Group #26137165 — subgroups

Subgroups — #26137165

Order 12032.

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

normal subgroups only

This group has 164 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	#20245663	C₂	1	1	yes	maximal, characteristic
2	#20241589	C₂	1	2	yes	maximal
3	C₄⋊(C₄₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#20241569	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689528	—	2	1	no	—
7	C₂×C₄₇⋊C₃₂	C₂²	1	2	yes	—
8	#15688949	—	2	1	no	—
9	#15688949	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆×C₄₇	—	2	1	no	—
12	C₄₇×#366	C₂²	1	1	yes	characteristic
13	C₄₇×#358	—	2	1	no	—
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	#13094024	—	4	1	no	—
16	#13093958	—	2	2	no	—
17	#13093958	—	4	1	no	—
18	C₄₇⋊C₃₂	—	2	2	no	—
19	C₂×Q₁₆×C₄₇	—	2	1	no	—
20	C₄×Q₈×C₄₇	—	4	1	no	—
21	Q₃₂×C₄₇	—	2	1	no	—
22	Q₃₂×C₄₇	Q₈	1	2	yes	complemented
23	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
25	#107×C₄₇	—	2	1	no	—
26	#103×C₄₇	—	4	1	no	—
27	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	—	4	1	no	—
29	C₂×Dic₉₄	—	8	1	no	—
30	Q₁₆×C₄₇	—	2	2	no	—
31	Q₁₆×C₄₇	—	4	1	no	—
32	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₄₇	—	4	1	no	—
35	C₄⋊C₄×C₄₇	—	8	1	no	—
36	C₄²×C₄₇	—	8	1	no	—
37	C₄²×C₄₇	D₈	1	1	yes	characteristic
38	C₄⋊Dic₄₇	—	4	2	no	—
39	C₄⋊Dic₄₇	—	8	1	no	—
40	C₁₆×C₄₇	—	2	1	no	—
41	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₁₆×C₄₇	#41	1	1	yes	characteristic
43	Q₈×C₄₇	—	4	2	no	—
44	Q₈×C₄₇	—	8	1	no	—
45	C₂×C₄×C₄₇	—	8	1	no	—
46	C₂×C₄×C₄₇	—	8	1	no	—
47	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
48	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₄₇	—	8	2	no	—
51	Dic₉₄	—	16	2	no	—
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₈×C₄₇	#137	1	1	yes	characteristic
55	#13704	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	—	8	2	no	—
59	C₄×C₄₇	—	16	1	no	—
60	C₄×C₄₇	SD₆₄	1	2	yes	—
61	C₄×C₄₇	#474	1	1	yes	characteristic
62	C₄×C₄₇	#510	1	1	yes	characteristic
63	Dic₄₇	—	16	2	no	—
64	#2252	—	47	1	no	—
65	C₄×Q₃₂	D₄₇	1	1	yes	characteristic
66	#1424	—	47	2	no	—
67	C₄⋊C₃₂	—	47	1	no	—
68	#1416	—	47	2	no	—
69	C₂×C₄₇	#2224	1	1	yes	characteristic
70	C₂×C₄₇	C₂×SD₆₄	1	1	yes	characteristic
71	C₂×C₄₇	#2264	1	1	yes	characteristic
72	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
73	#500	—	94	1	no	—
74	C₄×Q₁₆	—	2	1	no	—
75	C₂×C₃₂	—	47	2	no	—
76	#366	—	47	2	no	—
77	#366	—	94	1	no	—
78	#366	C₂×D₄₇	1	1	yes	characteristic
79	#358	—	2	1	no	—
80	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
81	C₄₇	#13704	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₁₆	—	2	1	no	—
83	C₄⋊Q₈	—	188	1	no	—
84	C₄×Q₈	—	4	1	no	—
85	Q₃₂	—	2	1	no	—
86	Q₃₂	Dic₉₄	1	2	yes	complemented
87	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
88	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	#107	—	94	2	no	—
91	#107	—	188	1	no	—
92	#103	—	4	1	no	—
93	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
94	C₃₂	—	94	2	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	376	1	no	—
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	4	1	no	—
99	C₂×C₈	#10625334	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	376	1	no	—
104	C₄⋊C₄	—	188	2	no	—
105	C₄²	—	8	1	no	—
106	C₄²	#10625334	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×Dic₉₄	1	1	yes	characteristic
109	C₁₆	#10625350	1	1	yes	characteristic
110	Q₈	—	4	2	no	—
111	Q₈	—	8	1	no	—
112	Q₈	—	752	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	376	2	no	—
116	C₂×C₄	#13093965	1	1	yes	characteristic
117	C₂×C₄	#13093965	1	1	yes	characteristic
118	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	#13094003	1	1	yes	characteristic
121	C₈	#13094054	1	1	yes	characteristic
122	C₂²	C₂×#13093965	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	8	2	no	—
125	C₄	—	16	1	no	—
126	C₄	—	752	2	no	—
127	C₄	#15688960	1	2	yes	—
128	C₄	#15689403	1	1	yes	characteristic
129	C₄	#15689563	1	1	yes	characteristic
130	C₂	#20245525	1	1	yes	characteristic
131	C₂	C₂×#15688960	1	1	yes	characteristic
132	C₂	#20245705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic

#20241589

C₂

yes

maximal

C₄⋊(C₄₇⋊C₃₂)

C₂

yes

maximal, characteristic

#20241569

C₂

yes

maximal

C₄×Q₃₂×C₄₇

C₂

yes

maximal, characteristic

#15689528

—

C₂×C₄₇⋊C₃₂

C₂²

yes

—

#15688949

—

#15688949

C₂²

yes

—

C₂×Q₃₂×C₄₇

C₂²

yes

characteristic

C₄×Q₁₆×C₄₇

—

C₄₇×#366

C₂²

yes

characteristic

C₄₇×#358

—

C₄×C₁₆×C₄₇

C₂²

yes

—

—

—

C₄₇⋊C₃₂

—

C₂×Q₁₆×C₄₇

—

C₄×Q₈×C₄₇

—

Q₃₂×C₄₇

—

Q₃₂×C₄₇

Q₈

yes

complemented

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×Q₈

yes

characteristic

C₁₆×C₄₇

#41

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊Q₈

yes

C₈×C₄₇

yes

—

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

SD₆₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#2252

—

C₄×Q₃₂

D₄₇

yes

characteristic

#1424

—

C₄⋊C₃₂

—

#1416

—

C₂×C₄₇

#2224

yes

characteristic

C₂×C₄₇

C₂×SD₆₄

yes

characteristic

C₂×C₄₇

#2264

yes

characteristic

C₂×Q₃₂

C₂×D₄₇

yes

characteristic

#500

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#366

—

#366

C₂×D₄₇

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13704

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

188

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

Dic₉₄

yes

complemented

C₂×C₁₆

C₂²⋊D₄₇

yes

characteristic

C₂×C₁₆

C₂²×D₄₇

yes

—

—

—

188

—

#103

—

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

376

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#10625334

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

376

—

104

C₄⋊C₄

—

188

—

105

C₄²

—

106

C₄²

#10625334

yes

characteristic

107

C₁₆

—

108

C₁₆

C₂×Dic₉₄

yes

characteristic

109

C₁₆

#10625350

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

752

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

376

—

116

C₂×C₄

#13093965

yes

characteristic

117

C₂×C₄

#13093965

yes

characteristic

118

C₂×C₄

C₂×#10625334

yes

characteristic

119

C₈

—

120

C₈

#13094003

yes

characteristic

121

C₈

#13094054

yes

characteristic

122

C₂²

C₂×#13093965

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

752

—

127

C₄

#15688960

yes

—

128

C₄

#15689403

yes

characteristic

129

C₄

#15689563

yes

characteristic

130

C₂

#20245525

yes

characteristic

131

C₂

C₂×#15688960

yes

characteristic

132

C₂

#20245705

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13093965
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄×Q₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₄₇
hypocenter	C₄₇	#13704
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#13093965

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄×Q₃₂×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₄₇

hypocenter

C₄₇

#13704

LCS

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
22. Q₃₂×C₄₇	Q₈	semidirect
81. C₄₇	#13704	semidirect
86. Q₃₂	Dic₉₄	semidirect

H ≅ G/N

type

22. Q₃₂×C₄₇

Q₈

semidirect

81. C₄₇

#13704

semidirect

86. Q₃₂

Dic₉₄

semidirect

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

subgroup	index
#20245663	2
#20241589	2
C₄⋊(C₄₇⋊C₃₂)	2
#20241569	2
C₄×Q₃₂×C₄₇	2
#13704	47

subgroup

index

C₄⋊(C₄₇⋊C₃₂)

C₄×Q₃₂×C₄₇

Subgroups — #26137165

Conjugacy classes of subgroups

Complements

Maximal subgroups