Group #26127874 — subgroups

Subgroups — #26127874

Order 12032.

220 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 143 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	C₂×#15688944	C₂	1	2	yes	maximal
2	#20241464	C₂	1	4	yes	maximal, complemented
3	C₄×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	2	no	—
5	#15688944	—	2	2	no	—
6	#15688944	C₂²	1	4	yes	—
7	C₂×M₃₂×C₄₇	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#346	C₄	1	2	yes	—
11	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
12	#13093964	—	2	4	no	—
13	#13093964	—	4	2	no	—
14	C₂×C₄₇⋊C₁₆	—	4	2	no	—
15	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
18	M₃₂×C₄₇	—	2	2	no	—
19	M₃₂×C₄₇	C₈	1	4	yes	—
20	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₄₇	D₈	1	2	yes	—
33	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
34	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₄₇	—	2	1	no	—
37	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₄₇	—	2	2	no	—
40	C₁₆×C₄₇	C₂×C₈	1	4	yes	—
41	C₄₇⋊C₁₆	—	4	4	no	—
42	C₂³×C₄₇	#108	1	1	yes	characteristic
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₄₇	#107	1	2	yes	—
48	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
51	C₈×C₄₇	#107	1	2	yes	—
52	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
56	#12011	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	—	4	1	no	—
58	C₂²×C₄₇	#335	1	2	yes	—
59	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	#335	1	2	yes	—
63	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
65	C₂×#364	—	47	2	no	—
66	C₄×M₃₂	D₄₇	1	1	yes	characteristic
67	#1373	—	47	4	no	—
68	C₂×C₄₇	—	2	1	no	—
69	C₂×C₄₇	—	2	2	no	—
70	C₂×C₄₇	C₂×#335	1	1	yes	characteristic
71	C₂×M₃₂	—	94	2	no	—
72	C₂×M₃₂	Dic₄₇	1	1	yes	characteristic
73	C₂×M₃₂	Dic₄₇	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
75	#364	—	47	4	no	—
76	#364	—	94	2	no	—
77	#346	Dic₄₇	1	2	yes	—
78	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
79	C₄₇	#12011	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
82	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
83	M₃₂	—	2	2	no	—
84	M₃₂	—	94	4	no	—
85	M₃₂	—	188	2	no	—
86	M₃₂	C₄₇⋊C₈	1	4	yes	—
87	C₂×C₁₆	—	188	2	no	—
88	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
89	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
90	C₄×C₈	Dic₉₄	1	1	yes	characteristic
91	C₄×C₈	D₁₈₈	1	1	yes	characteristic
92	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₈	D₃₇₆	1	2	yes	—
97	C₂×C₈	Dic₁₈₈	1	2	yes	—
98	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
99	C₂×C₈	#10625329	1	2	yes	—
100	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
101	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
102	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
103	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
106	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	188	4	no	—
109	C₁₆	C₂×C₄₇⋊C₈	1	4	yes	—
110	C₂³	#13093960	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
115	C₂×C₄	#13093958	1	2	yes	—
116	C₂×C₄	#13093960	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
118	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
119	C₈	#13093958	1	2	yes	—
120	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
121	C₈	C₂×#10625329	1	1	yes	characteristic
122	C₈	C₂×D₃₇₆	1	1	yes	characteristic
123	C₈	C₂×Dic₁₈₈	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	#15688876	1	2	yes	—
126	C₂²	C₂×#13093960	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	#15688876	1	2	yes	—
130	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
131	C₄	C₂×#13093958	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×#15688876	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688944

C₂

yes

maximal

#20241464

C₂

yes

maximal, complemented

C₄×M₃₂×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093964

—

#15688944

—

#15688944

C₂²

yes

—

C₂×M₃₂×C₄₇

C₄

yes

characteristic

C₂×M₃₂×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

C₄

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

C₈

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#108

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

#108

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

#107

yes

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

C₂×D₈

yes

characteristic

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

#12011

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#335

yes

—

C₂²×C₄₇

C₂×#108

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#335

yes

—

C₄×C₄₇

C₂×C₄⋊C₈

yes

C₄×C₄₇

yes

—

C₄×M₃₂

D₄₇

yes

characteristic

#1373

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#335

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

Dic₄₇

yes

characteristic

C₂×M₃₂

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

Dic₄₇

yes

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₄₇

#12011

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₈₈

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

Dic₉₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

188

—

M₃₂

C₄₇⋊C₈

yes

—

C₂×C₁₆

—

188

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

D₃₇₆

yes

—

C₂×C₈

Dic₁₈₈

yes

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

#10625329

yes

—

100

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

101

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

102

C₂×C₈

C₂×D₁₈₈

yes

characteristic

103

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

104

C₄²

—

105

C₄²

C₄⋊Dic₄₇

yes

characteristic

106

C₄²

C₂×Dic₉₄

yes

characteristic

107

C₁₆

—

108

C₁₆

—

188

—

109

C₁₆

C₂×C₄₇⋊C₈

yes

—

110

C₂³

#13093960

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

115

C₂×C₄

#13093958

yes

—

116

C₂×C₄

#13093960

yes

characteristic

117

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

118

C₈

C₄⋊(C₄₇⋊C₈)

yes

—

119

C₈

#13093958

yes

—

120

C₈

C₂²×C₄₇⋊C₈

yes

characteristic

121

C₈

C₂×#10625329

yes

characteristic

122

C₈

C₂×D₃₇₆

yes

characteristic

123

C₈

C₂×Dic₁₈₈

yes

characteristic

124

C₂²

—

125

C₂²

#15688876

yes

—

126

C₂²

C₂×#13093960

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

#15688876

yes

—

130

C₄

C₂×C₄⋊(C₄₇⋊C₈)

yes

characteristic

131

C₄

C₂×#13093958

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

C₂×#15688876

yes

characteristic

	subgroup	G/H
center	C₈	C₂×D₃₇₆
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄×M₃₂×C₄₇	C₂
socle	C₂×C₄₇	C₂×#335
prime radical	C₂³×C₄₇	#108
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₄×M₃₂	D₄₇
hypocenter	C₄₇	#12011
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₈

C₂×D₃₇₆

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄×M₃₂×C₄₇

C₂

C₂×C₄₇

C₂³×C₄₇

C₄×C₈×C₄₇

C₂³

hypercenter

C₄×M₃₂

D₄₇

hypocenter

C₄₇

#12011

LCS

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #20241464	C₂	semidirect
79. C₄₇	#12011	semidirect

H ≅ G/N

type

2. #20241464

C₂

semidirect

79. C₄₇

#12011

semidirect

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

subgroup	index
C₂×#15688944	2
#20241464	2
C₄×M₃₂×C₄₇	2
#12011	47

subgroup

index

C₂×#15688944

#20241464

C₄×M₃₂×C₄₇

#12011

Subgroups — #26127874

Conjugacy classes of subgroups

Complements

Maximal subgroups