Group #26130932 — subgroups

Subgroups — #26130932

Order 12032.

306 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245638	C₂	1	1	yes	maximal, characteristic
2	#20245613	C₂	1	2	yes	maximal, complemented
3	#20242981	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₅₂	—	2	2	no	—
7	#15689500	C₂²	1	2	yes	—
8	#15689497	—	2	2	no	—
9	#15689471	—	2	2	no	—
10	#15688923	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄₇×#500	C₂²	1	2	yes	—
14	C₄₇×#366	—	2	2	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#10625336	—	4	4	no	—
17	C₂×D₃₇₆	—	2	2	no	—
18	C₂×D₃₇₆	—	2	2	no	—
19	#13094021	—	2	2	no	—
20	D₇₅₂	—	4	4	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
22	#13093944	—	4	4	no	—
23	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₄₇	—	2	2	no	—
26	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
27	#107×C₄₇	—	2	2	no	—
28	#107×C₄₇	—	2	2	no	—
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₄₇	—	4	2	no	—
31	C₂×D₁₈₈	—	4	2	no	—
32	C₂×D₁₈₈	—	4	2	no	—
33	C₂×D₁₈₈	—	4	2	no	—
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₄₇	—	4	2	no	—
37	C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
39	#10625336	—	8	8	no	—
40	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
41	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
42	D₃₇₆	—	4	4	no	—
43	D₃₇₆	—	4	4	no	—
44	C₁₆×C₄₇	SD₁₆	1	4	yes	complemented
45	C₄₇⋊C₁₆	—	2	2	no	—
46	C₂²×D₄₇	—	8	2	no	—
47	Q₈×C₄₇	—	8	4	no	—
48	C₂×C₄×C₄₇	—	8	2	no	—
49	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
52	D₁₈₈	—	8	2	no	—
53	D₁₈₈	—	8	2	no	—
54	D₁₈₈	—	8	4	no	—
55	D₁₈₈	—	8	4	no	—
56	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
57	C₈×C₄₇	#135	1	2	yes	—
58	C₄₇⋊C₈	—	2	2	no	—
59	C₄₇⋊C₈	D₁₆	1	4	yes	complemented
60	#12503	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	#493	1	1	yes	characteristic
62	C₂×D₄₇	—	16	2	no	—
63	C₂×D₄₇	—	16	2	no	—
64	C₂×D₄₇	—	16	2	no	—
65	C₄×C₄₇	—	16	2	no	—
66	C₄×C₄₇	#492	1	1	yes	characteristic
67	C₄×C₄₇	#495	1	1	yes	characteristic
68	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
69	C₄×C₄₇	#508	1	2	yes	—
70	#2252	D₄₇	1	1	yes	characteristic, complemented
71	#2246	—	47	1	no	—
72	#2240	—	47	2	no	—
73	#1706	—	47	2	no	—
74	C₈×C₁₆	—	47	1	no	—
75	C₂×C₄₇	#1706	1	1	yes	characteristic
76	C₂×C₄₇	#2246	1	1	yes	characteristic
77	C₂×C₄₇	#2249	1	1	yes	characteristic
78	D₄₇	—	32	2	no	—
79	C₂×D₁₆	—	94	2	no	—
80	#500	C₂×D₄₇	1	2	yes	—
81	#493	—	47	2	no	—
82	#492	—	94	2	no	—
83	#486	—	94	2	no	—
84	#366	—	2	2	no	—
85	#357	—	94	4	no	—
86	C₄×C₁₆	—	47	1	no	—
87	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
88	C₈²	—	47	1	no	—
89	C₄₇	#12503	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	188	4	no	—
91	C₂×D₈	—	94	2	no	—
92	C₂×D₈	—	94	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	#127	—	94	2	no	—
95	D₁₆	—	188	4	no	—
96	C₂×C₁₆	—	47	2	no	—
97	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#102	—	188	4	no	—
101	C₄×C₈	—	47	1	no	—
102	C₄×C₈	—	47	1	no	—
103	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×D₄	—	188	2	no	—
106	C₂×D₄	—	188	2	no	—
107	C₂×D₄	—	188	2	no	—
108	SD₁₆	—	376	8	no	—
109	D₈	—	188	4	no	—
110	D₈	—	188	4	no	—
111	C₂×C₈	—	47	2	no	—
112	C₂×C₈	—	47	2	no	—
113	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	D₄×D₄₇	1	1	yes	characteristic
118	C₁₆	—	94	2	no	—
119	C₁₆	#10625336	1	4	yes	complemented
120	C₂³	—	376	2	no	—
121	Q₈	—	8	4	no	—
122	D₄	—	376	2	no	—
123	D₄	—	376	2	no	—
124	D₄	—	376	4	no	—
125	D₄	—	376	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	#13094020	1	1	yes	characteristic
128	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
129	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
130	C₈	—	47	4	no	—
131	C₈	—	94	2	no	—
132	C₈	C₂×#10625336	1	2	yes	—
133	C₈	#13094047	1	2	yes	—
134	C₂²	—	752	2	no	—
135	C₂²	—	752	2	no	—
136	C₂²	—	752	2	no	—
137	C₂²	#15689498	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	#15689497	1	1	yes	characteristic
140	C₄	#15689508	1	1	yes	characteristic
141	C₄	D₁₆×D₄₇	1	2	yes	—
142	C₄	#15689556	1	2	yes	—
143	C₂	—	1504	2	no	—
144	C₂	#20242981	1	1	yes	characteristic
145	C₂	#20245636	1	1	yes	characteristic
146	C₂	#20245650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245638

C₂

yes

maximal, characteristic

#20245613

C₂

yes

maximal, complemented

#20242981

C₂

yes

maximal, complemented

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#2252

C₂

yes

maximal, characteristic, complemented

C₂×D₇₅₂

—

#15689500

C₂²

yes

—

#15689497

—

#15689471

—

#15688923

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄₇×#500

C₂²

yes

—

C₄₇×#366

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#10625336

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

—

D₇₅₂

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093944

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

#10625336

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₂×C₄₇⋊C₈

D₈

yes

—

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₁₆

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

#135

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

D₁₆

yes

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

#508

yes

—

#2252

D₄₇

yes

characteristic, complemented

#2246

—

#2240

—

#1706

—

C₈×C₁₆

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×D₁₆

—

#500

C₂×D₄₇

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12503

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

188

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

188

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#107

—

#107

—

100

#102

—

188

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₄₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×D₄

—

188

—

106

C₂×D₄

—

188

—

107

C₂×D₄

—

188

—

108

SD₁₆

—

376

—

109

D₈

—

188

—

110

D₈

—

188

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₄₇

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

D₄×D₄₇

yes

characteristic

118

C₁₆

—

119

C₁₆

#10625336

yes

complemented

120

C₂³

—

376

—

121

Q₈

—

122

D₄

—

376

—

123

D₄

—

376

—

124

D₄

—

376

—

125

D₄

—

376

—

126

C₂×C₄

—

127

C₂×C₄

#13094020

yes

characteristic

128

C₂×C₄

D₈×D₄₇

yes

characteristic

129

C₂×C₄

D₈×D₄₇

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

C₂×#10625336

yes

—

133

C₈

#13094047

yes

—

134

C₂²

—

752

—

135

C₂²

—

752

—

136

C₂²

—

752

—

137

C₂²

#15689498

yes

characteristic

138

C₄

—

139

C₄

#15689497

yes

characteristic

140

C₄

#15689508

yes

characteristic

141

C₄

D₁₆×D₄₇

yes

—

142

C₄

#15689556

yes

—

143

C₂

—

1504

—

144

C₂

#20242981

yes

characteristic

145

C₂

#20245636

yes

characteristic

146

C₂

#20245650

yes

characteristic

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2252	C₂
socle	C₂²×C₄₇	#493
prime radical	#20245638	C₂
hypercenter	#2252	D₄₇
hypocenter	C₄₇	#12503
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2252

C₂

C₂²×C₄₇

C₂

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
2. #20245613	C₂	semidirect
3. #20242981	C₂	semidirect
4. C₁₆×C₄₇⋊C₈	C₂	semidirect
5. C₄₇×#2252	C₂	semidirect
44. C₁₆×C₄₇	SD₁₆	semidirect
59. C₄₇⋊C₈	D₁₆	semidirect
70. #2252	D₄₇	semidirect
89. C₄₇	#12503	semidirect
119. C₁₆	#10625336	semidirect

H ≅ G/N

type

2. #20245613

C₂

semidirect

3. #20242981

C₂

semidirect

4. C₁₆×C₄₇⋊C₈

C₂

semidirect

5. C₄₇×#2252

C₂

semidirect

44. C₁₆×C₄₇

SD₁₆

semidirect

59. C₄₇⋊C₈

D₁₆

semidirect

70. #2252

D₄₇

semidirect

89. C₄₇

#12503

semidirect

119. C₁₆

#10625336

semidirect

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

subgroup	index
#20245638	2
#20245613	2
#20242981	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2252	2
#12503	47

subgroup

index

#20245638

#20245613

#20242981

C₁₆×C₄₇⋊C₈

C₄₇×#2252

#12503

Subgroups — #26130932

Conjugacy classes of subgroups

Complements

Maximal subgroups