Group #26181842 — subgroups

Subgroups — C₄₇⋊#17309 #26181842

Order 12032.

1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 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	#20241658	C₂	1	12	yes	maximal
3	C₄₇×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₉₄	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	C₂²×Dic₉₄	—	4	12	no	—
10	C₂²×Dic₉₄	—	4	12	no	—
11	C₂²×Dic₉₄	Q₈	1	16	yes	complemented
12	C₂³×Dic₄₇	C₂³	1	8	yes	—
13	C₄₇⋊#95	—	4	12	no	—
14	C₄₇⋊#95	—	4	12	no	—
15	C₂²×Q₈×C₄₇	—	4	6	no	—
16	C₂²×Q₈×C₄₇	Q₈	1	4	yes	complemented
17	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
18	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
19	#95×C₄₇	—	4	6	no	—
20	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
21	C₂×Q₈×C₄₇	—	4	12	no	—
22	C₂×Q₈×C₄₇	—	4	16	no	—
23	C₂²×C₄×C₄₇	—	2	12	no	—
24	C₂²×C₄×C₄₇	—	2	6	no	—
25	C₂²×C₄×C₄₇	—	2	12	no	—
26	C₂²×C₄×C₄₇	—	2	12	no	—
27	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
29	C₂²×Dic₄₇	—	2	24	no	—
30	C₂²×Dic₄₇	—	2	24	no	—
31	C₂²×Dic₄₇	C₂×Q₈	1	16	yes	—
32	C₂×Dic₉₄	—	4	48	no	—
33	C₂×Dic₉₄	—	4	48	no	—
34	C₂×Dic₉₄	—	4	16	no	—
35	C₂³×C₄₇	C₂²×Q₈	1	4	yes	—
36	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
37	C₂³×C₄₇	H(5,𝔽₂)	1	6	yes	—
38	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
39	Q₈×C₄₇	—	4	16	no	—
40	C₂×C₄×C₄₇	—	2	24	no	—
41	C₂×C₄×C₄₇	—	2	12	no	—
42	C₂×C₄×C₄₇	—	2	6	no	—
43	C₂×C₄×C₄₇	—	4	12	no	—
44	C₂×C₄×C₄₇	—	4	4	no	—
45	C₂×C₄×C₄₇	—	4	12	no	—
46	C₂×Dic₄₇	—	2	48	no	—
47	C₂×Dic₄₇	—	4	24	no	—
48	C₂×Dic₄₇	—	4	8	no	—
49	Dic₉₄	—	4	64	no	—
50	#17309	—	47	1	no	maximal, Sylow
51	C₂²×C₄₇	#543	1	12	yes	—
52	C₂²×C₄₇	#543	1	3	yes	—
53	C₂²×C₄₇	Q₈²	1	6	yes	—
54	C₂²×C₄₇	Q₈²	1	4	yes	—
55	C₂²×C₄₇	H(3,𝔽₄)	1	6	yes	—
56	C₂²×C₄₇	H(3,𝔽₄)	1	4	yes	—
57	C₄×C₄₇	—	4	8	no	—
58	C₄×C₄₇	—	4	6	no	—
59	Dic₄₇	—	4	16	no	—
60	#1446	—	47	12	no	—
61	#1446	—	47	2	no	—
62	#1446	D₄₇	1	1	yes	characteristic
63	C₂×C₄₇	#2811	1	4	yes	—
64	C₂×C₄₇	#2811	1	6	yes	—
65	C₂×C₄₇	#2840	1	4	yes	—
66	C₂×C₄₇	#2840	1	1	yes	characteristic
67	C₂³×Q₈	—	47	4	no	—
68	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
69	C₂×#95	—	47	12	no	—
70	C₂×#95	—	47	12	no	—
71	C₂×#95	C₂×D₄₇	1	6	yes	—
72	C₄₇	#17309	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	4	6	no	—
74	C₂²×Q₈	—	47	16	no	—
75	C₂²×Q₈	—	188	12	no	—
76	C₂²×Q₈	—	188	12	no	—
77	C₂²×Q₈	Dic₉₄	1	4	yes	complemented
78	C₂³×C₄	—	47	8	no	—
79	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
80	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
81	#95	—	4	6	no	—
82	#95	—	188	12	no	—
83	#95	—	188	12	no	—
84	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
85	C₂×Q₈	—	4	12	no	—
86	C₂×Q₈	—	4	16	no	—
87	C₂×Q₈	—	188	48	no	—
88	C₂×Q₈	—	188	48	no	—
89	C₂×Q₈	—	188	16	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	2	12	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	12	no	—
94	C₂²×C₄	—	47	16	no	—
95	C₂²×C₄	—	94	24	no	—
96	C₂²×C₄	—	94	24	no	—
97	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
98	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
99	C₂³	C₂×Q₈×D₄₇	1	4	yes	—
100	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
101	C₂³	#13094069	1	6	yes	—
102	C₂³	#13094070	1	4	yes	—
103	Q₈	—	4	16	no	—
104	Q₈	—	188	64	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	4	12	no	—
109	C₂×C₄	—	4	12	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	94	48	no	—
112	C₂×C₄	—	188	24	no	—
113	C₂×C₄	—	188	8	no	—
114	C₂²	#15689748	1	12	yes	—
115	C₂²	#15689749	1	3	yes	—
116	C₂²	#15689853	1	6	yes	—
117	C₂²	Q₈×Dic₉₄	1	4	yes	—
118	C₂²	#15689865	1	6	yes	—
119	C₂²	#15689866	1	4	yes	—
120	C₄	—	4	8	no	—
121	C₄	—	4	6	no	—
122	C₄	—	188	16	no	—
123	C₂	#20251380	1	4	yes	—
124	C₂	#20251381	1	6	yes	—
125	C₂	#20251640	1	4	yes	—
126	C₂	#20251641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241659

C₂

yes

maximal

#20241658

C₂

yes

maximal

C₄₇×#1446

C₂

yes

maximal, characteristic

C₂³×Dic₉₄

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂³×Q₈×C₄₇

C₂²

yes

characteristic

C₂×#95×C₄₇

C₂²

yes

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

Q₈

yes

complemented

C₂³×Dic₄₇

C₂³

yes

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

Q₈

yes

complemented

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

C₂×Q₈

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂³×C₄₇

C₂²×Q₈

yes

—

C₂³×C₄₇

C₂²×Q₈

yes

characteristic

C₂³×C₄₇

H(5,𝔽₂)

yes

—

C₂³×C₄₇

H(5,𝔽₂)

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

—

Dic₉₄

—

#17309

—

maximal, Sylow

C₂²×C₄₇

#543

yes

—

C₂²×C₄₇

#543

yes

—

C₂²×C₄₇

Q₈²

yes

—

C₂²×C₄₇

Q₈²

yes

—

C₂²×C₄₇

H(3,𝔽₄)

yes

—

C₂²×C₄₇

H(3,𝔽₄)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#1446

—

#1446

—

#1446

D₄₇

yes

characteristic

C₂×C₄₇

#2811

yes

—

C₂×C₄₇

#2811

yes

—

C₂×C₄₇

#2840

yes

—

C₂×C₄₇

#2840

yes

characteristic

C₂³×Q₈

—

C₂³×Q₈

C₂×D₄₇

yes

characteristic

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₄₇

yes

—

C₄₇

#17309

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

188

—

C₂²×Q₈

—

188

—

C₂²×Q₈

Dic₉₄

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

—

#95

—

#95

—

188

—

#95

—

188

—

C₂⁴

C₂³×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₉₄

yes

—

C₂²×C₄

Q₈×D₄₇

yes

—

C₂³

C₂×Q₈×D₄₇

yes

—

100

C₂³

C₂²×Dic₉₄

yes

characteristic

101

C₂³

#13094069

yes

—

102

C₂³

#13094070

yes

—

103

Q₈

—

104

Q₈

—

188

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

188

—

113

C₂×C₄

—

188

—

114

C₂²

#15689748

yes

—

115

C₂²

#15689749

yes

—

116

C₂²

#15689853

yes

—

117

C₂²

Q₈×Dic₉₄

yes

—

118

C₂²

#15689865

yes

—

119

C₂²

#15689866

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

188

—

123

C₂

#20251380

yes

—

124

C₂

#20251381

yes

—

125

C₂

#20251640

yes

—

126

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	#17309
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
11. C₂²×Dic₉₄	Q₈	semidirect
16. C₂²×Q₈×C₄₇	Q₈	semidirect
72. C₄₇	#17309	semidirect
77. C₂²×Q₈	Dic₉₄	semidirect

H ≅ G/N

type

11. C₂²×Dic₉₄

Q₈

semidirect

16. C₂²×Q₈×C₄₇

Q₈

semidirect

72. C₄₇

#17309

semidirect

77. C₂²×Q₈

Dic₉₄

semidirect

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

subgroup	index
#20241659	2
#20241658	2
C₄₇×#1446	2
#17309	47

subgroup

index

#20241659

#20241658

C₄₇×#1446

#17309

Subgroups — C₄₇⋊#17309 #26181842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47⋊#17309 #26181842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇⋊#17309 #26181842