Subgroups — C₂²×#63304 #11765412

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
2	C₂²×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63304	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂²×D₃₅	C₄	1	8	yes	complemented
7	C₂²×D₃₅	C₂²	1	6	yes	—
8	C₂×C₇×Dic₅	C₂²	1	6	yes	complemented
9	C₂×C₅×Dic₇	C₂²	1	6	yes	complemented
10	#63304	C₂²	1	16	yes	direct product factor
11	C₂²×C₄×D₇	—	5	1	no	maximal, Hall
12	C₂²×C₄×D₅	—	7	1	no	maximal, Hall
13	C₂²×C₅×C₇	C₂×C₄	1	4	yes	complemented
14	C₂²×C₅×C₇	C₂³	1	3	yes	—
15	C₂×D₃₅	C₂×C₄	1	24	yes	complemented
16	C₂×D₃₅	C₂³	1	4	yes	—
17	C₅×Dic₇	C₂³	1	4	yes	complemented
18	C₇×Dic₅	C₂³	1	4	yes	complemented
19	C₂³×D₇	—	5	1	no	—
20	C₂²×C₄×C₇	—	5	1	no	—
21	C₂²×Dic₇	D₅	1	1	yes	characteristic, complemented
22	C₂×C₄×D₇	—	5	12	no	—
23	C₂³×D₅	—	7	1	no	—
24	C₂²×C₄×C₅	—	7	1	no	—
25	C₂²×Dic₅	D₇	1	1	yes	characteristic, complemented
26	C₂×C₄×D₅	—	7	12	no	—
27	C₂×C₅×C₇	C₂²×C₄	1	6	yes	complemented
28	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
29	D₃₅	C₂²×C₄	1	8	yes	complemented
30	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
31	C₂²×D₇	—	5	8	no	—
32	C₂²×D₇	—	5	6	no	—
33	C₂×C₄×C₇	—	5	6	no	—
34	C₂×Dic₇	C₂×D₅	1	6	yes	complemented
35	C₄×D₇	—	5	16	no	—
36	C₂³×C₅	C₂×D₇	1	1	yes	characteristic
37	C₂²×D₅	—	7	8	no	—
38	C₂²×D₅	—	7	6	no	—
39	C₂×C₄×C₅	—	7	6	no	—
40	C₂×Dic₅	C₂×D₇	1	6	yes	complemented
41	C₄×D₅	—	7	16	no	—
42	C₅×C₇	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
43	C₂³×C₄	—	35	1	no	Sylow
44	C₂²×C₇	C₄×D₅	1	4	yes	complemented
45	C₂²×C₇	C₂²×D₅	1	3	yes	—
46	C₂×D₇	—	5	24	no	—
47	C₂×D₇	—	5	4	no	—
48	C₄×C₇	—	5	4	no	—
49	Dic₇	C₂²×D₅	1	4	yes	complemented
50	C₂²×C₅	C₄×D₇	1	4	yes	complemented
51	C₂²×C₅	C₂²×D₇	1	3	yes	—
52	C₂×D₅	—	7	24	no	—
53	C₂×D₅	—	7	4	no	—
54	C₄×C₅	—	7	4	no	—
55	Dic₅	C₂²×D₇	1	4	yes	complemented
56	C₂⁴	—	35	1	no	—
57	C₂²×C₄	—	5	1	no	—
58	C₂²×C₄	—	7	1	no	—
59	C₂²×C₄	—	35	12	no	—
60	C₂×C₇	C₂×C₄×D₅	1	6	yes	complemented
61	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
62	D₇	—	5	8	no	—
63	C₂×C₅	C₂×C₄×D₇	1	6	yes	complemented
64	C₂×C₅	C₂³×D₇	1	1	yes	characteristic
65	D₅	—	7	8	no	—
66	C₂³	—	35	8	no	—
67	C₂³	—	35	6	no	—
68	C₂³	D₅×D₇	1	1	yes	characteristic
69	C₂×C₄	—	5	6	no	—
70	C₂×C₄	—	7	6	no	—
71	C₂×C₄	—	35	16	no	—
72	C₇	C₂²×C₄×D₅	1	1	yes	characteristic, complemented, Sylow
73	C₅	C₂²×C₄×D₇	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	35	24	no	—
75	C₂²	—	35	4	no	—
76	C₂²	#63304	1	4	yes	direct product factor
77	C₂²	C₂×D₅×D₇	1	3	yes	—
78	C₄	—	5	4	no	—
79	C₄	—	7	4	no	—
80	C₂	—	35	8	no	—
81	C₂	C₂×#63304	1	6	yes	direct product factor
82	C₂	C₂²×D₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×D₇
commutator, hypocenter	C₅×C₇	C₂³×C₄
Frattini	C₂	C₂²×D₅×D₇
Fitting, socle	C₂³×C₅×C₇	C₂²
prime radical	C₂³×D₃₅	C₂
cosocle	C₂×C₅×C₇	C₂⁴
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×Dic₅	C₂	semidirect
3. C₂²×C₅×Dic₇	C₂	semidirect
4. C₂×#63304	C₂	direct (N × H)
6. C₂²×D₃₅	C₄	semidirect
8. C₂×C₇×Dic₅	C₂²	semidirect
9. C₂×C₅×Dic₇	C₂²	semidirect
10. #63304	C₂²	direct (N × H)
13. C₂²×C₅×C₇	C₂×C₄	semidirect
15. C₂×D₃₅	C₂×C₄	semidirect
17. C₅×Dic₇	C₂³	semidirect
18. C₇×Dic₅	C₂³	semidirect
21. C₂²×Dic₇	D₅	semidirect
25. C₂²×Dic₅	D₇	semidirect
27. C₂×C₅×C₇	C₂²×C₄	semidirect
29. D₃₅	C₂²×C₄	semidirect
34. C₂×Dic₇	C₂×D₅	semidirect
40. C₂×Dic₅	C₂×D₇	semidirect
42. C₅×C₇	C₂³×C₄	semidirect
44. C₂²×C₇	C₄×D₅	semidirect
49. Dic₇	C₂²×D₅	semidirect
50. C₂²×C₅	C₄×D₇	semidirect
55. Dic₅	C₂²×D₇	semidirect
60. C₂×C₇	C₂×C₄×D₅	semidirect
63. C₂×C₅	C₂×C₄×D₇	semidirect
72. C₇	C₂²×C₄×D₅	semidirect
73. C₅	C₂²×C₄×D₇	semidirect
76. C₂²	#63304	direct (N × H)
81. C₂	C₂×#63304	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
C₂²×C₇×Dic₅	2
C₂²×C₅×Dic₇	2
C₂×#63304	2
C₂²×C₄×D₇	5
C₂²×C₄×D₅	7

Subgroups — C22×#63304 #11765412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#63304 #11765412