Subgroups — C₂²×#63306 #11765414

Order 1120.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63306	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂²×D₃₅	—	2	4	no	—
7	C₂²×D₃₅	C₂²	1	6	yes	complemented
8	C₂²×C₅×D₇	—	2	4	no	—
9	C₂²×C₅×D₇	C₂²	1	6	yes	complemented
10	C₂×C₇×Dic₅	C₂²	1	6	yes	complemented
11	#63306	C₂²	1	16	yes	direct product factor
12	C₂²×D₂₈	—	5	1	no	maximal, Hall
13	C₂²×C₂²⋊D₅	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	D₄	1	4	yes	complemented
15	C₂²×C₅×C₇	C₂³	1	3	yes	—
16	C₂×D₃₅	—	2	12	no	—
17	C₂×D₃₅	C₂³	1	4	yes	complemented
18	C₂×C₅×D₇	—	2	12	no	—
19	C₂×C₅×D₇	C₂³	1	4	yes	complemented
20	C₇×Dic₅	C₂³	1	4	yes	complemented
21	C₂³×D₇	—	5	1	no	—
22	C₂³×D₇	D₅	1	1	yes	characteristic, complemented
23	C₂²×C₄×C₇	—	5	1	no	—
24	C₂×D₂₈	—	5	12	no	—
25	C₂⁴×C₅	—	7	1	no	—
26	C₂³×D₅	—	7	1	no	—
27	C₂×C₂²⋊D₅	—	7	12	no	—
28	C₂²×Dic₅	D₇	1	1	yes	characteristic, complemented
29	C₂×C₅×C₇	C₂×D₄	1	6	yes	complemented
30	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
31	D₃₅	—	2	4	no	—
32	C₅×D₇	—	2	4	no	—
33	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
34	C₂²×D₇	—	2	4	no	—
35	C₂²×D₇	—	5	6	no	—
36	C₂²×D₇	—	10	4	no	—
37	C₂²×D₇	C₂×D₅	1	6	yes	complemented
38	C₂×C₄×C₇	—	5	6	no	—
39	D₂₈	—	5	16	no	—
40	C₂³×C₅	—	7	6	no	—
41	C₂³×C₅	—	14	4	no	—
42	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
43	C₂²×D₅	—	7	6	no	—
44	C₂²×D₅	—	14	4	no	—
45	C₂²⋊D₅	—	7	16	no	—
46	C₂×Dic₅	C₂×D₇	1	6	yes	complemented
47	C₅×C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
48	C₂²×D₄	—	35	1	no	Sylow
49	C₂²×C₇	C₂²⋊D₅	1	4	yes	complemented
50	C₂²×C₇	C₂²×D₅	1	3	yes	—
51	C₂×D₇	—	2	12	no	—
52	C₂×D₇	—	5	4	no	—
53	C₂×D₇	—	10	12	no	—
54	C₂×D₇	C₂²×D₅	1	4	yes	complemented
55	C₄×C₇	—	5	4	no	—
56	C₂²×C₅	—	7	4	no	—
57	C₂²×C₅	—	14	12	no	—
58	C₂²×C₅	D₂₈	1	4	yes	complemented
59	C₂²×C₅	C₂²×D₇	1	3	yes	—
60	C₂×D₅	—	7	4	no	—
61	C₂×D₅	—	14	12	no	—
62	Dic₅	C₂²×D₇	1	4	yes	complemented
63	C₂⁴	—	7	1	no	—
64	C₂⁴	—	35	1	no	—
65	C₂×D₄	—	35	12	no	—
66	C₂²×C₄	—	5	1	no	—
67	C₂×C₇	C₂×C₂²⋊D₅	1	6	yes	complemented
68	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
69	D₇	—	2	4	no	—
70	D₇	—	10	4	no	—
71	C₂×C₅	—	14	4	no	—
72	C₂×C₅	C₂×D₂₈	1	6	yes	complemented
73	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
74	D₅	—	14	4	no	—
75	C₂³	—	7	6	no	—
76	C₂³	—	14	4	no	—
77	C₂³	—	35	6	no	—
78	C₂³	—	70	4	no	—
79	C₂³	D₅×D₇	1	1	yes	characteristic
80	D₄	—	35	16	no	—
81	C₂×C₄	—	5	6	no	—
82	C₇	C₂²×C₂²⋊D₅	1	1	yes	characteristic, complemented, Sylow
83	C₅	C₂²×D₂₈	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	7	4	no	—
85	C₂²	—	14	12	no	—
86	C₂²	—	35	4	no	—
87	C₂²	—	70	12	no	—
88	C₂²	#63306	1	4	yes	direct product factor
89	C₂²	C₂×D₅×D₇	1	3	yes	—
90	C₄	—	5	4	no	—
91	C₂	—	14	4	no	—
92	C₂	—	70	4	no	—
93	C₂	C₂×#63306	1	6	yes	direct product factor
94	C₂	C₂²×D₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×D₇
commutator, cosocle	C₂×C₅×C₇	C₂⁴
Frattini	C₂	C₂²×D₅×D₇
Fitting, socle	C₂³×C₅×C₇	C₂²
hypocenter	C₅×C₇	C₂²×D₄
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₅	C₂	semidirect
2. C₂³×C₅×D₇	C₂	semidirect
3. C₂²×C₇×Dic₅	C₂	semidirect
4. C₂×#63306	C₂	direct (N × H)
7. C₂²×D₃₅	C₂²	semidirect
9. C₂²×C₅×D₇	C₂²	semidirect
10. C₂×C₇×Dic₅	C₂²	semidirect
11. #63306	C₂²	direct (N × H)
14. C₂²×C₅×C₇	D₄	semidirect
17. C₂×D₃₅	C₂³	semidirect
19. C₂×C₅×D₇	C₂³	semidirect
20. C₇×Dic₅	C₂³	semidirect
22. C₂³×D₇	D₅	semidirect
28. C₂²×Dic₅	D₇	semidirect
29. C₂×C₅×C₇	C₂×D₄	semidirect
37. C₂²×D₇	C₂×D₅	semidirect
46. C₂×Dic₅	C₂×D₇	semidirect
47. C₅×C₇	C₂²×D₄	semidirect
49. C₂²×C₇	C₂²⋊D₅	semidirect
54. C₂×D₇	C₂²×D₅	semidirect
58. C₂²×C₅	D₂₈	semidirect
62. Dic₅	C₂²×D₇	semidirect
67. C₂×C₇	C₂×C₂²⋊D₅	semidirect
72. C₂×C₅	C₂×D₂₈	semidirect
82. C₇	C₂²×C₂²⋊D₅	semidirect
83. C₅	C₂²×D₂₈	semidirect
88. C₂²	#63306	direct (N × H)
93. C₂	C₂×#63306	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃₅	2
C₂³×C₅×D₇	2
C₂²×C₇×Dic₅	2
C₂×#63306	2
C₂²×D₂₈	5
C₂²×C₂²⋊D₅	7

Subgroups — C22×#63306 #11765414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63306 #11765414