Subgroups — #11764893

Order 1120.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₄₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63306	C₂	1	4	yes	maximal, complemented
4	C₄×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₅	C₂²	1	2	yes	—
6	C₂²×C₅×D₇	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	D₁₄₀	—	2	2	no	—
9	C₂×C₇×Dic₅	C₂²	1	2	yes	—
10	C₅×D₂₈	—	2	2	no	—
11	#63306	—	2	8	no	—
12	#6437	—	5	1	no	maximal, Hall
13	#4086	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₂×D₃₅	—	2	4	no	—
16	C₂×D₃₅	—	2	2	no	—
17	C₂×C₅×D₇	—	2	4	no	—
18	C₂×C₅×D₇	—	2	2	no	—
19	C₄×C₅×C₇	D₄	1	2	yes	complemented
20	C₇×Dic₅	D₄	1	4	yes	complemented
21	C₂×D₂₈	—	5	4	no	—
22	C₂×D₂₈	—	5	1	no	—
23	C₂×D₂₈	D₅	1	1	yes	characteristic, complemented
24	C₄²×C₇	—	5	1	no	—
25	C₂×C₅×D₄	—	7	1	no	—
26	C₂×C₂²⋊D₅	—	7	4	no	—
27	C₂×D₂₀	—	7	1	no	—
28	C₄×Dic₅	D₇	1	1	yes	characteristic, complemented
29	C₂×C₅×C₇	C₂×D₄	1	2	yes	—
30	C₂×C₅×C₇	C₂×D₄	1	1	yes	characteristic
31	D₃₅	—	4	2	no	—
32	C₅×D₇	—	4	2	no	—
33	C₂²×D₇	—	5	2	no	—
34	C₂²×D₇	C₂×D₅	1	2	yes	—
35	C₂×C₄×C₇	—	5	2	no	—
36	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
37	D₂₈	—	2	2	no	—
38	D₂₈	—	10	8	no	—
39	D₂₈	—	10	2	no	—
40	C₂³×C₅	—	7	2	no	—
41	C₂²×D₅	—	7	2	no	—
42	C₅×D₄	—	14	2	no	—
43	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
44	C₂²⋊D₅	—	14	8	no	—
45	C₂×Dic₅	C₂×D₇	1	2	yes	—
46	D₂₀	—	14	2	no	—
47	C₅×C₇	#127	1	1	yes	characteristic, complemented, Hall
48	#127	—	35	1	no	Sylow
49	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
50	C₂×D₇	—	2	4	no	—
51	C₂×D₇	—	2	2	no	—
52	C₂×D₇	—	10	4	no	—
53	C₂×D₇	—	10	2	no	—
54	C₄×C₇	—	5	4	no	—
55	C₄×C₇	C₂²⋊D₅	1	2	yes	complemented
56	C₂²×C₅	—	14	4	no	—
57	C₂²×C₅	—	14	2	no	—
58	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
59	C₂×D₅	—	14	2	no	—
60	C₂×D₅	—	14	4	no	—
61	C₄×C₅	D₂₈	1	2	yes	complemented
62	Dic₅	D₂₈	1	4	yes	complemented
63	C₂×D₄	—	7	1	no	—
64	C₂×D₄	—	35	4	no	—
65	C₂×D₄	—	35	1	no	—
66	C₄²	—	5	1	no	—
67	C₂×C₇	D₄×D₅	1	2	yes	—
68	C₂×C₇	C₂×C₂²⋊D₅	1	1	yes	characteristic
69	D₇	—	4	2	no	—
70	D₇	—	20	2	no	—
71	C₂×C₅	—	28	2	no	—
72	C₂×C₅	C₂×D₂₈	1	2	yes	—
73	C₂×C₅	C₂×D₂₈	1	1	yes	characteristic
74	D₅	—	28	2	no	—
75	C₂³	—	7	2	no	—
76	C₂³	—	35	2	no	—
77	D₄	—	14	2	no	—
78	D₄	—	70	8	no	—
79	D₄	—	70	2	no	—
80	C₂×C₄	—	5	2	no	—
81	C₂×C₄	D₅×D₇	1	1	yes	characteristic
82	C₇	#4086	1	1	yes	characteristic, complemented, Sylow
83	C₅	#6437	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	14	4	no	—
85	C₂²	—	14	2	no	—
86	C₂²	—	70	4	no	—
87	C₂²	—	70	2	no	—
88	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
89	C₄	—	5	4	no	—
90	C₄	#63306	1	2	yes	complemented
91	C₂	—	28	2	no	—
92	C₂	—	140	2	no	—
93	C₂	D₅×D₂₈	1	2	yes	—
94	C₂	C₂×#63306	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×D₇
commutator, socle, cosocle	C₂²×C₅×C₇	C₂³
Fitting	C₂×C₄×C₅×C₇	C₂²
hypercenter	C₂×C₄	D₅×D₇
hypocenter	C₅×C₇	#127
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₄₀	C₂	semidirect
2. C₂×C₅×D₂₈	C₂	semidirect
3. C₂×#63306	C₂	semidirect
4. C₄×C₇×Dic₅	C₂	semidirect
19. C₄×C₅×C₇	D₄	semidirect
20. C₇×Dic₅	D₄	semidirect
23. C₂×D₂₈	D₅	semidirect
28. C₄×Dic₅	D₇	semidirect
47. C₅×C₇	#127	semidirect
55. C₄×C₇	C₂²⋊D₅	semidirect
61. C₄×C₅	D₂₈	semidirect
62. Dic₅	D₂₈	semidirect
82. C₇	#4086	semidirect
83. C₅	#6437	semidirect
90. C₄	#63306	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₄₀	2
C₂×C₅×D₂₈	2
C₂×#63306	2
C₄×C₇×Dic₅	2
#6437	5
#4086	7