Group #41013620 — subgroups

Subgroups — #41013620

Order 27008.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#27213913	C₂	1	1	yes	maximal, characteristic
2	#27213195	C₂	1	2	yes	maximal, complemented
3	#27213113	C₂	1	1	yes	maximal, characteristic
4	#27212910	C₂	1	2	yes	maximal, complemented
5	#413×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#20269984	—	2	4	no	—
7	C₂×#16809946	—	2	2	no	—
8	C₂×#16809946	C₂²	1	1	yes	characteristic
9	C₂×#16809946	C₂²	1	2	yes	—
10	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	#20269913	—	2	4	no	—
12	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
13	C₂×#41×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
15	#101×C₂₁₁	C₄	1	2	yes	—
16	#100×C₂₁₁	C₄	1	2	yes	—
17	#16809946	—	2	4	no	—
18	#16809946	—	2	2	no	—
19	#16809946	—	2	2	no	—
20	#16809946	D₄	1	4	yes	—
21	C₂×C₂₁₁⋊C₈	—	2	4	no	—
22	C₂×C₂₁₁⋊C₈	—	2	2	no	—
23	C₂×C₂₁₁⋊C₈	—	2	1	no	—
24	C₂×C₂₁₁⋊C₈	—	2	1	no	—
25	C₂×C₂₁₁⋊C₈	—	2	2	no	—
26	#41×C₂₁₁	—	2	2	no	—
27	#41×C₂₁₁	—	2	2	no	—
28	C₂×Q₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₂₁₁	Q₈	1	2	yes	—
31	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
32	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₂₁₁	—	2	2	no	—
34	M₁₆×C₂₁₁	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₂₁₁	—	2	2	no	—
36	C₂₁₁⋊C₈	—	2	4	no	—
37	C₂₁₁⋊C₈	—	4	1	no	—
38	C₂₁₁⋊C₈	—	4	1	no	—
39	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₂₁₁	—	2	2	no	—
42	D₄×C₂₁₁	—	2	4	no	—
43	D₄×C₂₁₁	—	2	2	no	—
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	—	2	2	no	—
46	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	#41	1	2	yes	—
51	C₈×C₂₁₁	—	2	4	no	—
52	C₂²×C₂₁₁	—	2	2	no	—
53	C₂²×C₂₁₁	—	4	1	no	—
54	C₂²×C₂₁₁	—	4	2	no	—
55	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₂₁₁	—	2	2	no	—
58	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
59	C₄×C₂₁₁	#120	1	1	yes	characteristic
60	C₄×C₂₁₁	#123	1	1	yes	characteristic
61	C₂×C₂₁₁	—	2	1	no	—
62	C₂×C₂₁₁	—	2	2	no	—
63	C₂×C₂₁₁	—	4	2	no	—
64	C₂×C₂₁₁	#385	1	1	yes	characteristic
65	C₂₁₁	#1858	1	1	yes	characteristic, complemented, Sylow
66	#1858	—	211	1	no	maximal, Sylow
67	#568	—	211	1	no	—
68	#430	—	211	2	no	—
69	#413	D₂₁₁	1	1	yes	characteristic
70	#408	—	211	1	no	—
71	#341	—	211	2	no	—
72	C₂×#41	C₂×D₂₁₁	1	1	yes	characteristic
73	#131	—	422	4	no	—
74	C₂×M₁₆	—	422	2	no	—
75	C₂×M₁₆	—	211	1	no	—
76	C₂×M₁₆	—	211	2	no	—
77	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
78	C₂²×C₈	—	211	1	no	—
79	#108	—	422	4	no	—
80	#101	Dic₂₁₁	1	2	yes	—
81	#100	Dic₂₁₁	1	2	yes	—
82	C₂²⋊C₈	—	422	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₂×Dic₂₁₁	1	1	yes	characteristic
86	C₂×D₄	C₂×Dic₂₁₁	1	1	yes	characteristic
87	C₂×D₄	Dic₄₂₂	1	2	yes	—
88	C₂²×C₄	D₈₄₄	1	2	yes	—
89	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	422	2	no	—
92	M₁₆	—	422	4	no	—
93	M₁₆	—	422	2	no	—
94	M₁₆	—	211	4	no	—
95	M₁₆	C₂×Dic₂₁₁	1	4	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	422	1	no	—
98	C₂×C₈	—	422	1	no	—
99	C₂×C₈	—	422	2	no	—
100	C₂×C₈	—	422	4	no	—
101	C₂×C₈	—	422	2	no	—
102	C₂³	C₄⋊Dic₂₁₁	1	2	yes	—
103	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
110	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
112	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
113	C₂×C₄	#16809961	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	844	1	no	—
116	C₈	—	844	1	no	—
117	C₈	—	422	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
122	C₂²	#20269954	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	D₄×Dic₂₁₁	1	2	yes	—
125	C₄	#20269947	1	1	yes	characteristic
126	C₄	#20269961	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#27213021	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213913

C₂

yes

maximal, characteristic

#27213195

C₂

yes

maximal, complemented

#27213113

C₂

yes

maximal, characteristic

#27212910

C₂

yes

maximal, complemented

#413×C₂₁₁

C₂

yes

maximal, characteristic

#20269984

—

C₂×#16809946

—

C₂×#16809946

C₂²

yes

characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

#20269913

—

C₂²⋊(C₂₁₁⋊C₈)

—

C₂×#41×C₂₁₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₁₁

C₂²

yes

—

#101×C₂₁₁

C₄

yes

—

#100×C₂₁₁

C₄

yes

—

—

—

—

D₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

#41×C₂₁₁

—

#41×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

characteristic

C₂×D₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂×D₄×C₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#385

yes

characteristic

C₂₁₁

#1858

yes

characteristic, complemented, Sylow

#1858

—

211

maximal, Sylow

#568

—

211

—

#430

—

211

—

#413

D₂₁₁

yes

characteristic

#408

—

211

—

#341

—

211

—

C₂×#41

C₂×D₂₁₁

yes

characteristic

#131

—

422

—

C₂×M₁₆

—

422

—

C₂×M₁₆

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

—

211

—

#108

—

422

—

#101

Dic₂₁₁

yes

—

#100

Dic₂₁₁

yes

—

C₂²⋊C₈

—

422

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₂₁₁

yes

characteristic

C₂×D₄

C₂×Dic₂₁₁

yes

characteristic

C₂×D₄

Dic₄₂₂

yes

—

C₂²×C₄

D₈₄₄

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

422

—

M₁₆

—

422

—

M₁₆

—

211

—

M₁₆

C₂×Dic₂₁₁

yes

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

100

C₂×C₈

—

422

—

101

C₂×C₈

—

422

—

102

C₂³

C₄⋊Dic₂₁₁

yes

—

103

C₂³

C₂×Dic₄₂₂

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊Dic₂₁₁

yes

—

110

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

111

C₂×C₄

D₄×D₂₁₁

yes

—

112

C₂×C₄

C₂×D₈₄₄

yes

characteristic

113

C₂×C₄

#16809961

yes

—

114

C₈

—

115

C₈

—

844

—

116

C₈

—

844

—

117

C₈

—

422

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₄⋊Dic₂₁₁

yes

characteristic

122

C₂²

#20269954

yes

—

123

C₄

—

124

C₄

D₄×Dic₂₁₁

yes

—

125

C₄

#20269947

yes

characteristic

126

C₄

#20269961

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#27213021

yes

characteristic

	subgroup	G/H
center	C₄	#20269947
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#413×C₂₁₁	C₂
socle	C₂×C₂₁₁	#385
prime radical	C₂×#41×C₂₁₁	C₂²
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#413	D₂₁₁
hypocenter	C₂₁₁	#1858
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#20269947

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#413×C₂₁₁

C₂

socle

C₂×C₂₁₁

#385

prime radical

C₂×#41×C₂₁₁

C₂²

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. #27213195	C₂	semidirect
4. #27212910	C₂	semidirect
65. C₂₁₁	#1858	semidirect

H ≅ G/N

type

2. #27213195

C₂

semidirect

4. #27212910

C₂

semidirect

65. C₂₁₁

#1858

semidirect

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

subgroup	index
#27213913	2
#27213195	2
#27213113	2
#27212910	2
#413×C₂₁₁	2
#1858	211

subgroup

index

#413×C₂₁₁

211

Subgroups — #41013620

Conjugacy classes of subgroups

Complements

Maximal subgroups