Group #26130656 — subgroups

Subgroups — #26130656

Order 12032.

188 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20245296	C₂	1	1	yes	maximal, characteristic
2	#20242789	C₂	1	2	yes	maximal
3	#20241462	C₂	1	1	yes	maximal, characteristic
4	#20241446	C₂	1	2	yes	maximal
5	C₄₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#15689415	—	2	2	no	—
7	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₄₇⋊C₁₆	—	2	1	no	—
10	#15688898	—	2	1	no	—
11	#15688876	C₂²	1	2	yes	—
12	#15688872	—	2	2	no	—
13	C₄₇×#500	C₂²	1	2	yes	—
14	C₄₇×#366	—	2	1	no	—
15	C₄₇×#366	—	2	1	no	—
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×Dic₉₄	D₄	1	1	yes	characteristic
18	C₂×C₄₇⋊C₁₆	—	2	1	no	—
19	#13093957	—	4	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₄₇⋊C₈	—	2	1	no	—
23	#13093947	—	4	2	no	—
24	C₈×Dic₄₇	—	2	1	no	—
25	C₄⋊Q₈×C₄₇	—	2	2	no	—
26	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
28	#107×C₄₇	—	2	2	no	—
29	#107×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₄₇	—	4	2	no	—
32	C₂×Dic₉₄	D₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄⋊C₄×C₄₇	—	4	2	no	—
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
39	Dic₄₇⋊C₄	—	2	1	no	—
40	C₄×Dic₄₇	—	2	1	no	—
41	C₂×C₄₇⋊C₈	—	2	1	no	—
42	C₂×C₄₇⋊C₈	—	4	2	no	—
43	C₁₆×C₄₇	—	2	1	no	—
44	C₁₆×C₄₇	Q₁₆	1	2	yes	—
45	C₄₇⋊C₁₆	—	4	1	no	—
46	Q₈×C₄₇	—	16	2	no	—
47	C₂×C₄×C₄₇	—	8	2	no	—
48	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	1	no	—
52	C₂×Dic₄₇	—	2	1	no	—
53	Dic₉₄	—	2	2	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₈×C₄₇	#137	1	1	yes	characteristic
57	C₄₇⋊C₈	—	4	1	no	—
58	C₄₇⋊C₈	—	8	2	no	—
59	#12461	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	#466	1	1	yes	characteristic
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	16	2	no	—
63	C₄×C₄₇	#462	1	1	yes	characteristic
64	C₄×C₄₇	#473	1	1	yes	characteristic
65	C₄×C₄₇	#510	1	2	yes	—
66	Dic₄₇	—	4	1	no	—
67	Dic₄₇	—	4	1	no	—
68	#2252	D₄₇	1	1	yes	characteristic
69	#2183	—	47	1	no	—
70	#1676	—	47	2	no	—
71	#1372	—	47	1	no	—
72	#1365	—	47	2	no	—
73	C₂×C₄₇	#1674	1	1	yes	characteristic
74	C₂×C₄₇	#2219	1	1	yes	characteristic
75	C₂×C₄₇	#2221	1	1	yes	characteristic
76	#500	C₂×D₄₇	1	2	yes	—
77	#477	—	94	2	no	—
78	C₈×Q₈	—	47	1	no	—
79	#366	—	2	1	no	—
80	#366	—	2	1	no	—
81	C₄⋊C₁₆	—	47	1	no	—
82	C₄⋊C₁₆	—	94	1	no	—
83	#346	—	94	1	no	—
84	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
85	#335	—	47	2	no	—
86	#333	—	94	2	no	—
87	C₄₇	#12461	1	1	yes	characteristic, complemented, Sylow
88	C₄⋊Q₈	—	2	2	no	—
89	C₄×Q₈	—	47	1	no	—
90	C₂×C₁₆	—	94	1	no	—
91	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
92	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
93	#107	—	2	2	no	—
94	#107	—	2	2	no	—
95	#107	—	188	2	no	—
96	C₄⋊C₈	—	47	1	no	—
97	C₄⋊C₈	—	94	1	no	—
98	C₄⋊C₈	—	94	2	no	—
99	#103	—	188	2	no	—
100	C₄×C₈	—	94	1	no	—
101	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	47	1	no	—
104	C₂×C₈	—	94	1	no	—
105	C₂×C₈	—	188	2	no	—
106	C₂×C₈	#10625352	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	47	1	no	—
111	C₄⋊C₄	—	94	1	no	—
112	C₄²	—	94	1	no	—
113	C₄²	D₄×D₄₇	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	188	1	no	—
116	C₁₆	#10625337	1	2	yes	—
117	Q₈	—	16	2	no	—
118	Q₈	—	94	2	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	94	1	no	—
121	C₂×C₄	—	94	1	no	—
122	C₂×C₄	#13093995	1	1	yes	characteristic
123	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
124	C₂×C₄	#13094043	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	376	2	no	—
127	C₈	—	188	1	no	—
128	C₈	C₂×#10625337	1	1	yes	characteristic
129	C₈	#13094055	1	1	yes	characteristic
130	C₂²	#15689356	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	16	2	no	—
133	C₄	—	188	1	no	—
134	C₄	—	188	1	no	—
135	C₄	#15689330	1	1	yes	characteristic
136	C₄	#15689395	1	1	yes	characteristic
137	C₄	#15689565	1	2	yes	—
138	C₂	#20242779	1	1	yes	characteristic
139	C₂	#20245493	1	1	yes	characteristic
140	C₂	#20245505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245296

C₂

yes

maximal, characteristic

#20242789

C₂

yes

maximal

#20241462

C₂

yes

maximal, characteristic

#20241446

C₂

yes

maximal

C₄₇×#2252

C₂

yes

maximal, characteristic

#15689415

—

C₈×Dic₉₄

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

characteristic

Dic₄₇⋊C₁₆

—

#15688898

—

#15688876

C₂²

yes

—

#15688872

—

C₄₇×#500

C₂²

yes

—

C₄₇×#366

—

C₄₇×#366

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄×Dic₉₄

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

#13093957

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

characteristic

Dic₄₇⋊C₈

—

#13093947

—

C₈×Dic₄₇

—

C₄⋊Q₈×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

D₈

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

Q₁₆

yes

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₈×C₄₇

#137

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

#2252

D₄₇

yes

—

—

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×D₄₇

yes

—

#477

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

#333

—

C₄₇

#12461

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₄₇

yes

—

—

—

188

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

188

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×C₈

—

105

C₂×C₈

—

188

—

106

C₂×C₈

#10625352

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

D₄×D₄₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

188

—

116

C₁₆

#10625337

yes

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#13093995

yes

characteristic

123

C₂×C₄

D₈×D₄₇

yes

characteristic

124

C₂×C₄

#13094043

yes

characteristic

125

C₈

—

126

C₈

—

376

—

127

C₈

—

188

—

128

C₈

C₂×#10625337

yes

characteristic

129

C₈

#13094055

yes

characteristic

130

C₂²

#15689356

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

188

—

134

C₄

—

188

—

135

C₄

#15689330

yes

characteristic

136

C₄

#15689395

yes

characteristic

137

C₄

#15689565

yes

—

138

C₂

#20242779

yes

characteristic

139

C₂

#20245493

yes

characteristic

140

C₂

#20245505

yes

characteristic

	subgroup	G/H
center	C₂²	#15689356
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2252	C₂
socle, prime radical	C₂²×C₄₇	#466
hypercenter	#2252	D₄₇
hypocenter	C₄₇	#12461
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#15689356

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2252

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
87. C₄₇	#12461	semidirect

H ≅ G/N

type

87. C₄₇

#12461

semidirect

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

subgroup	index
#20245296	2
#20242789	2
#20241462	2
#20241446	2
C₄₇×#2252	2
#12461	47

subgroup

index

C₄₇×#2252

Subgroups — #26130656

Conjugacy classes of subgroups

Complements

Maximal subgroups