Group #26128084 — subgroups

Subgroups — #26128084

Order 12032.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20241458	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal
4	C₈×C₄₇⋊C₁₆	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	1	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
19	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×Q₁₆×C₄₇	C₈	1	1	yes	characteristic
21	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
22	#107×C₄₇	C₈	1	1	yes	characteristic
23	C₄⋊C₈×C₄₇	—	2	2	no	—
24	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
25	#103×C₄₇	C₈	1	2	yes	—
26	C₄×C₈×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
31	Q₁₆×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₄₇⋊C₁₆	—	2	2	no	—
43	C₄₇⋊C₁₆	—	2	1	no	—
44	C₄₇⋊C₁₆	—	4	2	no	—
45	C₄₇⋊C₁₆	Q₁₆	1	2	yes	complemented
46	Q₈×C₄₇	—	2	2	no	—
47	Q₈×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	—	2	1	no	—
56	C₈×C₄₇	—	4	2	no	—
57	C₈×C₄₇	M₃₂	1	2	yes	—
58	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₄₇	#135	1	1	yes	characteristic
60	#12054	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	4	2	no	—
64	C₄×C₄₇	—	4	2	no	—
65	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
66	C₄×C₄₇	#443	1	1	yes	characteristic
67	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
68	C₄×C₄₇	#504	1	1	yes	characteristic
69	#2183	—	47	2	no	—
70	C₈×Q₁₆	D₄₇	1	1	yes	characteristic
71	#1370	—	47	1	no	—
72	Q₈⋊C₁₆	—	47	2	no	—
73	C₈×C₁₆	—	47	1	no	—
74	C₂×C₄₇	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₄₇	#2169	1	1	yes	characteristic
76	C₂×C₄₇	#2170	1	1	yes	characteristic
77	C₈×Q₈	C₂×D₄₇	1	2	yes	—
78	C₄×Q₁₆	Dic₄₇	1	1	yes	characteristic
79	C₄⋊C₁₆	—	47	2	no	—
80	C₄⋊C₁₆	—	94	2	no	—
81	#346	—	94	2	no	—
82	C₄×C₁₆	—	47	1	no	—
83	C₄×C₁₆	—	47	1	no	—
84	#335	Dic₄₇	1	1	yes	characteristic
85	Q₈⋊C₈	Dic₄₇	1	2	yes	—
86	C₈²	C₂×D₄₇	1	1	yes	characteristic
87	C₄₇	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	C₄₇⋊C₈	1	1	yes	characteristic
89	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
90	C₂×C₁₆	—	47	1	no	—
91	C₂×C₁₆	—	47	1	no	—
92	C₂×C₁₆	—	94	1	no	—
93	C₂×C₁₆	—	94	2	no	—
94	#107	C₄₇⋊C₈	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
97	#103	C₄₇⋊C₈	1	2	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
103	Q₁₆	—	2	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
108	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
109	C₂×C₈	#10625352	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
114	C₁₆	—	47	2	no	—
115	C₁₆	—	94	2	no	—
116	C₁₆	—	94	1	no	—
117	C₁₆	—	188	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
124	C₂×C₄	#13094031	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	4	2	no	—
129	C₈	#13093964	1	2	yes	—
130	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
131	C₈	#13094047	1	1	yes	characteristic
132	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	2	no	—
136	C₄	Q₁₆×Dic₄₇	1	1	yes	characteristic
137	C₄	#15689240	1	1	yes	characteristic
138	C₄	C₂×#13093964	1	1	yes	characteristic
139	C₄	#15689540	1	1	yes	characteristic
140	C₂	Q₁₆×C₄₇⋊C₈	1	1	yes	characteristic
141	C₂	#20245220	1	1	yes	characteristic
142	C₂	#20245225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245297

C₂

yes

maximal

#20241458

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₁₆)

C₂

yes

maximal

C₈×C₄₇⋊C₁₆

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

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

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

—

C₄₇⋊C₁₆

Q₁₆

yes

complemented

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

C₂×Q₁₆

yes

C₈×C₄₇

yes

—

C₂²×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×Q₁₆

yes

characteristic

C₄×C₄₇

#443

yes

characteristic

C₄×C₄₇

C₂×M₃₂

yes

C₄×C₄₇

yes

—

C₈×Q₁₆

D₄₇

yes

characteristic

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₄₇

C₈×Q₁₆

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

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₄₇⋊C₈

yes

characteristic

C₄×Q₈

C₂×Dic₄₇

yes

—

C₂×C₁₆

—

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

100

C₄×C₈

C₂×Dic₄₇

yes

characteristic

101

C₄×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×Q₈

C₂×C₄₇⋊C₈

yes

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

107

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

108

C₂×C₈

D₄×D₄₇

yes

characteristic

109

C₂×C₈

#10625352

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×C₄₇⋊C₈

yes

—

112

C₄²

—

113

C₄²

C₂²×Dic₄₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

188

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×Dic₄₇

yes

characteristic

123

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

124

C₂×C₄

#13094031

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#13093964

yes

—

130

C₈

Q₁₆×D₄₇

yes

characteristic

131

C₈

#13094047

yes

characteristic

132

C₂²

D₄×C₄₇⋊C₈

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₁₆×Dic₄₇

yes

characteristic

137

C₄

#15689240

yes

characteristic

138

C₄

C₂×#13093964

yes

characteristic

139

C₄

#15689540

yes

characteristic

140

C₂

Q₁₆×C₄₇⋊C₈

yes

characteristic

141

C₂

#20245220

yes

characteristic

142

C₂

#20245225

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	#12054
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

#12054

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
45. C₄₇⋊C₁₆	Q₁₆	semidirect
87. C₄₇	#12054	semidirect

H ≅ G/N

type

45. C₄₇⋊C₁₆

Q₁₆

semidirect

87. C₄₇

#12054

semidirect

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

subgroup	index
#20245297	2
#20241458	2
Q₈⋊(C₄₇⋊C₁₆)	2
C₈×C₄₇⋊C₁₆	2
C₈×Q₁₆×C₄₇	2
#12054	47

subgroup

index

#20245297

#20241458

Q₈⋊(C₄₇⋊C₁₆)

C₈×C₄₇⋊C₁₆

C₈×Q₁₆×C₄₇

#12054

Subgroups — #26128084

Conjugacy classes of subgroups

Complements

Maximal subgroups