Group #26132698 — subgroups

Subgroups — #26132698

Order 12032.

342 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 203 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	4	yes	maximal
2	#20245144	C₂	1	2	yes	maximal
3	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	C₂²	1	4	yes	—
5	#15688944	—	2	8	no	—
6	C₄⋊(C₄₇⋊C₁₆)	—	2	4	no	—
7	C₂×#97×C₄₇	C₄	1	2	yes	—
8	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#376	C₄	1	1	yes	characteristic
12	#13093964	—	2	8	no	—
13	C₂×C₄₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
19	#97×C₄₇	—	2	4	no	—
20	#97×C₄₇	C₈	1	4	yes	—
21	C₄×C₈×C₄₇	D₄	1	2	yes	—
22	C₄×C₈×C₄₇	D₄	1	4	yes	—
23	C₄×C₈×C₄₇	Q₈	1	4	yes	—
24	C₄×C₈×C₄₇	Q₈	1	2	yes	—
25	#95×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
34	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
40	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₄²×C₄₇	C₂×C₈	1	2	yes	—
42	C₄₇⋊C₁₆	—	4	4	no	—
43	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
52	C₈×C₄₇	—	2	2	no	—
53	C₈×C₄₇	C₄⋊C₈	1	8	yes	—
54	C₈×C₄₇	#127	1	1	yes	characteristic
55	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
56	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
57	#12901	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂²×C₄₇	#423	1	2	yes	—
61	C₄×C₄₇	—	2	4	no	—
62	C₄×C₄₇	—	4	4	no	—
63	C₄×C₄₇	#384	1	1	yes	characteristic
64	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
65	C₄×C₄₇	#423	1	1	yes	characteristic
66	C₂×#364	—	47	4	no	—
67	#2150	—	47	2	no	—
68	#1726	D₄₇	1	1	yes	characteristic
69	C₂×C₄₇	—	2	2	no	—
70	C₂×C₄₇	#1842	1	1	yes	characteristic
71	C₂×C₄₇	#2153	1	2	yes	—
72	C₂×M₃₂	—	47	4	no	—
73	C₂×#97	Dic₄₇	1	2	yes	—
74	C₂×#97	Dic₄₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
76	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
77	#376	Dic₄₇	1	1	yes	characteristic
78	#364	—	94	8	no	—
79	C₄⋊C₁₆	—	94	4	no	—
80	C₄₇	#12901	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
83	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
85	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
86	M₃₂	—	94	8	no	—
87	C₂×C₁₆	—	94	4	no	—
88	#97	—	2	4	no	—
89	#97	C₄₇⋊C₈	1	4	yes	—
90	C₄×C₈	Dic₉₄	1	4	yes	—
91	C₄×C₈	Dic₉₄	1	2	yes	—
92	C₄×C₈	D₁₈₈	1	4	yes	—
93	C₄×C₈	D₁₈₈	1	2	yes	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#10625329	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
100	C₂×C₈	#10625329	1	4	yes	—
101	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
102	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
103	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
105	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
106	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
107	C₄²	—	2	2	no	—
108	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
109	C₄²	C₄⋊Dic₄₇	1	4	yes	—
110	C₄²	C₄⋊Dic₄₇	1	2	yes	—
111	C₁₆	—	188	4	no	—
112	C₂³	C₂×#10625329	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
120	C₂×C₄	C₂×#10625329	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
123	C₈	#13094021	1	1	yes	characteristic
124	C₈	#13094024	1	2	yes	—
125	C₈	#13094024	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
128	C₂²	#15689147	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#15688993	1	1	yes	characteristic
132	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
133	C₄	#15689147	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#20243540	1	1	yes	characteristic
136	C₂	#20245154	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688944

C₂

yes

maximal

#20245144

C₂

yes

maximal

C₄₇×#1726

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

#15688944

—

C₄⋊(C₄₇⋊C₁₆)

—

C₂×#97×C₄₇

C₄

yes

—

C₂×#97×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#376

C₄

yes

characteristic

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#97×C₄₇

—

#97×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

#127

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

—

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

#12901

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#423

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#384

yes

characteristic

C₄×C₄₇

C₂×C₄⋊C₈

yes

—

C₄×C₄₇

yes

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1842

yes

characteristic

C₂×C₄₇

#2153

yes

—

C₂×M₃₂

—

C₂×#97

Dic₄₇

yes

—

C₂×#97

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#376

Dic₄₇

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄₇

#12901

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

C₄₇⋊C₈

yes

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

D₁₈₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#10625329

yes

—

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

100

C₂×C₈

#10625329

yes

—

101

C₂×C₈

C₄⋊Dic₄₇

yes

—

102

C₂×C₈

C₄⋊Dic₄₇

yes

—

103

C₂×C₈

C₂×Dic₉₄

yes

characteristic

104

C₂×C₈

C₂×Dic₉₄

yes

—

105

C₂×C₈

C₂×D₁₈₈

yes

—

106

C₂×C₈

C₂×D₁₈₈

yes

characteristic

107

C₄²

—

108

C₄²

C₂×C₄₇⋊C₈

yes

—

109

C₄²

C₄⋊Dic₄₇

yes

—

110

C₄²

C₄⋊Dic₄₇

yes

—

111

C₁₆

—

188

—

112

C₂³

C₂×#10625329

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

117

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

119

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

120

C₂×C₄

C₂×#10625329

yes

—

121

C₈

—

122

C₈

C₄⋊(C₄₇⋊C₈)

yes

—

123

C₈

#13094021

yes

characteristic

124

C₈

#13094024

yes

—

125

C₈

#13094024

yes

characteristic

126

C₂²

—

127

C₂²

C₂×C₄⋊(C₄₇⋊C₈)

yes

characteristic

128

C₂²

#15689147

yes

—

129

C₄

—

130

C₄

—

131

C₄

#15688993

yes

characteristic

132

C₄

C₂×C₄⋊(C₄₇⋊C₈)

yes

—

133

C₄

#15689147

yes

characteristic

134

C₂

—

135

C₂

#20243540

yes

characteristic

136

C₂

#20245154

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1726	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂³×C₄₇	C₂×M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#12901
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1726

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₈

prime radical

C₂³×C₄₇

C₂×M₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
80. C₄₇	#12901	semidirect

H ≅ G/N

type

80. C₄₇

#12901

semidirect

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

subgroup	index
C₂×#15688944	2
#20245144	2
C₄₇×#1726	2
#12901	47

subgroup

index

C₂×#15688944

#20245144

C₄₇×#1726

#12901

Subgroups — #26132698

Conjugacy classes of subgroups

Complements

Maximal subgroups