Group #26128128 — subgroups

Subgroups — #26128128

Order 12032.

200 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20245293	C₂	1	2	yes	maximal
2	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20241366	C₂	1	2	yes	maximal
4	#20241306	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1367	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₉₄	C₂²	1	2	yes	—
7	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
8	Dic₄₇⋊C₁₆	—	2	2	no	—
9	#15688898	—	2	2	no	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15688876	C₄	1	1	yes	characteristic
13	#15688855	C₄	1	2	yes	—
14	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₈₈	C₈	1	1	yes	characteristic
17	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
18	C₂×C₄₇⋊C₁₆	—	2	1	no	—
19	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
20	#13093958	C₈	1	1	yes	characteristic
21	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
22	Dic₄₇⋊C₈	—	2	2	no	—
23	#13093949	C₈	1	2	yes	—
24	C₈×Dic₄₇	—	2	2	no	—
25	C₂×C₁₆×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×Dic₉₄	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
35	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
37	Dic₄₇⋊C₄	—	2	2	no	—
38	C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	Dic₁₈₈	—	2	2	no	—
41	C₁₆×C₄₇	—	4	2	no	—
42	C₄₇⋊C₁₆	—	2	2	no	—
43	C₄₇⋊C₁₆	—	4	1	no	—
44	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
47	C₂×Dic₄₇	—	2	2	no	—
48	C₂×Dic₄₇	—	2	2	no	—
49	Dic₉₄	—	2	2	no	—
50	Dic₉₄	—	2	2	no	—
51	C₈×C₄₇	—	2	1	no	—
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	—	4	1	no	—
54	C₈×C₄₇	M₃₂	1	2	yes	—
55	C₈×C₄₇	#137	1	2	yes	—
56	C₄₇⋊C₈	—	4	2	no	—
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	#441	1	1	yes	characteristic
61	C₄×C₄₇	#444	1	1	yes	characteristic
62	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
63	C₄×C₄₇	#504	1	1	yes	characteristic
64	Dic₄₇	—	4	2	no	—
65	Dic₄₇	—	4	2	no	—
66	#2183	—	47	2	no	—
67	C₈×Q₁₆	—	47	1	no	—
68	#1367	D₄₇	1	1	yes	characteristic
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	—	47	2	no	—
75	C₄×Q₁₆	—	47	1	no	—
76	C₄⋊C₁₆	—	94	2	no	—
77	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
78	#346	—	94	2	no	—
79	C₄×C₁₆	—	47	1	no	—
80	C₄×C₁₆	—	47	1	no	—
81	#335	—	47	1	no	—
82	Q₈⋊C₈	—	47	2	no	—
83	C₈²	C₂×D₄₇	1	1	yes	characteristic
84	C₄₇	#12061	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	47	1	no	—
86	C₄×Q₈	—	47	2	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	47	2	no	—
89	C₂×C₁₆	—	94	1	no	—
90	#107	—	47	1	no	—
91	C₄⋊C₈	—	47	2	no	—
92	C₄⋊C₈	—	94	2	no	—
93	#103	—	47	2	no	—
94	C₄×C₈	—	94	2	no	—
95	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
96	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	C₂×Q₈	—	47	2	no	—
99	Q₁₆	—	94	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	94	2	no	—
102	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
103	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
104	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
105	C₂×C₈	#10625354	1	1	yes	characteristic
106	C₄⋊C₄	—	47	2	no	—
107	C₄⋊C₄	—	94	2	no	—
108	C₄²	—	94	2	no	—
109	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
110	C₁₆	—	4	2	no	—
111	C₁₆	—	94	2	no	—
112	C₁₆	—	188	1	no	—
113	Q₈	—	94	2	no	—
114	Q₈	—	94	2	no	—
115	C₂×C₄	—	94	2	no	—
116	C₂×C₄	—	94	2	no	—
117	C₂×C₄	#13093986	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
119	C₂×C₄	#13094032	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	188	2	no	—
124	C₈	#13093963	1	2	yes	—
125	C₈	#13094053	1	2	yes	—
126	C₂²	#15689188	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	188	2	no	—
129	C₄	—	188	2	no	—
130	C₄	#15689229	1	1	yes	characteristic
131	C₄	#15689244	1	1	yes	characteristic
132	C₄	C₂×#13093963	1	1	yes	characteristic
133	C₄	#15689541	1	1	yes	characteristic
134	C₂	#20242206	1	1	yes	characteristic
135	C₂	#20245219	1	1	yes	characteristic
136	C₂	#20245229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245293

C₂

yes

maximal

C₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20241366

C₂

yes

maximal

#20241306

C₂

yes

maximal, characteristic

C₄₇×#1367

C₂

yes

maximal, characteristic

C₈×Dic₉₄

C₂²

yes

—

C₄×Dic₁₈₈

C₄

yes

characteristic

Dic₄₇⋊C₁₆

—

#15688898

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15688876

C₄

yes

characteristic

#15688855

C₄

yes

—

C₄₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₈

yes

characteristic

C₄×Dic₉₄

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

C₈

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

Dic₄₇⋊C₈

—

#13093949

C₈

yes

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

C₂×C₈

yes

—

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

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₄₇

C₂×C₈

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

M₃₂

yes

—

C₈×C₄₇

#137

yes

—

C₄₇⋊C₈

—

#12061

—

maximal, Sylow

C₂²×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×M₃₂

yes

characteristic

C₄×C₄₇

#504

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2183

—

C₈×Q₁₆

—

#1367

D₄₇

yes

characteristic

Q₈⋊C₁₆

—

#1312

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₄₇

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12061

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₈×D₄₇

yes

characteristic

103

C₂×C₈

C₈×D₄₇

yes

characteristic

104

C₂×C₈

D₄×D₄₇

yes

characteristic

105

C₂×C₈

#10625354

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

C₂×C₄×D₄₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

188

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#13093986

yes

characteristic

118

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

119

C₂×C₄

#13094032

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

188

—

124

C₈

#13093963

yes

—

125

C₈

#13094053

yes

—

126

C₂²

#15689188

yes

characteristic

127

C₄

—

128

C₄

—

188

—

129

C₄

—

188

—

130

C₄

#15689229

yes

characteristic

131

C₄

#15689244

yes

characteristic

132

C₄

C₂×#13093963

yes

characteristic

133

C₄

#15689541

yes

characteristic

134

C₂

#20242206

yes

characteristic

135

C₂

#20245219

yes

characteristic

136

C₂

#20245229

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1367	C₂
socle, prime radical	C₂²×C₄₇	C₈×D₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1367	D₄₇
hypocenter	C₄₇	#12061
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1367

C₂

socle, prime radical

C₂²×C₄₇

C₈×D₄

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

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
#20245293	2
C₈×Dic₁₈₈	2
#20241366	2
#20241306	2
C₄₇×#1367	2
#12061	47

subgroup

index

C₈×Dic₁₈₈

C₄₇×#1367

Subgroups — #26128128

Conjugacy classes of subgroups

Complements

Maximal subgroups