Group #21456248 — subgroups

Subgroups — #21456248

Order 7808.

572 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14519800	C₂	1	4	yes	maximal, complemented
2	#16840313	C₂	1	2	yes	maximal, complemented
3	#16840147	C₂	1	4	yes	maximal, complemented
4	#16840139	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840136	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758226	C₂²	1	4	yes	—
9	#14519800	—	2	16	no	—
10	C₂×#11758201	C₂²	1	8	yes	—
11	C₂×D₄₈₈	C₂²	1	4	yes	—
12	C₂×C₈×D₆₁	C₂²	1	2	yes	—
13	#14519774	C₂²	1	2	yes	—
14	#14519766	C₂²	1	4	yes	—
15	#14519752	—	2	4	no	—
16	#14519739	—	2	4	no	—
17	C₄²×D₆₁	C₂²	1	1	yes	characteristic
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₁	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₆₁	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	C₂³	1	4	yes	—
24	#11758226	—	2	16	no	—
25	C₂×D₂₄₄	—	2	4	no	—
26	C₂×D₂₄₄	—	2	4	no	—
27	C₂×D₂₄₄	C₂³	1	4	yes	—
28	C₂×C₄×D₆₁	—	2	4	no	—
29	C₂×C₄×D₆₁	D₄	1	2	yes	—
30	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
31	Q₁₆×C₆₁	—	2	8	no	—
32	C₂×C₈×C₆₁	C₂³	1	2	yes	—
33	C₄⋊C₄×C₆₁	—	2	4	no	—
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#11758201	—	2	16	no	—
36	#11758198	—	2	8	no	—
37	C₄×Dic₆₁	D₄	1	2	yes	—
38	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
39	C₂×#92418	C₂³	1	2	yes	—
40	D₄₈₈	—	2	8	no	—
41	C₈×D₆₁	D₄	1	8	yes	complemented
42	C₂²×D₆₁	—	2	4	no	—
43	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₆₁	—	2	8	no	—
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
49	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
50	D₂₄₄	—	2	8	no	—
51	D₂₄₄	—	4	8	no	—
52	D₂₄₄	—	4	8	no	—
53	C₄×D₆₁	—	2	4	no	—
54	C₄×D₆₁	—	4	8	no	—
55	C₄×D₆₁	C₂×D₄	1	4	yes	—
56	C₈×C₆₁	C₂×D₄	1	4	yes	—
57	#92418	C₂×D₄	1	4	yes	—
58	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₆₁	—	2	2	no	—
60	C₂×D₆₁	—	4	4	no	—
61	C₂×D₆₁	—	4	8	no	—
62	C₂×D₆₁	#127	1	2	yes	—
63	C₄×C₆₁	—	4	4	no	—
64	C₄×C₆₁	#135	1	4	yes	complemented
65	C₄×C₆₁	C₂²×D₄	1	2	yes	—
66	Dic₆₁	—	2	2	no	—
67	Dic₆₁	#127	1	2	yes	—
68	#3147	—	61	1	no	maximal, Sylow
69	C₂×C₆₁	C₂×#127	1	1	yes	characteristic
70	C₂×C₆₁	C₂×#135	1	2	yes	—
71	D₆₁	—	2	2	no	—
72	D₆₁	—	8	4	no	—
73	C₂×#135	—	61	4	no	—
74	#532	—	61	2	no	—
75	#495	—	61	4	no	—
76	#494	D₆₁	1	1	yes	characteristic, complemented
77	#493	—	61	1	no	—
78	#492	—	61	2	no	—
79	C₂×C₄×C₈	—	61	1	no	—
80	C₆₁	#3147	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	61	4	no	—
82	#135	—	122	16	no	—
83	C₂×Q₁₆	C₂×D₆₁	1	4	yes	—
84	C₂×SD₁₆	—	61	8	no	—
85	C₂×D₈	—	61	4	no	—
86	C₂²×C₈	—	61	2	no	—
87	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
88	#127	—	61	2	no	—
89	#124	—	61	4	no	—
90	#121	—	122	4	no	—
91	C₄×D₄	—	122	4	no	—
92	C₂×C₄²	—	61	1	no	—
93	C₄×C₈	—	61	2	no	—
94	C₄×C₈	—	61	1	no	—
95	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
96	#41	—	122	16	no	—
97	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
98	C₂×D₄	—	61	4	no	—
99	C₂×D₄	—	122	4	no	—
100	C₂×D₄	—	122	4	no	—
101	C₂²×C₄	—	61	1	no	—
102	C₂²×C₄	—	61	2	no	—
103	C₂²×C₄	—	122	4	no	—
104	Q₁₆	—	2	8	no	—
105	SD₁₆	—	122	16	no	—
106	D₈	—	122	8	no	—
107	C₂×C₈	—	61	2	no	—
108	C₂×C₈	—	61	8	no	—
109	C₂×C₈	C₂²×D₆₁	1	2	yes	—
110	C₄⋊C₄	—	2	4	no	—
111	C₂²⋊C₄	—	122	8	no	—
112	C₄²	—	61	2	no	—
113	C₄²	—	61	1	no	—
114	C₄²	C₂²×D₆₁	1	1	yes	characteristic
115	C₂³	—	61	1	no	—
116	C₂³	—	122	4	no	—
117	Q₈	—	2	8	no	—
118	D₄	—	122	8	no	—
119	D₄	—	244	8	no	—
120	D₄	—	244	8	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	61	4	no	—
124	C₂×C₄	—	61	2	no	—
125	C₂×C₄	—	122	4	no	—
126	C₂×C₄	—	244	8	no	—
127	C₂×C₄	D₄×D₆₁	1	2	yes	—
128	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
129	C₈	—	61	4	no	—
130	C₈	D₄×D₆₁	1	4	yes	—
131	C₂²	—	61	2	no	—
132	C₂²	—	122	2	no	—
133	C₂²	—	244	8	no	—
134	C₂²	—	244	4	no	—
135	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	61	2	no	—
138	C₄	—	122	2	no	—
139	C₄	#14519800	1	4	yes	complemented
140	C₄	C₂×D₄×D₆₁	1	2	yes	—
141	C₂	—	122	2	no	—
142	C₂	—	488	4	no	—
143	C₂	#127×D₆₁	1	1	yes	characteristic
144	C₂	C₂×#14519800	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519800

C₂

yes

maximal, complemented

#16840313

C₂

yes

maximal, complemented

#16840147

C₂

yes

maximal, complemented

#16840139

C₂

yes

maximal, characteristic, complemented

#16840136

C₂

yes

maximal, complemented

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#494

C₂

yes

maximal, characteristic, complemented

C₂×#11758226

C₂²

yes

—

#14519800

—

C₂×#11758201

C₂²

yes

—

C₂×D₄₈₈

C₂²

yes

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519774

C₂²

yes

—

#14519766

C₂²

yes

—

#14519752

—

#14519739

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

#11758226

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758201

—

#11758198

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

D₄₈₈

—

C₈×D₆₁

D₄

yes

complemented

C₂²×D₆₁

—

C₂²×D₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₄

yes

—

C₈×C₆₁

C₂×D₄

yes

—

#92418

C₂×D₄

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#127

yes

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

complemented

C₄×C₆₁

C₂²×D₄

yes

—

Dic₆₁

—

Dic₆₁

#127

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

—

D₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

#493

—

#492

—

C₂×C₄×C₈

—

C₆₁

#3147

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

122

—

C₂×Q₁₆

C₂×D₆₁

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#127

—

#124

—

#121

—

122

—

C₄×D₄

—

122

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

#41

—

122

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

122

—

100

C₂×D₄

—

122

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

122

—

104

Q₁₆

—

105

SD₁₆

—

122

—

106

D₈

—

122

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₆₁

yes

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

122

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₆₁

yes

characteristic

115

C₂³

—

116

C₂³

—

122

—

117

Q₈

—

118

D₄

—

122

—

119

D₄

—

244

—

120

D₄

—

244

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

244

—

127

C₂×C₄

D₄×D₆₁

yes

—

128

C₂×C₄

C₂³×D₆₁

yes

characteristic

129

C₈

—

130

C₈

D₄×D₆₁

yes

—

131

C₂²

—

132

C₂²

—

122

—

133

C₂²

—

244

—

134

C₂²

—

244

—

135

C₂²

C₂×D₄×D₆₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

122

—

139

C₄

#14519800

yes

complemented

140

C₄

C₂×D₄×D₆₁

yes

—

141

C₂

—

122

—

142

C₂

—

488

—

143

C₂

#127×D₆₁

yes

characteristic

144

C₂

C₂×#14519800

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#494	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#494	D₆₁
hypocenter	C₆₁	#3147
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#494

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#14519800	C₂	semidirect
2. #16840313	C₂	semidirect
3. #16840147	C₂	semidirect
4. #16840139	C₂	semidirect
5. #16840136	C₂	semidirect
6. C₄×C₈×D₆₁	C₂	semidirect
7. C₆₁×#494	C₂	semidirect
41. C₈×D₆₁	D₄	semidirect
64. C₄×C₆₁	#135	semidirect
76. #494	D₆₁	semidirect
80. C₆₁	#3147	semidirect
139. C₄	#14519800	semidirect

H ≅ G/N

type

1. C₂×#14519800

C₂

semidirect

2. #16840313

C₂

semidirect

3. #16840147

C₂

semidirect

4. #16840139

C₂

semidirect

5. #16840136

C₂

semidirect

6. C₄×C₈×D₆₁

C₂

semidirect

7. C₆₁×#494

C₂

semidirect

41. C₈×D₆₁

D₄

semidirect

64. C₄×C₆₁

#135

semidirect

76. #494

D₆₁

semidirect

80. C₆₁

#3147

semidirect

139. C₄

#14519800

semidirect

Subgroups — #21456248

Conjugacy classes of subgroups

Complements