Subgroups — #11728272

Order 896.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×#6396	C₂	1	4	yes	maximal
3	#89696	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×M₁₆	C₂²	1	2	yes	—
6	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂×#1136	C₂²	1	2	yes	—
8	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
9	C₇×C₄⋊C₈	—	2	2	no	—
10	#6396	—	2	8	no	—
11	C₄⋊(C₇⋊C₈)	—	2	2	no	—
12	C₄×C₇⋊C₈	C₄	1	2	yes	—
13	C₄×C₇⋊C₈	C₄	1	2	yes	—
14	#1848	—	7	1	no	maximal, Sylow
15	C₂²×C₄×C₇	Q₈	1	2	yes	—
16	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₇×M₁₆	—	2	4	no	—
18	C₂×C₇×C₈	—	2	2	no	—
19	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₇	D₄	1	2	yes	—
22	#1136	—	2	4	no	—
23	C₂×C₇⋊C₈	—	2	2	no	—
24	C₂×C₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂×C₇⋊C₈	D₄	1	4	yes	—
26	C₂×C₇⋊C₈	Q₈	1	4	yes	—
27	C₂×#108	—	7	4	no	—
28	#423	—	7	1	no	—
29	#423	D₇	1	1	yes	characteristic
30	C₂×C₄×C₈	—	7	1	no	—
31	C₂³×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇	—	2	4	no	—
33	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
39	C₇×C₈	—	4	2	no	—
40	C₇⋊C₈	—	4	2	no	—
41	C₇⋊C₈	C₄⋊C₄	1	8	yes	—
42	C₂×M₁₆	—	7	2	no	—
43	C₂×M₁₆	C₂×D₇	1	2	yes	—
44	C₂²×C₈	—	7	2	no	—
45	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
46	#108	—	14	8	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	14	2	no	—
49	C₄×C₈	—	7	2	no	—
50	C₄×C₈	—	7	2	no	—
51	C₂²×C₇	—	2	2	no	—
52	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₇	C₄⋊Q₈	1	2	yes	—
54	C₄×C₇	—	2	2	no	—
55	C₄×C₇	#108	1	4	yes	complemented
56	C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₄×C₇	#127	1	1	yes	characteristic
58	C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂²×C₄	Dic₁₄	1	2	yes	—
60	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
61	M₁₆	—	2	4	no	—
62	M₁₆	—	14	4	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	7	4	no	—
65	C₂×C₈	—	7	4	no	—
66	C₂×C₈	—	7	4	no	—
67	C₂×C₈	—	14	2	no	—
68	C₄²	C₄×D₇	1	1	yes	characteristic
69	C₄²	C₄×D₇	1	1	yes	characteristic
70	C₄²	C₂²⋊D₇	1	2	yes	—
71	C₂×C₇	—	2	2	no	—
72	C₂×C₇	#384	1	1	yes	characteristic
73	C₂×C₇	C₂×#108	1	2	yes	—
74	C₂³	C₂×Dic₁₄	1	1	yes	characteristic
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	Dic₇⋊C₄	1	2	yes	—
77	C₂×C₄	Dic₇⋊C₄	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
79	C₂×C₄	D₄×D₇	1	2	yes	—
80	C₂×C₄	Q₈×D₇	1	2	yes	—
81	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
82	C₈	—	4	2	no	—
83	C₈	—	7	8	no	—
84	C₈	—	28	2	no	—
85	C₇	#1848	1	1	yes	characteristic, complemented, Sylow
86	C₂²	—	2	2	no	—
87	C₂²	#6451	1	2	yes	—
88	C₂²	C₂×Dic₇⋊C₄	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	#6396	1	4	yes	complemented
91	C₄	D₇×C₄⋊C₄	1	2	yes	—
92	C₄	#6503	1	1	yes	characteristic
93	C₄	#6507	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	#89748	1	1	yes	characteristic
96	C₂	C₂×#6396	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#423	C₂
socle	C₂²×C₇	C₂×C₄⋊C₄
prime radical	C₂³×C₇	C₂×Q₈
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	#423	D₇
hypocenter	C₇	#1848
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
55. C₄×C₇	#108	semidirect
85. C₇	#1848	semidirect
90. C₄	#6396	semidirect

Maximal subgroups

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

subgroup	index
C₇×#423	2
C₂×#6396	2
#89696	2
C₂×C₄×C₇⋊C₈	2
#1848	7