Group #41014549 — subgroups

Subgroups — #41014549

Order 27008.

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

normal subgroups only

This group has 258 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	C₂×#20269969	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269901	C₂	1	2	yes	maximal
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	#20269969	—	2	4	no	—
8	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	C₂×Dic₂₁₁⋊C₄	—	2	2	no	—
10	#20269901	—	2	4	no	—
11	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	#95×C₂₁₁	—	2	2	no	—
15	C₂²×Dic₂₁₁	—	2	2	no	—
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	2	no	—
20	C₄⋊Dic₂₁₁	D₄	1	4	yes	—
21	Dic₂₁₁⋊C₄	—	2	8	no	—
22	C₂×Q₈×C₂₁₁	—	2	1	no	—
23	C₂×Q₈×C₂₁₁	—	2	2	no	—
24	C₂×Q₈×C₂₁₁	—	2	2	no	—
25	C₂×Q₈×C₂₁₁	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₁₁	—	2	1	no	—
27	C₂²×C₄×C₂₁₁	—	2	1	no	—
28	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
29	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₁₁	—	2	2	no	—
31	C₄²×C₂₁₁	—	2	2	no	—
32	C₂×Dic₂₁₁	—	2	4	no	—
33	C₂×Dic₂₁₁	—	4	2	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂₁₁⋊C₈	—	4	2	no	—
36	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₂₁₁	—	2	4	no	—
38	Q₈×C₂₁₁	—	4	2	no	—
39	C₂×C₄×C₂₁₁	—	2	2	no	—
40	C₂×C₄×C₂₁₁	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	2	2	no	—
42	C₂×C₄×C₂₁₁	—	2	2	no	—
43	C₂×C₄×C₂₁₁	—	4	1	no	—
44	C₂×C₄×C₂₁₁	—	4	1	no	—
45	C₂×C₄×C₂₁₁	—	4	1	no	—
46	C₂×C₄×C₂₁₁	—	4	2	no	—
47	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₂₁₁	#41	1	2	yes	—
50	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
51	C₂²×C₂₁₁	#120	1	1	yes	characteristic
52	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₂₁₁	#124	1	1	yes	characteristic
54	C₂²×C₂₁₁	#135	1	2	yes	—
55	C₂²×C₂₁₁	#137	1	2	yes	—
56	C₄×C₂₁₁	—	4	2	no	—
57	C₄×C₂₁₁	—	4	2	no	—
58	C₄×C₂₁₁	—	8	1	no	—
59	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₂₁₁	#124	1	2	yes	—
61	Dic₂₁₁	—	4	4	no	—
62	C₂×C₂₁₁	#393	1	1	yes	characteristic
63	C₂×C₂₁₁	#452	1	2	yes	—
64	C₂×C₂₁₁	#479	1	2	yes	—
65	C₂×C₂₁₁	#488	1	1	yes	characteristic
66	C₂×C₂₁₁	#491	1	1	yes	characteristic
67	C₂₁₁	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	211	1	no	maximal, Sylow
69	C₂×#125	—	211	1	no	—
70	C₂×#103	—	211	2	no	—
71	#391	D₂₁₁	1	1	yes	characteristic
72	#340	—	211	2	no	—
73	#336	—	211	1	no	—
74	C₂²×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
75	C₂²×C₈	—	211	2	no	—
76	#125	—	422	4	no	—
77	C₂×C₄⋊C₄	—	422	2	no	—
78	C₂×C₄⋊C₄	—	211	2	no	—
79	C₂×C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
81	#103	—	422	4	no	—
82	#95	—	2	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	Dic₄₂₂	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	422	2	no	—
90	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
92	C₂×C₈	—	422	2	no	—
93	C₂×C₈	—	422	2	no	—
94	C₂×C₈	—	422	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	422	8	no	—
97	C₄⋊C₄	—	422	2	no	—
98	C₄⋊C₄	—	211	4	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	844	2	no	—
112	C₂×C₄	—	844	2	no	—
113	C₂×C₄	—	422	4	no	—
114	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
115	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
116	C₂×C₄	#16809961	1	2	yes	—
117	C₂×C₄	#16809962	1	1	yes	characteristic
118	C₈	—	844	2	no	—
119	C₂²	#20269945	1	1	yes	characteristic
120	C₂²	#20269956	1	1	yes	characteristic
121	C₂²	#20269964	1	1	yes	characteristic
122	C₂²	#20269997	1	2	yes	—
123	C₂²	#20270008	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	—	8	1	no	—
127	C₄	—	844	4	no	—
128	C₄	#20269954	1	2	yes	—
129	C₄	#20269963	1	2	yes	—
130	C₂	#27213058	1	1	yes	characteristic
131	C₂	#27213306	1	2	yes	—
132	C₂	#27213453	1	2	yes	—
133	C₂	#27213500	1	1	yes	characteristic
134	C₂	#27213514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269969

C₂

yes

maximal, characteristic

C₂×#20269901

C₂

yes

maximal

#27212908

C₂

yes

maximal

#27212901

C₂

yes

maximal, characteristic

#391×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

#20269969

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×Dic₂₁₁⋊C₄

—

#20269901

—

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

D₄

yes

—

Dic₂₁₁⋊C₄

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

Q₈

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

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₂²⋊Q₈

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#124

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

#393

yes

characteristic

C₂×C₂₁₁

#452

yes

—

C₂×C₂₁₁

#479

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#2026

—

211

maximal, Sylow

C₂×#125

—

211

—

C₂×#103

—

211

—

#391

D₂₁₁

yes

characteristic

#340

—

211

—

#336

—

211

—

C₂²×Q₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

#125

—

422

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₄₂₂

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

—

C₄⋊C₄

—

211

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

844

—

112

C₂×C₄

—

844

—

113

C₂×C₄

—

422

—

114

C₂×C₄

D₄×D₂₁₁

yes

—

115

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

116

C₂×C₄

#16809961

yes

—

117

C₂×C₄

#16809962

yes

characteristic

118

C₈

—

844

—

119

C₂²

#20269945

yes

characteristic

120

C₂²

#20269956

yes

characteristic

121

C₂²

#20269964

yes

characteristic

122

C₂²

#20269997

yes

—

123

C₂²

#20270008

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

844

—

128

C₄

#20269954

yes

—

129

C₄

#20269963

yes

—

130

C₂

#27213058

yes

characteristic

131

C₂

#27213306

yes

—

132

C₂

#27213453

yes

—

133

C₂

#27213500

yes

characteristic

134

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	#2026
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
67. C₂₁₁	#2026	semidirect

H ≅ G/N

type

67. C₂₁₁

#2026

semidirect

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

subgroup	index
C₂×#20269969	2
C₂×#20269901	2
#27212908	2
#27212901	2
#391×C₂₁₁	2
#2026	211

subgroup

index

#391×C₂₁₁

211

Subgroups — #41014549

Conjugacy classes of subgroups

Complements

Maximal subgroups