Subgroups — #15689704

Order 3008.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625350	C₂	1	2	yes	maximal, complemented
2	#13094004	C₂	1	4	yes	maximal, complemented
3	#13093999	C₂	1	4	yes	maximal, complemented
4	#13093998	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
7	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₄₇	C₂²	1	4	yes	—
9	#10625350	—	2	8	no	—
10	C₂×D₁₈₈	C₂²	1	2	yes	—
11	C₂×D₁₈₈	C₂²	1	4	yes	—
12	C₂×C₄×D₄₇	C₂²	1	4	yes	—
13	C₂×Dic₉₄	C₂²	1	2	yes	—
14	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
15	(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
16	C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂³×C₄₇	C₂³	1	1	yes	characteristic
18	C₂²×D₄₇	C₂³	1	4	yes	—
19	C₂×C₄×C₄₇	—	4	2	no	—
20	C₂×C₄×C₄₇	D₄	1	4	yes	complemented
21	C₂×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂²⋊D₄₇	—	2	8	no	—
24	C₂×Dic₄₇	C₂³	1	4	yes	—
25	D₁₈₈	—	2	4	no	—
26	D₁₈₈	—	4	4	no	—
27	C₄×D₄₇	—	2	8	no	—
28	Dic₉₄	—	2	4	no	—
29	C₂²×C₄₇	—	2	1	no	—
30	C₂²×C₄₇	—	2	1	no	—
31	C₂²×C₄₇	C₂×D₄	1	2	yes	—
32	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
33	C₂×D₄₇	—	2	4	no	—
34	C₂×D₄₇	—	2	4	no	—
35	C₂×D₄₇	—	2	4	no	—
36	C₄×C₄₇	—	2	4	no	—
37	C₄×C₄₇	C₂×D₄	1	4	yes	—
38	Dic₄₇	—	2	4	no	—
39	C₂×C₄₇	—	2	2	no	—
40	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
42	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
43	D₄₇	—	4	4	no	—
44	#537	—	47	1	no	maximal, Sylow
45	C₄₇	#537	1	1	yes	characteristic, complemented, Sylow
46	C₂×#41	—	47	2	no	—
47	C₂²⋊Q₈	—	47	4	no	—
48	#121	—	47	4	no	—
49	#121	—	47	4	no	—
50	C₂×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
51	#41	—	94	8	no	—
52	C₂×Q₈	—	47	2	no	—
53	C₂×D₄	—	47	4	no	—
54	C₂×D₄	—	47	2	no	—
55	C₂×D₄	—	47	4	no	—
56	C₂²×C₄	—	47	4	no	—
57	C₂²×C₄	C₂×D₄₇	1	2	yes	—
58	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
59	C₄⋊C₄	—	47	4	no	—
60	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
61	C₂²⋊C₄	—	47	8	no	—
62	C₂³	—	47	4	no	—
63	C₂³	C₂²×D₄₇	1	1	yes	characteristic
64	Q₈	—	94	4	no	—
65	D₄	—	94	8	no	—
66	D₄	—	94	4	no	—
67	D₄	—	188	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	47	4	no	—
70	C₂×C₄	—	94	8	no	—
71	C₂×C₄	D₁₈₈	1	4	yes	complemented
72	C₂×C₄	C₂²×D₄₇	1	4	yes	—
73	C₂×C₄	C₂²×D₄₇	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	94	4	no	—
77	C₂²	—	94	4	no	—
78	C₂²	—	94	4	no	—
79	C₂²	C₂×D₁₈₈	1	2	yes	—
80	C₂²	C₂³×D₄₇	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	94	4	no	—
83	C₄	C₂×D₁₈₈	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	188	4	no	—
86	C₂	C₂²×D₁₈₈	1	1	yes	characteristic
87	C₂	#13094069	1	1	yes	characteristic
88	C₂	#13094072	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₂×C₄⋊C₄×C₄₇	C₂
hypercenter	C₂×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#537
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10625350	C₂	semidirect
2. #13094004	C₂	semidirect
3. #13093999	C₂	semidirect
4. #13093998	C₂	semidirect
5. C₂×C₄⋊C₄×C₄₇	C₂	semidirect
20. C₂×C₄×C₄₇	D₄	semidirect
45. C₄₇	#537	semidirect
50. C₂×C₄⋊C₄	D₄₇	semidirect
71. C₂×C₄	D₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625350	2
#13094004	2
#13093999	2
#13093998	2
C₂×C₄⋊C₄×C₄₇	2
#537	47