Group #26108846 — subgroups

Subgroups — #26108846

Order 12032.

656 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20244366	C₂	1	3	yes	maximal
2	#20242126	C₂	1	3	yes	maximal
3	C₄₇×#3556	C₂	1	1	yes	maximal, characteristic
4	#15689147	C₂²	1	3	yes	—
5	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
6	#15688882	—	2	6	no	—
7	#15688869	—	2	6	no	—
8	C₄₇×#564	C₄	1	4	yes	complemented
9	C₄₇×#557	—	2	6	no	—
10	C₄₇×#541	—	2	6	no	—
11	C₄₇×#533	C₂²	1	3	yes	—
12	C₂×#10625329	—	2	3	no	—
13	C₂×C₄×Dic₄₇	—	2	3	no	—
14	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
15	C₄⋊Q₈×C₄₇	—	2	6	no	—
16	C₄⋊Q₈×C₄₇	D₄	1	6	yes	—
17	#126×C₄₇	—	2	8	no	—
18	#125×C₄₇	—	2	6	no	—
19	#125×C₄₇	—	2	24	no	—
20	#125×C₄₇	C₂×C₄	1	6	yes	—
21	C₂²⋊Q₈×C₄₇	—	2	6	no	—
22	C₄×Q₈×C₄₇	—	2	12	no	—
23	C₄×Q₈×C₄₇	—	2	6	no	—
24	#117×C₄₇	D₄	1	6	yes	—
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₄₇	—	2	6	no	—
27	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
28	C₂²×Dic₄₇	—	4	1	no	—
29	C₄⋊C₄×C₄₇	—	2	24	no	—
30	C₄⋊C₄×C₄₇	—	2	12	no	—
31	C₄⋊C₄×C₄₇	—	2	6	no	—
32	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	12	yes	—
33	C₂²⋊C₄×C₄₇	—	2	6	no	—
34	C₄²×C₄₇	—	2	8	no	—
35	C₄²×C₄₇	—	2	6	no	—
36	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×D₄	1	3	yes	—
38	C₄×Dic₄₇	—	2	6	no	—
39	C₄×Dic₄₇	—	4	3	no	—
40	#10625329	—	4	6	no	—
41	C₂×C₄₇⋊C₈	—	4	3	no	—
42	C₂³×C₄₇	#120	1	1	yes	characteristic
43	Q₈×C₄₇	—	4	6	no	—
44	Q₈×C₄₇	—	4	6	no	—
45	C₂×C₄×C₄₇	—	2	3	no	—
46	C₂×C₄×C₄₇	—	2	3	no	—
47	C₂×C₄×C₄₇	—	2	12	no	—
48	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
49	C₂×C₄×C₄₇	#120	1	3	yes	—
50	C₂×Dic₄₇	—	4	6	no	—
51	C₄₇⋊C₈	—	8	3	no	—
52	#8533	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	—	2	3	no	—
54	C₂²×C₄₇	#379	1	1	yes	characteristic
55	C₄×C₄₇	—	2	3	no	—
56	C₄×C₄₇	—	4	12	no	—
57	C₄×C₄₇	#454	1	3	yes	—
58	C₄×C₄₇	#456	1	3	yes	—
59	Dic₄₇	—	4	4	no	—
60	#3556	D₄₇	1	1	yes	characteristic
61	#1994	—	47	3	no	—
62	#1561	—	47	3	no	—
63	C₂×C₄₇	—	4	1	no	—
64	C₂×C₄₇	#1793	1	3	yes	—
65	#564	Dic₄₇	1	4	yes	complemented
66	#557	—	2	6	no	—
67	#541	—	2	6	no	—
68	#533	C₂×D₄₇	1	3	yes	—
69	#423	—	47	3	no	—
70	C₄³	—	47	1	no	—
71	#339	—	94	6	no	—
72	#332	—	94	6	no	—
73	C₄₇	#8533	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₁₆	—	94	3	no	—
75	C₄⋊Q₈	—	2	6	no	—
76	C₄⋊Q₈	C₂²⋊D₄₇	1	6	yes	—
77	#126	—	2	8	no	—
78	#125	—	2	6	no	—
79	#125	—	2	24	no	—
80	#125	C₂×Dic₄₇	1	6	yes	—
81	C₂²⋊Q₈	—	2	6	no	—
82	C₄×Q₈	—	2	12	no	—
83	C₄×Q₈	—	2	6	no	—
84	#117	C₂²⋊D₄₇	1	6	yes	—
85	C₂×C₄²	—	94	3	no	—
86	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
87	C₄⋊C₈	—	94	6	no	—
88	C₂×Q₈	—	2	6	no	—
89	C₂²×C₄	—	188	1	no	—
90	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
91	M₁₆	—	188	6	no	—
92	C₂×C₈	—	188	3	no	—
93	C₄⋊C₄	—	2	24	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	C₂²⋊Dic₄₇	1	12	yes	—
97	C₂²⋊C₄	—	2	6	no	—
98	C₄²	—	2	8	no	—
99	C₄²	—	2	6	no	—
100	C₄²	—	94	6	no	—
101	C₄²	—	188	3	no	—
102	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
103	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
104	C₂³	#13093993	1	1	yes	characteristic
105	Q₈	—	4	6	no	—
106	Q₈	—	4	6	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	188	6	no	—
111	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
112	C₂×C₄	#13093993	1	3	yes	—
113	C₈	—	376	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	#15688979	1	1	yes	characteristic
116	C₄	—	2	3	no	—
117	C₄	—	4	12	no	—
118	C₄	—	188	4	no	—
119	C₄	#15689296	1	3	yes	—
120	C₄	#15689306	1	3	yes	—
121	C₂	—	4	1	no	—
122	C₂	#20243347	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244366

C₂

yes

maximal

#20242126

C₂

yes

maximal

C₄₇×#3556

C₂

yes

maximal, characteristic

#15689147

C₂²

yes

—

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688882

—

#15688869

—

C₄₇×#564

C₄

yes

complemented

C₄₇×#557

—

C₄₇×#541

—

C₄₇×#533

C₂²

yes

—

C₂×#10625329

—

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#126×C₄₇

—

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

#8533

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#379

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#454

yes

—

C₄×C₄₇

#456

yes

—

Dic₄₇

—

#3556

D₄₇

yes

characteristic

#1994

—

#1561

—

C₂×C₄₇

—

C₂×C₄₇

#1793

yes

—

#564

Dic₄₇

yes

—

—

C₂×D₄₇

yes

—

#423

—

C₄³

—

#339

—

#332

—

C₄₇

#8533

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

—

—

—

C₂×Dic₄₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂²⋊D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

M₁₆

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊Dic₄₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

—

188

—

102

C₄²

C₂²×Dic₄₇

yes

characteristic

103

C₄²

C₂×C₂²⋊D₄₇

yes

—

104

C₂³

#13093993

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

188

—

111

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

112

C₂×C₄

#13093993

yes

—

113

C₈

—

376

—

114

C₂²

—

115

C₂²

#15688979

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

188

—

119

C₄

#15689296

yes

—

120

C₄

#15689306

yes

—

121

C₂

—

122

C₂

#20243347

yes

—

	subgroup	G/H
center	C₂²	#15688979
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3556	C₂
socle	C₂²×C₄₇	#379
prime radical	C₂³×C₄₇	#120
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3556	D₄₇
hypocenter	C₄₇	#8533
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

subgroup

G/H

center

C₂²

#15688979

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3556

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

N	H ≅ G/N	type
8. C₄₇×#564	C₄	semidirect
65. #564	Dic₄₇	semidirect
73. C₄₇	#8533	semidirect

H ≅ G/N

type

8. C₄₇×#564

C₄

semidirect

65. #564

Dic₄₇

semidirect

73. C₄₇

#8533

semidirect

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

subgroup	index
#20244366	2
#20242126	2
C₄₇×#3556	2
#8533	47

subgroup

index

#20244366

#20242126

C₄₇×#3556

#8533

Subgroups — #26108846

Conjugacy classes of subgroups

Complements

Maximal subgroups