Group #26181841 — subgroups

Subgroups — #26181841

Order 12032.

1034 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241659	C₂	1	2	yes	maximal
2	#20241644	C₂	1	12	yes	maximal
3	C₄₇×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
5	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
6	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
7	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×#95×C₄₇	C₂²	1	6	yes	—
9	C₂³×Dic₄₇	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₄₇	—	4	12	no	—
11	C₄₇⋊#95	—	2	24	no	—
12	C₄₇⋊#95	—	4	12	no	—
13	C₂²×Q₈×C₄₇	—	4	6	no	—
14	C₂²×Q₈×C₄₇	Q₈	1	4	yes	—
15	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
16	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
17	#95×C₄₇	—	4	6	no	—
18	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
19	C₂×Q₈×C₄₇	—	4	12	no	—
20	C₂×Q₈×C₄₇	—	4	16	no	—
21	C₂²×C₄×C₄₇	—	2	12	no	—
22	C₂²×C₄×C₄₇	—	2	6	no	—
23	C₂²×C₄×C₄₇	—	2	12	no	—
24	C₂²×C₄×C₄₇	—	2	12	no	—
25	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
26	C₂²×C₄×C₄₇	#41	1	8	yes	—
27	C₂²×Dic₄₇	—	2	24	no	—
28	C₂²×Dic₄₇	—	2	24	no	—
29	C₂²×Dic₄₇	—	2	8	no	—
30	C₄⋊Dic₄₇	—	4	16	no	—
31	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
32	C₂³×C₄₇	C₂×#41	1	4	yes	—
33	C₂³×C₄₇	H(5,𝔽₂)	1	6	yes	—
34	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
35	Q₈×C₄₇	—	4	16	no	—
36	C₂×C₄×C₄₇	—	2	24	no	—
37	C₂×C₄×C₄₇	—	2	12	no	—
38	C₂×C₄×C₄₇	—	2	6	no	—
39	C₂×C₄×C₄₇	—	4	4	no	—
40	C₂×C₄×C₄₇	—	4	12	no	—
41	C₂×C₄×C₄₇	—	4	12	no	—
42	C₂×Dic₄₇	—	4	24	no	—
43	C₂×Dic₄₇	—	4	24	no	—
44	C₂×Dic₄₇	—	4	8	no	—
45	#17308	—	47	1	no	maximal, Sylow
46	C₂²×C₄₇	#539	1	12	yes	—
47	C₂²×C₄₇	#543	1	3	yes	—
48	C₂²×C₄₇	#555	1	6	yes	—
49	C₂²×C₄₇	#557	1	4	yes	—
50	C₂²×C₄₇	H(3,𝔽₄)	1	4	yes	—
51	C₂²×C₄₇	#563	1	6	yes	—
52	C₄×C₄₇	—	4	8	no	—
53	C₄×C₄₇	—	4	6	no	—
54	Dic₄₇	—	8	8	no	—
55	#1446	—	47	2	no	—
56	#1446	D₄₇	1	1	yes	characteristic
57	#1442	—	47	12	no	—
58	C₂×C₄₇	#2785	1	4	yes	—
59	C₂×C₄₇	#2809	1	6	yes	—
60	C₂×C₄₇	#2835	1	4	yes	—
61	C₂×C₄₇	#2840	1	1	yes	characteristic
62	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
63	C₂²×C₄⋊C₄	—	47	4	no	—
64	C₂×#95	—	47	12	no	—
65	C₂×#95	—	47	12	no	—
66	C₂×#95	C₂×D₄₇	1	6	yes	—
67	C₄₇	#17308	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	—	4	6	no	—
69	C₂²×Q₈	Dic₉₄	1	4	yes	—
70	C₂³×C₄	—	47	8	no	—
71	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
72	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
73	C₂×C₄⋊C₄	—	188	12	no	—
74	#95	—	4	6	no	—
75	#95	—	94	24	no	—
76	#95	—	188	12	no	—
77	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
78	C₂×Q₈	—	4	12	no	—
79	C₂×Q₈	—	4	16	no	—
80	C₂²×C₄	—	2	12	no	—
81	C₂²×C₄	—	2	6	no	—
82	C₂²×C₄	—	2	12	no	—
83	C₂²×C₄	—	2	12	no	—
84	C₂²×C₄	—	94	24	no	—
85	C₂²×C₄	—	94	24	no	—
86	C₂²×C₄	—	94	8	no	—
87	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
88	C₂²×C₄	#10625354	1	8	yes	—
89	C₄⋊C₄	—	188	16	no	—
90	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
91	C₂³	C₂×#10625354	1	4	yes	—
92	C₂³	#13094069	1	6	yes	—
93	C₂³	#13094071	1	4	yes	—
94	Q₈	—	4	16	no	—
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	12	no	—
97	C₂×C₄	—	2	24	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	4	12	no	—
100	C₂×C₄	—	4	12	no	—
101	C₂×C₄	—	188	24	no	—
102	C₂×C₄	—	188	8	no	—
103	C₂×C₄	—	188	24	no	—
104	C₂²	#15689722	1	12	yes	—
105	C₂²	#15689749	1	3	yes	—
106	C₂²	#15689839	1	6	yes	—
107	C₂²	#15689852	1	4	yes	—
108	C₂²	#15689866	1	4	yes	—
109	C₂²	#15689873	1	6	yes	—
110	C₄	—	4	8	no	—
111	C₄	—	4	6	no	—
112	C₄	—	376	8	no	—
113	C₂	#20251096	1	4	yes	—
114	C₂	#20251376	1	6	yes	—
115	C₂	#20251579	1	4	yes	—
116	C₂	#20251641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241659

C₂

yes

maximal

#20241644

C₂

yes

maximal

C₄₇×#1446

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂³×Q₈×C₄₇

C₂²

yes

characteristic

C₂×#95×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂×C₄⋊Dic₄₇

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

Q₈

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

#95×C₄₇

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

C₂²×Q₈

yes

characteristic

C₂³×C₄₇

C₂×#41

yes

—

C₂³×C₄₇

H(5,𝔽₂)

yes

—

C₂³×C₄₇

2₋¹⁺⁴

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#17308

—

maximal, Sylow

C₂²×C₄₇

#539

yes

—

C₂²×C₄₇

#543

yes

—

C₂²×C₄₇

#555

yes

—

C₂²×C₄₇

#557

yes

—

C₂²×C₄₇

H(3,𝔽₄)

yes

—

C₂²×C₄₇

#563

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#1446

—

#1446

D₄₇

yes

characteristic

#1442

—

C₂×C₄₇

#2785

yes

—

C₂×C₄₇

#2809

yes

—

C₂×C₄₇

#2835

yes

—

C₂×C₄₇

#2840

yes

characteristic

C₂³×Q₈

C₂×D₄₇

yes

characteristic

C₂²×C₄⋊C₄

—

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₄₇

yes

—

C₄₇

#17308

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

Dic₉₄

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂×C₄⋊C₄

—

188

—

#95

—

#95

—

#95

—

188

—

C₂⁴

C₂³×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₉₄

yes

—

C₂²×C₄

#10625354

yes

—

C₄⋊C₄

—

188

—

C₂³

C₂²×Dic₉₄

yes

characteristic

C₂³

C₂×#10625354

yes

—

C₂³

#13094069

yes

—

C₂³

#13094071

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

188

—

102

C₂×C₄

—

188

—

103

C₂×C₄

—

188

—

104

C₂²

#15689722

yes

—

105

C₂²

#15689749

yes

—

106

C₂²

#15689839

yes

—

107

C₂²

#15689852

yes

—

108

C₂²

#15689866

yes

—

109

C₂²

#15689873

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

376

—

113

C₂

#20251096

yes

—

114

C₂

#20251376

yes

—

115

C₂

#20251579

yes

—

116

C₂

#20251641

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator, socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
Fitting	C₄₇×#1446	C₂
hypercenter	#1446	D₄₇
hypocenter	C₄₇	#17308
LCS	G ▷ C₂⁴×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator, socle, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

Fitting

C₄₇×#1446

C₂

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
67. C₄₇	#17308	semidirect

H ≅ G/N

type

67. C₄₇

#17308

semidirect

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

subgroup	index
#20241659	2
#20241644	2
C₄₇×#1446	2
#17308	47

subgroup

index

#20241659

#20241644

C₄₇×#1446

#17308

Subgroups — #26181841

Conjugacy classes of subgroups

Complements

Maximal subgroups