Group #41013571 — subgroups

Subgroups — #41013571

Order 27008.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#20269913	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269912	C₂	1	2	yes	maximal
3	#27213169	C₂	1	2	yes	maximal
4	C₂×#20269887	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	#20269913	—	2	2	no	—
10	#20269912	—	2	4	no	—
11	Dic₂₁₁⋊C₈	—	2	4	no	—
12	#20269887	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	#108×C₂₁₁	—	2	2	no	—
16	C₂²×Dic₂₁₁	Q₈	1	2	yes	—
17	#16809946	—	2	2	no	—
18	#16809946	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
22	C₂×C₂₁₁⋊C₈	Q₈	1	2	yes	—
23	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₁₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₁₁	—	2	2	no	—
27	M₁₆×C₂₁₁	—	2	2	no	—
28	C₂×C₈×C₂₁₁	—	2	2	no	—
29	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
31	C₂×C₈×C₂₁₁	Q₈	1	2	yes	—
32	C₂×Dic₂₁₁	—	4	2	no	—
33	C₂×Dic₂₁₁	C₄⋊C₄	1	4	yes	—
34	C₂₁₁⋊C₈	—	4	2	no	—
35	C₂₁₁⋊C₈	C₄⋊C₄	1	4	yes	—
36	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₂₁₁	—	4	2	no	—
44	C₈×C₂₁₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₂₁₁	—	2	2	no	—
46	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₂₁₁	#127	1	1	yes	characteristic
52	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₂₁₁	—	2	4	no	—
54	C₂×C₂₁₁	—	2	1	no	—
55	C₂×C₂₁₁	—	2	1	no	—
56	C₂×C₂₁₁	#384	1	1	yes	characteristic
57	C₂×C₂₁₁	#430	1	2	yes	—
58	C₂₁₁	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	211	1	no	maximal, Sylow
60	C₂×#108	—	211	2	no	—
61	C₂×#108	—	211	1	no	—
62	C₂×#108	D₂₁₁	1	1	yes	characteristic
63	#423	—	211	2	no	—
64	C₂×#97	—	211	1	no	—
65	C₂×M₁₆	—	211	2	no	—
66	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
67	C₂²×C₈	—	211	1	no	—
68	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
69	C₂×C₄²	—	211	1	no	—
70	#108	—	2	2	no	—
71	#108	—	422	4	no	—
72	#108	—	422	2	no	—
73	C₄⋊C₈	—	422	4	no	—
74	#97	—	211	4	no	—
75	C₂²×C₄	—	211	2	no	—
76	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	422	2	no	—
80	M₁₆	—	422	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	422	2	no	—
83	C₂×C₈	—	211	2	no	—
84	C₂×C₈	—	211	2	no	—
85	C₂×C₈	—	211	2	no	—
86	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
87	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
88	C₂×C₈	Dic₄₂₂	1	2	yes	—
89	C₄²	—	211	2	no	—
90	C₄²	—	211	2	no	—
91	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
92	C₂×C₄	—	844	2	no	—
93	C₂×C₄	—	211	4	no	—
94	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
95	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
96	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	844	2	no	—
102	C₈	—	211	4	no	—
103	C₈	Dic₂₁₁⋊C₄	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
106	C₂²	#20269975	1	1	yes	characteristic
107	C₂²	#20269976	1	1	yes	characteristic
108	C₄	—	422	4	no	—
109	C₄	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
110	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
111	C₄	#20269973	1	1	yes	characteristic
112	C₄	#20269976	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#27213015	1	1	yes	characteristic
116	C₂	#27213193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269913

C₂

yes

maximal, characteristic

C₂×#20269912

C₂

yes

maximal

#27213169

C₂

yes

maximal

C₂×#20269887

C₂

yes

maximal, characteristic

C₂×#108×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

#20269913

—

#20269912

—

Dic₂₁₁⋊C₈

—

#20269887

C₄

yes

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#108×C₂₁₁

—

C₂²×Dic₂₁₁

Q₈

yes

—

#16809946

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂×C₂₁₁⋊C₈

Q₈

yes

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

Q₈

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

C₄⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₁₁

#127

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#384

yes

characteristic

C₂×C₂₁₁

#430

yes

—

C₂₁₁

#1849

yes

characteristic, complemented, Sylow

#1849

—

211

maximal, Sylow

C₂×#108

—

211

—

C₂×#108

—

211

—

C₂×#108

D₂₁₁

yes

characteristic

#423

—

211

—

C₂×#97

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

—

211

—

#108

—

#108

—

422

—

#108

—

422

—

C₄⋊C₈

—

422

—

#97

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₂×C₈

C₂²⋊D₂₁₁

yes

—

C₂×C₈

Dic₄₂₂

yes

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

Q₈×D₂₁₁

yes

characteristic

C₂×C₄

—

844

—

C₂×C₄

—

211

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

D₄×D₂₁₁

yes

characteristic

C₂×C₄

D₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

100

C₈

—

101

C₈

—

844

—

102

C₈

—

211

—

103

C₈

Dic₂₁₁⋊C₄

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

106

C₂²

#20269975

yes

characteristic

107

C₂²

#20269976

yes

characteristic

108

C₄

—

422

—

109

C₄

C₄⋊C₄×D₂₁₁

yes

characteristic

110

C₄

C₂×Dic₂₁₁⋊C₄

yes

characteristic

111

C₄

#20269973

yes

characteristic

112

C₄

#20269976

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#27213015

yes

characteristic

116

C₂

#27213193

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#108×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#108	D₂₁₁
hypocenter	C₂₁₁	#1849
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×#108×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₁₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
58. C₂₁₁	#1849	semidirect

H ≅ G/N

type

58. C₂₁₁

#1849

semidirect

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

subgroup	index
C₂×#20269913	2
C₂×#20269912	2
#27213169	2
C₂×#20269887	2
C₂×#108×C₂₁₁	2
#1849	211

subgroup

index

C₂×#108×C₂₁₁

211

Subgroups — #41013571

Conjugacy classes of subgroups

Complements

Maximal subgroups