Group #26110050 — subgroups

Subgroups — #26110050

Order 12032.

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

normal subgroups only

This group has 422 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	#20243545	C₂	1	1	yes	maximal, characteristic
2	#20243515	C₂	1	4	yes	maximal
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093956	—	2	2	no	—
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
7	#15688993	C₂²	1	2	yes	—
8	#15688885	—	2	8	no	—
9	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
10	C₂×#106×C₄₇	—	2	2	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#384	C₂²	1	2	yes	—
13	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	—	2	2	no	—
16	#13093956	—	4	4	no	—
17	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₄₇	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
23	#106×C₄₇	—	4	4	no	—
24	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₇	—	4	2	no	—
27	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
29	C₂²×Dic₄₇	—	4	2	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
32	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
33	C₄⋊C₄×C₄₇	—	4	4	no	—
34	C₄⋊C₄×C₄₇	—	4	8	no	—
35	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
38	C₄⋊Dic₄₇	—	4	4	no	—
39	C₄⋊Dic₄₇	—	4	8	no	—
40	C₂×C₄₇⋊C₈	—	2	8	no	—
41	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
42	C₂³×C₄₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₄₇	—	8	2	no	—
44	C₂×C₄×C₄₇	—	8	4	no	—
45	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₄₇	#135	1	8	yes	—
52	C₂×Dic₄₇	—	8	2	no	—
53	C₂×Dic₄₇	—	8	4	no	—
54	C₈×C₄₇	#106	1	8	yes	—
55	C₄₇⋊C₈	—	2	8	no	—
56	#8815	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	#384	1	1	yes	characteristic
58	C₂²×C₄₇	#492	1	1	yes	characteristic
59	C₂²×C₄₇	#492	1	1	yes	characteristic
60	C₂²×C₄₇	#495	1	4	yes	—
61	C₄×C₄₇	—	16	2	no	—
62	C₄×C₄₇	C₂×#106	1	2	yes	—
63	C₄×C₄₇	#427	1	4	yes	—
64	C₄×C₄₇	#498	1	2	yes	—
65	C₄×C₄₇	#499	1	4	yes	—
66	Dic₄₇	—	16	2	no	—
67	#1844	—	47	1	no	—
68	#1844	D₄₇	1	1	yes	characteristic
69	#1834	—	47	4	no	—
70	C₂×C₈²	—	47	1	no	—
71	C₂×C₄₇	#1708	1	2	yes	—
72	C₂×C₄₇	#1708	1	2	yes	—
73	C₂×C₄₇	#1844	1	1	yes	characteristic
74	C₂×C₄₇	#1846	1	2	yes	—
75	C₂×#106	—	2	2	no	—
76	C₂×#106	—	94	2	no	—
77	C₂×C₄×C₈	—	47	2	no	—
78	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
79	#384	—	47	2	no	—
80	#384	C₂×D₄₇	1	2	yes	—
81	#340	—	94	8	no	—
82	C₈²	—	47	4	no	—
83	C₄₇	#8815	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	47	4	no	—
85	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	94	4	no	—
89	C₂×C₄⋊C₄	—	94	2	no	—
90	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	188	4	no	—
93	C₄×C₈	—	47	4	no	—
94	C₄×C₈	—	47	4	no	—
95	C₄×C₈	Dic₉₄	1	2	yes	—
96	C₄×C₈	C₄×D₄₇	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	188	2	no	—
99	C₂²×C₄	D₄×D₄₇	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
101	C₂×C₈	—	47	8	no	—
102	C₂×C₈	—	94	8	no	—
103	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
104	C₂×C₈	#10625335	1	4	yes	—
105	C₂×C₈	#10625336	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	188	4	no	—
109	C₄⋊C₄	—	188	8	no	—
110	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
111	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
112	C₄²	Q₈×D₄₇	1	2	yes	—
113	C₂³	#13094020	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	376	2	no	—
117	C₂×C₄	—	376	4	no	—
118	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
119	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
120	C₂×C₄	#13094022	1	1	yes	characteristic
121	C₂×C₄	#13094023	1	2	yes	—
122	C₂×C₄	C₂×#10625336	1	2	yes	—
123	C₂×C₄	C₂×#10625335	1	2	yes	—
124	C₂×C₄	#13094045	1	8	yes	—
125	C₈	—	94	8	no	—
126	C₈	#13093955	1	8	yes	—
127	C₂²	#15688992	1	1	yes	characteristic
128	C₂²	#15689495	1	1	yes	characteristic
129	C₂²	#15689497	1	1	yes	characteristic
130	C₂²	#15689504	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	752	2	no	—
133	C₄	C₂×#13093955	1	2	yes	—
134	C₄	#15689158	1	4	yes	—
135	C₄	#15689519	1	2	yes	—
136	C₄	#15689523	1	4	yes	—
137	C₂	#20242987	1	2	yes	—
138	C₂	#20242990	1	2	yes	—
139	C₂	#20243544	1	1	yes	characteristic
140	C₂	#20243549	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243545

C₂

yes

maximal, characteristic

#20243515

C₂

yes

maximal

C₂×C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#1844

C₂

yes

maximal, characteristic

C₂×#13093956

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

—

#15688885

—

C₈×C₄₇⋊C₈

C₄

yes

complemented

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#106

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#106

yes

—

C₄×C₄₇

#427

yes

—

C₄×C₄₇

#498

yes

—

C₄×C₄₇

#499

yes

—

Dic₄₇

—

#1844

—

#1844

D₄₇

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₄₇

#1708

yes

—

C₂×C₄₇

#1708

yes

—

C₂×C₄₇

#1844

yes

characteristic

C₂×C₄₇

#1846

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#384

—

#384

C₂×D₄₇

yes

—

#340

—

C₈²

—

C₄₇

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#106

—

#106

—

188

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₄×D₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

D₄×D₄₇

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

Dic₄₇⋊C₄

yes

—

104

C₂×C₈

#10625335

yes

—

105

C₂×C₈

#10625336

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

188

—

109

C₄⋊C₄

—

188

—

110

C₄²

C₂×Dic₉₄

yes

characteristic

111

C₄²

C₂×C₄×D₄₇

yes

characteristic

112

C₄²

Q₈×D₄₇

yes

—

113

C₂³

#13094020

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

376

—

117

C₂×C₄

—

376

—

118

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

119

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

120

C₂×C₄

#13094022

yes

characteristic

121

C₂×C₄

#13094023

yes

—

122

C₂×C₄

C₂×#10625336

yes

—

123

C₂×C₄

C₂×#10625335

yes

—

124

C₂×C₄

#13094045

yes

—

125

C₈

—

126

C₈

#13093955

yes

—

127

C₂²

#15688992

yes

characteristic

128

C₂²

#15689495

yes

characteristic

129

C₂²

#15689497

yes

characteristic

130

C₂²

#15689504

yes

—

131

C₄

—

132

C₄

—

752

—

133

C₄

C₂×#13093955

yes

—

134

C₄

#15689158

yes

—

135

C₄

#15689519

yes

—

136

C₄

#15689523

yes

—

137

C₂

#20242987

yes

—

138

C₂

#20242990

yes

—

139

C₂

#20243544

yes

characteristic

140

C₂

#20243549

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#8815
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1844

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
9. C₈×C₄₇⋊C₈	C₄	semidirect
83. C₄₇	#8815	semidirect

H ≅ G/N

type

9. C₈×C₄₇⋊C₈

C₄

semidirect

83. C₄₇

#8815

semidirect

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

subgroup	index
#20243545	2
#20243515	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1844	2
#8815	47

subgroup

index

#20243545

#20243515

C₂×C₈×C₄₇⋊C₈

C₄₇×#1844

#8815

Subgroups — #26110050

Conjugacy classes of subgroups

Complements

Maximal subgroups