Group #21462467 — subgroups

Subgroups — #21462467

Order 7808.

484 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 204 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×#14519865	C₂	1	2	yes	maximal
2	#16840769	C₂	1	6	yes	maximal, complemented
3	#16840767	C₂	1	3	yes	maximal, complemented
4	D₆₁⋊(C₄⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#11758226	C₂	1	1	yes	maximal, characteristic
6	#14519865	—	2	8	no	—
7	C₂×#11758213	C₂²	1	6	yes	—
8	C₂×D₆₁⋊C₈	C₂²	1	6	yes	—
9	#14519844	C₂²	1	1	yes	characteristic
10	#14519844	C₂²	1	3	yes	—
11	#14519844	C₂²	1	6	yes	—
12	C₄⋊#92420	C₂²	1	6	yes	—
13	C₂×#11758226	C₂²	1	1	yes	characteristic
14	Q₈×Dic₆₁	C₄	1	1	yes	characteristic
15	C₄×D₂₄₄	C₄	1	3	yes	—
16	#14519739	C₄	1	3	yes	—
17	#14519735	C₂²	1	3	yes	—
18	C₄²×D₆₁	C₂²	1	3	yes	—
19	C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	#11758226	D₄	1	4	yes	—
22	#11758226	Q₈	1	4	yes	—
23	C₂×D₂₄₄	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₆₁	C₂³	1	3	yes	—
27	C₂×#92420	C₂³	1	2	yes	—
28	C₂×#92420	C₂³	1	6	yes	—
29	#11758213	—	2	12	no	—
30	D₆₁⋊C₈	—	2	12	no	—
31	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
32	C₄²×C₆₁	C₂×C₄	1	3	yes	—
33	#11758198	C₂×C₄	1	6	yes	—
34	C₄⋊Dic₆₁	C₂×C₄	1	3	yes	—
35	C₄×Dic₆₁	C₂×C₄	1	3	yes	—
36	C₄×Dic₆₁	C₂³	1	3	yes	—
37	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
38	C₂²×D₆₁	C₂²×C₄	1	3	yes	—
39	Q₈×C₆₁	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
43	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
44	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
46	D₂₄₄	C₄⋊C₄	1	12	yes	—
47	C₄×D₆₁	—	2	6	no	—
48	C₄×D₆₁	—	4	3	no	—
49	C₄×D₆₁	C₂×D₄	1	6	yes	—
50	C₄×D₆₁	C₂×Q₈	1	6	yes	—
51	#92420	—	2	2	no	—
52	#92420	—	2	6	no	—
53	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
54	C₂×D₆₁	—	2	3	no	—
55	C₂×D₆₁	—	2	3	no	—
56	C₂×D₆₁	C₂×C₄⋊C₄	1	6	yes	—
57	C₄×C₆₁	—	2	3	no	—
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
60	Dic₆₁	—	2	3	no	—
61	Dic₆₁	#131	1	2	yes	complemented
62	Dic₆₁	C₂²×D₄	1	1	yes	characteristic
63	Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
64	#2906	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
67	C₂×C₆₁	#568	1	1	yes	characteristic
68	D₆₁	—	2	6	no	—
69	C₂×#131	—	61	2	no	—
70	C₄×#41	—	61	1	no	—
71	#424	—	61	6	no	—
72	#423	—	61	3	no	—
73	C₂×C₄⋊C₈	—	61	3	no	—
74	C₆₁	#2906	1	1	yes	characteristic, complemented, Sylow
75	C₂×#41	—	61	1	no	—
76	#131	—	122	8	no	—
77	C₂×M₁₆	—	61	6	no	—
78	C₂²×C₈	—	61	6	no	—
79	C₄×Q₈	—	61	1	no	—
80	C₄×Q₈	#6923	1	1	yes	characteristic
81	C₄×D₄	—	61	3	no	—
82	C₄×D₄	—	61	3	no	—
83	#117	—	61	3	no	—
84	C₂×C₄²	—	61	3	no	—
85	C₄⋊C₈	—	61	6	no	—
86	C₄⋊C₈	—	61	1	no	—
87	C₄⋊C₈	—	61	3	no	—
88	C₄⋊C₈	—	61	6	no	—
89	#41	—	61	4	no	—
90	#41	—	61	4	no	—
91	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
92	C₂×D₄	—	61	3	no	—
93	C₂²×C₄	—	61	3	no	—
94	C₂²×C₄	—	61	3	no	—
95	C₂²×C₄	—	61	3	no	—
96	M₁₆	—	122	12	no	—
97	C₂×C₈	—	61	2	no	—
98	C₂×C₈	—	61	6	no	—
99	C₂×C₈	—	122	12	no	—
100	C₄⋊C₄	—	61	3	no	—
101	C₄⋊C₄	C₂×#6923	1	3	yes	—
102	C₂²⋊C₄	—	61	6	no	—
103	C₄²	—	61	1	no	—
104	C₄²	—	61	3	no	—
105	C₄²	—	61	3	no	—
106	C₄²	C₂×#6923	1	3	yes	—
107	C₂³	—	61	3	no	—
108	Q₈	C₄⋊#6923	1	4	yes	—
109	D₄	—	61	12	no	—
110	C₂×C₄	—	61	1	no	—
111	C₂×C₄	—	61	1	no	—
112	C₂×C₄	—	61	6	no	—
113	C₂×C₄	—	61	6	no	—
114	C₂×C₄	—	61	3	no	—
115	C₂×C₄	—	122	6	no	—
116	C₂×C₄	—	244	3	no	—
117	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
118	C₂×C₄	C₂²×#6923	1	3	yes	—
119	C₂×C₄	C₂²×#6923	1	3	yes	—
120	C₈	—	122	2	no	—
121	C₈	—	122	6	no	—
122	C₂²	—	61	6	no	—
123	C₂²	—	122	3	no	—
124	C₂²	—	122	3	no	—
125	C₂²	C₂³×#6923	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	2	1	no	—
128	C₄	—	61	1	no	—
129	C₄	—	61	1	no	—
130	C₄	—	61	2	no	—
131	C₄	—	122	3	no	—
132	C₄	C₂×C₄⋊#6923	1	6	yes	—
133	C₂	—	122	6	no	—
134	C₂	C₂²×C₄⋊#6923	1	1	yes	characteristic
135	C₂	D₆₁⋊#131	1	1	yes	characteristic
136	C₂	#16840842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519865

C₂

yes

maximal

#16840769

C₂

yes

maximal, complemented

#16840767

C₂

yes

maximal, complemented

D₆₁⋊(C₄⋊C₈)

C₂

yes

maximal, complemented

C₄×#11758226

C₂

yes

maximal, characteristic

#14519865

—

C₂×#11758213

C₂²

yes

—

C₂×D₆₁⋊C₈

C₂²

yes

—

#14519844

C₂²

yes

characteristic

#14519844

C₂²

yes

—

#14519844

C₂²

yes

—

C₄⋊#92420

C₂²

yes

—

C₂×#11758226

C₂²

yes

characteristic

Q₈×Dic₆₁

C₄

yes

characteristic

C₄×D₂₄₄

C₄

yes

—

#14519739

C₄

yes

—

#14519735

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

#11758226

D₄

yes

—

#11758226

Q₈

yes

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

#11758213

—

D₆₁⋊C₈

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

#11758198

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²×C₄

yes

—

Q₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂⁴

yes

characteristic

D₂₄₄

C₄⋊C₄

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₄

yes

—

C₄×D₆₁

C₂×Q₈

yes

—

#92420

—

#92420

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

Dic₆₁

—

Dic₆₁

#131

yes

complemented

Dic₆₁

C₂²×D₄

yes

characteristic

Dic₆₁

C₂²×Q₈

yes

characteristic

#2906

—

maximal, Sylow

C₂×C₆₁

C₂²×C₄⋊C₄

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×#131

—

C₄×#41

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₆₁

#2906

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

122

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#6923

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₂×#6923

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×#6923

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#6923

yes

—

107

C₂³

—

108

Q₈

C₄⋊#6923

yes

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

122

—

116

C₂×C₄

—

244

—

117

C₂×C₄

C₂²×#6923

yes

characteristic

118

C₂×C₄

C₂²×#6923

yes

—

119

C₂×C₄

C₂²×#6923

yes

—

120

C₈

—

122

—

121

C₈

—

122

—

122

C₂²

—

123

C₂²

—

122

—

124

C₂²

—

122

—

125

C₂²

C₂³×#6923

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

122

—

132

C₄

C₂×C₄⋊#6923

yes

—

133

C₂

—

122

—

134

C₂

C₂²×C₄⋊#6923

yes

characteristic

135

C₂

D₆₁⋊#131

yes

characteristic

136

C₂

#16840842

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×Q₈×C₆₁	C₄
hypercenter	C₄×Q₈	#6923
hypocenter	C₆₁	#2906
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6923

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₄×Q₈×C₆₁

C₄

C₄×Q₈

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #16840769	C₂	semidirect
3. #16840767	C₂	semidirect
4. D₆₁⋊(C₄⋊C₈)	C₂	semidirect
61. Dic₆₁	#131	semidirect
74. C₆₁	#2906	semidirect

H ≅ G/N

type

2. #16840769

C₂

semidirect

3. #16840767

C₂

semidirect

4. D₆₁⋊(C₄⋊C₈)

C₂

semidirect

61. Dic₆₁

#131

semidirect

74. C₆₁

#2906

semidirect

Subgroups — #21462467

Conjugacy classes of subgroups

Complements