Group #26130949 — subgroups

Subgroups — #26130949

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	#20245638	C₂	1	1	yes	maximal, characteristic
2	#20245613	C₂	1	2	yes	maximal, complemented
3	#20242981	C₂	1	2	yes	maximal, complemented
4	#20241304	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2254	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×#367	—	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	2	yes	—
24	C₄⋊Q₈×C₄₇	—	2	2	no	—
25	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
26	#107×C₄₇	—	2	2	no	—
27	#107×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₄₇	—	4	2	no	—
30	C₂×D₁₈₈	—	4	2	no	—
31	C₂×D₁₈₈	—	4	2	no	—
32	C₂×D₁₈₈	—	4	2	no	—
33	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄⋊C₄×C₄₇	—	4	2	no	—
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	#10625336	—	8	8	no	—
39	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
40	D₃₇₆	—	4	4	no	—
41	D₃₇₆	—	4	4	no	—
42	C₁₆×C₄₇	SD₁₆	1	4	yes	complemented
43	C₄₇⋊C₁₆	—	2	2	no	—
44	C₂²×D₄₇	—	8	2	no	—
45	Q₈×C₄₇	—	8	4	no	—
46	C₂×C₄×C₄₇	—	8	2	no	—
47	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
49	D₁₈₈	—	8	2	no	—
50	D₁₈₈	—	8	2	no	—
51	D₁₈₈	—	8	4	no	—
52	D₁₈₈	—	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	#493	1	1	yes	characteristic
58	C₂×D₄₇	—	16	2	no	—
59	C₂×D₄₇	—	16	2	no	—
60	C₂×D₄₇	—	16	2	no	—
61	C₄×C₄₇	—	16	2	no	—
62	C₄×C₄₇	#492	1	1	yes	characteristic
63	C₄×C₄₇	#495	1	1	yes	characteristic
64	C₄×C₄₇	#509	1	4	yes	—
65	#2254	D₄₇	1	1	yes	characteristic, complemented
66	#2246	—	47	1	no	—
67	#2240	—	47	2	no	—
68	#1706	—	47	2	no	—
69	#1311	—	47	1	no	—
70	C₂×C₄₇	#1706	1	1	yes	characteristic
71	C₂×C₄₇	#2250	1	2	yes	—
72	D₄₇	—	32	2	no	—
73	C₂×D₁₆	—	94	2	no	—
74	#500	C₂×D₄₇	1	2	yes	—
75	#493	—	47	2	no	—
76	#492	—	94	2	no	—
77	#486	—	94	2	no	—
78	#367	—	2	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	—	94	2	no	—
86	C₂×D₈	—	94	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#127	—	94	2	no	—
89	D₁₆	—	188	4	no	—
90	C₂×C₁₆	—	47	2	no	—
91	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	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	—	4	2	no	—
98	C₂×D₄	—	188	2	no	—
99	C₂×D₄	—	188	2	no	—
100	C₂×D₄	—	188	2	no	—
101	SD₁₆	—	376	8	no	—
102	D₈	—	188	4	no	—
103	D₈	—	188	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	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₄₇	1	1	yes	characteristic
110	C₁₆	—	94	2	no	—
111	C₁₆	#10625336	1	4	yes	complemented
112	C₂³	—	376	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	376	2	no	—
115	D₄	—	376	2	no	—
116	D₄	—	376	4	no	—
117	D₄	—	376	4	no	—
118	C₂×C₄	—	8	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×#10625336	1	2	yes	—
123	C₈	#13094047	1	2	yes	—
124	C₂²	—	752	2	no	—
125	C₂²	—	752	2	no	—
126	C₂²	—	752	2	no	—
127	C₂²	#15689498	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#15689497	1	1	yes	characteristic
130	C₄	#15689508	1	1	yes	characteristic
131	C₄	#15689560	1	4	yes	—
132	C₂	—	1504	2	no	—
133	C₂	#20242981	1	1	yes	characteristic
134	C₂	#20245653	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245638

C₂

yes

maximal, characteristic

#20245613

C₂

yes

maximal, complemented

#20242981

C₂

yes

maximal, complemented

#20241304

C₂

yes

maximal, characteristic, complemented

C₄₇×#2254

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×#367

—

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

—

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

—

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

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

#135

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#2254

D₄₇

yes

characteristic, complemented

—

—

—

—

C₂×C₄₇

#1706

yes

characteristic

C₂×C₄₇

#2250

yes

—

D₄₇

—

C₂×D₁₆

—

#500

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

—

188

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

#107

—

#107

—

#102

—

188

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

100

C₂×D₄

—

188

—

101

SD₁₆

—

376

—

102

D₈

—

188

—

103

D₈

—

188

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₄₇

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₄₇

yes

characteristic

110

C₁₆

—

111

C₁₆

#10625336

yes

complemented

112

C₂³

—

376

—

113

Q₈

—

114

D₄

—

376

—

115

D₄

—

376

—

116

D₄

—

376

—

117

D₄

—

376

—

118

C₂×C₄

—

119

C₂×C₄

#13094020

yes

characteristic

120

C₂×C₄

D₈×D₄₇

yes

—

121

C₈

—

122

C₈

C₂×#10625336

yes

—

123

C₈

#13094047

yes

—

124

C₂²

—

752

—

125

C₂²

—

752

—

126

C₂²

—

752

—

127

C₂²

#15689498

yes

characteristic

128

C₄

—

129

C₄

#15689497

yes

characteristic

130

C₄

#15689508

yes

characteristic

131

C₄

#15689560

yes

—

132

C₂

—

1504

—

133

C₂

#20242981

yes

characteristic

134

C₂

#20245653

yes

—

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2254	C₂
socle	C₂²×C₄₇	#493
prime radical	#20245638	C₂
hypercenter	#2254	D₄₇
hypocenter	C₄₇	#12506
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#15689498

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2254

C₂

C₂²×C₄₇

C₂

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #20245613	C₂	semidirect
3. #20242981	C₂	semidirect
4. #20241304	C₂	semidirect
5. C₄₇×#2254	C₂	semidirect
42. C₁₆×C₄₇	SD₁₆	semidirect
65. #2254	D₄₇	semidirect
83. C₄₇	#12506	semidirect
111. C₁₆	#10625336	semidirect

H ≅ G/N

type

2. #20245613

C₂

semidirect

3. #20242981

C₂

semidirect

4. #20241304

C₂

semidirect

5. C₄₇×#2254

C₂

semidirect

42. C₁₆×C₄₇

SD₁₆

semidirect

65. #2254

D₄₇

semidirect

83. C₄₇

#12506

semidirect

111. 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
#20241304	2
C₄₇×#2254	2
#12506	47

subgroup

index

C₄₇×#2254

Subgroups — #26130949

Conjugacy classes of subgroups

Complements

Maximal subgroups