Group #41014816 — subgroups

Subgroups — #41014816

Order 27008.

222 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20269901	C₂	1	2	yes	maximal
2	#27213012	C₂	1	1	yes	maximal, characteristic
3	#27212908	C₂	1	2	yes	maximal
4	#27212901	C₂	1	1	yes	maximal, characteristic
5	#391×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	#20269901	—	2	4	no	—
9	C₂₁₁⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	#95×C₂₁₁	—	2	2	no	—
14	C₂²×Dic₂₁₁	—	2	2	no	—
15	C₂×C₂₁₁⋊C₈	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₄⋊Dic₂₁₁	—	2	2	no	—
19	C₄⋊Dic₂₁₁	—	2	2	no	—
20	C₂×Q₈×C₂₁₁	—	2	1	no	—
21	C₂×Q₈×C₂₁₁	—	2	2	no	—
22	C₂×Q₈×C₂₁₁	—	2	2	no	—
23	C₂×Q₈×C₂₁₁	D₄	1	2	yes	—
24	C₂²×C₄×C₂₁₁	—	2	1	no	—
25	C₂²×C₄×C₂₁₁	—	2	1	no	—
26	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
27	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₁₁	—	2	2	no	—
29	C₄²×C₂₁₁	—	2	2	no	—
30	C₂×Dic₂₁₁	—	4	2	no	—
31	C₂×Dic₂₁₁	—	4	2	no	—
32	C₂×Dic₂₁₁	—	4	2	no	—
33	C₂₁₁⋊C₈	—	4	2	no	—
34	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	Q₈×C₂₁₁	—	2	4	no	—
36	Q₈×C₂₁₁	—	4	2	no	—
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	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	4	1	no	—
42	C₂×C₄×C₂₁₁	—	4	1	no	—
43	C₂×C₄×C₂₁₁	—	4	1	no	—
44	C₂×C₄×C₂₁₁	—	4	2	no	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	#41	1	2	yes	—
47	C₂×C₄×C₂₁₁	#41	1	2	yes	—
48	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
49	C₂²×C₂₁₁	#121	1	1	yes	characteristic
50	C₂²×C₂₁₁	#123	1	1	yes	characteristic
51	C₂²×C₂₁₁	#124	1	1	yes	characteristic
52	C₂²×C₂₁₁	#135	1	2	yes	—
53	C₂²×C₂₁₁	#137	1	2	yes	—
54	C₄×C₂₁₁	—	4	2	no	—
55	C₄×C₂₁₁	—	4	2	no	—
56	C₄×C₂₁₁	—	8	1	no	—
57	C₄×C₂₁₁	#123	1	2	yes	—
58	C₄×C₂₁₁	#126	1	2	yes	—
59	Dic₂₁₁	—	8	2	no	—
60	C₂×C₂₁₁	#397	1	1	yes	characteristic
61	C₂×C₂₁₁	#464	1	2	yes	—
62	C₂×C₂₁₁	#485	1	2	yes	—
63	C₂×C₂₁₁	#488	1	1	yes	characteristic
64	C₂×C₂₁₁	#491	1	1	yes	characteristic
65	C₂₁₁	#2074	1	1	yes	characteristic, complemented, Sylow
66	#2074	—	211	1	no	maximal, Sylow
67	C₂×#103	—	211	2	no	—
68	#391	D₂₁₁	1	1	yes	characteristic
69	#382	—	211	1	no	—
70	#340	—	211	2	no	—
71	#336	—	211	1	no	—
72	C₂²×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
73	C₂²×C₈	—	211	2	no	—
74	C₂×C₄⋊C₄	—	211	2	no	—
75	C₂×C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
77	#103	—	422	4	no	—
78	#95	—	2	2	no	—
79	#95	—	422	2	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	D₈₄₄	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	422	2	no	—
87	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
88	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
89	C₂×C₈	—	422	2	no	—
90	C₂×C₈	—	422	2	no	—
91	C₂×C₈	—	422	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	422	2	no	—
94	C₄⋊C₄	—	422	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
97	Q₈	—	2	4	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	844	2	no	—
108	C₂×C₄	—	844	2	no	—
109	C₂×C₄	—	844	2	no	—
110	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
111	C₂×C₄	#16809961	1	2	yes	—
112	C₂×C₄	#16809961	1	2	yes	—
113	C₂×C₄	#16809962	1	1	yes	characteristic
114	C₈	—	844	2	no	—
115	C₂²	#20269951	1	1	yes	characteristic
116	C₂²	#20269958	1	1	yes	characteristic
117	C₂²	#20269964	1	1	yes	characteristic
118	C₂²	#20269997	1	2	yes	—
119	C₂²	#20270008	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	1	no	—
123	C₄	—	1688	2	no	—
124	C₄	#20269961	1	2	yes	—
125	C₄	#20269970	1	2	yes	—
126	C₂	#27213071	1	1	yes	characteristic
127	C₂	#27213368	1	2	yes	—
128	C₂	#27213484	1	2	yes	—
129	C₂	#27213500	1	1	yes	characteristic
130	C₂	#27213514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269901

C₂

yes

maximal

#27213012

C₂

yes

maximal, characteristic

#27212908

C₂

yes

maximal

#27212901

C₂

yes

maximal, characteristic

#391×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269901

—

C₂₁₁⋊#95

—

C₂²×Q₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#95×C₂₁₁

—

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#123

yes

—

C₄×C₂₁₁

#126

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

#397

yes

characteristic

C₂×C₂₁₁

#464

yes

—

C₂×C₂₁₁

#485

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#2074

—

211

maximal, Sylow

C₂×#103

—

211

—

#391

D₂₁₁

yes

characteristic

#382

—

211

—

#340

—

211

—

#336

—

211

—

C₂²×Q₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

#95

—

#95

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₈₄₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

—

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

844

—

108

C₂×C₄

—

844

—

109

C₂×C₄

—

844

—

110

C₂×C₄

C₂×D₈₄₄

yes

characteristic

111

C₂×C₄

#16809961

yes

—

112

C₂×C₄

#16809961

yes

—

113

C₂×C₄

#16809962

yes

characteristic

114

C₈

—

844

—

115

C₂²

#20269951

yes

characteristic

116

C₂²

#20269958

yes

characteristic

117

C₂²

#20269964

yes

characteristic

118

C₂²

#20269997

yes

—

119

C₂²

#20270008

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

1688

—

124

C₄

#20269961

yes

—

125

C₄

#20269970

yes

—

126

C₂

#27213071

yes

characteristic

127

C₂

#27213368

yes

—

128

C₂

#27213484

yes

—

129

C₂

#27213500

yes

characteristic

130

C₂

#27213514

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#391×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
hypercenter	#391	D₂₁₁
hypocenter	C₂₁₁	#2074
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator, cosocle

C₂²×C₄×C₂₁₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#391×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

D₂₁₁

C₂₁₁

G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
65. C₂₁₁	#2074	semidirect

H ≅ G/N

type

65. C₂₁₁

#2074

semidirect

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

subgroup	index
C₂×#20269901	2
#27213012	2
#27212908	2
#27212901	2
#391×C₂₁₁	2
#2074	211

subgroup

index

#391×C₂₁₁

211

Subgroups — #41014816

Conjugacy classes of subgroups

Complements

Maximal subgroups