Subgroups — C₂²×#1157 #90606

Order 448.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
3	C₂×#1157	C₂	1	24	yes	maximal, direct product factor
4	C₂²×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₇	C₂	1	2	yes	maximal, complemented
6	C₂²×Dic₁₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₇	C₂²	1	1	yes	characteristic
8	C₂³×D₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₇	C₂²	1	12	yes	complemented
12	C₂×C₂²⋊D₇	C₂²	1	24	yes	complemented
13	C₂²×Dic₇	C₂²	1	2	yes	—
14	#1157	C₂²	1	64	yes	direct product factor
15	C₂×D₂₈	C₂²	1	12	yes	complemented
16	C₂×C₄×D₇	C₂²	1	24	yes	complemented
17	C₂×Dic₁₄	C₂²	1	12	yes	complemented
18	C₂²×#41	—	7	1	no	maximal, Sylow
19	C₂³×C₇	—	2	4	no	—
20	C₂³×C₇	C₂³	1	6	yes	—
21	C₂³×C₇	C₂³	1	1	yes	characteristic
22	C₂²×D₇	—	2	8	no	—
23	C₂²×D₇	C₂³	1	12	yes	—
24	C₂×C₄×C₇	C₂³	1	6	yes	—
25	C₂×C₄×C₇	C₂³	1	6	yes	—
26	C₂×C₄×C₇	C₂³	1	16	yes	complemented
27	C₂²⋊D₇	C₂³	1	32	yes	complemented
28	C₂×Dic₇	C₂³	1	12	yes	—
29	D₂₈	C₂³	1	16	yes	complemented
30	C₄×D₇	C₂³	1	32	yes	complemented
31	Dic₁₄	C₂³	1	16	yes	complemented
32	C₂×#41	—	7	24	no	—
33	C₂²×Q₈	—	7	1	no	—
34	C₂²×D₄	—	7	2	no	—
35	C₂²×D₄	—	7	1	no	—
36	C₂³×C₄	—	7	2	no	—
37	C₂³×C₄	D₇	1	1	yes	characteristic, complemented
38	C₂²×C₇	—	2	12	no	—
39	C₂²×C₇	#41	1	4	yes	complemented
40	C₂²×C₇	C₂⁴	1	4	yes	—
41	C₂²×C₇	C₂⁴	1	3	yes	—
42	C₂×D₇	—	2	24	no	—
43	C₂×D₇	C₂⁴	1	8	yes	—
44	C₄×C₇	C₂⁴	1	4	yes	—
45	C₄×C₇	C₂⁴	1	4	yes	—
46	Dic₇	C₂⁴	1	8	yes	—
47	C₂⁴	—	7	2	no	—
48	C₂⁴	C₂×D₇	1	1	yes	characteristic
49	#41	—	7	64	no	—
50	C₂×Q₈	—	7	12	no	—
51	C₂×D₄	—	7	24	no	—
52	C₂×D₄	—	7	12	no	—
53	C₂²×C₄	—	7	2	no	—
54	C₂²×C₄	—	7	24	no	—
55	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₇	1	12	yes	complemented
58	C₂×C₇	—	2	4	no	—
59	C₂×C₇	C₂×#41	1	6	yes	complemented
60	C₂×C₇	C₂⁵	1	1	yes	characteristic
61	D₇	—	2	8	no	—
62	C₂³	—	2	4	no	—
63	C₂³	—	7	12	no	—
64	C₂³	—	14	8	no	—
65	C₂³	C₂²×D₇	1	6	yes	—
66	C₂³	C₂²×D₇	1	1	yes	characteristic
67	Q₈	—	7	16	no	—
68	D₄	—	7	32	no	—
69	D₄	—	7	16	no	—
70	C₂×C₄	—	7	12	no	—
71	C₂×C₄	—	7	32	no	—
72	C₂×C₄	C₂²×D₇	1	6	yes	—
73	C₂×C₄	C₂²×D₇	1	6	yes	—
74	C₂×C₄	C₂²×D₇	1	16	yes	complemented
75	C₇	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	2	12	no	—
77	C₂²	—	7	8	no	—
78	C₂²	—	14	24	no	—
79	C₂²	#1157	1	4	yes	direct product factor
80	C₂²	C₂³×D₇	1	4	yes	—
81	C₂²	C₂³×D₇	1	3	yes	—
82	C₄	—	7	8	no	—
83	C₄	C₂³×D₇	1	4	yes	—
84	C₄	C₂³×D₇	1	4	yes	—
85	C₂	—	2	4	no	—
86	C₂	—	14	8	no	—
87	C₂	C₂×#1157	1	6	yes	direct product factor
88	C₂	C₂⁴×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇
commutator, cosocle	C₂×C₇	C₂⁵
Frattini	C₂	C₂⁴×D₇
Fitting	C₂³×C₄×C₇	C₂
socle	C₂³×C₇	C₂³
hypercenter	C₂³×C₄	D₇
hypocenter	C₇	C₂²×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄×C₇	C₂	semidirect
2. C₂²×C₂²⋊D₇	C₂	semidirect
3. C₂×#1157	C₂	direct (N × H)
4. C₂²×D₂₈	C₂	semidirect
5. C₂²×C₄×D₇	C₂	semidirect
6. C₂²×Dic₁₄	C₂	semidirect
11. C₂²×C₄×C₇	C₂²	semidirect
12. C₂×C₂²⋊D₇	C₂²	semidirect
14. #1157	C₂²	direct (N × H)
15. C₂×D₂₈	C₂²	semidirect
16. C₂×C₄×D₇	C₂²	semidirect
17. C₂×Dic₁₄	C₂²	semidirect
26. C₂×C₄×C₇	C₂³	semidirect
27. C₂²⋊D₇	C₂³	semidirect
29. D₂₈	C₂³	semidirect
30. C₄×D₇	C₂³	semidirect
31. Dic₁₄	C₂³	semidirect
37. C₂³×C₄	D₇	semidirect
39. C₂²×C₇	#41	semidirect
57. C₂²×C₄	C₂×D₇	semidirect
59. C₂×C₇	C₂×#41	semidirect
74. C₂×C₄	C₂²×D₇	semidirect
75. C₇	C₂²×#41	semidirect
79. C₂²	#1157	direct (N × H)
87. C₂	C₂×#1157	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×C₄×C₇	2
C₂²×C₂²⋊D₇	2
C₂×#1157	2
C₂²×D₂₈	2
C₂²×C₄×D₇	2
C₂²×Dic₁₄	2
C₂²×#41	7

Subgroups — C22×#1157 #90606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1157 #90606