Group #26131694 — subgroups

Subgroups — #26131694

Order 12032.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#15688953	C₂	1	2	yes	maximal
2	C₄×#13093960	C₂	1	2	yes	maximal
3	#20241482	C₂	1	2	yes	maximal
4	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093960	C₂²	1	4	yes	—
6	C₄×#10625329	—	2	2	no	—
7	#15688953	C₄	1	8	yes	complemented
8	#15688888	—	2	2	no	—
9	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#346	C₄	1	4	yes	—
13	C₂×#10625329	—	2	4	no	—
14	#13093960	—	2	4	no	—
15	#13093960	C₂×C₄	1	8	yes	—
16	#13093935	—	4	2	no	—
17	C₄×C₄₇⋊C₈	—	4	2	no	—
18	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₄₇	—	2	2	no	—
22	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
24	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₄₇	Q₈	1	2	yes	—
27	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₄₇	#41	1	2	yes	—
35	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
36	C₄²×C₄₇	D₈	1	2	yes	—
37	#10625329	—	2	8	no	—
38	#10625329	—	4	4	no	—
39	C₂×C₄₇⋊C₈	—	4	4	no	—
40	C₁₆×C₄₇	—	2	2	no	—
41	C₁₆×C₄₇	C₄²	1	4	yes	—
42	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₄₇	—	4	2	no	—
44	C₂×C₄×C₄₇	#107	1	4	yes	—
45	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#135	1	2	yes	—
49	C₈×C₄₇	—	2	2	no	—
50	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
51	C₈×C₄₇	#117	1	1	yes	characteristic
52	C₈×C₄₇	C₄×Q₈	1	2	yes	—
53	C₄₇⋊C₈	—	4	8	no	—
54	#12676	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
57	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
58	C₄×C₄₇	—	2	4	no	—
59	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₄₇	#427	1	1	yes	characteristic
61	C₄×C₄₇	C₄×D₈	1	2	yes	—
62	C₂×#368	—	47	2	no	—
63	C₂×#346	D₄₇	1	1	yes	characteristic
64	C₄×#108	—	47	2	no	—
65	#1381	—	47	2	no	—
66	C₂×C₄₇	—	2	2	no	—
67	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
68	C₂×C₄₇	#2161	1	2	yes	—
69	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
71	C₂×#108	—	47	4	no	—
72	C₄×M₁₆	—	94	2	no	—
73	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
74	#368	—	47	8	no	—
75	#346	Dic₄₇	1	4	yes	—
76	#341	—	94	2	no	—
77	C₄₇	#12676	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	94	4	no	—
79	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
81	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
84	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
86	#108	—	47	8	no	—
87	#108	—	94	4	no	—
88	#97	—	188	2	no	—
89	C₄×C₈	—	188	2	no	—
90	C₄×C₈	Dic₉₄	1	2	yes	—
91	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
92	C₂²×C₄	Dic₁₈₈	1	2	yes	—
93	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
94	M₁₆	—	94	8	no	—
95	M₁₆	—	188	4	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	188	4	no	—
98	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
99	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
101	C₂×C₈	#10625350	1	2	yes	—
102	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
103	C₄²	D₃₇₆	1	2	yes	—
104	C₄²	C₄⋊Dic₄₇	1	2	yes	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄×Dic₄₇	1	4	yes	—
107	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#13093958	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
111	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
112	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
113	C₂×C₄	#13094046	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	188	8	no	—
116	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
117	C₈	#13093983	1	1	yes	characteristic
118	C₈	C₄×Dic₉₄	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#13093958	1	1	yes	characteristic
121	C₂²	C₄×Dic₁₈₈	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
124	C₄	#15689161	1	1	yes	characteristic
125	C₄	C₄×D₃₇₆	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₄×#13093958	1	1	yes	characteristic
128	C₂	#20245183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688953

C₂

yes

maximal

C₄×#13093960

C₂

yes

maximal

#20241482

C₂

yes

maximal

C₂×C₄₇×#346

C₂

yes

maximal, characteristic

C₂×#13093960

C₂²

yes

—

C₄×#10625329

—

#15688953

C₄

yes

complemented

#15688888

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

C₄

yes

—

C₂×#10625329

—

#13093960

—

#13093960

C₂×C₄

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₄²

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄²

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

C₄×Q₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₇

#427

yes

characteristic

C₄×C₄₇

C₄×D₈

yes

—

C₂×#368

—

C₂×#346

D₄₇

yes

characteristic

C₄×#108

—

#1381

—

C₂×C₄₇

—

C₂×C₄₇

C₄×#107

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#368

—

#346

Dic₄₇

yes

—

#341

—

C₄₇

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

D₁₈₈

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

#108

—

#108

—

#97

—

188

—

C₄×C₈

—

188

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

Dic₁₈₈

yes

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₂×Dic₉₄

yes

characteristic

100

C₂×C₈

C₂×C₄×D₄₇

yes

—

101

C₂×C₈

#10625350

yes

—

102

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

103

C₄²

D₃₇₆

yes

—

104

C₄²

C₄⋊Dic₄₇

yes

—

105

C₁₆

—

106

C₁₆

C₄×Dic₄₇

yes

—

107

C₂³

C₂×Dic₁₈₈

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#13093958

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

111

C₂×C₄

C₄×D₁₈₈

yes

—

112

C₂×C₄

C₂×D₃₇₆

yes

characteristic

113

C₂×C₄

#13094046

yes

—

114

C₈

—

115

C₈

—

188

—

116

C₈

C₂×C₄×Dic₄₇

yes

characteristic

117

C₈

#13093983

yes

characteristic

118

C₈

C₄×Dic₉₄

yes

—

119

C₂²

—

120

C₂²

C₂×#13093958

yes

characteristic

121

C₂²

C₄×Dic₁₈₈

yes

—

122

C₄

—

123

C₄

C₄×C₄⋊Dic₄₇

yes

characteristic

124

C₄

#15689161

yes

characteristic

125

C₄

C₄×D₃₇₆

yes

—

126

C₂

—

127

C₂

C₄×#13093958

yes

characteristic

128

C₂

#20245183

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#12676
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₃₇₆

commutator

C₈×C₄₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#346

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×C₄₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
7. #15688953	C₄	semidirect
77. C₄₇	#12676	semidirect

H ≅ G/N

type

7. #15688953

C₄

semidirect

77. C₄₇

#12676

semidirect

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

subgroup	index
C₂×#15688953	2
C₄×#13093960	2
#20241482	2
C₂×C₄₇×#346	2
#12676	47

subgroup

index

C₂×C₄₇×#346

Subgroups — #26131694

Conjugacy classes of subgroups

Complements

Maximal subgroups