Group #26103150 — subgroups

Subgroups — #26103150

Order 12032.

222 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20242136	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689176	—	2	2	no	—
4	#15688876	C₄	1	4	yes	—
5	#15688874	—	2	2	no	—
6	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₄₇	—	2	2	no	—
9	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
11	C₄×C₄₇⋊C₈	—	4	2	no	—
12	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₄₇	—	2	2	no	—
16	C₂×C₁₆×C₄₇	—	2	2	no	—
17	C₂×C₁₆×C₄₇	C₈	1	4	yes	complemented
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	2	no	—
23	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
24	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
27	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
28	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₄₇	D₈	1	1	yes	characteristic
30	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
31	C₂×C₄₇⋊C₈	—	4	4	no	—
32	C₁₆×C₄₇	—	2	4	no	—
33	C₂³×C₄₇	#106	1	1	yes	characteristic
34	C₂×C₄×C₄₇	—	2	4	no	—
35	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
36	C₂×C₄×C₄₇	#102	1	2	yes	—
37	C₂×C₄×C₄₇	#103	1	2	yes	—
38	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
39	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
40	C₂×C₄×C₄₇	#108	1	2	yes	—
41	C₈×C₄₇	—	2	2	no	—
42	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
43	C₈×C₄₇	#97	1	1	yes	characteristic
44	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
45	C₄₇⋊C₈	—	8	4	no	—
46	#7155	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	—	2	2	no	—
48	C₂²×C₄₇	#334	1	2	yes	—
49	C₂²×C₄₇	#340	1	1	yes	characteristic
50	C₄×C₄₇	—	2	2	no	—
51	C₄×C₄₇	#335	1	2	yes	—
52	C₄×C₄₇	#336	1	1	yes	characteristic
53	C₄×C₄₇	#341	1	1	yes	characteristic
54	C₄×C₄₇	#359	1	4	yes	—
55	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
56	#1564	—	47	2	no	—
57	C₂×C₄₇	—	2	2	no	—
58	C₂×C₄₇	#1277	1	1	yes	characteristic
59	C₂×C₄₇	#1382	1	2	yes	—
60	C₂²×C₁₆	Dic₄₇	1	2	yes	—
61	#431	—	94	2	no	—
62	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
63	C₄×C₁₆	—	2	2	no	—
64	#335	—	47	4	no	—
65	#334	—	94	2	no	—
66	C₄₇	#7155	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
69	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₄₇⋊C₈	1	4	yes	complemented
73	C₄⋊C₈	—	94	4	no	—
74	C₂²⋊C₈	—	188	2	no	—
75	C₄×C₈	—	188	2	no	—
76	C₄×C₈	C₄×D₄₇	1	2	yes	—
77	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
78	C₂²×C₄	#10625325	1	2	yes	—
79	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	188	4	no	—
82	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
83	C₂×C₈	#10625329	1	2	yes	—
84	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
85	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
86	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
87	C₄²	D₃₇₆	1	1	yes	characteristic
88	C₄²	Dic₁₈₈	1	1	yes	characteristic
89	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
90	C₁₆	—	2	4	no	—
91	C₂³	#13093956	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
94	C₂×C₄	#13093946	1	2	yes	—
95	C₂×C₄	#13093949	1	2	yes	—
96	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
97	C₂×C₄	#13093958	1	1	yes	characteristic
98	C₂×C₄	#13093960	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	376	4	no	—
101	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
102	C₈	#13093935	1	1	yes	characteristic
103	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	#15688874	1	2	yes	—
106	C₂²	#15688886	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	#15688876	1	2	yes	—
109	C₄	#15688878	1	1	yes	characteristic
110	C₄	#15688888	1	1	yes	characteristic
111	C₄	#15688931	1	4	yes	—
112	C₂	—	2	2	no	—
113	C₂	#20241223	1	1	yes	characteristic
114	C₂	#20241484	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242136

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₇

C₂

yes

maximal, characteristic

#15689176

—

#15688876

C₄

yes

—

#15688874

—

C₂²×C₁₆×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₈

yes

complemented

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

Q₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

#107

yes

characteristic

C₂×C₄×C₄₇

#108

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₄×C₈

yes

characteristic

C₈×C₄₇

#97

yes

characteristic

C₈×C₄₇

C₂²⋊C₈

yes

—

C₄₇⋊C₈

—

#7155

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#334

yes

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#335

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄×C₁₆

D₄₇

yes

characteristic

#1564

—

C₂×C₄₇

—

C₂×C₄₇

#1277

yes

characteristic

C₂×C₄₇

#1382

yes

—

C₂²×C₁₆

Dic₄₇

yes

—

#431

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₁₆

—

#335

—

#334

—

C₄₇

#7155

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄₇⋊C₈

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

188

—

C₄×C₈

—

188

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

#10625325

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

#10625329

yes

—

C₂×C₈

C₄×Dic₄₇

yes

characteristic

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

C₄²

D₃₇₆

yes

characteristic

C₄²

Dic₁₈₈

yes

characteristic

C₄²

(C₂×D₄₇)⋊C₄

yes

—

C₁₆

—

C₂³

#13093956

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

#13093946

yes

—

C₂×C₄

#13093949

yes

—

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

C₂×C₄

#13093958

yes

characteristic

C₂×C₄

#13093960

yes

—

C₈

—

100

C₈

—

376

—

101

C₈

C₄×C₄₇⋊C₈

yes

characteristic

102

C₈

#13093935

yes

characteristic

103

C₈

C₂²⋊(C₄₇⋊C₈)

yes

—

104

C₂²

—

105

C₂²

#15688874

yes

—

106

C₂²

#15688886

yes

characteristic

107

C₄

—

108

C₄

#15688876

yes

—

109

C₄

#15688878

yes

characteristic

110

C₄

#15688888

yes

characteristic

111

C₄

#15688931

yes

—

112

C₂

—

113

C₂

#20241223

yes

characteristic

114

C₂

#20241484

yes

—

	subgroup	G/H
center	C₄²	D₃₇₆
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#106
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#7155
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₄²

D₃₇₆

commutator

C₈×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄×C₁₆×C₄₇

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

hypercenter

C₂×C₄×C₁₆

D₄₇

hypocenter

C₄₇

#7155

LCS

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
17. C₂×C₁₆×C₄₇	C₈	semidirect
66. C₄₇	#7155	semidirect
72. C₂×C₁₆	C₄₇⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₄₇

C₈

semidirect

66. C₄₇

#7155

semidirect

72. C₂×C₁₆

C₄₇⋊C₈

semidirect

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

subgroup	index
#20242136	2
C₂×C₄×C₁₆×C₄₇	2
#7155	47

subgroup

index

#20242136

C₂×C₄×C₁₆×C₄₇

#7155

Subgroups — #26103150

Conjugacy classes of subgroups

Complements

Maximal subgroups