Subgroups — #89787

Order 448.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#1141	C₂	1	2	yes	maximal, complemented
3	#6437	C₂	1	1	yes	maximal, characteristic, complemented
4	#6383	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	—	2	1	no	—
8	C₂×D₂₈	—	2	1	no	—
9	C₂×D₂₈	—	2	1	no	—
10	C₂×D₂₈	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₇×C₂²⋊C₄	—	2	1	no	—
13	C₄²×C₇	C₂²	1	1	yes	characteristic
14	#1141	—	2	4	no	—
15	C₂×C₇⋊C₈	C₂²	1	2	yes	—
16	#459	—	7	1	no	maximal, Sylow
17	C₂³×C₇	—	2	1	no	—
18	C₂²×D₇	—	2	2	no	—
19	C₇×D₄	—	2	1	no	—
20	C₇×D₄	D₄	1	2	yes	complemented
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	—	4	2	no	—
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	D₂₈	—	2	2	no	—
27	D₂₈	—	2	2	no	—
28	D₂₈	—	4	2	no	—
29	D₂₈	—	4	2	no	—
30	C₇⋊C₈	—	2	2	no	—
31	C₂×D₈	—	7	2	no	—
32	#127	—	7	1	no	—
33	C₄×D₄	D₇	1	1	yes	characteristic, complemented
34	C₄⋊C₈	—	7	1	no	—
35	#102	—	7	2	no	—
36	C₂²×C₇	—	2	2	no	—
37	C₂²×C₇	—	4	1	no	—
38	C₂²×C₇	—	4	1	no	—
39	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂×D₇	—	4	2	no	—
41	C₂×D₇	—	4	2	no	—
42	C₂×D₇	—	4	2	no	—
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	—	4	1	no	—
45	C₄×C₇	D₈	1	2	yes	complemented
46	C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₇	#41	1	1	yes	characteristic
48	C₂×D₄	—	7	2	no	—
49	C₂×D₄	—	14	1	no	—
50	C₂×D₄	—	14	1	no	—
51	C₂×D₄	C₂×D₇	1	1	yes	characteristic
52	C₂²×C₄	—	2	1	no	—
53	D₈	—	14	4	no	—
54	C₂×C₈	—	7	2	no	—
55	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
56	C₂²⋊C₄	—	2	1	no	—
57	C₄²	C₂×D₇	1	1	yes	characteristic
58	C₂×C₇	—	4	2	no	—
59	C₂×C₇	#121	1	1	yes	characteristic
60	C₂×C₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	D₇	—	8	2	no	—
63	C₂³	—	2	1	no	—
64	C₂³	—	14	2	no	—
65	D₄	—	2	1	no	—
66	D₄	—	14	2	no	—
67	D₄	—	14	2	no	—
68	D₄	—	28	2	no	—
69	D₄	—	28	2	no	—
70	D₄	D₂₈	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
76	C₈	—	14	2	no	—
77	C₇	#459	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	28	2	no	—
82	C₂²	—	28	2	no	—
83	C₂²	—	28	2	no	—
84	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	#1141	1	2	yes	complemented
88	C₄	C₂×D₂₈	1	1	yes	characteristic
89	C₄	#1157	1	1	yes	characteristic
90	C₂	—	4	2	no	—
91	C₂	—	56	2	no	—
92	C₂	#6493	1	1	yes	characteristic
93	C₂	C₂×#1141	1	1	yes	characteristic
94	C₂	#6512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×D₄	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	C₄×D₄	D₇
hypocenter	C₇	#459
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄×C₇×D₄	C₂	semidirect
2. C₂×#1141	C₂	semidirect
3. #6437	C₂	semidirect
4. #6383	C₂	semidirect
5. C₄⋊(C₇⋊C₈)	C₂	semidirect
20. C₇×D₄	D₄	semidirect
33. C₄×D₄	D₇	semidirect
45. C₄×C₇	D₈	semidirect
70. D₄	D₂₈	semidirect
77. C₇	#459	semidirect
87. C₄	#1141	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×D₄	2
C₂×#1141	2
#6437	2
#6383	2
C₄⋊(C₇⋊C₈)	2
#459	7