Group #26128129 — subgroups

Subgroups — #26128129

Order 12032.

200 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20245297	C₂	1	2	yes	maximal
2	#20241452	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal
4	#20241306	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
7	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
8	#15688899	—	2	2	no	—
9	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
13	C₄₇×#335	C₄	1	1	yes	characteristic
14	C₄₇×Q₈⋊C₈	C₄	1	2	yes	—
15	C₈²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	2	1	no	—
18	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
19	C₂×Q₁₆×C₄₇	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
21	#107×C₄₇	C₈	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	—	2	2	no	—
23	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
24	#103×C₄₇	C₈	1	2	yes	—
25	C₄×C₈×C₄₇	—	2	2	no	—
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
30	Q₁₆×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
39	C₄²×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₄₇⋊C₁₆	—	2	2	no	—
42	C₄₇⋊C₁₆	—	4	1	no	—
43	C₄₇⋊C₁₆	—	4	2	no	—
44	Q₈×C₄₇	—	2	2	no	—
45	Q₈×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
51	C₈×C₄₇	—	2	1	no	—
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	—	4	1	no	—
54	C₈×C₄₇	—	4	2	no	—
55	C₈×C₄₇	M₃₂	1	2	yes	—
56	C₈×C₄₇	#137	1	2	yes	—
57	#12061	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	4	2	no	—
61	C₄×C₄₇	—	4	2	no	—
62	C₄×C₄₇	#441	1	1	yes	characteristic
63	C₄×C₄₇	#444	1	1	yes	characteristic
64	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
65	C₄×C₄₇	#504	1	1	yes	characteristic
66	#2183	—	47	2	no	—
67	C₈×Q₁₆	D₄₇	1	1	yes	characteristic
68	#1367	—	47	1	no	—
69	Q₈⋊C₁₆	—	47	2	no	—
70	#1312	—	47	1	no	—
71	C₂×C₄₇	#1579	1	1	yes	characteristic
72	C₂×C₄₇	#2169	1	1	yes	characteristic
73	C₂×C₄₇	#2171	1	1	yes	characteristic
74	C₈×Q₈	C₂×D₄₇	1	2	yes	—
75	C₄×Q₁₆	Dic₄₇	1	1	yes	characteristic
76	C₄⋊C₁₆	—	47	2	no	—
77	C₄⋊C₁₆	—	94	2	no	—
78	#346	—	94	2	no	—
79	C₄×C₁₆	—	47	1	no	—
80	C₄×C₁₆	—	47	1	no	—
81	#335	Dic₄₇	1	1	yes	characteristic
82	Q₈⋊C₈	Dic₄₇	1	2	yes	—
83	C₈²	C₂×D₄₇	1	1	yes	characteristic
84	C₄₇	#12061	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₄₇⋊C₈	1	1	yes	characteristic
86	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
87	C₂×C₁₆	—	47	2	no	—
88	C₂×C₁₆	—	94	2	no	—
89	C₂×C₁₆	—	94	1	no	—
90	#107	C₄₇⋊C₈	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
93	#103	C₄₇⋊C₈	1	2	yes	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
99	Q₁₆	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
104	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
105	C₂×C₈	#10625352	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
110	C₁₆	—	94	2	no	—
111	C₁₆	—	188	1	no	—
112	C₁₆	—	188	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#13094031	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	#13093964	1	2	yes	—
125	C₈	#13094056	1	2	yes	—
126	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	#15689230	1	1	yes	characteristic
131	C₄	#15689245	1	1	yes	characteristic
132	C₄	C₂×#13093964	1	1	yes	characteristic
133	C₄	#15689540	1	1	yes	characteristic
134	C₂	#20242207	1	1	yes	characteristic
135	C₂	#20245220	1	1	yes	characteristic
136	C₂	#20245230	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245297

C₂

yes

maximal

#20241452

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₁₆)

C₂

yes

maximal

#20241306

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₄₇

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₁₆)

—

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

—

#15688899

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×Q₈×C₄₇

C₂²

yes

—

C₄×Q₁₆×C₄₇

C₄

yes

characteristic

C₄₇×#335

C₄

yes

characteristic

C₄₇×Q₈⋊C₈

C₄

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₄₇

C₈

yes

characteristic

C₄×Q₈×C₄₇

C₂×C₄

yes

—

#107×C₄₇

C₈

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂×C₄

yes

—

#103×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

C₂×C₈

yes

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

M₃₂

yes

—

C₈×C₄₇

#137

yes

—

#12061

—

maximal, Sylow

C₂²×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×M₃₂

yes

C₄×C₄₇

yes

—

C₈×Q₁₆

D₄₇

yes

characteristic

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₈×Q₈

C₂×D₄₇

yes

—

C₄×Q₁₆

Dic₄₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₄₇

yes

characteristic

Q₈⋊C₈

Dic₄₇

yes

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12061

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₄₇⋊C₈

yes

characteristic

C₄×Q₈

C₂×Dic₄₇

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

C₄₇⋊C₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₄₇

yes

—

#103

C₄₇⋊C₈

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

C₂×C₄₇⋊C₈

yes

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

103

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

104

C₂×C₈

D₄×D₄₇

yes

characteristic

105

C₂×C₈

#10625352

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×C₄₇⋊C₈

yes

—

108

C₄²

—

109

C₄²

C₂²×Dic₄₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

188

—

112

C₁₆

—

188

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×Dic₄₇

yes

characteristic

118

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

119

C₂×C₄

#13094031

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

#13093964

yes

—

125

C₈

#13094056

yes

—

126

C₂²

D₄×C₄₇⋊C₈

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#15689230

yes

characteristic

131

C₄

#15689245

yes

characteristic

132

C₄

C₂×#13093964

yes

characteristic

133

C₄

#15689540

yes

characteristic

134

C₂

#20242207

yes

characteristic

135

C₂

#20245220

yes

characteristic

136

C₂

#20245230

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×Q₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₈×D₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₈×Q₁₆	D₄₇
hypocenter	C₄₇	#12061
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₈×Q₁₆×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

C₈×D₄

cosocle

C₄×C₈×C₄₇

C₂³

hypercenter

C₈×Q₁₆

D₄₇

hypocenter

C₄₇

#12061

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
84. C₄₇	#12061	semidirect

H ≅ G/N

type

84. C₄₇

#12061

semidirect

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

subgroup	index
#20245297	2
#20241452	2
Q₈⋊(C₄₇⋊C₁₆)	2
#20241306	2
C₈×Q₁₆×C₄₇	2
#12061	47

subgroup

index

Q₈⋊(C₄₇⋊C₁₆)

C₈×Q₁₆×C₄₇

Subgroups — #26128129

Conjugacy classes of subgroups

Complements

Maximal subgroups