Group #26109985 — subgroups

Subgroups — #26109985

Order 12032.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20242151	C₂	1	2	yes	maximal
2	#20241250	C₂	1	4	yes	maximal
3	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic
4	#15689180	C₂²	1	4	yes	—
5	#15688876	—	2	2	no	—
6	#15688874	—	2	2	no	—
7	C₄₇×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#322	C₄	1	2	yes	—
10	C₈²×C₄₇	C₄	1	2	yes	—
11	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
13	#13093935	—	2	4	no	—
14	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	—	2	2	no	—
18	C₂²⋊C₈×C₄₇	—	2	2	no	—
19	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	#41	1	2	yes	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	—	2	4	no	—
29	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
33	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₄₇	#41	1	2	yes	—
36	C₂×C₄₇⋊C₈	—	4	4	no	—
37	C₂³×C₄₇	#125	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	1	no	—
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	—	4	2	no	—
41	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	#117	1	2	yes	—
43	C₂×C₄×C₄₇	#124	1	2	yes	—
44	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
45	C₈×C₄₇	—	2	1	no	—
46	C₈×C₄₇	—	2	1	no	—
47	C₈×C₄₇	—	4	4	no	—
48	C₈×C₄₇	#108	1	2	yes	—
49	C₈×C₄₇	#108	1	2	yes	—
50	C₄₇⋊C₈	—	8	4	no	—
51	#8796	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	#382	1	1	yes	characteristic
55	C₄×C₄₇	—	2	2	no	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
59	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
60	#1568	—	47	2	no	—
61	#1450	D₄₇	1	1	yes	characteristic
62	#1287	—	47	4	no	—
63	C₂×C₄₇	—	4	1	no	—
64	C₂×C₄₇	#1833	1	2	yes	—
65	C₂×C₄₇	#1837	1	1	yes	characteristic
66	#432	—	47	4	no	—
67	#431	C₂×D₄₇	1	2	yes	—
68	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
69	#335	—	94	2	no	—
70	#334	—	94	2	no	—
71	#322	Dic₄₇	1	2	yes	—
72	C₈²	Dic₄₇	1	2	yes	—
73	C₄₇	#8796	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
75	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
76	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	94	4	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	94	4	no	—
81	#97	—	94	4	no	—
82	C₄×C₈	D₁₈₈	1	1	yes	characteristic
83	C₄×C₈	D₁₈₈	1	1	yes	characteristic
84	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
85	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
88	C₂²×C₄	#10625350	1	2	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	188	4	no	—
93	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
97	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
98	C₄²	#10625350	1	2	yes	—
99	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
100	C₂³	#13094016	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
105	C₂×C₄	#13093983	1	2	yes	—
106	C₂×C₄	#13094012	1	2	yes	—
107	C₂×C₄	#13094016	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	4	no	—
111	C₈	—	376	4	no	—
112	C₈	#13093960	1	2	yes	—
113	C₈	#13093960	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#15688989	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×#13093960	1	1	yes	characteristic
121	C₄	C₂×#13093960	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	#20243512	1	2	yes	—
124	C₂	#20243526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal

#20241250

C₂

yes

maximal

C₄₇×#1450

C₂

yes

maximal, characteristic

#15689180

C₂²

yes

—

#15688876

—

#15688874

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₄

yes

—

C₈²×C₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#125

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#108

yes

—

C₈×C₄₇

#108

yes

—

C₄₇⋊C₈

—

#8796

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#382

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

characteristic

#1287

—

C₂×C₄₇

—

C₂×C₄₇

#1833

yes

—

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

Dic₄₇

yes

—

C₈²

Dic₄₇

yes

—

C₄₇

#8796

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

Dic₉₄

yes

characteristic

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

#10625350

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

C₄²

C₂×D₁₈₈

yes

characteristic

C₄²

#10625350

yes

—

C₄²

C₂²×Dic₄₇

yes

characteristic

100

C₂³

#13094016

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

105

C₂×C₄

#13093983

yes

—

106

C₂×C₄

#13094012

yes

—

107

C₂×C₄

#13094016

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

376

—

112

C₈

#13093960

yes

—

113

C₈

#13093960

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

#15688989

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#13093960

yes

characteristic

121

C₄

C₂×#13093960

yes

characteristic

122

C₂

—

123

C₂

#20243512

yes

—

124

C₂

#20243526

yes

characteristic

	subgroup	G/H
center	C₂²	#15688989
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#125
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#8796
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#15688989

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1450

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
73. C₄₇	#8796	semidirect

H ≅ G/N

type

73. C₄₇

#8796

semidirect

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

subgroup	index
#20242151	2
#20241250	2
C₄₇×#1450	2
#8796	47

subgroup

index

#20242151

#20241250

C₄₇×#1450

#8796

Subgroups — #26109985

Conjugacy classes of subgroups

Complements

Maximal subgroups