Group #26119355 — subgroups

Subgroups — #26119355

Order 12032.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20244211	C₂	1	3	yes	maximal
2	#20241269	C₂	1	3	yes	maximal, complemented
3	C₄₇×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13094021	C₂²	1	1	yes	characteristic
5	C₂×#13093944	—	2	6	no	—
6	C₂×#13093935	C₂²	1	3	yes	—
7	C₄₇×#398	—	2	6	no	—
8	C₄₇×#382	C₂²	1	3	yes	—
9	C₂²×D₁₈₈	—	2	3	no	—
10	C₂²×C₄₇⋊C₈	—	2	3	no	—
11	#13094021	—	2	3	no	—
12	#13094021	Q₈	1	2	yes	—
13	#13093944	—	4	12	no	—
14	#13093935	—	2	6	no	—
15	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
16	C₂×C₄⋊C₄×C₄₇	D₄	1	6	yes	—
17	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
18	#95×C₄₇	—	2	2	no	—
19	#95×C₄₇	—	2	6	no	—
20	C₂³×D₄₇	—	4	1	no	—
21	C₂²×C₄×C₄₇	—	2	6	no	—
22	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
23	C₂×D₁₈₈	—	2	6	no	—
24	C₂×D₁₈₈	—	4	6	no	—
25	C₂×D₁₈₈	—	4	6	no	—
26	C₂×D₁₈₈	—	4	3	no	—
27	C₄⋊C₄×C₄₇	—	4	6	no	—
28	C₄⋊C₄×C₄₇	—	8	6	no	—
29	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₇	#41	1	3	yes	—
31	C₂×C₄₇⋊C₈	—	2	6	no	—
32	C₂×C₄₇⋊C₈	—	4	3	no	—
33	C₂×C₄₇⋊C₈	—	4	3	no	—
34	C₂³×C₄₇	#120	1	1	yes	characteristic
35	C₂²×D₄₇	—	4	2	no	—
36	C₂²×D₄₇	—	8	3	no	—
37	C₂²×D₄₇	—	8	3	no	—
38	C₂×C₄×C₄₇	—	2	3	no	—
39	C₂×C₄×C₄₇	—	2	3	no	—
40	C₂×C₄×C₄₇	—	4	6	no	—
41	C₂×C₄×C₄₇	—	4	6	no	—
42	C₂×C₄×C₄₇	—	4	6	no	—
43	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₄₇	#124	1	3	yes	—
45	D₁₈₈	—	4	12	no	—
46	D₁₈₈	—	8	6	no	—
47	C₄₇⋊C₈	—	4	6	no	—
48	#10360	—	47	1	no	maximal, Sylow
49	C₂²×C₄₇	#393	1	1	yes	characteristic
50	C₂²×C₄₇	#453	1	3	yes	—
51	C₂²×C₄₇	#454	1	3	yes	—
52	C₂×D₄₇	—	8	6	no	—
53	C₂×D₄₇	—	16	3	no	—
54	C₂×D₄₇	—	16	1	no	—
55	C₄×C₄₇	—	2	3	no	—
56	C₄×C₄₇	—	2	3	no	—
57	C₄×C₄₇	—	8	6	no	—
58	#2845	D₄₇	1	1	yes	characteristic, complemented
59	#1961	—	47	3	no	—
60	#1295	—	47	3	no	—
61	C₂×C₄₇	#1656	1	3	yes	—
62	C₂×C₄₇	#1658	1	1	yes	characteristic
63	C₂×C₄₇	#2027	1	3	yes	—
64	D₄₇	—	16	2	no	—
65	C₂×#127	—	47	1	no	—
66	C₂×#102	—	94	6	no	—
67	C₂×#97	—	47	3	no	—
68	#398	—	2	6	no	—
69	#382	C₂×D₄₇	1	3	yes	—
70	C₄₇	#10360	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	94	3	no	—
72	C₂²×C₈	—	94	3	no	—
73	#127	—	47	2	no	—
74	#127	—	94	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	6	yes	—
77	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
78	#102	—	188	12	no	—
79	#97	—	94	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	188	1	no	—
83	C₂×D₄	—	94	6	no	—
84	C₂×D₄	—	188	6	no	—
85	C₂×D₄	—	188	6	no	—
86	C₂×D₄	—	188	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
89	C₂×C₈	—	94	6	no	—
90	C₂×C₈	—	188	3	no	—
91	C₂×C₈	—	188	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₄₇	1	1	yes	characteristic
95	C₄²	#10625354	1	3	yes	—
96	C₂³	—	376	3	no	—
97	C₂³	—	376	3	no	—
98	C₂³	—	188	2	no	—
99	C₂³	#13093993	1	1	yes	characteristic
100	D₄	—	376	6	no	—
101	D₄	—	188	12	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#13094000	1	3	yes	—
108	C₂×C₄	#13094013	1	3	yes	—
109	C₈	—	188	6	no	—
110	C₂²	—	376	6	no	—
111	C₂²	—	752	1	no	—
112	C₂²	—	752	3	no	—
113	C₂²	#15689034	1	1	yes	characteristic
114	C₂²	#15689291	1	3	yes	—
115	C₂²	#15689296	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	752	2	no	—
120	C₂	#20242681	1	3	yes	—
121	C₂	#20242690	1	1	yes	characteristic
122	C₂	#20244541	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244211

C₂

yes

maximal

#20241269

C₂

yes

maximal, complemented

C₄₇×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#13094021

C₂²

yes

characteristic

C₂×#13093944

—

C₂×#13093935

C₂²

yes

—

C₄₇×#398

—

C₄₇×#382

C₂²

yes

—

C₂²×D₁₈₈

—

C₂²×C₄₇⋊C₈

—

#13094021

—

#13094021

Q₈

yes

—

#13093944

—

#13093935

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

#95×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#454

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#2845

D₄₇

yes

characteristic, complemented

#1961

—

#1295

—

C₂×C₄₇

#1656

yes

—

C₂×C₄₇

#1658

yes

characteristic

C₂×C₄₇

#2027

yes

—

D₄₇

—

C₂×#127

—

C₂×#102

—

C₂×#97

—

#398

—

#382

C₂×D₄₇

yes

—

C₄₇

#10360

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#102

—

188

—

#97

—

#95

—

#95

—

C₂⁴

—

188

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₄₇

yes

characteristic

C₄²

#10625354

yes

—

C₂³

—

376

—

C₂³

—

376

—

C₂³

—

188

—

C₂³

#13093993

yes

characteristic

100

D₄

—

376

—

101

D₄

—

188

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#13094000

yes

—

108

C₂×C₄

#13094013

yes

—

109

C₈

—

188

—

110

C₂²

—

376

—

111

C₂²

—

752

—

112

C₂²

—

752

—

113

C₂²

#15689034

yes

characteristic

114

C₂²

#15689291

yes

—

115

C₂²

#15689296

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

752

—

120

C₂

#20242681

yes

—

121

C₂

#20242690

yes

characteristic

122

C₂

#20244541

yes

—

	subgroup	G/H
center	C₂³	#13093993
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2845	C₂
socle	C₂³×C₄₇	#120
prime radical	C₂×#13094021	C₂²
hypercenter	#2845	D₄₇
hypocenter	C₄₇	#10360
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#13093993

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2845

C₂

C₂³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #20241269	C₂	semidirect
3. C₄₇×#2845	C₂	semidirect
58. #2845	D₄₇	semidirect
70. C₄₇	#10360	semidirect

H ≅ G/N

type

2. #20241269

C₂

semidirect

3. C₄₇×#2845

C₂

semidirect

58. #2845

D₄₇

semidirect

70. C₄₇

#10360

semidirect

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

subgroup	index
#20244211	2
#20241269	2
C₄₇×#2845	2
#10360	47

subgroup

index

#20244211

#20241269

C₄₇×#2845

#10360

Subgroups — #26119355

Conjugacy classes of subgroups

Complements

Maximal subgroups