Group #21456315 — subgroups

Subgroups — #498×D₆₁ #21456315

Order 7808.

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

normal subgroups only

This group has 424 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	C₄⋊Q₈×D₆₁	C₂	1	2	yes	maximal
2	#16840160	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840159	C₂	1	4	yes	maximal, complemented
4	#16840158	C₂	1	2	yes	maximal, complemented
5	#106×D₆₁	C₂	1	4	yes	maximal
6	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	—	2	2	no	—
9	C₂×C₈×D₆₁	C₂²	1	2	yes	—
10	#14519777	C₂²	1	2	yes	—
11	#14519776	—	2	2	no	—
12	#14519776	C₂²	1	4	yes	—
13	#14519775	—	2	2	no	—
14	C₄⋊C₄×D₆₁	—	2	2	no	—
15	C₄⋊C₄×D₆₁	C₂²	1	4	yes	—
16	C₄²×D₆₁	C₂²	1	1	yes	characteristic
17	#14519709	C₂²	1	4	yes	—
18	#14519708	C₂²	1	8	yes	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₈×Dic₆₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
22	#106×C₆₁	C₂²	1	4	yes	—
23	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₆₁	—	2	2	no	—
25	Q₈×D₆₁	—	4	8	no	—
26	C₂×C₄×D₆₁	—	2	4	no	—
27	C₂×C₄×D₆₁	D₄	1	2	yes	—
28	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₂₂	—	2	2	no	—
30	C₂×Dic₁₂₂	—	2	4	no	—
31	C₂×C₈×C₆₁	C₂³	1	2	yes	—
32	C₄⋊C₄×C₆₁	—	2	2	no	—
33	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	C₂³	1	4	yes	—
37	#11758196	—	2	4	no	—
38	#11758196	—	2	4	no	—
39	C₄×Dic₆₁	D₄	1	2	yes	—
40	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	C₂×#92418	C₂³	1	2	yes	—
42	C₈×D₆₁	Q₈	1	8	yes	complemented
43	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₆₁	—	4	4	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	—	2	4	no	—
49	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
51	C₄×D₆₁	—	4	8	no	—
52	C₄×D₆₁	SD₁₆	1	8	yes	—
53	C₄×D₆₁	C₂×Q₈	1	4	yes	—
54	Dic₁₂₂	—	4	4	no	—
55	Dic₁₂₂	—	4	8	no	—
56	C₈×C₆₁	C₂×Q₈	1	4	yes	complemented
57	#92418	C₂×Q₈	1	4	yes	complemented
58	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₆₁	C₄⋊Q₈	1	2	yes	—
60	C₂×D₆₁	C₂×SD₁₆	1	4	yes	—
61	C₄×C₆₁	—	4	4	no	—
62	C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
63	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
64	Dic₆₁	—	4	4	no	—
65	Dic₆₁	C₄⋊Q₈	1	2	yes	—
66	Dic₆₁	C₂×SD₁₆	1	4	yes	—
67	C₂×#498	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₆₁	C₂²×SD₁₆	1	2	yes	—
70	D₆₁	#498	1	4	yes	direct product factor
71	C₂×C₄⋊Q₈	—	61	2	no	—
72	#498	—	61	2	no	—
73	#498	—	61	4	no	—
74	#498	—	61	1	no	—
75	#498	D₆₁	1	1	yes	characteristic, direct product factor
76	C₂×#106	—	61	4	no	—
77	C₂×C₄×C₈	—	61	1	no	—
78	C₆₁	C₂×#498	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	122	2	no	—
80	C₂²×C₈	—	61	2	no	—
81	C₄⋊Q₈	—	61	4	no	—
82	C₄⋊Q₈	—	61	2	no	—
83	C₄⋊Q₈	—	122	2	no	—
84	C₄⋊Q₈	—	122	2	no	—
85	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
86	C₂×C₄⋊C₄	—	61	4	no	—
87	C₂×C₄⋊C₄	—	122	2	no	—
88	C₂×C₄²	—	61	1	no	—
89	#106	—	61	4	no	—
90	#106	—	61	8	no	—
91	#106	C₂×D₆₁	1	4	yes	—
92	C₄×C₈	—	61	2	no	—
93	C₄×C₈	—	61	1	no	—
94	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	122	4	no	—
97	C₂×Q₈	—	122	2	no	—
98	C₂×Q₈	—	244	8	no	—
99	C₂²×C₄	—	61	2	no	—
100	C₂²×C₄	—	61	1	no	—
101	C₂²×C₄	—	122	4	no	—
102	C₂×C₈	—	61	2	no	—
103	C₂×C₈	—	61	8	no	—
104	C₂×C₈	C₂²×D₆₁	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	61	4	no	—
107	C₄⋊C₄	—	122	4	no	—
108	C₄⋊C₄	—	122	4	no	—
109	C₄⋊C₄	—	122	2	no	—
110	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
111	C₄²	—	61	2	no	—
112	C₄²	—	61	1	no	—
113	C₄²	C₂²×D₆₁	1	1	yes	characteristic
114	C₂³	—	61	1	no	—
115	Q₈	—	4	4	no	—
116	Q₈	—	244	8	no	—
117	Q₈	—	244	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	61	1	no	—
120	C₂×C₄	—	61	2	no	—
121	C₂×C₄	—	61	4	no	—
122	C₂×C₄	—	61	8	no	—
123	C₂×C₄	—	122	4	no	—
124	C₂×C₄	—	244	8	no	—
125	C₂×C₄	D₄×D₆₁	1	2	yes	—
126	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
127	C₈	—	61	4	no	—
128	C₈	Q₈×D₆₁	1	4	yes	complemented
129	C₂²	—	61	4	no	—
130	C₂²	—	61	2	no	—
131	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
132	C₄	—	4	4	no	—
133	C₄	—	61	4	no	—
134	C₄	—	61	2	no	—
135	C₄	—	244	4	no	—
136	C₄	SD₁₆×D₆₁	1	4	yes	—
137	C₄	C₂×Q₈×D₆₁	1	2	yes	—
138	C₂	—	61	4	no	—
139	C₂	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
140	C₂	C₂×SD₁₆×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₆₁

C₂

yes

maximal

#16840160

C₂

yes

maximal, characteristic, complemented

#16840159

C₂

yes

maximal, complemented

#16840158

C₂

yes

maximal, complemented

#106×D₆₁

C₂

yes

maximal

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic

C₆₁×#498

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₂²

yes

—

#14519776

—

#14519776

C₂²

yes

—

#14519775

—

C₄⋊C₄×D₆₁

—

C₄⋊C₄×D₆₁

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519709

C₂²

yes

—

#14519708

C₂²

yes

—

C₄×#92418

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₈×D₆₁

Q₈

yes

complemented

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

SD₁₆

yes

—

C₄×D₆₁

C₂×Q₈

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

complemented

#92418

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

C₄⋊Q₈

yes

—

C₂×D₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

Dic₆₁

C₄⋊Q₈

yes

—

Dic₆₁

C₂×SD₁₆

yes

—

C₂×#498

—

maximal, Sylow

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×SD₁₆

yes

—

D₆₁

#498

yes

direct product factor

C₂×C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic, direct product factor

C₂×#106

—

C₂×C₄×C₈

—

C₆₁

C₂×#498

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₆₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

122

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₆₁

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄⋊C₄

—

122

—

110

C₄⋊C₄

C₂²×D₆₁

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂²×D₆₁

yes

characteristic

114

C₂³

—

115

Q₈

—

116

Q₈

—

244

—

117

Q₈

—

244

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

244

—

125

C₂×C₄

D₄×D₆₁

yes

—

126

C₂×C₄

C₂³×D₆₁

yes

characteristic

127

C₈

—

128

C₈

Q₈×D₆₁

yes

complemented

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×D₄×D₆₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

244

—

136

C₄

SD₁₆×D₆₁

yes

—

137

C₄

C₂×Q₈×D₆₁

yes

—

138

C₂

—

139

C₂

C₄⋊Q₈×D₆₁

yes

characteristic

140

C₂

C₂×SD₁₆×D₆₁

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#498	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#498	D₆₁
hypocenter	C₆₁	C₂×#498
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#498

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #16840160	C₂	semidirect
3. #16840159	C₂	semidirect
4. #16840158	C₂	semidirect
7. C₆₁×#498	C₂	semidirect
42. C₈×D₆₁	Q₈	semidirect
56. C₈×C₆₁	C₂×Q₈	semidirect
57. #92418	C₂×Q₈	semidirect
70. D₆₁	#498	direct (N × H)
75. #498	D₆₁	direct (N × H)
78. C₆₁	C₂×#498	semidirect
128. C₈	Q₈×D₆₁	semidirect

H ≅ G/N

type

2. #16840160

C₂

semidirect

3. #16840159

C₂

semidirect

4. #16840158

C₂

semidirect

7. C₆₁×#498

C₂

semidirect

42. C₈×D₆₁

Q₈

semidirect

56. C₈×C₆₁

C₂×Q₈

semidirect

57. #92418

C₂×Q₈

semidirect

70. D₆₁

#498

direct (N × H)

75. #498

D₆₁

direct (N × H)

78. C₆₁

C₂×#498

semidirect

128. C₈

Q₈×D₆₁

semidirect

Subgroups — #498×D₆₁ #21456315

Conjugacy classes of subgroups

Complements

Subgroups — #498×D61 #21456315

Conjugacy classes of subgroups

Complements

Subgroups — #498×D₆₁ #21456315