Group #21446548 — subgroups

Subgroups — #21446548

Order 7808.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14519709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839661	C₂	1	2	yes	maximal
4	#16839524	C₂	1	2	yes	maximal
5	C₂×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#92418	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	—	2	2	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
9	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	#14519709	—	2	2	no	—
11	C₈×Dic₆₁	—	2	2	no	—
12	#14519684	—	2	2	no	—
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
15	#106×C₆₁	C₄	1	4	yes	complemented
16	C₂²×C₄×C₆₁	—	2	2	no	—
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	2	no	—
19	C₂²×Dic₆₁	D₄	1	2	yes	—
20	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₆₁	D₄	1	2	yes	—
22	C₂×C₈×C₆₁	Q₈	1	2	yes	—
23	C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	—	4	4	no	—
28	C₄×Dic₆₁	—	2	1	no	—
29	C₄×Dic₆₁	—	2	1	no	—
30	C₂×#92418	—	2	1	no	—
31	C₂×#92418	—	2	1	no	—
32	C₂×#92418	Q₈	1	2	yes	—
33	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	4	no	—
35	C₂×C₄×C₆₁	—	4	2	no	—
36	C₂×C₄×C₆₁	—	4	2	no	—
37	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	—	2	2	no	—
45	C₂×Dic₆₁	—	4	2	no	—
46	C₂×Dic₆₁	—	4	2	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	SD₁₆	1	4	yes	—
49	C₈×C₆₁	C₄⋊C₄	1	4	yes	complemented
50	#92418	—	2	2	no	—
51	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₆₁	#121	1	1	yes	characteristic
53	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₆₁	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₆₁	#135	1	2	yes	—
56	C₄×C₆₁	—	4	4	no	—
57	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₆₁	#125	1	1	yes	characteristic
61	Dic₆₁	—	2	4	no	—
62	Dic₆₁	—	8	2	no	—
63	#1941	—	61	1	no	maximal, Sylow
64	C₂×C₆₁	#389	1	1	yes	characteristic
65	C₂×C₆₁	C₄×SD₁₆	1	2	yes	—
66	C₂×C₆₁	#465	1	2	yes	—
67	C₂×C₆₁	#498	1	1	yes	characteristic
68	C₂×C₆₁	#499	1	1	yes	characteristic
69	C₂×#106	—	61	1	no	—
70	C₂×#106	D₆₁	1	1	yes	characteristic
71	C₂×C₄×C₈	—	61	1	no	—
72	#389	—	61	2	no	—
73	#340	—	61	2	no	—
74	C₆₁	#1941	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	61	1	no	—
76	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	61	2	no	—
78	C₂×C₄⋊C₄	—	122	2	no	—
79	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
80	C₂×C₄²	—	61	1	no	—
81	#106	—	122	2	no	—
82	#106	Dic₆₁	1	4	yes	complemented
83	C₄×C₈	—	122	2	no	—
84	#95	—	122	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	61	2	no	—
87	C₂²×C₄	—	122	2	no	—
88	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
89	C₂×C₈	—	61	2	no	—
90	C₂×C₈	—	122	1	no	—
91	C₂×C₈	—	122	1	no	—
92	C₂×C₈	Dic₁₂₂	1	2	yes	—
93	C₂×C₈	D₂₄₄	1	2	yes	—
94	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	122	2	no	—
97	C₄⋊C₄	—	122	2	no	—
98	C₄⋊C₄	—	244	4	no	—
99	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
100	C₄²	—	122	1	no	—
101	C₄²	—	122	1	no	—
102	C₂³	D₄×D₆₁	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	61	4	no	—
107	C₂×C₄	—	122	2	no	—
108	C₂×C₄	—	122	2	no	—
109	C₂×C₄	—	244	2	no	—
110	C₂×C₄	—	244	2	no	—
111	C₂×C₄	—	244	2	no	—
112	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
113	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
114	C₂×C₄	#11758224	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
116	C₂×C₄	#11758226	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
118	C₈	—	122	2	no	—
119	C₈	C₄⋊Dic₆₁	1	4	yes	complemented
120	C₂²	D₄×Dic₆₁	1	1	yes	characteristic
121	C₂²	#14519752	1	1	yes	characteristic
122	C₂²	#14519776	1	1	yes	characteristic
123	C₂²	SD₁₆×D₆₁	1	2	yes	—
124	C₂²	#14519798	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	122	4	no	—
127	C₄	—	488	2	no	—
128	C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
129	C₄	Q₈×Dic₆₁	1	1	yes	characteristic
130	C₄	#14519757	1	1	yes	characteristic
131	C₄	#14519768	1	1	yes	characteristic
132	C₂	#16839661	1	1	yes	characteristic
133	C₂	SD₁₆×Dic₆₁	1	2	yes	—
134	C₂	#16839992	1	2	yes	—
135	C₂	#16840159	1	1	yes	characteristic
136	C₂	#16840162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519709

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#16839661

C₂

yes

maximal

#16839524

C₂

yes

maximal

C₂×#106×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519709

—

C₈×Dic₆₁

—

#14519684

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

#106×C₆₁

C₄

yes

complemented

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

Q₈

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

SD₁₆

yes

—

C₈×C₆₁

C₄⋊C₄

yes

complemented

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

—

C₂²×C₆₁

#135

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₄×Q₈

yes

characteristic

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

#125

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₄×SD₁₆

yes

—

C₂×C₆₁

#465

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

yes

characteristic

C₂×C₄×C₈

—

#389

—

#340

—

C₆₁

#1941

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

#106

—

122

—

#106

Dic₆₁

yes

complemented

C₄×C₈

—

122

—

#95

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

D₂₄₄

yes

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

100

C₄²

—

122

—

101

C₄²

—

122

—

102

C₂³

D₄×D₆₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

122

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

244

—

110

C₂×C₄

—

244

—

111

C₂×C₄

—

244

—

112

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

113

C₂×C₄

C₂×D₂₄₄

yes

characteristic

114

C₂×C₄

#11758224

yes

characteristic

115

C₂×C₄

Q₈×D₆₁

yes

characteristic

116

C₂×C₄

#11758226

yes

characteristic

117

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

118

C₈

—

122

—

119

C₈

C₄⋊Dic₆₁

yes

complemented

120

C₂²

D₄×Dic₆₁

yes

characteristic

121

C₂²

#14519752

yes

characteristic

122

C₂²

#14519776

yes

characteristic

123

C₂²

SD₁₆×D₆₁

yes

—

124

C₂²

#14519798

yes

—

125

C₄

—

126

C₄

—

122

—

127

C₄

—

488

—

128

C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

129

C₄

Q₈×Dic₆₁

yes

characteristic

130

C₄

#14519757

yes

characteristic

131

C₄

#14519768

yes

characteristic

132

C₂

#16839661

yes

characteristic

133

C₂

SD₁₆×Dic₆₁

yes

—

134

C₂

#16839992

yes

—

135

C₂

#16840159

yes

characteristic

136

C₂

#16840162

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#106×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#106	D₆₁
hypocenter	C₆₁	#1941
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#106×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
15. #106×C₆₁	C₄	semidirect
49. C₈×C₆₁	C₄⋊C₄	semidirect
74. C₆₁	#1941	semidirect
82. #106	Dic₆₁	semidirect
119. C₈	C₄⋊Dic₆₁	semidirect

H ≅ G/N

type

15. #106×C₆₁

C₄

semidirect

49. C₈×C₆₁

C₄⋊C₄

semidirect

74. C₆₁

#1941

semidirect

82. #106

Dic₆₁

semidirect

119. C₈

C₄⋊Dic₆₁

semidirect

Subgroups — #21446548

Conjugacy classes of subgroups

Complements