Subgroups — #11731801

Order 896.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6451	C₂	1	6	yes	maximal
2	C₇×#384	C₂	1	1	yes	maximal, characteristic
3	#89748	C₂	1	6	yes	maximal
4	#89707	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄	C₂²	1	4	yes	—
7	C₂×C₇×C₄⋊C₄	C₂²	1	6	yes	—
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	C₂²	1	6	yes	—
10	C₂×Dic₇⋊C₄	C₂²	1	12	yes	—
11	C₂×C₄×Dic₇	C₂²	1	3	yes	—
12	C₂×C₄×Dic₇	C₂²	1	3	yes	—
13	#6451	—	2	24	no	—
14	#2868	—	7	1	no	maximal, Sylow
15	C₂²×C₄×C₇	C₂³	1	4	yes	—
16	C₂²×C₄×C₇	C₂³	1	3	yes	—
17	C₂²×Dic₇	C₂³	1	4	yes	—
18	C₂²×Dic₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₇	C₂³	1	3	yes	—
20	C₂×Dic₁₄	—	2	24	no	—
21	C₇×C₄⋊C₄	—	2	12	no	—
22	C₄²×C₇	Q₈	1	4	yes	complemented
23	C₄⋊Dic₇	—	2	12	no	—
24	Dic₇⋊C₄	—	2	24	no	—
25	C₄×Dic₇	D₄	1	12	yes	—
26	C₄×Dic₇	Q₈	1	12	yes	complemented
27	C₂×C₄⋊Q₈	—	7	6	no	—
28	#384	—	7	6	no	—
29	#384	—	7	1	no	—
30	#384	D₇	1	1	yes	characteristic
31	C₄³	—	7	1	no	—
32	C₂³×C₇	C₂⁴	1	1	yes	characteristic
33	C₂×C₄×C₇	—	2	12	no	—
34	C₂×C₄×C₇	C₂×D₄	1	6	yes	—
35	C₂×C₄×C₇	C₂×Q₈	1	6	yes	—
36	C₂×C₄×C₇	C₂×Q₈	1	6	yes	—
37	C₂×Dic₇	—	2	12	no	—
38	C₂×Dic₇	C₂×D₄	1	6	yes	—
39	C₂×Dic₇	C₂×D₄	1	6	yes	—
40	C₂×Dic₇	C₂×Q₈	1	12	yes	—
41	Dic₁₄	—	4	16	no	—
42	C₂²×Q₈	—	7	4	no	—
43	C₄⋊Q₈	—	14	24	no	—
44	C₂×C₄⋊C₄	—	7	12	no	—
45	C₂×C₄⋊C₄	—	7	6	no	—
46	C₂×C₄⋊C₄	C₂×D₇	1	6	yes	—
47	C₂×C₄²	—	7	3	no	—
48	C₂×C₄²	—	7	3	no	—
49	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
50	C₂²×C₇	C₂²×D₄	1	3	yes	—
51	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₇	C₂²×Q₈	1	3	yes	—
53	C₄×C₇	—	4	4	no	—
54	C₄×C₇	C₄⋊Q₈	1	12	yes	complemented
55	Dic₇	—	4	4	no	—
56	Dic₇	#127	1	4	yes	—
57	Dic₇	C₄⋊Q₈	1	12	yes	complemented
58	C₂×Q₈	—	14	24	no	—
59	C₂²×C₄	—	7	3	no	—
60	C₂²×C₄	—	7	1	no	—
61	C₂²×C₄	—	7	4	no	—
62	C₂²×C₄	C₂²×D₇	1	4	yes	—
63	C₂²×C₄	C₂²×D₇	1	3	yes	—
64	C₄⋊C₄	—	2	12	no	—
65	C₄⋊C₄	—	14	24	no	—
66	C₄⋊C₄	—	14	12	no	—
67	C₄²	—	7	12	no	—
68	C₄²	—	7	12	no	—
69	C₄²	Dic₁₄	1	4	yes	complemented
70	C₂×C₇	C₂×#127	1	1	yes	characteristic
71	C₂×C₇	C₂×C₄⋊Q₈	1	3	yes	—
72	C₂×C₇	C₂×C₄⋊Q₈	1	3	yes	—
73	C₂³	C₂³×D₇	1	1	yes	characteristic
74	Q₈	—	28	16	no	—
75	C₂×C₄	—	2	12	no	—
76	C₂×C₄	—	7	6	no	—
77	C₂×C₄	—	7	6	no	—
78	C₂×C₄	—	7	12	no	—
79	C₂×C₄	—	14	12	no	—
80	C₂×C₄	C₂×Dic₁₄	1	6	yes	—
81	C₂×C₄	D₄×D₇	1	6	yes	—
82	C₂×C₄	Q₈×D₇	1	6	yes	—
83	C₇	#2868	1	1	yes	characteristic, complemented, Sylow
84	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
85	C₂²	C₂×D₄×D₇	1	3	yes	—
86	C₂²	C₂×Q₈×D₇	1	3	yes	—
87	C₄	—	4	4	no	—
88	C₄	—	7	4	no	—
89	C₄	—	7	12	no	—
90	C₄	—	28	4	no	—
91	C₄	#6451	1	12	yes	complemented
92	C₂	C₂×#6451	1	3	yes	—
93	C₂	D₇×#127	1	1	yes	characteristic
94	C₂	D₇×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#384	C₂
hypercenter	#384	D₇
hypocenter	C₇	#2868
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
22. C₄²×C₇	Q₈	semidirect
26. C₄×Dic₇	Q₈	semidirect
54. C₄×C₇	C₄⋊Q₈	semidirect
57. Dic₇	C₄⋊Q₈	semidirect
69. C₄²	Dic₁₄	semidirect
83. C₇	#2868	semidirect
91. C₄	#6451	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6451	2
C₇×#384	2
#89748	2
#89707	2
C₄²×Dic₇	2
#2868	7