Group #26110079 — subgroups

Subgroups — #26110079

Order 12032.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	2	yes	maximal
2	#20242144	C₂	1	4	yes	maximal
3	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093960	—	2	4	no	—
5	#15689147	C₂²	1	4	yes	—
6	#15688874	—	2	8	no	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
9	C₈²×C₄₇	C₄	1	1	yes	characteristic
10	C₈²×C₄₇	C₄	1	2	yes	—
11	C₈²×C₄₇	C₄	1	1	yes	characteristic
12	C₂×#10625329	—	2	4	no	—
13	#13093960	—	4	8	no	—
14	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
15	C₂²×C₈×C₄₇	D₄	1	2	yes	—
16	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
17	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₇	D₄	1	4	yes	—
23	C₄×C₈×C₄₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
26	C₂×C₈×C₄₇	—	2	8	no	—
27	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₄₇	SD₁₆	1	8	yes	—
32	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₄₇	C₂×D₄	1	2	yes	—
34	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
35	#10625329	—	4	8	no	—
36	C₂×C₄₇⋊C₈	—	4	4	no	—
37	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	4	no	—
39	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
43	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
44	C₈×C₄₇	—	2	4	no	—
45	C₈×C₄₇	#106	1	4	yes	—
46	C₈×C₄₇	#106	1	4	yes	—
47	C₈×C₄₇	#108	1	8	yes	complemented
48	C₄₇⋊C₈	—	8	4	no	—
49	#8822	—	47	1	no	maximal, Sylow
50	C₂²×C₄₇	—	2	2	no	—
51	C₂²×C₄₇	#384	1	1	yes	characteristic
52	C₂²×C₄₇	#498	1	2	yes	—
53	C₄×C₄₇	—	2	2	no	—
54	C₄×C₄₇	C₂×#106	1	2	yes	—
55	C₄×C₄₇	C₂×#108	1	4	yes	—
56	C₄×C₄₇	#492	1	2	yes	—
57	#1848	—	47	2	no	—
58	#1566	—	47	4	no	—
59	C₂×C₈²	D₄₇	1	1	yes	characteristic
60	C₂×C₄₇	—	2	2	no	—
61	C₂×C₄₇	#1844	1	1	yes	characteristic
62	C₂×C₄₇	#1848	1	2	yes	—
63	C₂×#108	—	94	4	no	—
64	#423	—	47	4	no	—
65	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
67	#334	—	94	8	no	—
68	C₈²	Dic₄₇	1	1	yes	characteristic
69	C₈²	Dic₄₇	1	2	yes	—
70	C₈²	Dic₄₇	1	1	yes	characteristic
71	C₄₇	#8822	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	94	4	no	—
73	C₂²×C₈	Dic₉₄	1	4	yes	—
74	C₂²×C₈	D₁₈₈	1	2	yes	—
75	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
76	#108	—	188	8	no	—
77	C₄⋊C₈	—	94	8	no	—
78	C₄×C₈	Dic₉₄	1	2	yes	—
79	C₄×C₈	D₁₈₈	1	4	yes	—
80	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
81	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
82	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
84	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
85	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
86	M₁₆	—	188	8	no	—
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	188	4	no	—
89	C₂×C₈	#10625325	1	8	yes	—
90	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
91	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
92	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
93	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
94	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
95	C₄²	C₂×D₁₈₈	1	2	yes	—
96	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
97	C₂³	#13094024	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
101	C₂×C₄	#13094021	1	1	yes	characteristic
102	C₂×C₄	#13094024	1	2	yes	—
103	C₂×C₄	C₂×#10625325	1	4	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	376	4	no	—
106	C₈	#13093956	1	4	yes	—
107	C₈	#13093956	1	4	yes	—
108	C₈	#13093960	1	8	yes	complemented
109	C₂²	—	2	2	no	—
110	C₂²	#15688993	1	1	yes	characteristic
111	C₂²	#15689521	1	2	yes	—
112	C₄	—	2	2	no	—
113	C₄	C₂×#13093956	1	2	yes	—
114	C₄	C₂×#13093960	1	4	yes	—
115	C₄	#15689496	1	2	yes	—
116	C₂	—	2	2	no	—
117	C₂	#20243545	1	1	yes	characteristic
118	C₂	#20243560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal

#20242144

C₂

yes

maximal

C₂×C₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

#15688874

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₄

yes

characteristic

C₈²×C₄₇

C₄

yes

—

C₈²×C₄₇

C₄

yes

characteristic

C₂×#10625329

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

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

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#108

yes

complemented

C₄₇⋊C₈

—

#8822

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#498

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#106

yes

—

C₄×C₄₇

C₂×#108

yes

—

C₄×C₄₇

#492

yes

—

#1848

—

#1566

—

C₂×C₈²

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1844

yes

characteristic

C₂×C₄₇

#1848

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

#334

—

C₈²

Dic₄₇

yes

characteristic

C₈²

Dic₄₇

yes

—

C₈²

Dic₄₇

yes

characteristic

C₄₇

#8822

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

Dic₉₄

yes

—

C₂²×C₈

D₁₈₈

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#108

—

188

—

C₄⋊C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

#10625325

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₄²

C₂×Dic₉₄

yes

characteristic

C₄²

C₂×D₁₈₈

yes

—

C₄²

C₂²×Dic₄₇

yes

characteristic

C₂³

#13094024

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

100

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

101

C₂×C₄

#13094021

yes

characteristic

102

C₂×C₄

#13094024

yes

—

103

C₂×C₄

C₂×#10625325

yes

—

104

C₈

—

105

C₈

—

376

—

106

C₈

#13093956

yes

—

107

C₈

#13093956

yes

—

108

C₈

#13093960

yes

complemented

109

C₂²

—

110

C₂²

#15688993

yes

characteristic

111

C₂²

#15689521

yes

—

112

C₄

—

113

C₄

C₂×#13093956

yes

—

114

C₄

C₂×#13093960

yes

—

115

C₄

#15689496

yes

—

116

C₂

—

117

C₂

#20243545

yes

characteristic

118

C₂

#20243560

yes

—

	subgroup	G/H
center	C₂×C₄	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#8822
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₈²×C₄₇

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₂×C₈²

D₄₇

hypocenter

C₄₇

#8822

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
47. C₈×C₄₇	#108	semidirect
71. C₄₇	#8822	semidirect
108. C₈	#13093960	semidirect

H ≅ G/N

type

47. C₈×C₄₇

#108

semidirect

71. C₄₇

#8822

semidirect

108. C₈

#13093960

semidirect

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

subgroup	index
#20243560	2
#20242144	2
C₂×C₈²×C₄₇	2
#8822	47

subgroup

index

#20243560

#20242144

C₂×C₈²×C₄₇

#8822

Subgroups — #26110079

Conjugacy classes of subgroups

Complements

Maximal subgroups