Subgroups — #90201

Order 448.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1157	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#6493	C₂	1	4	yes	maximal, complemented
4	#6460	C₂	1	4	yes	maximal, complemented
5	#6458	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	2	yes	—
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₇	C₂²	1	4	yes	—
9	#1157	—	2	8	no	—
10	C₂×D₂₈	C₂²	1	2	yes	—
11	C₂×D₂₈	C₂²	1	4	yes	—
12	C₂×C₄×D₇	C₂²	1	4	yes	—
13	C₂×Dic₁₄	C₂²	1	2	yes	—
14	C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	(C₂×D₇)⋊C₄	C₂²	1	8	yes	—
16	C₄⋊Dic₇	C₂²	1	4	yes	—
17	#537	—	7	1	no	maximal, Sylow
18	C₂³×C₇	C₂³	1	1	yes	characteristic
19	C₂²×D₇	C₂³	1	4	yes	—
20	C₂×C₄×C₇	—	4	2	no	—
21	C₂×C₄×C₇	D₄	1	4	yes	complemented
22	C₂×C₄×C₇	C₂³	1	4	yes	—
23	C₂×C₄×C₇	C₂³	1	2	yes	—
24	C₂²⋊D₇	—	2	8	no	—
25	C₂×Dic₇	C₂³	1	4	yes	—
26	D₂₈	—	2	4	no	—
27	D₂₈	—	4	4	no	—
28	C₄×D₇	—	2	8	no	—
29	Dic₁₄	—	2	4	no	—
30	C₂×#41	—	7	2	no	—
31	C₂²⋊Q₈	—	7	4	no	—
32	#121	—	7	4	no	—
33	#121	—	7	4	no	—
34	C₂×C₄⋊C₄	D₇	1	1	yes	characteristic, complemented
35	C₂²×C₇	—	2	1	no	—
36	C₂²×C₇	—	2	1	no	—
37	C₂²×C₇	C₂×D₄	1	2	yes	—
38	C₂²×C₇	C₂⁴	1	1	yes	characteristic
39	C₂×D₇	—	2	4	no	—
40	C₂×D₇	—	2	4	no	—
41	C₂×D₇	—	2	4	no	—
42	C₄×C₇	—	2	4	no	—
43	C₄×C₇	C₂×D₄	1	4	yes	—
44	Dic₇	—	2	4	no	—
45	#41	—	14	8	no	—
46	C₂×Q₈	—	7	2	no	—
47	C₂×D₄	—	7	4	no	—
48	C₂×D₄	—	7	2	no	—
49	C₂×D₄	—	7	4	no	—
50	C₂²×C₄	—	7	4	no	—
51	C₂²×C₄	C₂×D₇	1	2	yes	—
52	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
53	C₄⋊C₄	—	7	4	no	—
54	C₄⋊C₄	C₂×D₇	1	4	yes	—
55	C₂²⋊C₄	—	7	8	no	—
56	C₂×C₇	—	2	2	no	—
57	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
59	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
60	D₇	—	4	4	no	—
61	C₂³	—	7	4	no	—
62	C₂³	C₂²×D₇	1	1	yes	characteristic
63	Q₈	—	14	4	no	—
64	D₄	—	14	8	no	—
65	D₄	—	14	4	no	—
66	D₄	—	28	4	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	7	4	no	—
69	C₂×C₄	—	14	8	no	—
70	C₂×C₄	D₂₈	1	4	yes	complemented
71	C₂×C₄	C₂²×D₇	1	4	yes	—
72	C₂×C₄	C₂²×D₇	1	2	yes	—
73	C₇	#537	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	14	4	no	—
77	C₂²	—	14	4	no	—
78	C₂²	—	14	4	no	—
79	C₂²	C₂×D₂₈	1	2	yes	—
80	C₂²	C₂³×D₇	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	14	4	no	—
83	C₄	C₂×D₂₈	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	28	4	no	—
86	C₂	C₂²×D₂₈	1	1	yes	characteristic
87	C₂	#6548	1	1	yes	characteristic
88	C₂	#6551	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₂×C₇×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	D₇
hypocenter	C₇	#537
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#1157	C₂	semidirect
2. C₂×C₇×C₄⋊C₄	C₂	semidirect
3. #6493	C₂	semidirect
4. #6460	C₂	semidirect
5. #6458	C₂	semidirect
21. C₂×C₄×C₇	D₄	semidirect
34. C₂×C₄⋊C₄	D₇	semidirect
70. C₂×C₄	D₂₈	semidirect
73. C₇	#537	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1157	2
C₂×C₇×C₄⋊C₄	2
#6493	2
#6460	2
#6458	2
#537	7