Group #41012474 — subgroups

Subgroups — #41012474

Order 27008.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27213876	C₂	1	1	yes	maximal, characteristic, complemented
2	#27213343	C₂	1	2	yes	maximal
3	#27212895	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212878	C₂	1	2	yes	maximal, complemented
5	#492×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809950	—	2	2	no	—
7	#20269976	—	2	2	no	—
8	#20269975	—	2	2	no	—
9	C₄×Dic₄₂₂	C₂²	1	2	yes	—
10	#20269942	—	2	2	no	—
11	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	2	yes	—
12	#20269898	—	2	2	no	—
13	C₂×SD₁₆×C₂₁₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	#127×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₄₂₂	D₄	1	2	yes	—
19	#16809950	—	4	4	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₁₁	D₄	1	2	yes	—
22	Dic₂₁₁⋊C₄	—	2	4	no	—
23	Dic₂₁₁⋊C₄	—	2	2	no	—
24	C₄×Dic₂₁₁	—	2	2	no	—
25	C₄×Dic₂₁₁	—	2	2	no	—
26	C₂×Q₈×C₂₁₁	D₄	1	2	yes	—
27	C₂×D₄×C₂₁₁	—	2	1	no	—
28	C₂×D₄×C₂₁₁	—	2	2	no	—
29	SD₁₆×C₂₁₁	—	4	4	no	—
30	C₂×C₈×C₂₁₁	—	2	1	no	—
31	C₄⋊C₄×C₂₁₁	—	2	2	no	—
32	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
33	Dic₄₂₂	—	2	2	no	—
34	Dic₄₂₂	—	8	2	no	—
35	Dic₄₂₂	SD₁₆	1	4	yes	complemented
36	C₂×Dic₂₁₁	—	2	2	no	—
37	C₂×Dic₂₁₁	—	2	2	no	—
38	C₂₁₁⋊C₈	—	4	2	no	—
39	C₂³×C₂₁₁	—	4	1	no	—
40	Q₈×C₂₁₁	—	4	2	no	—
41	D₄×C₂₁₁	—	4	2	no	—
42	D₄×C₂₁₁	—	4	2	no	—
43	D₄×C₂₁₁	—	4	2	no	—
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₁₁	—	2	2	no	—
48	C₂²×C₂₁₁	—	8	2	no	—
49	C₂²×C₂₁₁	—	8	1	no	—
50	C₂²×C₂₁₁	#120	1	1	yes	characteristic
51	C₄×C₂₁₁	—	2	1	no	—
52	C₄×C₂₁₁	—	4	2	no	—
53	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₁₁	#137	1	2	yes	—
55	Dic₂₁₁	—	2	4	no	—
56	Dic₂₁₁	—	4	2	no	—
57	C₂×C₂₁₁	—	16	1	no	—
58	C₂×C₂₁₁	#448	1	2	yes	—
59	C₂×C₂₁₁	#453	1	1	yes	characteristic
60	C₂₁₁	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	211	1	no	maximal, Sylow
62	Q₈²	—	211	1	no	—
63	#492	D₂₁₁	1	1	yes	characteristic
64	#460	—	211	2	no	—
65	#333	—	211	1	no	—
66	Q₈⋊C₈	—	211	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	422	2	no	—
69	C₄⋊Q₈	—	422	2	no	—
70	C₄⋊Q₈	—	422	2	no	—
71	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
72	#127	C₂×D₂₁₁	1	1	yes	characteristic
73	C₄×Q₈	—	422	2	no	—
74	C₄×Q₈	—	211	2	no	—
75	C₄⋊C₈	—	211	2	no	—
76	#102	—	422	2	no	—
77	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
78	C₂×Q₈	—	422	2	no	—
79	C₂×Q₈	—	211	2	no	—
80	C₂×Q₈	C₂²⋊D₂₁₁	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	844	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	422	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	422	4	no	—
89	C₄⋊C₄	—	422	2	no	—
90	C₄⋊C₄	—	211	2	no	—
91	C₄²	—	422	2	no	—
92	C₄²	—	422	2	no	—
93	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	1688	2	no	—
97	Q₈	—	422	2	no	—
98	Q₈	—	211	4	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	422	2	no	—
104	C₂×C₄	—	422	2	no	—
105	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	844	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	2	no	—
111	C₂²	#20269945	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	844	2	no	—
115	C₄	—	422	4	no	—
116	C₄	SD₁₆×D₂₁₁	1	2	yes	—
117	C₄	#20270006	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#27213283	1	2	yes	—
120	C₂	#27213313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213876

C₂

yes

maximal, characteristic, complemented

#27213343

C₂

yes

maximal

#27212895

C₂

yes

maximal, characteristic, complemented

#27212878

C₂

yes

maximal, complemented

#492×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809950

—

#20269976

—

#20269975

—

C₄×Dic₄₂₂

C₂²

yes

—

#20269942

—

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

—

#20269898

—

C₂×SD₁₆×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

#127×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

D₄

yes

—

#16809950

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

D₄

yes

—

Dic₂₁₁⋊C₄

—

Dic₂₁₁⋊C₄

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

SD₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

SD₁₆

yes

complemented

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

—

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#120

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×SD₁₆

yes

—

C₄×C₂₁₁

#137

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#448

yes

—

C₂×C₂₁₁

#453

yes

characteristic

C₂₁₁

#1623

yes

characteristic, complemented, Sylow

#1623

—

211

maximal, Sylow

Q₈²

—

211

—

#492

D₂₁₁

yes

characteristic

#460

—

211

—

#333

—

211

—

Q₈⋊C₈

—

211

—

C₂×SD₁₆

—

C₂×SD₁₆

—

422

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

#127

C₂×D₂₁₁

yes

characteristic

C₄×Q₈

—

422

—

C₄×Q₈

—

211

—

C₄⋊C₈

—

211

—

#102

—

422

—

C₄×C₈

C₂×D₂₁₁

yes

characteristic

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂×Q₈

C₂²⋊D₂₁₁

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

844

—

C₂×C₈

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

211

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

1688

—

Q₈

—

422

—

Q₈

—

211

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

422

—

104

C₂×C₄

—

422

—

105

C₂×C₄

D₄×D₂₁₁

yes

—

106

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

107

C₈

—

108

C₈

—

844

—

109

C₂²

—

110

C₂²

—

111

C₂²

#20269945

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

844

—

115

C₄

—

422

—

116

C₄

SD₁₆×D₂₁₁

yes

—

117

C₄

#20270006

yes

—

118

C₂

—

119

C₂

#27213283

yes

—

120

C₂

#27213313

yes

characteristic

	subgroup	G/H
center	C₂²	#20269945
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#492×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#120
prime radical	#127×C₂₁₁	C₂²
hypercenter	#492	D₂₁₁
hypocenter	C₂₁₁	#1623
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20269945

commutator, cosocle

C₄²×C₂₁₁

C₂³

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#492×C₂₁₁

C₂

socle

C₂²×C₂₁₁

#120

prime radical

#127×C₂₁₁

C₂²

D₂₁₁

C₂₁₁

G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27213876	C₂	semidirect
3. #27212895	C₂	semidirect
4. #27212878	C₂	semidirect
35. Dic₄₂₂	SD₁₆	semidirect
60. C₂₁₁	#1623	semidirect

H ≅ G/N

type

1. #27213876

C₂

semidirect

3. #27212895

C₂

semidirect

4. #27212878

C₂

semidirect

35. Dic₄₂₂

SD₁₆

semidirect

60. C₂₁₁

#1623

semidirect

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

subgroup	index
#27213876	2
#27213343	2
#27212895	2
#27212878	2
#492×C₂₁₁	2
#1623	211

subgroup

index

#492×C₂₁₁

211

Subgroups — #41012474

Conjugacy classes of subgroups

Complements

Maximal subgroups