Group #41013542 — subgroups

Subgroups — #41013542

Order 27008.

196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 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	#27212992	C₂	1	1	yes	maximal, characteristic
2	#27212901	C₂	1	1	yes	maximal, characteristic
3	#27212900	C₂	1	2	yes	maximal
4	#27212900	C₂	1	2	yes	maximal
5	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	C₂₁₁⋊#95	C₄	1	2	yes	—
10	C₂₁₁⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₄×Dic₂₁₁	—	2	1	no	—
20	C₄×Dic₂₁₁	—	2	2	no	—
21	C₄×Dic₂₁₁	—	2	1	no	—
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	—	2	2	no	—
26	C₂×C₈×C₂₁₁	—	2	2	no	—
27	C₄²×C₂₁₁	—	2	2	no	—
28	C₂×Dic₂₁₁	—	2	1	no	—
29	C₂×Dic₂₁₁	—	2	1	no	—
30	C₂×Dic₂₁₁	—	2	2	no	—
31	C₂×Dic₂₁₁	—	2	2	no	—
32	C₂×Dic₂₁₁	—	2	2	no	—
33	C₂×Dic₂₁₁	—	2	2	no	—
34	C₂×Dic₂₁₁	—	2	2	no	—
35	C₂₁₁⋊C₈	—	4	2	no	—
36	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	—	2	2	no	—
38	C₂×C₄×C₂₁₁	—	2	2	no	—
39	C₂×C₄×C₂₁₁	—	2	2	no	—
40	C₂×C₄×C₂₁₁	#41	1	2	yes	—
41	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	#41	1	2	yes	—
43	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
44	C₈×C₂₁₁	—	4	2	no	—
45	C₂²×C₂₁₁	#117	1	1	yes	characteristic
46	C₂²×C₂₁₁	#117	1	1	yes	characteristic
47	C₂²×C₂₁₁	#117	1	1	yes	characteristic
48	C₂²×C₂₁₁	#131	1	2	yes	—
49	C₂²×C₂₁₁	#131	1	2	yes	—
50	C₄×C₂₁₁	—	4	2	no	—
51	C₄×C₂₁₁	#126	1	2	yes	—
52	C₄×C₂₁₁	#126	1	2	yes	—
53	Dic₂₁₁	—	4	1	no	—
54	Dic₂₁₁	—	4	2	no	—
55	Dic₂₁₁	—	4	1	no	—
56	C₂×C₂₁₁	#383	1	1	yes	characteristic
57	C₂×C₂₁₁	#433	1	2	yes	—
58	C₂×C₂₁₁	#433	1	1	yes	characteristic
59	C₂×C₂₁₁	#433	1	2	yes	—
60	C₂×C₂₁₁	#433	1	1	yes	characteristic
61	C₂₁₁	#1840	1	1	yes	characteristic, complemented, Sylow
62	#1840	—	211	1	no	maximal, Sylow
63	#376	—	211	1	no	—
64	#336	—	211	2	no	—
65	#336	—	211	2	no	—
66	#336	—	211	1	no	—
67	#336	D₂₁₁	1	1	yes	characteristic
68	C₂²×C₈	—	211	2	no	—
69	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
70	C₂×C₄²	—	211	1	no	—
71	C₂×C₄²	—	211	1	no	—
72	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
73	#95	—	211	2	no	—
74	#95	—	211	2	no	—
75	C₂²×C₄	—	211	1	no	—
76	C₂²×C₄	—	211	1	no	—
77	C₂²×C₄	—	211	2	no	—
78	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
79	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	422	2	no	—
84	C₂×C₈	—	422	2	no	—
85	C₂×C₈	—	422	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	422	1	no	—
88	C₄²	—	422	2	no	—
89	C₄²	—	422	1	no	—
90	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	422	1	no	—
95	C₂×C₄	—	422	1	no	—
96	C₂×C₄	—	422	2	no	—
97	C₂×C₄	—	422	2	no	—
98	C₂×C₄	—	422	2	no	—
99	C₂×C₄	—	422	2	no	—
100	C₂×C₄	—	422	2	no	—
101	C₂×C₄	#16809961	1	1	yes	characteristic
102	C₂×C₄	#16809961	1	2	yes	—
103	C₂×C₄	#16809962	1	2	yes	—
104	C₂×C₄	#16809963	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	844	2	no	—
107	C₂²	#20269933	1	1	yes	characteristic
108	C₂²	#20269934	1	1	yes	characteristic
109	C₂²	#20269935	1	1	yes	characteristic
110	C₂²	#20269983	1	2	yes	—
111	C₂²	#20269985	1	2	yes	—
112	C₄	—	4	2	no	—
113	C₄	—	844	1	no	—
114	C₄	—	844	2	no	—
115	C₄	—	844	1	no	—
116	C₄	#20269970	1	2	yes	—
117	C₄	#20269972	1	2	yes	—
118	C₂	#27213013	1	1	yes	characteristic
119	C₂	#27213204	1	2	yes	—
120	C₂	#27213205	1	1	yes	characteristic
121	C₂	#27213206	1	2	yes	—
122	C₂	#27213208	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27212992

C₂

yes

maximal, characteristic

#27212901

C₂

yes

maximal, characteristic

#27212900

C₂

yes

maximal

#27212900

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

C₂₁₁⋊#95

C₄

yes

—

C₂₁₁⋊#95

C₄

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₂²×C₂₁₁

#131

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#126

yes

—

C₄×C₂₁₁

#126

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#383

yes

characteristic

C₂×C₂₁₁

#433

yes

—

C₂×C₂₁₁

#433

yes

characteristic

C₂×C₂₁₁

#433

yes

—

C₂×C₂₁₁

#433

yes

characteristic

C₂₁₁

#1840

yes

characteristic, complemented, Sylow

#1840

—

211

maximal, Sylow

#376

—

211

—

#336

—

211

—

#336

—

211

—

#336

—

211

—

#336

D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#95

—

211

—

#95

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₄×D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₄²

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

—

422

—

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

100

C₂×C₄

—

422

—

101

C₂×C₄

#16809961

yes

characteristic

102

C₂×C₄

#16809961

yes

—

103

C₂×C₄

#16809962

yes

—

104

C₂×C₄

#16809963

yes

characteristic

105

C₈

—

106

C₈

—

844

—

107

C₂²

#20269933

yes

characteristic

108

C₂²

#20269934

yes

characteristic

109

C₂²

#20269935

yes

characteristic

110

C₂²

#20269983

yes

—

111

C₂²

#20269985

yes

—

112

C₄

—

113

C₄

—

844

—

114

C₄

—

844

—

115

C₄

—

844

—

116

C₄

#20269970

yes

—

117

C₄

#20269972

yes

—

118

C₂

#27213013

yes

characteristic

119

C₂

#27213204

yes

—

120

C₂

#27213205

yes

characteristic

121

C₂

#27213206

yes

—

122

C₂

#27213208

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator, socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
Fitting	#336×C₂₁₁	C₂
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#336	D₂₁₁
hypocenter	C₂₁₁	#1840
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator, socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

Fitting

#336×C₂₁₁

C₂

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
61. C₂₁₁	#1840	semidirect

H ≅ G/N

type

61. C₂₁₁

#1840

semidirect

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

subgroup	index
#27212992	2
#27212901	2
#27212900	2
#336×C₂₁₁	2
#1840	211

subgroup

index

#336×C₂₁₁

211

Subgroups — #41013542

Conjugacy classes of subgroups

Complements

Maximal subgroups