Subgroups — #90018

Order 448.

142 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
2	#6514	C₂	1	2	yes	maximal, complemented
3	#6507	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×Dic₁₄	C₂²	1	1	yes	characteristic
12	#1144	—	2	2	no	—
13	#1142	—	2	2	no	—
14	Dic₇⋊C₄	—	2	2	no	—
15	C₄×Dic₇	C₂²	1	1	yes	characteristic
16	#1136	C₂²	1	2	yes	—
17	#456	—	7	1	no	maximal, Sylow
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×Q₈	D₄	1	2	yes	—
21	C₇×D₄	—	2	2	no	—
22	C₇×D₄	—	2	2	no	—
23	C₇×D₄	D₄	1	2	yes	—
24	C₂×C₄×C₇	—	2	4	no	—
25	C₂×C₄×C₇	—	2	2	no	—
26	C₂×C₄×C₇	D₄	1	2	yes	—
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇	—	2	1	no	—
29	C₂×Dic₇	—	2	1	no	—
30	Dic₁₄	—	2	2	no	—
31	C₇⋊C₈	—	2	2	no	—
32	2₋¹⁺⁴	D₇	1	1	yes	characteristic
33	#137	—	7	2	no	—
34	C₄⋊Q₈	—	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	C₄×C₇	—	2	2	no	—
40	C₄×C₇	—	2	2	no	—
41	C₄×C₇	C₂×D₄	1	2	yes	—
42	Dic₇	—	2	2	no	—
43	Dic₇	—	4	1	no	—
44	#41	—	2	2	no	—
45	#41	—	2	2	no	—
46	#41	C₂×D₇	1	2	yes	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	—	7	1	no	—
49	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
50	Q₁₆	—	14	2	no	—
51	SD₁₆	—	14	2	no	—
52	M₁₆	—	7	2	no	—
53	C₄⋊C₄	—	14	2	no	—
54	C₄²	—	7	1	no	—
55	C₂×C₇	—	2	1	no	—
56	C₂×C₇	—	4	2	no	—
57	C₂×C₇	#120	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	14	2	no	—
61	Q₈	C₂²⋊D₇	1	2	yes	—
62	D₄	—	2	2	no	—
63	D₄	—	2	2	no	—
64	D₄	C₂²⋊D₇	1	2	yes	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	14	1	no	—
68	C₂×C₄	—	14	1	no	—
69	C₂×C₄	C₂²⋊D₇	1	2	yes	—
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₈	—	14	2	no	—
72	C₇	#456	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	2	2	no	—
74	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	14	2	no	—
78	C₄	—	28	1	no	—
79	C₄	C₂×C₂²⋊D₇	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	#6516	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6516
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₇×2₋¹⁺⁴	C₂
socle	C₂×C₇	#120
hypercenter	2₋¹⁺⁴	D₇
hypocenter	C₇	#456
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
2. #6514	C₂	semidirect
3. #6507	C₂	semidirect
4. #6411	C₂	semidirect
5. #6410	C₂	semidirect
72. C₇	#456	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
#6514	2
#6507	2
#6411	2
#6410	2
#456	7