C₂²	1	2	yes	—
37	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₈×Dic₇	C₂²	1	2	yes	—
39	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
41	#2993	—	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	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	—	4	2	no	—
50	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂³	1	2	yes	—
53	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
59	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₇	C₂×C₄	1	2	yes	—
62	C₄⋊Dic₇	C₂×C₄	1	2	yes	—
63	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
64	Dic₇⋊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	C₄×Dic₇	C₂³	1	2	yes	—
68	C₄×Dic₇	C₂³	1	1	yes	characteristic
69	#1136	—	4	2	no	—
70	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇⋊C₈	C₂³	1	2	yes	—
72	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
73	#1131	—	4	2	no	—
74	#529	—	7	1	no	—
75	#446	—	7	1	no	—
76	C₈×Q₈	—	7	1	no	—
77	#436	—	7	1	no	—
78	#436	—	7	2	no	—
79	#435	—	7	2	no	—
80	C₈×D₄	D₇	1	1	yes	characteristic, complemented
81	#433	—	7	2	no	—
82	#431	—	7	1	no	—
83	#424	—	7	2	no	—
84	#423	—	7	1	no	—
85	C₂³×C₇	C₂²×C₄	1	2	yes	—
86	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
87	C₇×D₄	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₇	—	4	2	no	—
95	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₇	C₂²×C₄	1	2	yes	—
99	C₂×Dic₇	C₂²×C₄	1	2	yes	—
100	C₄×D₇	—	2	2	no	—
101	C₄×D₇	—	2	1	no	—
102	C₄×D₇	—	2	1	no	—
103	C₄×D₇	—	4	1	no	—
104	Dic₁₄	—	2	2	no	—
105	C₇×C₈	—	2	2	no	—
106	C₇×C₈	—	2	1	no	—
107	C₇×C₈	—	2	1	no	—
108	C₇⋊C₈	—	2	2	no	—
109	C₇⋊C₈	—	2	1	no	—
110	C₇⋊C₈	—	2	1	no	—
111	C₂×M₁₆	—	7	2	no	—
112	C₂×M₁₆	—	7	2	no	—
113	C₂²×C₈	C₂×D₇	1	2	yes	—
114	#126	—	7	2	no	—
115	#125	—	7	1	no	—
116	#124	—	7	1	no	—
117	#123	—	7	2	no	—
118	C₂²⋊Q₈	—	7	1	no	—
119	#121	—	7	1	no	—
120	C₄×Q₈	—	7	1	no	—
121	C₄×D₄	—	7	2	no	—
122	C₄×D₄	C₂×D₇	1	1	yes	characteristic
123	#117	—	7	2	no	—
124	C₂×C₄²	—	7	1	no	—
125	C₄⋊C₈	—	7	2	no	—
126	C₄⋊C₈	—	7	1	no	—
127	C₄⋊C₈	—	7	2	no	—
128	C₄⋊C₈	—	7	2	no	—
129	C₄⋊C₈	—	7	1	no	—
130	C₄⋊C₈	—	7	1	no	—
131	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
132	C₂²⋊C₈	—	7	2	no	—
133	C₂²⋊C₈	—	7	1	no	—
134	C₂²⋊C₈	—	7	1	no	—
135	C₂²⋊C₈	C₂×D₇	1	2	yes	—
136	#97	—	7	2	no	—
137	C₄×C₈	—	7	2	no	—
138	C₄×C₈	—	7	1	no	—
139	C₄×C₈	C₂×D₇	1	1	yes	characteristic
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	—	2	2	no	—
142	C₂²×C₇	—	2	2	no	—
143	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×D₇	—	2	1	no	—
145	C₂×D₇	—	2	1	no	—
146	C₂×D₇	—	2	1	no	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	#131	1	2	yes	—
150	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₇	—	2	1	no	—
153	Dic₇	—	2	2	no	—
154	Dic₇	—	2	1	no	—
155	Dic₇	—	2	1	no	—
156	Dic₇	—	2	2	no	—
157	C₂×Q₈	—	7	1	no	—
158	C₂×D₄	—	7	2	no	—
159	C₂×D₄	C₄×D₇	1	1	yes	characteristic
160	C₂²×C₄	—	7	1	no	—
161	C₂²×C₄	—	7	1	no	—
162	C₂²×C₄	—	7	1	no	—
163	C₂²×C₄	C₂²×D₇	1	2	yes	—
164	M₁₆	—	28	2	no	—
165	M₁₆	—	28	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	7	1	no	—
168	C₂×C₈	—	7	2	no	—
169	C₂×C₈	—	7	1	no	—
170	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₇	1	2	yes	—
173	C₄⋊C₄	—	7	2	no	—
174	C₄⋊C₄	—	7	2	no	—
175	C₄⋊C₄	—	7	2	no	—
176	C₄⋊C₄	—	7	2	no	—
177	C₄⋊C₄	—	7	1	no	—
178	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
179	C₂²⋊C₄	—	7	2	no	—
180	C₂²⋊C₄	—	7	2	no	—
181	C₂²⋊C₄	—	7	2	no	—
182	C₂²⋊C₄	—	7	2	no	—
183	C₂²⋊C₄	C₄×D₇	1	2	yes	—
184	C₄²	—	7	1	no	—
185	C₄²	—	7	1	no	—
186	C₄²	—	7	2	no	—
187	C₄²	—	7	1	no	—
188	C₄²	C₂²×D₇	1	1	yes	characteristic
189	C₂×C₇	—	4	2	no	—
190	C₂×C₇	#520	1	1	yes	characteristic
191	C₂×C₇	C₂×#131	1	1	yes	characteristic
192	C₂×C₇	#568	1	1	yes	characteristic
193	D₇	—	4	1	no	—
194	C₂³	—	7	1	no	—
195	C₂³	C₂×C₄×D₇	1	2	yes	—
196	Q₈	—	14	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	28	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	7	1	no	—
202	C₂×C₄	—	7	1	no	—
203	C₂×C₄	—	7	1	no	—
204	C₂×C₄	—	7	2	no	—
205	C₂×C₄	—	7	2	no	—
206	C₂×C₄	—	14	2	no	—
207	C₂×C₄	—	14	1	no	—
208	C₂×C₄	—	14	1	no	—
209	C₂×C₄	—	28	1	no	—
210	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	14	2	no	—
218	C₈	—	14	1	no	—
219	C₈	—	14	1	no	—
220	C₇	#2993	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	14	1	no	—
225	C₂²	—	14	1	no	—
226	C₂²	—	14	1	no	—
227	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	14	1	no	—
231	C₄	—	14	2	no	—
232	C₄	—	14	1	no	—
233	C₄	—	14	2	no	—
234	C₄	—	14	1	no	—
235	C₄	#6464	1	2	yes	—
236	C₄	#6548	1	1	yes	characteristic
237	C₄	#6554	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	28	1	no	—
240	C₂	#90237	1	1	yes	characteristic
241	C₂	C₂×#6464	1	1	yes	characteristic
242	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	#6501	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₈×D₄	D₇
hypocenter	C₇	#2993
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. #89874	C₂	semidirect
5. #89784	C₂	semidirect
6. #89609	C₂	semidirect
7. #89603	C₂	semidirect
8. #89501	C₂	semidirect
9. #89492	C₂	semidirect
10. #89457	C₂	semidirect
11. C₈×Dic₁₄	C₂	semidirect
80. C₈×D₄	D₇	semidirect
220. C₇	#2993	semidirect

Maximal subgroups

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

subgroup	index
#90248	2
C₇×C₈×D₄	2
#89883	2
#89874	2
#89784	2
#89609	2
#89603	2
#89501	2
#89492	2
#89457	2
C₈×Dic₁₄	2
#2993	7