Subgroups — #15688984

Order 3008.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#95	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
7	C₂²×Dic₄₇	—	2	2	no	—
8	C₂²×Dic₄₇	—	2	2	no	—
9	C₂²×Dic₄₇	C₄	1	4	yes	complemented
10	C₂²×Dic₄₇	C₂²	1	2	yes	—
11	C₂²×Dic₄₇	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₄₇	—	2	2	no	—
13	C₂²⋊Dic₄₇	—	2	2	no	—
14	C₂³×C₄₇	—	2	4	no	—
15	C₂³×C₄₇	C₂×C₄	1	2	yes	—
16	C₂³×C₄₇	D₄	1	2	yes	—
17	C₂³×C₄₇	D₄	1	2	yes	—
18	C₂³×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₄₇	—	2	2	no	—
20	C₂×C₄×C₄₇	—	2	4	no	—
21	C₂×Dic₄₇	—	2	8	no	—
22	C₂×Dic₄₇	—	2	2	no	—
23	C₂×Dic₄₇	—	2	2	no	—
24	C₂×Dic₄₇	—	2	2	no	—
25	C₂×Dic₄₇	—	2	4	no	—
26	C₂×Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇	—	2	4	no	—
28	C₂²×C₄₇	—	2	4	no	—
29	C₂²×C₄₇	—	2	4	no	—
30	C₂²×C₄₇	C₂²⋊C₄	1	4	yes	—
31	C₂²×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄₇	#41	1	4	yes	—
35	C₄×C₄₇	—	4	2	no	—
36	Dic₄₇	—	2	4	no	—
37	Dic₄₇	—	4	2	no	—
38	C₂×C₄₇	—	2	4	no	—
39	C₂×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄₇	#117	1	2	yes	—
41	C₂×C₄₇	#123	1	2	yes	—
42	C₂×C₄₇	#123	1	2	yes	—
43	#381	—	47	1	no	maximal, Sylow
44	C₄₇	#381	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₄	—	47	1	no	—
46	C₂×C₂²⋊C₄	—	47	1	no	—
47	C₂×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
48	#95	—	47	4	no	—
49	C₂⁴	C₂×D₄₇	1	1	yes	characteristic
50	C₂²×C₄	—	47	4	no	—
51	C₂²×C₄	—	47	2	no	—
52	C₂²×C₄	—	47	2	no	—
53	C₂²×C₄	—	94	2	no	—
54	C₂²×C₄	—	94	2	no	—
55	C₂²×C₄	C₂×D₄₇	1	2	yes	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂²⋊C₄	—	94	2	no	—
58	C₂³	—	2	4	no	—
59	C₂³	C₄×D₄₇	1	2	yes	—
60	C₂³	D₁₈₈	1	2	yes	—
61	C₂³	C₂²⋊D₄₇	1	2	yes	—
62	C₂³	C₂²×D₄₇	1	1	yes	characteristic
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	47	4	no	—
66	C₂×C₄	—	94	8	no	—
67	C₂×C₄	—	94	2	no	—
68	C₂×C₄	—	94	2	no	—
69	C₂×C₄	—	94	2	no	—
70	C₂×C₄	—	94	4	no	—
71	C₂²	—	2	4	no	—
72	C₂²	—	2	4	no	—
73	C₂²	—	2	4	no	—
74	C₂²	(C₂×D₄₇)⋊C₄	1	4	yes	—
75	C₂²	C₂×C₄×D₄₇	1	1	yes	characteristic
76	C₂²	C₂×D₁₈₈	1	1	yes	characteristic
77	C₂²	#10625352	1	4	yes	—
78	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	94	4	no	—
81	C₄	—	188	2	no	—
82	C₂	—	2	4	no	—
83	C₂	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
84	C₂	#13093980	1	2	yes	—
85	C₂	#13094005	1	2	yes	—
86	C₂	#13094008	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄
Fitting	C₂×C₂²⋊C₄×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂³
prime radical	C₂⁴×C₄₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#381
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₄₇⋊#95	C₂	semidirect
9. C₂²×Dic₄₇	C₄	semidirect
44. C₄₇	#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Dic₄₇	2
C₂×C₂²⋊Dic₄₇	2
C₄₇⋊#95	2
C₂×C₂²⋊C₄×C₄₇	2
#381	47