Group #41013025 — subgroups

Subgroups — #41013025

Order 27008.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#27213542	C₂	1	1	yes	maximal, characteristic, complemented
2	#27213522	C₂	1	1	yes	maximal, characteristic
3	#27213518	C₂	1	1	yes	maximal, characteristic, complemented
4	#27213494	C₂	1	1	yes	maximal, characteristic, complemented
5	#27213493	C₂	1	1	yes	maximal, characteristic, complemented
6	#27212868	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16809950	—	2	2	no	—
9	C₂×#16809947	—	2	2	no	—
10	#20269977	C₂²	1	1	yes	characteristic
11	#20269974	C₂²	1	1	yes	characteristic
12	#20269908	—	2	2	no	—
13	#20269899	—	2	2	no	—
14	#20269897	—	2	2	no	—
15	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₂₁₁	—	2	2	no	—
18	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	#127×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₂₂	—	2	1	no	—
22	C₂×D₈₄₄	—	2	1	no	—
23	C₂×D₈₄₄	—	2	1	no	—
24	C₂×D₈₄₄	—	2	1	no	—
25	#16809950	—	4	4	no	—
26	#16809947	—	4	4	no	—
27	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
28	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₂₁₁	—	2	1	no	—
30	C₄⋊Dic₂₁₁	—	2	1	no	—
31	C₂×Q₈×C₂₁₁	—	2	1	no	—
32	C₂×D₄×C₂₁₁	—	2	1	no	—
33	C₂×D₄×C₂₁₁	—	2	1	no	—
34	C₂×D₄×C₂₁₁	—	2	1	no	—
35	SD₁₆×C₂₁₁	—	4	4	no	—
36	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
37	C₄⋊C₄×C₂₁₁	—	2	1	no	—
38	C₄⋊C₄×C₂₁₁	—	2	1	no	—
39	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
40	C₂²×D₂₁₁	—	4	1	no	—
41	Dic₄₂₂	—	4	2	no	—
42	D₈₄₄	—	4	2	no	—
43	D₈₄₄	—	4	2	no	—
44	D₈₄₄	—	4	2	no	—
45	C₂×Dic₂₁₁	—	4	1	no	—
46	C₂₁₁⋊C₈	—	2	2	no	—
47	C₂₁₁⋊C₈	SD₁₆	1	4	yes	complemented
48	C₂³×C₂₁₁	—	4	1	no	—
49	Q₈×C₂₁₁	—	4	2	no	—
50	D₄×C₂₁₁	—	4	2	no	—
51	D₄×C₂₁₁	—	4	2	no	—
52	D₄×C₂₁₁	—	4	2	no	—
53	C₂×C₄×C₂₁₁	—	4	1	no	—
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
57	C₈×C₂₁₁	—	2	2	no	—
58	C₂²×C₂₁₁	—	8	1	no	—
59	C₂²×C₂₁₁	—	8	1	no	—
60	C₂²×C₂₁₁	—	8	1	no	—
61	C₂²×C₂₁₁	#127	1	1	yes	characteristic
62	C₂×D₂₁₁	—	8	1	no	—
63	C₂×D₂₁₁	—	8	1	no	—
64	C₂×D₂₁₁	—	8	1	no	—
65	C₄×C₂₁₁	—	8	1	no	—
66	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
67	C₄×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₂₁₁	—	8	1	no	—
70	C₂×C₂₁₁	—	16	1	no	—
71	C₂×C₂₁₁	#492	1	1	yes	characteristic
72	C₂×C₂₁₁	#496	1	1	yes	characteristic
73	C₂×C₂₁₁	#496	1	1	yes	characteristic
74	D₂₁₁	—	16	1	no	—
75	C₂₁₁	#1715	1	1	yes	characteristic, complemented, Sylow
76	#1715	—	211	1	no	maximal, Sylow
77	#498	—	211	1	no	—
78	#493	—	211	1	no	—
79	#492	—	211	1	no	—
80	#492	D₂₁₁	1	1	yes	characteristic, complemented
81	#486	—	211	1	no	—
82	#486	—	211	1	no	—
83	#322	—	211	1	no	—
84	C₂×SD₁₆	—	2	2	no	—
85	C₂×SD₁₆	—	422	2	no	—
86	C₂×D₈	—	422	2	no	—
87	C₄⋊Q₈	—	211	1	no	—
88	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
89	#127	—	211	1	no	—
90	#127	C₂×D₂₁₁	1	1	yes	characteristic
91	#106	—	422	2	no	—
92	#102	—	422	2	no	—
93	#102	—	422	2	no	—
94	C₄×C₈	—	211	1	no	—
95	C₄×C₈	—	211	1	no	—
96	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	422	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	422	1	no	—
103	C₂×D₄	—	422	1	no	—
104	C₂×D₄	—	422	1	no	—
105	SD₁₆	—	4	4	no	—
106	SD₁₆	—	844	4	no	—
107	D₈	—	844	4	no	—
108	C₂×C₈	—	211	2	no	—
109	C₂×C₈	—	211	2	no	—
110	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	422	1	no	—
114	C₄⋊C₄	—	422	1	no	—
115	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	844	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	844	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	844	2	no	—
124	D₄	—	844	2	no	—
125	D₄	—	844	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	844	1	no	—
128	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	422	2	no	—
133	C₈	—	211	4	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	1688	1	no	—
138	C₂²	—	1688	1	no	—
139	C₂²	—	1688	1	no	—
140	C₂²	#20269973	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	1688	1	no	—
143	C₄	SD₁₆×D₂₁₁	1	2	yes	—
144	C₄	#20270003	1	2	yes	—
145	C₄	#20270004	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	3376	1	no	—
148	C₂	#27213517	1	1	yes	characteristic
149	C₂	#27213532	1	1	yes	characteristic
150	C₂	#27213533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213542

C₂

yes

maximal, characteristic, complemented

#27213522

C₂

yes

maximal, characteristic

#27213518

C₂

yes

maximal, characteristic, complemented

#27213494

C₂

yes

maximal, characteristic, complemented

#27213493

C₂

yes

maximal, characteristic, complemented

#27212868

C₂

yes

maximal, characteristic, complemented

#492×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×#16809950

—

C₂×#16809947

—

#20269977

C₂²

yes

characteristic

#20269974

C₂²

yes

—

—

—

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

characteristic

#127×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×D₈₄₄

—

C₂×D₈₄₄

—

C₂×D₈₄₄

—

#16809950

—

#16809947

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

SD₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂²×D₂₁₁

—

Dic₄₂₂

—

D₈₄₄

—

D₈₄₄

—

D₈₄₄

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

SD₁₆

yes

complemented

C₂³×C₂₁₁

—

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#127

yes

characteristic

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×SD₁₆

yes

—

C₄×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₁₁

AGL(1,ℤ/8)

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

D₂₁₁

—

C₂₁₁

#1715

yes

characteristic, complemented, Sylow

#1715

—

211

maximal, Sylow

#498

—

211

—

#493

—

211

—

#492

—

211

—

#492

D₂₁₁

yes

characteristic, complemented

#486

—

211

—

#486

—

211

—

#322

—

211

—

C₂×SD₁₆

—

C₂×SD₁₆

—

422

—

C₂×D₈

—

422

—

C₄⋊Q₈

—

211

—

C₄⋊Q₈

C₂×D₂₁₁

yes

characteristic

#127

—

211

—

#127

C₂×D₂₁₁

yes

characteristic

#106

—

422

—

#102

—

422

—

#102

—

422

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₄×C₈

C₂×D₂₁₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

422

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

422

—

103

C₂×D₄

—

422

—

104

C₂×D₄

—

422

—

105

SD₁₆

—

106

SD₁₆

—

844

—

107

D₈

—

844

—

108

C₂×C₈

—

211

—

109

C₂×C₈

—

211

—

110

C₂×C₈

C₂²⋊D₂₁₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

422

—

114

C₄⋊C₄

—

422

—

115

C₄²

C₂²×D₂₁₁

yes

characteristic

116

C₂³

—

117

C₂³

—

844

—

118

Q₈

—

119

Q₈

—

844

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

844

—

124

D₄

—

844

—

125

D₄

—

844

—

126

C₂×C₄

—

127

C₂×C₄

—

844

—

128

C₂×C₄

D₄×D₂₁₁

yes

characteristic

129

C₂×C₄

D₄×D₂₁₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

131

C₈

—

132

C₈

—

422

—

133

C₈

—

211

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

1688

—

138

C₂²

—

1688

—

139

C₂²

—

1688

—

140

C₂²

#20269973

yes

characteristic

141

C₄

—

142

C₄

—

1688

—

143

C₄

SD₁₆×D₂₁₁

yes

—

144

C₄

#20270003

yes

—

145

C₄

#20270004

yes

—

146

C₂

—

147

C₂

—

3376

—

148

C₂

#27213517

yes

characteristic

149

C₂

#27213532

yes

characteristic

150

C₂

#27213533

yes

characteristic

	subgroup	G/H
center	C₂²	#20269973
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#492×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#127
prime radical	#27213522	C₂
hypercenter	#492	D₂₁₁
hypocenter	C₂₁₁	#1715
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20269973

commutator, cosocle

C₄²×C₂₁₁

C₂³

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#492×C₂₁₁

C₂

C₂²×C₂₁₁

C₂

D₂₁₁

C₂₁₁

G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27213542	C₂	semidirect
3. #27213518	C₂	semidirect
4. #27213494	C₂	semidirect
5. #27213493	C₂	semidirect
6. #27212868	C₂	semidirect
7. #492×C₂₁₁	C₂	semidirect
47. C₂₁₁⋊C₈	SD₁₆	semidirect
75. C₂₁₁	#1715	semidirect
80. #492	D₂₁₁	semidirect

H ≅ G/N

type

1. #27213542

C₂

semidirect

3. #27213518

C₂

semidirect

4. #27213494

C₂

semidirect

5. #27213493

C₂

semidirect

6. #27212868

C₂

semidirect

7. #492×C₂₁₁

C₂

semidirect

47. C₂₁₁⋊C₈

SD₁₆

semidirect

75. C₂₁₁

#1715

semidirect

80. #492

D₂₁₁

semidirect

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

subgroup	index
#27213542	2
#27213522	2
#27213518	2
#27213494	2
#27213493	2
#27212868	2
#492×C₂₁₁	2
#1715	211

subgroup

index

#492×C₂₁₁

211

Subgroups — #41013025

Conjugacy classes of subgroups

Complements

Maximal subgroups