Group #26132628 — subgroups

Subgroups — #26132628

Order 12032.

322 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 171 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×C₄⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal
2	#20241466	C₂	1	4	yes	maximal
3	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
5	C₄⋊(C₄₇⋊C₁₆)	—	2	4	no	—
6	C₂×#97×C₄₇	C₄	1	2	yes	—
7	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#376	C₄	1	1	yes	characteristic
11	C₂×C₄₇⋊C₁₆	—	2	4	no	—
12	C₂×C₄₇⋊C₁₆	—	2	8	no	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
18	#97×C₄₇	—	2	4	no	—
19	#97×C₄₇	—	2	2	no	—
20	C₄×C₈×C₄₇	—	2	4	no	—
21	C₄×C₈×C₄₇	D₄	1	2	yes	—
22	C₄×C₈×C₄₇	Q₈	1	2	yes	—
23	#95×C₄₇	C₈	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
32	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₄₇	#41	1	4	yes	—
35	C₄²×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	—	2	2	no	—
37	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
38	C₄²×C₄₇	M₁₆	1	2	yes	—
39	C₄₇⋊C₁₆	—	4	4	no	—
40	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	2	4	no	—
43	C₂×C₄×C₄₇	—	2	8	no	—
44	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#117	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
49	C₈×C₄₇	—	2	2	no	—
50	C₈×C₄₇	—	2	4	no	—
51	C₈×C₄₇	#124	1	2	yes	—
52	C₈×C₄₇	#125	1	2	yes	—
53	#12880	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₄₇	#431	1	2	yes	—
56	C₂²×C₄₇	#504	1	4	yes	—
57	C₄×C₄₇	—	2	4	no	—
58	C₄×C₄₇	—	4	4	no	—
59	C₄×C₄₇	#382	1	1	yes	characteristic
60	C₄×C₄₇	#423	1	1	yes	characteristic
61	C₄×C₄₇	#432	1	2	yes	—
62	C₂×C₄⋊C₁₆	—	47	2	no	—
63	#1726	D₄₇	1	1	yes	characteristic
64	#1374	—	47	4	no	—
65	C₂×C₄₇	#1831	1	1	yes	characteristic
66	C₂×C₄₇	#2151	1	2	yes	—
67	C₂×C₄₇	#2164	1	4	yes	—
68	C₂²×C₁₆	—	47	4	no	—
69	C₂×#97	Dic₄₇	1	2	yes	—
70	C₂×#97	Dic₄₇	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
73	#376	Dic₄₇	1	1	yes	characteristic
74	C₄⋊C₁₆	—	94	4	no	—
75	C₄₇	#12880	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
77	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
80	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
81	C₂×C₁₆	—	94	4	no	—
82	C₂×C₁₆	—	94	8	no	—
83	#97	—	2	4	no	—
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	Dic₉₄	1	2	yes	—
87	C₄×C₈	D₁₈₈	1	2	yes	—
88	#95	C₄₇⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
90	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
91	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	#10625329	1	4	yes	—
96	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
97	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
99	C₂×C₈	#10625350	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	#10625329	1	2	yes	—
103	C₄²	C₄⋊Dic₄₇	1	2	yes	—
104	C₁₆	—	188	4	no	—
105	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
111	C₂×C₄	#13093983	1	2	yes	—
112	C₂×C₄	C₂×#10625329	1	2	yes	—
113	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#13094012	1	2	yes	—
117	C₈	#13094016	1	2	yes	—
118	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
119	C₂²	#15689176	1	2	yes	—
120	C₂²	#15689540	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#15688989	1	1	yes	characteristic
124	C₄	#15689147	1	1	yes	characteristic
125	C₄	#15689180	1	2	yes	—
126	C₂	#20243505	1	1	yes	characteristic
127	C₂	#20245147	1	2	yes	—
128	C₂	#20245195	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₁₆)

C₂

yes

maximal

#20241466

C₂

yes

maximal

C₄₇×#1726

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₄⋊(C₄₇⋊C₁₆)

—

C₂×#97×C₄₇

C₄

yes

—

C₂×#97×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#376

C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#97×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

#95×C₄₇

C₈

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#124

yes

—

C₈×C₄₇

#125

yes

—

#12880

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₂²×C₄₇

#504

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄⋊C₁₆

—

#1726

D₄₇

yes

characteristic

#1374

—

C₂×C₄₇

#1831

yes

characteristic

C₂×C₄₇

#2151

yes

—

C₂×C₄₇

#2164

yes

—

C₂²×C₁₆

—

C₂×#97

Dic₄₇

yes

—

C₂×#97

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#376

Dic₄₇

yes

characteristic

C₄⋊C₁₆

—

C₄₇

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

D₁₈₈

yes

—

#95

C₄₇⋊C₈

yes

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#10625329

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₂×Dic₉₄

yes

characteristic

C₂×C₈

C₂×D₁₈₈

yes

characteristic

C₂×C₈

#10625350

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

#10625329

yes

—

103

C₄²

C₄⋊Dic₄₇

yes

—

104

C₁₆

—

188

—

105

C₂³

C₂²×C₄₇⋊C₈

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

111

C₂×C₄

#13093983

yes

—

112

C₂×C₄

C₂×#10625329

yes

—

113

C₂×C₄

C₂×#10625329

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

#13094012

yes

—

117

C₈

#13094016

yes

—

118

C₂²

C₂×C₄⋊(C₄₇⋊C₈)

yes

characteristic

119

C₂²

#15689176

yes

—

120

C₂²

#15689540

yes

—

121

C₄

—

122

C₄

—

123

C₄

#15688989

yes

characteristic

124

C₄

#15689147

yes

characteristic

125

C₄

#15689180

yes

—

126

C₂

#20243505

yes

characteristic

127

C₂

#20245147

yes

—

128

C₂

#20245195

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂²×C₈
Fitting	C₄₇×#1726	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#12880
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂²×C₈

Fitting

C₄₇×#1726

C₂

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₈ ◁ #1726

N	H ≅ G/N	type
75. C₄₇	#12880	semidirect

H ≅ G/N

type

75. C₄₇

#12880

semidirect

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20241466	2
C₄₇×#1726	2
#12880	47

subgroup

index

C₂×C₄⋊(C₄₇⋊C₁₆)

#20241466

C₄₇×#1726

#12880

Subgroups — #26132628

Conjugacy classes of subgroups

Complements

Maximal subgroups