Subgroups — C₂²×#10625334 #15689892

Order 3008.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625334	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₄₇	—	2	1	no	—
6	C₂³×D₄₇	—	2	1	no	—
7	C₂×D₄×C₄₇	—	2	3	no	—
8	C₂×D₄×C₄₇	C₂²	1	6	yes	complemented
9	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×D₁₈₈	—	2	3	no	—
11	C₂×D₁₈₈	C₂²	1	6	yes	complemented
12	#10625334	C₂²	1	16	yes	direct product factor
13	C₂×C₄₇⋊C₈	C₂²	1	6	yes	complemented
14	C₂³×C₄₇	—	2	6	no	—
15	C₂³×C₄₇	—	4	4	no	—
16	C₂³×C₄₇	D₄	1	1	yes	characteristic
17	C₂²×D₄₇	—	2	6	no	—
18	C₂²×D₄₇	—	4	4	no	—
19	D₄×C₄₇	—	2	6	no	—
20	D₄×C₄₇	C₂³	1	4	yes	complemented
21	C₂×C₄×C₄₇	D₄	1	3	yes	—
22	C₂×C₄×C₄₇	C₂³	1	3	yes	—
23	D₁₈₈	—	2	6	no	—
24	D₁₈₈	C₂³	1	4	yes	complemented
25	C₄₇⋊C₈	C₂³	1	4	yes	complemented
26	C₂²×C₄₇	—	2	4	no	—
27	C₂²×C₄₇	—	4	12	no	—
28	C₂²×C₄₇	D₈	1	4	yes	complemented
29	C₂²×C₄₇	C₂×D₄	1	3	yes	—
30	C₂×D₄₇	—	2	4	no	—
31	C₂×D₄₇	—	4	12	no	—
32	C₄×C₄₇	C₂×D₄	1	3	yes	—
33	C₄×C₄₇	C₂⁴	1	1	yes	characteristic
34	C₂×C₄₇	—	4	4	no	—
35	C₂×C₄₇	C₂×D₈	1	6	yes	complemented
36	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
37	D₄₇	—	4	4	no	—
38	C₂²×D₈	—	47	1	no	maximal, Sylow
39	C₄₇	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
40	C₂²×D₄	—	47	1	no	—
41	C₂²×D₄	D₄₇	1	1	yes	characteristic, complemented
42	C₂×D₈	—	47	12	no	—
43	C₂²×C₈	—	47	1	no	—
44	C₂⁴	—	2	1	no	—
45	C₂⁴	—	94	1	no	—
46	C₂×D₄	—	2	3	no	—
47	C₂×D₄	—	47	6	no	—
48	C₂×D₄	—	94	3	no	—
49	C₂×D₄	C₂×D₄₇	1	6	yes	complemented
50	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
51	D₈	—	47	16	no	—
52	C₂×C₈	—	47	6	no	—
53	C₂³	—	2	6	no	—
54	C₂³	—	4	4	no	—
55	C₂³	—	94	6	no	—
56	C₂³	—	188	4	no	—
57	C₂³	C₂²⋊D₄₇	1	1	yes	characteristic
58	D₄	—	2	6	no	—
59	D₄	—	47	4	no	—
60	D₄	—	94	6	no	—
61	D₄	C₂²×D₄₇	1	4	yes	complemented
62	C₂×C₄	C₂²⋊D₄₇	1	3	yes	—
63	C₂×C₄	C₂²×D₄₇	1	3	yes	—
64	C₈	—	47	4	no	—
65	C₂²	—	2	4	no	—
66	C₂²	—	4	12	no	—
67	C₂²	—	94	4	no	—
68	C₂²	—	188	12	no	—
69	C₂²	#10625334	1	4	yes	direct product factor
70	C₂²	C₂×C₂²⋊D₄₇	1	3	yes	—
71	C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
72	C₄	C₂³×D₄₇	1	1	yes	characteristic
73	C₂	—	4	4	no	—
74	C₂	—	188	4	no	—
75	C₂	C₂×#10625334	1	6	yes	direct product factor
76	C₂	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₄₇
commutator, cosocle	C₄×C₄₇	C₂⁴
Frattini	C₄	C₂³×D₄₇
Fitting	C₂²×D₄×C₄₇	C₂
socle	C₂³×C₄₇	D₄
hypercenter	C₂²×D₄	D₄₇
hypocenter	C₄₇	C₂²×D₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₈₈	C₂	semidirect
2. C₂×#10625334	C₂	direct (N × H)
3. C₂²×C₄₇⋊C₈	C₂	semidirect
4. C₂²×D₄×C₄₇	C₂	semidirect
8. C₂×D₄×C₄₇	C₂²	semidirect
11. C₂×D₁₈₈	C₂²	semidirect
12. #10625334	C₂²	direct (N × H)
13. C₂×C₄₇⋊C₈	C₂²	semidirect
20. D₄×C₄₇	C₂³	semidirect
24. D₁₈₈	C₂³	semidirect
25. C₄₇⋊C₈	C₂³	semidirect
28. C₂²×C₄₇	D₈	semidirect
35. C₂×C₄₇	C₂×D₈	semidirect
39. C₄₇	C₂²×D₈	semidirect
41. C₂²×D₄	D₄₇	semidirect
49. C₂×D₄	C₂×D₄₇	semidirect
61. D₄	C₂²×D₄₇	semidirect
69. C₂²	#10625334	direct (N × H)
75. C₂	C₂×#10625334	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₈₈	2
C₂×#10625334	2
C₂²×C₄₇⋊C₈	2
C₂²×D₄×C₄₇	2
C₂²×D₈	47

Subgroups — C22×#10625334 #15689892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10625334 #15689892