Subgroups — #15689685

Order 3008.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₄₇	C₂	1	2	yes	maximal, complemented
2	#13094008	C₂	1	4	yes	maximal, complemented
3	#13093998	C₂	1	4	yes	maximal, complemented
4	#13093994	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄₇	C₂²	1	1	yes	characteristic
7	C₂³×D₄₇	C₂²	1	2	yes	complemented
8	C₂²×C₄×C₄₇	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₄₇	—	2	8	no	—
10	C₂×C₂²⋊D₄₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
12	C₂²×Dic₄₇	C₂²	1	2	yes	complemented
13	C₂×D₁₈₈	C₂²	1	4	yes	complemented
14	C₂²⋊C₄×C₄₇	C₂²	1	4	yes	complemented
15	(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	complemented
16	C₄⋊Dic₄₇	C₂²	1	4	yes	complemented
17	C₂³×C₄₇	—	4	2	no	—
18	C₂³×C₄₇	D₄	1	4	yes	complemented
19	C₂³×C₄₇	C₂³	1	2	yes	—
20	C₂³×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×D₄₇	—	2	4	no	—
22	C₂²×D₄₇	—	4	4	no	—
23	C₂²×D₄₇	C₂³	1	4	yes	—
24	C₂×C₄×C₄₇	—	4	2	no	—
25	C₂×C₄×C₄₇	C₂³	1	4	yes	—
26	C₂²⋊D₄₇	—	2	8	no	—
27	C₂²⋊D₄₇	—	2	8	no	—
28	C₂×Dic₄₇	—	2	4	no	—
29	C₂×Dic₄₇	C₂³	1	4	yes	—
30	D₁₈₈	—	4	4	no	—
31	C₂²×C₄₇	—	2	2	no	—
32	C₂²×C₄₇	—	2	4	no	—
33	C₂²×C₄₇	—	4	4	no	—
34	C₂²×C₄₇	C₂×D₄	1	4	yes	—
35	C₂²×C₄₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
37	C₂×D₄₇	—	2	4	no	—
38	C₂×D₄₇	—	2	8	no	—
39	C₂×D₄₇	—	4	8	no	—
40	C₄×C₄₇	—	2	4	no	—
41	Dic₄₇	—	2	4	no	—
42	C₂×C₄₇	—	2	4	no	—
43	C₂×C₄₇	—	2	2	no	—
44	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
45	C₂×C₄₇	H(5,𝔽₂)	1	2	yes	—
46	D₄₇	—	4	4	no	—
47	#534	—	47	1	no	maximal, Sylow
48	C₄₇	#534	1	1	yes	characteristic, complemented, Sylow
49	C₂²×D₄	—	47	2	no	—
50	#123	—	47	4	no	—
51	#121	—	47	4	no	—
52	#120	—	47	4	no	—
53	C₂×C₂²⋊C₄	D₄₇	1	1	yes	characteristic, complemented
54	C₂⁴	—	47	2	no	—
55	C₂⁴	C₂×D₄₇	1	1	yes	characteristic
56	C₂×D₄	—	47	4	no	—
57	C₂×D₄	—	47	4	no	—
58	C₂×D₄	—	47	4	no	—
59	C₂×D₄	—	94	8	no	—
60	C₂²×C₄	—	47	2	no	—
61	C₂²×C₄	C₂×D₄₇	1	2	yes	complemented
62	C₄⋊C₄	—	47	4	no	—
63	C₂²⋊C₄	—	47	8	no	—
64	C₂²⋊C₄	C₂×D₄₇	1	4	yes	complemented
65	C₂³	—	4	2	no	—
66	C₂³	—	47	4	no	—
67	C₂³	—	94	4	no	—
68	C₂³	—	188	4	no	—
69	C₂³	D₁₈₈	1	4	yes	complemented
70	C₂³	C₂²×D₄₇	1	2	yes	—
71	C₂³	C₂²×D₄₇	1	1	yes	characteristic
72	D₄	—	94	8	no	—
73	D₄	—	94	8	no	—
74	D₄	—	188	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	47	4	no	—
77	C₂×C₄	—	94	4	no	—
78	C₂×C₄	C₂²×D₄₇	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	4	no	—
82	C₂²	—	94	4	no	—
83	C₂²	—	94	8	no	—
84	C₂²	—	188	8	no	—
85	C₂²	C₂×D₁₈₈	1	4	yes	—
86	C₂²	C₂×D₁₈₈	1	2	yes	—
87	C₂²	C₂³×D₄₇	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	94	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	188	4	no	—
93	C₂	C₂²×D₁₈₈	1	1	yes	characteristic
94	C₂	#13094069	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₄₇	C₂
hypercenter	C₂×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#534
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₄₇	C₂	semidirect
2. #13094008	C₂	semidirect
3. #13093998	C₂	semidirect
4. #13093994	C₂	semidirect
5. C₂×C₂²⋊C₄×C₄₇	C₂	semidirect
7. C₂³×D₄₇	C₂²	semidirect
8. C₂²×C₄×C₄₇	C₂²	semidirect
11. C₂×C₂²⋊D₄₇	C₂²	semidirect
12. C₂²×Dic₄₇	C₂²	semidirect
13. C₂×D₁₈₈	C₂²	semidirect
14. C₂²⋊C₄×C₄₇	C₂²	semidirect
15. (C₂×D₄₇)⋊C₄	C₂²	semidirect
16. C₄⋊Dic₄₇	C₂²	semidirect
18. C₂³×C₄₇	D₄	semidirect
48. C₄₇	#534	semidirect
53. C₂×C₂²⋊C₄	D₄₇	semidirect
61. C₂²×C₄	C₂×D₄₇	semidirect
64. C₂²⋊C₄	C₂×D₄₇	semidirect
69. C₂³	D₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊D₄₇	2
#13094008	2
#13093998	2
#13093994	2
C₂×C₂²⋊C₄×C₄₇	2
#534	47