Subgroups — #90017

Order 448.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#6512	C₂	1	2	yes	maximal, complemented
3	#6510	C₂	1	1	yes	maximal, characteristic, complemented
4	#6411	C₂	1	2	yes	maximal, complemented
5	#6410	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	2	no	—
7	C₇×#41	—	2	2	no	—
8	C₇×#41	C₂²	1	2	yes	—
9	C₂×C₇×Q₈	—	2	2	no	—
10	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×D₂₈	C₂²	1	1	yes	characteristic
12	#1143	—	2	2	no	—
13	#1141	—	2	2	no	—
14	(C₂×D₇)⋊C₄	—	2	2	no	—
15	C₄×Dic₇	C₂²	1	1	yes	characteristic
16	#1136	C₂²	1	2	yes	—
17	#454	—	7	1	no	maximal, Sylow
18	C₂²×D₇	—	2	1	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×Q₈	—	2	2	no	—
21	C₇×Q₈	D₄	1	2	yes	—
22	C₇×D₄	—	2	2	no	—
23	C₇×D₄	—	2	2	no	—
24	C₇×D₄	D₄	1	2	yes	—
25	C₂×C₄×C₇	—	2	4	no	—
26	C₂×C₄×C₇	—	2	2	no	—
27	C₂×C₄×C₇	D₄	1	2	yes	—
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₇	—	2	1	no	—
30	D₂₈	—	2	2	no	—
31	C₇⋊C₈	—	2	2	no	—
32	2₋¹⁺⁴	D₇	1	1	yes	characteristic, complemented
33	AGL(1,ℤ/8)	—	7	2	no	—
34	#124	—	7	1	no	—
35	#104	—	7	2	no	—
36	#101	—	7	1	no	—
37	C₂²×C₇	—	2	2	no	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂×D₇	—	4	2	no	—
40	C₂×D₇	—	4	1	no	—
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	—	2	2	no	—
43	C₄×C₇	C₂×D₄	1	2	yes	—
44	Dic₇	—	4	1	no	—
45	#41	—	2	2	no	—
46	#41	—	2	2	no	—
47	#41	C₂×D₇	1	2	yes	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
50	C₂×D₄	—	7	1	no	—
51	SD₁₆	—	14	2	no	—
52	D₈	—	14	2	no	—
53	M₁₆	—	7	2	no	—
54	C₂²⋊C₄	—	14	2	no	—
55	C₄²	—	7	1	no	—
56	C₂×C₇	—	2	1	no	—
57	C₂×C₇	—	4	2	no	—
58	C₂×C₇	#120	1	1	yes	characteristic
59	D₇	—	8	1	no	—
60	C₂³	—	14	1	no	—
61	Q₈	—	2	2	no	—
62	Q₈	—	2	2	no	—
63	Q₈	C₂²⋊D₇	1	2	yes	—
64	D₄	—	2	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	14	2	no	—
67	D₄	C₂²⋊D₇	1	2	yes	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	C₂²⋊D₇	1	2	yes	—
72	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
73	C₈	—	14	2	no	—
74	C₇	#454	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	2	2	no	—
76	C₂²	—	28	2	no	—
77	C₂²	—	28	1	no	—
78	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	28	1	no	—
82	C₄	C₂×C₂²⋊D₇	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	4	2	no	—
85	C₂	—	56	1	no	—
86	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6516
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×2₋¹⁺⁴	C₂
socle	C₂×C₇	#120
hypercenter	2₋¹⁺⁴	D₇
hypocenter	C₇	#454
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₇×2₋¹⁺⁴	C₂	semidirect
2. #6512	C₂	semidirect
3. #6510	C₂	semidirect
4. #6411	C₂	semidirect
5. #6410	C₂	semidirect
32. 2₋¹⁺⁴	D₇	semidirect
74. C₇	#454	semidirect

Maximal subgroups

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

subgroup	index
C₇×2₋¹⁺⁴	2
#6512	2
#6510	2
#6411	2
#6410	2
#454	7