Group #41014178 — subgroups

Subgroups — #41014178

Order 27008.

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

normal subgroups only

This group has 196 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	C₂×#20269913	C₂	1	1	yes	maximal, characteristic
2	#27213189	C₂	1	2	yes	maximal
3	C₂×#20269909	C₂	1	1	yes	maximal, characteristic
4	#27212914	C₂	1	2	yes	maximal
5	#405×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	#20269913	C₄	1	4	yes	complemented
9	#20269910	—	2	2	no	—
10	#20269909	—	2	1	no	—
11	#20269909	—	2	1	no	—
12	#20269908	—	2	2	no	—
13	C₂×M₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	#117×C₂₁₁	C₂²	1	1	yes	characteristic
16	#97×C₂₁₁	—	2	1	no	—
17	C₄×C₈×C₂₁₁	—	2	1	no	—
18	C₂²×Dic₂₁₁	—	2	2	no	—
19	#16809946	—	2	2	no	—
20	#16809946	C₂×C₄	1	4	yes	—
21	C₂×C₂₁₁⋊C₈	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	2	no	—
23	C₄⋊Dic₂₁₁	—	2	2	no	—
24	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₁₁	—	2	1	no	—
26	M₁₆×C₂₁₁	Q₈	1	2	yes	—
27	C₂×C₈×C₂₁₁	—	2	1	no	—
28	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₁₁	Q₈	1	2	yes	—
32	C₄⋊C₄×C₂₁₁	D₄	1	2	yes	—
33	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
34	C₄²×C₂₁₁	—	2	1	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	8	2	no	—
37	C₂₁₁⋊C₈	—	2	4	no	—
38	C₂³×C₂₁₁	#41	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	—	2	2	no	—
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
46	C₈×C₂₁₁	—	2	2	no	—
47	C₈×C₂₁₁	C₄⋊C₄	1	2	yes	—
48	C₈×C₂₁₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₂₁₁	—	2	2	no	—
50	C₂²×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₂₁₁	#121	1	1	yes	characteristic
52	C₂²×C₂₁₁	#125	1	1	yes	characteristic
53	C₄×C₂₁₁	—	4	2	no	—
54	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
56	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₁₁	—	8	2	no	—
59	C₂×C₂₁₁	—	2	1	no	—
60	C₂×C₂₁₁	—	2	1	no	—
61	C₂×C₂₁₁	#389	1	1	yes	characteristic
62	C₂×C₂₁₁	#471	1	2	yes	—
63	C₂₁₁	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	211	1	no	maximal, Sylow
65	C₂×#108	—	211	1	no	—
66	#428	—	211	2	no	—
67	C₂×#106	—	211	1	no	—
68	#405	D₂₁₁	1	1	yes	characteristic
69	#343	—	211	2	no	—
70	C₂×M₁₆	—	211	2	no	—
71	C₂×M₁₆	C₂×D₂₁₁	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
73	#117	C₂×D₂₁₁	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	211	2	no	—
75	#108	—	211	4	no	—
76	#107	—	422	2	no	—
77	#106	—	422	1	no	—
78	#106	—	422	1	no	—
79	#106	—	422	2	no	—
80	#97	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₂²×C₄	—	422	2	no	—
83	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	422	2	no	—
86	M₁₆	—	211	4	no	—
87	M₁₆	Dic₄₂₂	1	2	yes	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	422	2	no	—
90	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
91	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
93	C₂×C₈	Dic₄₂₂	1	2	yes	—
94	C₄⋊C₄	—	422	2	no	—
95	C₄⋊C₄	—	422	2	no	—
96	C₄⋊C₄	D₈₄₄	1	2	yes	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₂³	#16809962	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	844	2	no	—
102	C₂×C₄	—	1688	2	no	—
103	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
105	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
108	C₂×C₄	#16809962	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	422	4	no	—
111	C₈	Dic₂₁₁⋊C₄	1	2	yes	—
112	C₈	Dic₂₁₁⋊C₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄×Dic₄₂₂	1	1	yes	characteristic
115	C₂²	#20269951	1	1	yes	characteristic
116	C₂²	#20269969	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	1688	2	no	—
119	C₄	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
120	C₄	C₄×D₈₄₄	1	1	yes	characteristic
121	C₄	#20269956	1	1	yes	characteristic
122	C₄	#20269977	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#27213044	1	1	yes	characteristic
126	C₂	#27213409	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269913

C₂

yes

maximal, characteristic

#27213189

C₂

yes

maximal

C₂×#20269909

C₂

yes

maximal, characteristic

#27212914

C₂

yes

maximal

#405×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269913

C₄

yes

—

—

—

—

C₂×M₁₆×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

C₂²

yes

characteristic

#97×C₂₁₁

—

C₄×C₈×C₂₁₁

—

C₂²×Dic₂₁₁

—

#16809946

—

#16809946

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

Q₈

yes

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

Q₈

yes

—

C₄⋊C₄×C₂₁₁

D₄

yes

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

—

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×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

C₄⋊C₄

yes

—

C₈×C₂₁₁

C₄⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₄×Q₈

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

characteristic

C₄×C₂₁₁

C₂²⋊Q₈

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#389

yes

characteristic

C₂×C₂₁₁

#471

yes

—

C₂₁₁

#1951

yes

characteristic, complemented, Sylow

#1951

—

211

maximal, Sylow

C₂×#108

—

211

—

#428

—

211

—

C₂×#106

—

211

—

#405

D₂₁₁

yes

characteristic

#343

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#117

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

#108

—

211

—

#107

—

422

—

#106

—

422

—

#106

—

422

—

#106

—

422

—

#97

—

C₄×C₈

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

211

—

M₁₆

Dic₄₂₂

yes

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₂×C₈

C₂²⋊D₂₁₁

yes

characteristic

C₂×C₈

C₂²⋊D₂₁₁

yes

characteristic

C₂×C₈

Dic₄₂₂

yes

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

D₈₄₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#16809962

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

844

—

102

C₂×C₄

—

1688

—

103

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

105

C₂×C₄

C₂×D₈₄₄

yes

characteristic

106

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

107

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

108

C₂×C₄

#16809962

yes

characteristic

109

C₈

—

110

C₈

—

422

—

111

C₈

Dic₂₁₁⋊C₄

yes

—

112

C₈

Dic₂₁₁⋊C₄

yes

—

113

C₂²

—

114

C₂²

C₄×Dic₄₂₂

yes

characteristic

115

C₂²

#20269951

yes

characteristic

116

C₂²

#20269969

yes

characteristic

117

C₄

—

118

C₄

—

1688

—

119

C₄

C₂×Dic₂₁₁⋊C₄

yes

characteristic

120

C₄

C₄×D₈₄₄

yes

characteristic

121

C₄

#20269956

yes

characteristic

122

C₄

#20269977

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#27213044

yes

characteristic

126

C₂

#27213409

yes

—

	subgroup	G/H
center	C₂²	#20269951
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#405×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#121
prime radical	C₂³×C₂₁₁	#41
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#405	D₂₁₁
hypocenter	C₂₁₁	#1951
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#20269951

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#405×C₂₁₁

C₂

C₂²×C₂₁₁

C₂³×C₂₁₁

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #20269913	C₄	semidirect
63. C₂₁₁	#1951	semidirect

H ≅ G/N

type

8. #20269913

C₄

semidirect

63. C₂₁₁

#1951

semidirect

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

subgroup	index
C₂×#20269913	2
#27213189	2
C₂×#20269909	2
#27212914	2
#405×C₂₁₁	2
#1951	211

subgroup

index

#405×C₂₁₁

211

Subgroups — #41014178

Conjugacy classes of subgroups

Complements

Maximal subgroups