Group #21456232 — subgroups

Subgroups — #494×D₆₁ #21456232

Order 7808.

552 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 165 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×Q₁₆×D₆₁	C₂	1	4	yes	maximal
2	C₄⋊Q₈×D₆₁	C₂	1	2	yes	maximal
3	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840141	C₂	1	2	yes	maximal, complemented
5	#16840140	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	C₂²	1	4	yes	—
9	C₂×Dic₂₄₄	C₂²	1	4	yes	—
10	C₂×#11758202	C₂²	1	8	yes	—
11	Q₁₆×D₆₁	—	2	16	no	—
12	C₂×C₈×D₆₁	C₂²	1	2	yes	—
13	#14519777	C₂²	1	2	yes	—
14	#14519776	—	2	4	no	—
15	#14519775	C₂²	1	4	yes	—
16	C₄⋊C₄×D₆₁	—	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	Q₈×D₆₁	—	2	16	no	—
25	C₂×C₄×D₆₁	—	2	4	no	—
26	C₂×C₄×D₆₁	D₄	1	2	yes	—
27	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₂₂	—	2	4	no	—
29	C₂×Dic₁₂₂	C₂³	1	4	yes	—
30	Q₁₆×C₆₁	—	2	8	no	—
31	C₂×C₈×C₆₁	C₂³	1	2	yes	—
32	C₄⋊C₄×C₆₁	—	2	4	no	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#11758202	—	2	16	no	—
35	C₄⋊Dic₆₁	—	2	4	no	—
36	#11758196	—	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	Dic₂₄₄	—	2	8	no	—
41	C₈×D₆₁	D₄	1	8	yes	—
42	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₆₁	—	2	8	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	4	no	—
48	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
50	C₄×D₆₁	—	4	8	no	—
51	C₄×D₆₁	Q₁₆	1	8	yes	complemented
52	C₄×D₆₁	C₂×D₄	1	4	yes	—
53	Dic₁₂₂	—	2	8	no	—
54	Dic₁₂₂	—	4	8	no	—
55	C₈×C₆₁	C₂×D₄	1	4	yes	—
56	#92418	C₂×D₄	1	4	yes	—
57	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
58	C₂×D₆₁	#127	1	2	yes	—
59	C₂×D₆₁	C₂×Q₁₆	1	4	yes	—
60	C₄×C₆₁	—	4	4	no	—
61	C₄×C₆₁	C₂×Q₁₆	1	4	yes	complemented
62	C₄×C₆₁	C₂²×D₄	1	2	yes	—
63	Dic₆₁	—	4	4	no	—
64	Dic₆₁	#127	1	2	yes	—
65	Dic₆₁	C₂×Q₁₆	1	4	yes	complemented
66	C₂×#494	—	61	1	no	maximal, Sylow
67	C₂×C₆₁	C₂×#127	1	1	yes	characteristic
68	C₂×C₆₁	C₂²×Q₁₆	1	2	yes	—
69	D₆₁	#494	1	4	yes	direct product factor
70	C₂²×Q₁₆	—	61	4	no	—
71	C₂×C₄⋊Q₈	—	61	2	no	—
72	#494	—	61	4	no	—
73	#494	—	61	2	no	—
74	#494	—	61	1	no	—
75	#494	D₆₁	1	1	yes	characteristic, direct product factor
76	C₂×C₄×C₈	—	61	1	no	—
77	C₆₁	C₂×#494	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	61	4	no	—
79	C₂×Q₁₆	—	61	8	no	—
80	C₂×Q₁₆	—	61	4	no	—
81	C₂×Q₁₆	—	122	16	no	—
82	C₂×Q₁₆	C₂×D₆₁	1	4	yes	—
83	C₂²×C₈	—	61	2	no	—
84	C₄⋊Q₈	—	61	4	no	—
85	C₄⋊Q₈	—	61	2	no	—
86	C₄⋊Q₈	—	122	4	no	—
87	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
88	C₂×C₄⋊C₄	—	122	4	no	—
89	C₂×C₄²	—	61	1	no	—
90	C₄×C₈	—	61	2	no	—
91	C₄×C₈	—	61	1	no	—
92	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
93	C₂×Q₈	—	61	4	no	—
94	C₂×Q₈	—	122	4	no	—
95	C₂×Q₈	—	122	16	no	—
96	C₂×Q₈	C₂²×D₆₁	1	4	yes	—
97	C₂²×C₄	—	61	2	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	—	122	4	no	—
100	Q₁₆	—	2	8	no	—
101	Q₁₆	—	122	16	no	—
102	Q₁₆	—	122	8	no	—
103	C₂×C₈	—	61	2	no	—
104	C₂×C₈	—	61	8	no	—
105	C₂×C₈	C₂²×D₆₁	1	2	yes	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	122	8	no	—
108	C₄⋊C₄	—	122	4	no	—
109	C₄²	—	61	2	no	—
110	C₄²	—	61	1	no	—
111	C₄²	C₂²×D₆₁	1	1	yes	characteristic
112	C₂³	—	61	1	no	—
113	Q₈	—	2	8	no	—
114	Q₈	—	122	8	no	—
115	Q₈	—	244	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	61	2	no	—
118	C₂×C₄	—	61	1	no	—
119	C₂×C₄	—	61	4	no	—
120	C₂×C₄	—	61	8	no	—
121	C₂×C₄	—	122	4	no	—
122	C₂×C₄	—	244	8	no	—
123	C₂×C₄	D₄×D₆₁	1	2	yes	—
124	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
125	C₈	—	61	4	no	—
126	C₈	D₄×D₆₁	1	4	yes	—
127	C₂²	—	61	4	no	—
128	C₂²	—	61	2	no	—
129	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	61	4	no	—
132	C₄	—	61	2	no	—
133	C₄	—	244	4	no	—
134	C₄	Q₁₆×D₆₁	1	4	yes	complemented
135	C₄	C₂×D₄×D₆₁	1	2	yes	—
136	C₂	—	61	4	no	—
137	C₂	#127×D₆₁	1	1	yes	characteristic
138	C₂	C₂×Q₁₆×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₆₁

C₂

yes

maximal

C₄⋊Q₈×D₆₁

C₂

yes

maximal

#16840142

C₂

yes

maximal, characteristic, complemented

#16840141

C₂

yes

maximal, complemented

#16840140

C₂

yes

maximal, complemented

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic

C₆₁×#494

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

—

C₂×#11758202

C₂²

yes

—

Q₁₆×D₆₁

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₂²

yes

—

#14519776

—

#14519775

C₂²

yes

—

C₄⋊C₄×D₆₁

—

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

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

Dic₂₄₄

—

C₈×D₆₁

D₄

yes

—

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×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₄×D₆₁

—

C₄×D₆₁

Q₁₆

yes

complemented

C₄×D₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₄

yes

—

#92418

C₂×D₄

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

#127

yes

—

C₂×D₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

complemented

C₄×C₆₁

C₂²×D₄

yes

—

Dic₆₁

—

Dic₆₁

#127

yes

—

Dic₆₁

C₂×Q₁₆

yes

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×Q₁₆

yes

—

D₆₁

#494

yes

direct product factor

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic, direct product factor

C₂×C₄×C₈

—

C₆₁

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

C₂×D₆₁

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

100

Q₁₆

—

101

Q₁₆

—

122

—

102

Q₁₆

—

122

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×D₆₁

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₆₁

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

122

—

115

Q₈

—

244

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

—

122

C₂×C₄

—

244

—

123

C₂×C₄

D₄×D₆₁

yes

—

124

C₂×C₄

C₂³×D₆₁

yes

characteristic

125

C₈

—

126

C₈

D₄×D₆₁

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×D₄×D₆₁

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

244

—

134

C₄

Q₁₆×D₆₁

yes

complemented

135

C₄

C₂×D₄×D₆₁

yes

—

136

C₂

—

137

C₂

#127×D₆₁

yes

characteristic

138

C₂

C₂×Q₁₆×D₆₁

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	C₂×#494
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
3. #16840142	C₂	semidirect
4. #16840141	C₂	semidirect
5. #16840140	C₂	semidirect
7. C₆₁×#494	C₂	semidirect
51. C₄×D₆₁	Q₁₆	semidirect
61. C₄×C₆₁	C₂×Q₁₆	semidirect
65. Dic₆₁	C₂×Q₁₆	semidirect
69. D₆₁	#494	direct (N × H)
75. #494	D₆₁	direct (N × H)
77. C₆₁	C₂×#494	semidirect
134. C₄	Q₁₆×D₆₁	semidirect

H ≅ G/N

type

3. #16840142

C₂

semidirect

4. #16840141

C₂

semidirect

5. #16840140

C₂

semidirect

7. C₆₁×#494

C₂

semidirect

51. C₄×D₆₁

Q₁₆

semidirect

61. C₄×C₆₁

C₂×Q₁₆

semidirect

65. Dic₆₁

C₂×Q₁₆

semidirect

69. D₆₁

#494

direct (N × H)

75. #494

D₆₁

direct (N × H)

77. C₆₁

C₂×#494

semidirect

134. C₄

Q₁₆×D₆₁

semidirect

Subgroups — #494×D₆₁ #21456232

Conjugacy classes of subgroups

Complements

Subgroups — #494×D61 #21456232

Conjugacy classes of subgroups

Complements

Subgroups — #494×D₆₁ #21456232