Group #41012800 — subgroups

Subgroups — #41012800

Order 27008.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27213549	C₂	1	1	yes	maximal, characteristic
2	#27213441	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27212898	C₂	1	1	yes	maximal, characteristic
5	#27212895	C₂	1	1	yes	maximal, characteristic
6	#27212894	C₂	1	1	yes	maximal, characteristic
7	#477×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	#20269977	C₂²	1	1	yes	characteristic
9	#20269910	—	2	1	no	—
10	#20269910	—	2	1	no	—
11	#20269910	—	2	1	no	—
12	C₄⋊(C₂₁₁⋊C₈)	—	2	1	no	—
13	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#20269901	—	2	1	no	—
16	C₄×C₂₁₁⋊C₈	—	2	1	no	—
17	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	#107×C₂₁₁	—	2	1	no	—
21	C₄⋊C₈×C₂₁₁	C₂²	1	1	yes	characteristic
22	#103×C₂₁₁	—	2	1	no	—
23	C₂×Dic₄₂₂	—	2	1	no	—
24	C₂×C₂₁₁⋊C₈	—	2	1	no	—
25	C₂×C₂₁₁⋊C₈	—	2	1	no	—
26	C₂×C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₂₁₁	—	2	1	no	—
29	C₄⋊Dic₂₁₁	—	2	1	no	—
30	C₂×Q₈×C₂₁₁	—	2	1	no	—
31	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₁₁	—	2	1	no	—
33	C₄⋊C₄×C₂₁₁	—	2	1	no	—
34	C₄⋊C₄×C₂₁₁	—	2	1	no	—
35	C₄⋊C₄×C₂₁₁	—	2	1	no	—
36	C₄⋊C₄×C₂₁₁	D₄	1	1	yes	characteristic
37	C₄²×C₂₁₁	—	2	1	no	—
38	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
39	Dic₄₂₂	—	8	1	no	—
40	C₂×Dic₂₁₁	—	4	1	no	—
41	C₂₁₁⋊C₈	—	2	1	no	—
42	C₂₁₁⋊C₈	—	4	1	no	—
43	C₂₁₁⋊C₈	—	4	1	no	—
44	C₂₁₁⋊C₈	Q₁₆	1	2	yes	—
45	Q₈×C₂₁₁	—	2	1	no	—
46	Q₈×C₂₁₁	—	8	1	no	—
47	Q₈×C₂₁₁	Q₁₆	1	2	yes	—
48	C₂×C₄×C₂₁₁	—	2	1	no	—
49	C₂×C₄×C₂₁₁	—	2	1	no	—
50	C₂×C₄×C₂₁₁	—	4	1	no	—
51	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
54	C₈×C₂₁₁	—	4	1	no	—
55	C₂²×C₂₁₁	#121	1	1	yes	characteristic
56	C₄×C₂₁₁	—	2	2	no	—
57	C₄×C₂₁₁	—	2	1	no	—
58	C₄×C₂₁₁	—	4	1	no	—
59	C₄×C₂₁₁	—	8	1	no	—
60	C₄×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₂₁₁	#135	1	1	yes	characteristic
63	C₄×C₂₁₁	#137	1	1	yes	characteristic
64	Dic₂₁₁	—	8	1	no	—
65	C₂×C₂₁₁	#462	1	1	yes	characteristic
66	C₂×C₂₁₁	#467	1	1	yes	characteristic
67	C₂×C₂₁₁	#472	1	1	yes	characteristic
68	C₂₁₁	#1676	1	1	yes	characteristic, complemented, Sylow
69	#1676	—	211	1	no	maximal, Sylow
70	#500	—	211	1	no	—
71	#477	—	211	1	no	—
72	#477	D₂₁₁	1	1	yes	characteristic
73	C₈×Q₈	—	211	1	no	—
74	#335	—	211	1	no	—
75	#333	—	211	1	no	—
76	#333	—	211	1	no	—
77	C₄⋊Q₈	—	211	1	no	—
78	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
79	C₄×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
80	#107	—	2	1	no	—
81	#107	—	422	1	no	—
82	#107	—	422	1	no	—
83	#107	—	422	1	no	—
84	C₄⋊C₈	—	422	1	no	—
85	C₄⋊C₈	—	211	1	no	—
86	C₄⋊C₈	—	211	1	no	—
87	C₄⋊C₈	C₂×D₂₁₁	1	1	yes	characteristic
88	#103	—	2	1	no	—
89	#103	—	422	1	no	—
90	C₄×C₈	—	422	1	no	—
91	C₄×C₈	—	211	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	422	1	no	—
94	C₂×Q₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	422	1	no	—
97	C₂×C₈	—	422	1	no	—
98	C₂×C₈	—	211	1	no	—
99	C₂×C₈	—	211	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	422	1	no	—
104	C₄⋊C₄	—	422	1	no	—
105	C₄⋊C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
108	Q₈	—	2	1	no	—
109	Q₈	—	8	1	no	—
110	Q₈	—	1688	1	no	—
111	Q₈	#16809952	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	844	1	no	—
116	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
118	C₂×C₄	#16809961	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	844	1	no	—
121	C₈	—	844	1	no	—
122	C₈	—	422	1	no	—
123	C₈	—	211	2	no	—
124	C₂²	#20269949	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	4	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	1688	1	no	—
130	C₄	Q₁₆×D₂₁₁	1	1	yes	characteristic
131	C₄	C₂×#16809952	1	1	yes	characteristic
132	C₄	#20269997	1	1	yes	characteristic
133	C₄	#20270009	1	1	yes	characteristic
134	C₂	#27213354	1	1	yes	characteristic
135	C₂	#27213385	1	1	yes	characteristic
136	C₂	#27213414	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213549

C₂

yes

maximal, characteristic

#27213441

C₂

yes

maximal, characteristic

Q₈×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27212898

C₂

yes

maximal, characteristic

#27212895

C₂

yes

maximal, characteristic

#27212894

C₂

yes

maximal, characteristic

#477×C₂₁₁

C₂

yes

maximal, characteristic

#20269977

C₂²

yes

—

—

—

C₄⋊(C₂₁₁⋊C₈)

—

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₁₁⋊C₈)

C₂²

yes

characteristic

#20269901

—

C₄×C₂₁₁⋊C₈

—

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

C₄×Q₈×C₂₁₁

C₂²

yes

characteristic

#107×C₂₁₁

—

C₄⋊C₈×C₂₁₁

C₂²

yes

characteristic

#103×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

characteristic

C₂×C₂₁₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

Q₁₆

yes

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×Q₁₆

yes

characteristic

C₄×C₂₁₁

C₂×Q₁₆

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1676

—

211

maximal, Sylow

#500

—

211

—

#477

—

211

—

#477

D₂₁₁

yes

characteristic

C₈×Q₈

—

211

—

#335

—

211

—

#333

—

211

—

#333

—

211

—

C₄⋊Q₈

—

211

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

C₄×Q₈

C₂×D₂₁₁

yes

characteristic

#107

—

#107

—

422

—

#107

—

422

—

#107

—

422

—

C₄⋊C₈

—

422

—

C₄⋊C₈

—

211

—

C₄⋊C₈

—

211

—

C₄⋊C₈

C₂×D₂₁₁

yes

characteristic

#103

—

#103

—

422

—

C₄×C₈

—

422

—

C₄×C₈

—

211

—

C₂×Q₈

—

C₂×Q₈

—

422

—

C₂×Q₈

C₂²⋊D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

422

—

104

C₄⋊C₄

—

422

—

105

C₄⋊C₄

C₂²⋊D₂₁₁

yes

characteristic

106

C₄²

—

107

C₄²

C₂²×D₂₁₁

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

1688

—

111

Q₈

#16809952

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

844

—

116

C₂×C₄

D₄×D₂₁₁

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

118

C₂×C₄

#16809961

yes

characteristic

119

C₈

—

120

C₈

—

844

—

121

C₈

—

844

—

122

C₈

—

422

—

123

C₈

—

211

—

124

C₂²

#20269949

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

1688

—

130

C₄

Q₁₆×D₂₁₁

yes

characteristic

131

C₄

C₂×#16809952

yes

characteristic

132

C₄

#20269997

yes

characteristic

133

C₄

#20270009

yes

characteristic

134

C₂

#27213354

yes

characteristic

135

C₂

#27213385

yes

characteristic

136

C₂

#27213414

yes

characteristic

	subgroup	G/H
center	C₂²	#20269949
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#477×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	#121
hypercenter	#477	D₂₁₁
hypocenter	C₂₁₁	#1676
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#20269949

commutator, cosocle

C₄²×C₂₁₁

C₂³

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#477×C₂₁₁

C₂

C₂²×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
68. C₂₁₁	#1676	semidirect

H ≅ G/N

type

68. C₂₁₁

#1676

semidirect

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

subgroup	index
#27213549	2
#27213441	2
Q₈×C₂₁₁⋊C₈	2
#27212898	2
#27212895	2
#27212894	2
#477×C₂₁₁	2
#1676	211

subgroup

index

Q₈×C₂₁₁⋊C₈

#477×C₂₁₁

211

Subgroups — #41012800

Conjugacy classes of subgroups

Complements

Maximal subgroups