Group #26130945 — subgroups

Subgroups — #26130945

Order 12032.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20245670	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245611	C₂	1	2	yes	maximal, complemented
3	#20242977	C₂	1	2	yes	maximal, complemented
4	#20241304	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2246	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689495	—	2	2	no	—
8	#15689469	—	2	2	no	—
9	#15688951	—	2	2	no	—
10	#15688924	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₄₇	—	2	2	no	—
14	C₄₇×#493	C₂²	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#10625335	—	4	4	no	—
17	#13094024	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
19	#13093958	—	2	2	no	—
20	#13093958	—	2	2	no	—
21	#13093945	—	4	4	no	—
22	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₂×D₈×C₄₇	—	2	2	no	—
24	C₂×D₈×C₄₇	—	2	2	no	—
25	#127×C₄₇	—	2	2	no	—
26	D₁₆×C₄₇	—	4	4	no	—
27	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₄₇	—	4	2	no	—
30	C₂×D₄×C₄₇	—	4	2	no	—
31	C₂×D₄×C₄₇	—	4	2	no	—
32	C₂×Dic₉₄	—	4	2	no	—
33	D₈×C₄₇	—	4	4	no	—
34	D₈×C₄₇	—	4	4	no	—
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	#10625335	—	8	8	no	—
39	C₄⋊Dic₄₇	—	4	2	no	—
40	C₄⋊Dic₄₇	—	4	2	no	—
41	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
42	C₁₆×C₄₇	SD₁₆	1	4	yes	complemented
43	C₄₇⋊C₁₆	—	2	2	no	—
44	C₂³×C₄₇	—	8	2	no	—
45	D₄×C₄₇	—	8	2	no	—
46	D₄×C₄₇	—	8	2	no	—
47	D₄×C₄₇	—	8	4	no	—
48	D₄×C₄₇	—	8	4	no	—
49	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
51	C₂×Dic₄₇	—	8	2	no	—
52	Dic₉₄	—	8	4	no	—
53	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
54	C₈×C₄₇	#135	1	2	yes	—
55	C₄₇⋊C₈	—	2	4	no	—
56	#12506	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	—	16	2	no	—
58	C₂²×C₄₇	—	16	2	no	—
59	C₂²×C₄₇	—	16	2	no	—
60	C₂²×C₄₇	#493	1	1	yes	characteristic
61	C₄×C₄₇	#492	1	1	yes	characteristic
62	C₄×C₄₇	#495	1	1	yes	characteristic
63	C₄×C₄₇	#509	1	4	yes	—
64	Dic₄₇	—	16	2	no	—
65	#2254	—	47	1	no	—
66	#2246	D₄₇	1	1	yes	characteristic
67	#2240	—	47	2	no	—
68	#1706	—	47	2	no	—
69	#1311	—	47	1	no	—
70	C₂×C₄₇	—	32	2	no	—
71	C₂×C₄₇	#1706	1	1	yes	characteristic
72	C₂×C₄₇	#2250	1	2	yes	—
73	C₂×D₁₆	—	2	2	no	—
74	#500	—	47	2	no	—
75	#493	C₂×D₄₇	1	2	yes	—
76	#492	—	94	2	no	—
77	#486	—	94	2	no	—
78	#367	—	94	2	no	—
79	#357	—	94	4	no	—
80	C₄×C₁₆	—	47	1	no	—
81	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
82	C₈²	—	47	1	no	—
83	C₄₇	#12506	1	1	yes	characteristic, complemented, Sylow
84	C₂×SD₁₆	—	188	4	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₄⋊Q₈	—	94	2	no	—
88	#127	—	2	2	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	47	2	no	—
91	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
92	#107	—	94	2	no	—
93	#107	—	94	2	no	—
94	#102	—	188	4	no	—
95	C₄×C₈	—	47	2	no	—
96	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
97	C₂×Q₈	—	188	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	2	no	—
101	SD₁₆	—	376	8	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	47	4	no	—
105	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
107	C₄⋊C₄	—	188	2	no	—
108	C₄⋊C₄	—	188	2	no	—
109	C₄²	D₄×D₄₇	1	1	yes	characteristic
110	C₁₆	—	94	2	no	—
111	C₁₆	#10625335	1	4	yes	complemented
112	C₂³	—	8	2	no	—
113	Q₈	—	376	4	no	—
114	D₄	—	8	2	no	—
115	D₄	—	8	2	no	—
116	D₄	—	8	4	no	—
117	D₄	—	8	4	no	—
118	C₂×C₄	—	376	2	no	—
119	C₂×C₄	#13094020	1	1	yes	characteristic
120	C₂×C₄	D₈×D₄₇	1	2	yes	—
121	C₈	—	94	4	no	—
122	C₈	C₂×#10625335	1	2	yes	—
123	C₈	#13094043	1	2	yes	—
124	C₂²	—	16	2	no	—
125	C₂²	—	16	2	no	—
126	C₂²	—	16	2	no	—
127	C₂²	#15689498	1	1	yes	characteristic
128	C₄	—	752	2	no	—
129	C₄	#15689495	1	1	yes	characteristic
130	C₄	#15689506	1	1	yes	characteristic
131	C₄	#15689557	1	4	yes	—
132	C₂	—	32	2	no	—
133	C₂	#20242977	1	1	yes	characteristic
134	C₂	#20245651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245670

C₂

yes

maximal, characteristic, complemented

#20245611

C₂

yes

maximal, complemented

#20242977

C₂

yes

maximal, complemented

#20241304

C₂

yes

maximal, characteristic, complemented

C₄₇×#2246

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×D₁₆×C₄₇

—

C₄₇×#493

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#10625335

—

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

#13093945

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

#127×C₄₇

—

D₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₁₆×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

#135

yes

—

C₄₇⋊C₈

—

#12506

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2254

—

#2246

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1706

yes

characteristic

C₂×C₄₇

#2250

yes

—

C₂×D₁₆

—

#500

—

#493

C₂×D₄₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12506

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

188

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#107

—

#107

—

#102

—

188

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

SD₁₆

—

376

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₄₇

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

107

C₄⋊C₄

—

188

—

108

C₄⋊C₄

—

188

—

109

C₄²

D₄×D₄₇

yes

characteristic

110

C₁₆

—

111

C₁₆

#10625335

yes

complemented

112

C₂³

—

113

Q₈

—

376

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

376

—

119

C₂×C₄

#13094020

yes

characteristic

120

C₂×C₄

D₈×D₄₇

yes

—

121

C₈

—

122

C₈

C₂×#10625335

yes

—

123

C₈

#13094043

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#15689498

yes

characteristic

128

C₄

—

752

—

129

C₄

#15689495

yes

characteristic

130

C₄

#15689506

yes

characteristic

131

C₄

#15689557

yes

—

132

C₂

—

133

C₂

#20242977

yes

characteristic

134

C₂

#20245651

yes

—

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting, prime radical	C₄₇×#2246	C₂
socle	C₂²×C₄₇	#493
hypercenter	#2246	D₄₇
hypocenter	C₄₇	#12506
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting, prime radical

C₄₇×#2246

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #20245670	C₂	semidirect
2. #20245611	C₂	semidirect
3. #20242977	C₂	semidirect
4. #20241304	C₂	semidirect
42. C₁₆×C₄₇	SD₁₆	semidirect
83. C₄₇	#12506	semidirect
111. C₁₆	#10625335	semidirect

H ≅ G/N

type

1. #20245670

C₂

semidirect

2. #20245611

C₂

semidirect

3. #20242977

C₂

semidirect

4. #20241304

C₂

semidirect

42. C₁₆×C₄₇

SD₁₆

semidirect

83. C₄₇

#12506

semidirect

111. C₁₆

#10625335

semidirect

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

subgroup	index
#20245670	2
#20245611	2
#20242977	2
#20241304	2
C₄₇×#2246	2
#12506	47

subgroup

index

C₄₇×#2246

Subgroups — #26130945

Conjugacy classes of subgroups

Complements

Maximal subgroups