Group #26110087 — subgroups

Subgroups — #26110087

Order 12032.

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

normal subgroups only

This group has 338 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	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20242143	C₂	1	4	yes	maximal
3	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093960	—	2	2	no	—
6	#15689147	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688873	—	2	8	no	—
9	#15688844	C₄	1	4	yes	—
10	C₂×#108×C₄₇	—	2	2	no	—
11	C₄₇×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#10625329	—	2	2	no	—
14	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
15	#13093960	—	4	4	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
17	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
19	C₂×M₁₆×C₄₇	—	2	2	no	—
20	C₂²×C₈×C₄₇	D₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	#108×C₄₇	—	4	4	no	—
23	C₄⋊C₈×C₄₇	—	2	4	no	—
24	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
28	M₁₆×C₄₇	—	4	4	no	—
29	C₂×C₈×C₄₇	—	4	2	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
32	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
33	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₄₇	C₂×D₄	1	2	yes	—
35	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	#10625329	—	4	4	no	—
37	C₂×C₄₇⋊C₈	—	4	4	no	—
38	C₂×C₄₇⋊C₈	—	4	2	no	—
39	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
40	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
48	C₈×C₄₇	—	8	2	no	—
49	C₈×C₄₇	#106	1	8	yes	—
50	C₄₇⋊C₈	—	2	8	no	—
51	C₄₇⋊C₈	—	8	2	no	—
52	#8825	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	#384	1	1	yes	characteristic
55	C₂²×C₄₇	#498	1	2	yes	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	C₂×#106	1	2	yes	—
58	C₄×C₄₇	#430	1	4	yes	—
59	C₄×C₄₇	#492	1	1	yes	characteristic
60	C₄×C₄₇	#492	1	1	yes	characteristic
61	#1848	—	47	1	no	—
62	#1848	D₄₇	1	1	yes	characteristic
63	#1566	—	47	4	no	—
64	C₂×#322	—	47	1	no	—
65	C₂×C₄₇	—	2	2	no	—
66	C₂×C₄₇	#1844	1	1	yes	characteristic
67	C₂×C₄₇	#1849	1	2	yes	—
68	C₂×#108	—	2	2	no	—
69	C₂×#108	—	94	2	no	—
70	#423	—	47	2	no	—
71	#423	C₂×D₄₇	1	2	yes	—
72	C₂×C₄×C₈	—	47	2	no	—
73	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
74	#334	—	94	8	no	—
75	#322	—	47	4	no	—
76	C₄₇	#8825	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	94	2	no	—
79	C₂²×C₈	—	47	4	no	—
80	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
81	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
82	#108	—	4	4	no	—
83	#108	—	188	4	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	94	4	no	—
86	C₄×C₈	—	47	4	no	—
87	C₄×C₈	—	47	4	no	—
88	C₄×C₈	Dic₉₄	1	2	yes	—
89	C₄×C₈	C₄×D₄₇	1	2	yes	—
90	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
91	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
92	M₁₆	—	4	4	no	—
93	M₁₆	—	188	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	47	8	no	—
96	C₂×C₈	—	188	4	no	—
97	C₂×C₈	—	188	2	no	—
98	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
99	C₂×C₈	#10625335	1	4	yes	—
100	C₂×C₈	#10625336	1	4	yes	—
101	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
102	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
103	C₄²	D₄×D₄₇	1	2	yes	—
104	C₂³	#13094022	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
108	C₂×C₄	#13094020	1	1	yes	characteristic
109	C₂×C₄	#13094023	1	2	yes	—
110	C₂×C₄	C₂×#10625336	1	2	yes	—
111	C₂×C₄	C₂×#10625335	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	94	8	no	—
114	C₈	—	376	2	no	—
115	C₈	#13093955	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#15688992	1	1	yes	characteristic
118	C₂²	#15689519	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#13093955	1	2	yes	—
121	C₄	#15689170	1	4	yes	—
122	C₄	#15689495	1	1	yes	characteristic
123	C₄	#15689497	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#20243544	1	1	yes	characteristic
126	C₂	#20243561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal, characteristic

#20242143

C₂

yes

maximal

C₂×#15688844

C₂

yes

maximal, characteristic

C₄₇×#1848

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688873

—

#15688844

C₄

yes

—

C₂×#108×C₄₇

—

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

Q₈

yes

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

#106

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8825

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#498

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#106

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

characteristic

#1566

—

C₂×#322

—

C₂×C₄₇

—

C₂×C₄₇

#1844

yes

characteristic

C₂×C₄₇

#1849

yes

—

—

—

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#334

—

#322

—

C₄₇

#8825

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#108

—

#108

—

188

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₄×D₄₇

yes

—

C₂²×C₄

Q₈×D₄₇

yes

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

#10625335

yes

—

100

C₂×C₈

#10625336

yes

—

101

C₄²

C₂×Dic₉₄

yes

characteristic

102

C₄²

C₂×C₄×D₄₇

yes

characteristic

103

C₄²

D₄×D₄₇

yes

—

104

C₂³

#13094022

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

107

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

108

C₂×C₄

#13094020

yes

characteristic

109

C₂×C₄

#13094023

yes

—

110

C₂×C₄

C₂×#10625336

yes

—

111

C₂×C₄

C₂×#10625335

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

376

—

115

C₈

#13093955

yes

—

116

C₂²

—

117

C₂²

#15688992

yes

characteristic

118

C₂²

#15689519

yes

—

119

C₄

—

120

C₄

C₂×#13093955

yes

—

121

C₄

#15689170

yes

—

122

C₄

#15689495

yes

characteristic

123

C₄

#15689497

yes

characteristic

124

C₂

—

125

C₂

#20243544

yes

characteristic

126

C₂

#20243561

yes

—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#8825
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1848

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
76. C₄₇	#8825	semidirect

H ≅ G/N

type

76. C₄₇

#8825

semidirect

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

subgroup	index
#20243560	2
#20242143	2
C₂×#15688844	2
C₄₇×#1848	2
#8825	47

subgroup

index

C₄₇×#1848

Subgroups — #26110087

Conjugacy classes of subgroups

Complements

Maximal subgroups