Group #26130660 — subgroups

Subgroups — #26130660

Order 12032.

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

normal subgroups only

This group has 188 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	#20245296	C₂	1	1	yes	maximal, characteristic
3	#20242793	C₂	1	2	yes	maximal
4	#20241447	C₂	1	2	yes	maximal
5	C₄₇×#1372	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689419	—	2	2	no	—
8	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
9	#15688949	—	2	1	no	—
10	#15688949	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₄₇⋊C₁₆	—	2	1	no	—
13	#15688898	—	2	1	no	—
14	#15688871	—	2	2	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#335	C₂²	1	2	yes	—
17	#13094024	—	2	2	no	—
18	C₄×Dic₉₄	D₄	1	1	yes	characteristic
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	#13093958	—	2	2	no	—
21	#13093958	—	2	2	no	—
22	#13093958	—	4	2	no	—
23	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₄₇⋊C₈	—	2	1	no	—
25	#13093949	—	4	2	no	—
26	C₈×Dic₄₇	—	2	1	no	—
27	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₉₄	—	4	2	no	—
32	C₂×Dic₉₄	D₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₄⋊Dic₄₇	—	4	2	no	—
39	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
40	Dic₄₇⋊C₄	—	2	1	no	—
41	C₄×Dic₄₇	—	2	1	no	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₁₆×C₄₇	—	2	1	no	—
44	C₁₆×C₄₇	Q₁₆	1	2	yes	—
45	C₄₇⋊C₁₆	—	4	1	no	—
46	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
49	C₂×Dic₄₇	—	2	1	no	—
50	C₂×Dic₄₇	—	2	1	no	—
51	C₂×Dic₄₇	—	8	2	no	—
52	Dic₉₄	—	2	2	no	—
53	Dic₉₄	—	16	2	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	—	8	2	no	—
56	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₄₇	#137	1	1	yes	characteristic
58	C₄₇⋊C₈	—	4	1	no	—
59	#12461	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	#466	1	1	yes	characteristic
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	#462	1	1	yes	characteristic
63	C₄×C₄₇	#473	1	1	yes	characteristic
64	C₄×C₄₇	#510	1	2	yes	—
65	Dic₄₇	—	4	1	no	—
66	Dic₄₇	—	4	1	no	—
67	Dic₄₇	—	16	2	no	—
68	#2252	—	47	1	no	—
69	#2183	—	47	1	no	—
70	#1676	—	47	2	no	—
71	#1372	D₄₇	1	1	yes	characteristic
72	#1365	—	47	2	no	—
73	C₂×C₄₇	#1674	1	1	yes	characteristic
74	C₂×C₄₇	#2219	1	1	yes	characteristic
75	C₂×C₄₇	#2221	1	1	yes	characteristic
76	#500	—	47	2	no	—
77	#477	—	94	2	no	—
78	C₈×Q₈	—	47	1	no	—
79	#366	—	94	1	no	—
80	#366	—	94	1	no	—
81	C₄⋊C₁₆	—	47	1	no	—
82	C₄⋊C₁₆	—	94	1	no	—
83	#346	—	94	1	no	—
84	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
85	#335	C₂×D₄₇	1	2	yes	—
86	#333	—	94	2	no	—
87	C₄₇	#12461	1	1	yes	characteristic, complemented, Sylow
88	C₄⋊Q₈	—	94	2	no	—
89	C₄×Q₈	—	47	1	no	—
90	C₂×C₁₆	—	94	1	no	—
91	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
92	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
93	#107	—	94	2	no	—
94	#107	—	94	2	no	—
95	#107	—	188	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	47	1	no	—
98	C₄⋊C₈	—	94	1	no	—
99	#103	—	188	2	no	—
100	C₄×C₈	—	94	1	no	—
101	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	47	1	no	—
103	C₂×Q₈	—	188	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	94	1	no	—
106	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
107	C₂×C₈	#10625354	1	1	yes	characteristic
108	C₄⋊C₄	—	47	1	no	—
109	C₄⋊C₄	—	94	1	no	—
110	C₄⋊C₄	—	188	2	no	—
111	C₄⋊C₄	—	188	2	no	—
112	C₄²	—	94	1	no	—
113	C₄²	D₄×D₄₇	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	188	1	no	—
116	C₁₆	Dic₁₈₈	1	2	yes	—
117	Q₈	—	94	2	no	—
118	Q₈	—	752	2	no	—
119	C₂×C₄	—	94	1	no	—
120	C₂×C₄	—	94	1	no	—
121	C₂×C₄	—	376	2	no	—
122	C₂×C₄	#13093999	1	1	yes	characteristic
123	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
124	C₂×C₄	#13094047	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	188	1	no	—
128	C₈	C₂×Dic₁₈₈	1	1	yes	characteristic
129	C₈	#13094057	1	1	yes	characteristic
130	C₂²	#15689360	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	188	1	no	—
133	C₄	—	188	1	no	—
134	C₄	—	752	2	no	—
135	C₄	#15689334	1	1	yes	characteristic
136	C₄	#15689399	1	1	yes	characteristic
137	C₄	#15689562	1	2	yes	—
138	C₂	#20242783	1	1	yes	characteristic
139	C₂	#20245497	1	1	yes	characteristic
140	C₂	#20245509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic

#20245296

C₂

yes

maximal, characteristic

#20242793

C₂

yes

maximal

#20241447

C₂

yes

maximal

C₄₇×#1372

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

#15689419

—

C₈×Dic₉₄

C₂²

yes

characteristic

#15688949

—

#15688949

—

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

characteristic

Dic₄₇⋊C₁₆

—

#15688898

—

#15688871

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#335

C₂²

yes

—

#13094024

—

C₄×Dic₉₄

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

#13093958

—

#13093958

—

#13093958

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

characteristic

Dic₄₇⋊C₈

—

#13093949

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

D₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₈

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

Q₁₆

yes

—

C₄₇⋊C₁₆

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₈×C₄₇

#137

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1365

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

C₂×D₄₇

yes

—

#333

—

C₄₇

#12461

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

D₁₈₈

yes

characteristic

C₂×C₁₆

D₁₈₈

yes

—

—

—

188

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

188

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

188

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×D₁₈₈

yes

characteristic

107

C₂×C₈

#10625354

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

188

—

111

C₄⋊C₄

—

188

—

112

C₄²

—

113

C₄²

D₄×D₄₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

188

—

116

C₁₆

Dic₁₈₈

yes

—

117

Q₈

—

118

Q₈

—

752

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

376

—

122

C₂×C₄

#13093999

yes

characteristic

123

C₂×C₄

D₈×D₄₇

yes

characteristic

124

C₂×C₄

#13094047

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

188

—

128

C₈

C₂×Dic₁₈₈

yes

characteristic

129

C₈

#13094057

yes

characteristic

130

C₂²

#15689360

yes

characteristic

131

C₄

—

132

C₄

—

188

—

133

C₄

—

188

—

134

C₄

—

752

—

135

C₄

#15689334

yes

characteristic

136

C₄

#15689399

yes

characteristic

137

C₄

#15689562

yes

—

138

C₂

#20242783

yes

characteristic

139

C₂

#20245497

yes

characteristic

140

C₂

#20245509

yes

characteristic

	subgroup	G/H
center	C₂²	#15689360
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1372	C₂
socle, prime radical	C₂²×C₄₇	#466
hypercenter	#1372	D₄₇
hypocenter	C₄₇	#12461
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#15689360

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1372

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
87. C₄₇	#12461	semidirect

H ≅ G/N

type

87. C₄₇

#12461

semidirect

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

subgroup	index
#20245663	2
#20245296	2
#20242793	2
#20241447	2
C₄₇×#1372	2
#12461	47

subgroup

index

C₄₇×#1372

Subgroups — #26130660

Conjugacy classes of subgroups

Complements

Maximal subgroups