Group #26131014 — subgroups

Subgroups — #26131014

Order 12032.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20245661	C₂	1	2	yes	maximal
3	#20243005	C₂	1	2	yes	maximal
4	C₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2252	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689526	—	2	2	no	—
8	#15689502	—	2	2	no	—
9	#15688949	—	2	2	no	—
10	#15688948	—	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×#10625337	—	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	#13093957	—	4	4	no	—
22	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
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×Dic₉₄	—	4	2	no	—
31	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₄⋊C₄×C₄₇	—	4	2	no	—
35	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
36	#10625337	—	8	8	no	—
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₄⋊Dic₄₇	—	4	2	no	—
39	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
40	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
41	C₁₆×C₄₇	Q₁₆	1	4	yes	complemented
42	C₄₇⋊C₁₆	Q₁₆	1	4	yes	complemented
43	Q₈×C₄₇	—	8	4	no	—
44	C₂×C₄×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₄₇	—	8	2	no	—
49	Dic₉₄	—	8	4	no	—
50	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
51	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
52	C₄₇⋊C₈	D₁₆	1	4	yes	—
53	C₄₇⋊C₈	Q₃₂	1	4	yes	—
54	#12521	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	#493	1	1	yes	characteristic
56	C₄×C₄₇	—	16	2	no	—
57	C₄×C₄₇	#494	1	1	yes	characteristic
58	C₄×C₄₇	#494	1	1	yes	characteristic
59	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
60	C₄×C₄₇	C₂×Q₃₂	1	2	yes	—
61	Dic₄₇	—	16	2	no	—
62	#2252	—	47	2	no	—
63	#2252	—	47	1	no	—
64	#2252	D₄₇	1	1	yes	characteristic
65	#1713	—	47	2	no	—
66	C₈×C₁₆	—	47	1	no	—
67	C₂×C₄₇	#1713	1	1	yes	characteristic
68	C₂×C₄₇	#2246	1	1	yes	characteristic
69	C₂×C₄₇	#2247	1	1	yes	characteristic
70	#500	—	47	2	no	—
71	#500	—	94	2	no	—
72	#500	C₂×D₄₇	1	2	yes	—
73	#494	—	94	2	no	—
74	#366	—	2	2	no	—
75	#366	—	94	4	no	—
76	#366	—	94	2	no	—
77	C₄×C₁₆	—	47	1	no	—
78	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
79	C₈²	—	47	1	no	—
80	C₄₇	#12521	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	188	4	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	94	2	no	—
84	C₂×C₁₆	—	47	2	no	—
85	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	94	2	no	—
89	#107	—	94	2	no	—
90	#107	—	188	4	no	—
91	C₄×C₈	—	47	1	no	—
92	C₄×C₈	—	47	1	no	—
93	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	188	2	no	—
96	Q₁₆	—	376	8	no	—
97	C₂×C₈	—	47	2	no	—
98	C₂×C₈	—	47	2	no	—
99	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	188	2	no	—
104	C₄⋊C₄	—	188	2	no	—
105	C₄²	D₄×D₄₇	1	1	yes	characteristic
106	C₁₆	—	47	4	no	—
107	C₁₆	#10625337	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	376	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	376	2	no	—
112	C₂×C₄	#13094020	1	1	yes	characteristic
113	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
114	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
115	C₈	—	47	4	no	—
116	C₈	—	47	4	no	—
117	C₈	Q₁₆×D₄₇	1	2	yes	—
118	C₈	C₂×#10625337	1	2	yes	—
119	C₂²	#15689498	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	752	2	no	—
122	C₄	#15689501	1	1	yes	characteristic
123	C₄	#15689502	1	1	yes	characteristic
124	C₄	D₁₆×D₄₇	1	2	yes	—
125	C₄	Q₃₂×D₄₇	1	2	yes	—
126	C₂	#20243005	1	1	yes	characteristic
127	C₂	#20245636	1	1	yes	characteristic
128	C₂	#20245639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245663

C₂

yes

maximal, characteristic

#20245661

C₂

yes

maximal

#20243005

C₂

yes

maximal

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#2252

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

—

—

—

—

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×#10625337

—

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

#13093957

—

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×Dic₉₄

—

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

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₁₆×C₄₇

Q₁₆

yes

complemented

C₄₇⋊C₁₆

Q₁₆

yes

complemented

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

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₄₇⋊C₈

D₁₆

yes

—

C₄₇⋊C₈

Q₃₂

yes

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

C₂×Q₃₂

yes

—

Dic₄₇

—

#2252

—

#2252

—

#2252

D₄₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×D₄₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₈²

—

C₄₇

#12521

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₄₇

yes

—

—

—

—

—

—

188

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

Q₁₆

—

376

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₄₇

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

188

—

104

C₄⋊C₄

—

188

—

105

C₄²

D₄×D₄₇

yes

characteristic

106

C₁₆

—

107

C₁₆

#10625337

yes

complemented

108

Q₈

—

109

Q₈

—

376

—

110

C₂×C₄

—

111

C₂×C₄

—

376

—

112

C₂×C₄

#13094020

yes

characteristic

113

C₂×C₄

D₈×D₄₇

yes

characteristic

114

C₂×C₄

D₈×D₄₇

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₄₇

yes

—

118

C₈

C₂×#10625337

yes

—

119

C₂²

#15689498

yes

characteristic

120

C₄

—

121

C₄

—

752

—

122

C₄

#15689501

yes

characteristic

123

C₄

#15689502

yes

characteristic

124

C₄

D₁₆×D₄₇

yes

—

125

C₄

Q₃₂×D₄₇

yes

—

126

C₂

#20243005

yes

characteristic

127

C₂

#20245636

yes

characteristic

128

C₂

#20245639

yes

characteristic

	subgroup	G/H
center	C₂²	#15689498
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2252	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#2252	D₄₇
hypocenter	C₄₇	#12521
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

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
41. C₁₆×C₄₇	Q₁₆	semidirect
42. C₄₇⋊C₁₆	Q₁₆	semidirect
80. C₄₇	#12521	semidirect
107. C₁₆	#10625337	semidirect

H ≅ G/N

type

41. C₁₆×C₄₇

Q₁₆

semidirect

42. C₄₇⋊C₁₆

Q₁₆

semidirect

80. C₄₇

#12521

semidirect

107. C₁₆

#10625337

semidirect

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

subgroup	index
#20245663	2
#20245661	2
#20243005	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2252	2
#12521	47

subgroup

index

#20245663

#20245661

#20243005

C₁₆×C₄₇⋊C₈

C₄₇×#2252

#12521

Subgroups — #26131014

Conjugacy classes of subgroups

Complements

Maximal subgroups