Subgroups — #90212

Order 448.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	#6489	C₂	1	1	yes	maximal, characteristic, complemented
4	#6456	C₂	1	4	yes	maximal, complemented
5	D₇×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₄×D₂₈	C₂	1	4	yes	maximal, complemented
7	C₂³×D₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×D₂₈	C₄	1	8	yes	complemented
10	C₂×D₂₈	C₂²	1	2	yes	—
11	C₂×D₂₈	C₂²	1	2	yes	—
12	C₂×C₄×D₇	C₂²	1	8	yes	complemented
13	C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₄²×C₇	C₂²	1	2	yes	complemented
16	C₂²⋊Dic₇	C₂²	1	2	yes	—
17	(C₂×D₇)⋊C₄	C₂²	1	8	yes	complemented
18	C₄⋊Dic₇	C₂²	1	2	yes	complemented
19	C₄×Dic₇	C₂²	1	2	yes	complemented
20	#518	—	7	1	no	maximal, Sylow
21	C₂³×C₇	C₂³	1	1	yes	characteristic
22	C₂²×D₇	—	4	4	no	—
23	C₂²×D₇	C₂×C₄	1	8	yes	complemented
24	C₂²×D₇	C₂³	1	4	yes	—
25	C₂×C₄×C₇	C₂×C₄	1	4	yes	complemented
26	C₂×C₄×C₇	C₂³	1	4	yes	—
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₇	C₂³	1	4	yes	—
30	D₂₈	C₂×C₄	1	8	yes	complemented
31	D₂₈	C₂×C₄	1	8	yes	complemented
32	C₄×D₇	—	4	8	no	—
33	C₂²×D₄	—	7	1	no	—
34	C₄×D₄	—	7	4	no	—
35	C₄×D₄	—	7	4	no	—
36	#117	—	7	1	no	—
37	#117	D₇	1	1	yes	characteristic, complemented
38	C₂×C₂²⋊C₄	—	7	4	no	—
39	C₂²×C₇	—	2	2	no	—
40	C₂²×C₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₇	C₂⁴	1	1	yes	characteristic
42	C₂×D₇	—	2	8	no	—
43	C₂×D₇	—	4	8	no	—
44	C₂×D₇	C₂²×C₄	1	8	yes	—
45	C₄×C₇	—	2	4	no	—
46	C₄×C₇	C₂²×C₄	1	2	yes	—
47	C₄×C₇	C₂²×C₄	1	2	yes	—
48	Dic₇	—	2	4	no	—
49	C₂⁴	—	7	2	no	—
50	C₂×D₄	—	7	8	no	—
51	C₂×D₄	—	7	2	no	—
52	C₂×D₄	—	7	2	no	—
53	C₂²×C₄	—	7	8	no	—
54	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
55	C₄⋊C₄	—	7	2	no	—
56	C₄⋊C₄	C₂×D₇	1	2	yes	complemented
57	C₂²⋊C₄	—	7	2	no	—
58	C₂²⋊C₄	—	7	8	no	—
59	C₂²⋊C₄	C₂×D₇	1	2	yes	—
60	C₄²	—	7	2	no	—
61	C₄²	C₂×D₇	1	2	yes	complemented
62	C₂×C₇	—	2	2	no	—
63	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
65	D₇	—	2	8	no	—
66	C₂³	—	7	4	no	—
67	C₂³	—	7	8	no	—
68	C₂³	—	28	4	no	—
69	C₂³	C₂²×D₇	1	1	yes	characteristic
70	D₄	—	7	8	no	—
71	D₄	—	7	8	no	—
72	C₂×C₄	—	7	4	no	—
73	C₂×C₄	—	28	8	no	—
74	C₂×C₄	C₄×D₇	1	4	yes	complemented
75	C₂×C₄	C₂²×D₇	1	4	yes	—
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
78	C₇	#518	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	2	2	no	—
80	C₂²	—	7	8	no	—
81	C₂²	—	14	8	no	—
82	C₂²	—	28	8	no	—
83	C₂²	C₂×C₄×D₇	1	2	yes	—
84	C₂²	C₂³×D₇	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	14	4	no	—
87	C₄	C₂×C₄×D₇	1	2	yes	—
88	C₄	C₂×C₄×D₇	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	14	8	no	—
91	C₂	C₂²×C₄×D₇	1	1	yes	characteristic
92	C₂	#6553	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator	C₂×C₇	C₂³×C₄
Fitting	C₇×#117	C₂
socle, cosocle	C₂²×C₇	C₂⁴
prime radical	C₂²×D₂₈	C₂
hypercenter	#117	D₇
hypocenter	C₇	#518
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₇×#117	C₂	semidirect
3. #6489	C₂	semidirect
4. #6456	C₂	semidirect
5. D₇×C₂²⋊C₄	C₂	semidirect
6. C₄×D₂₈	C₂	semidirect
9. C₂×D₂₈	C₄	semidirect
12. C₂×C₄×D₇	C₂²	semidirect
13. C₇×C₄⋊C₄	C₂²	semidirect
15. C₄²×C₇	C₂²	semidirect
17. (C₂×D₇)⋊C₄	C₂²	semidirect
18. C₄⋊Dic₇	C₂²	semidirect
19. C₄×Dic₇	C₂²	semidirect
23. C₂²×D₇	C₂×C₄	semidirect
25. C₂×C₄×C₇	C₂×C₄	semidirect
30. D₂₈	C₂×C₄	semidirect
31. D₂₈	C₂×C₄	semidirect
37. #117	D₇	semidirect
56. C₄⋊C₄	C₂×D₇	semidirect
61. C₄²	C₂×D₇	semidirect
74. C₂×C₄	C₄×D₇	semidirect
78. C₇	#518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₈	2
C₇×#117	2
#6489	2
#6456	2
D₇×C₂²⋊C₄	2
C₄×D₂₈	2
#518	7