Group #16840484 — subgroups

Subgroups — #16840484

Order 3904.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14519775	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519772	C₂	1	2	yes	maximal, complemented
3	#14519771	C₂	1	2	yes	maximal, complemented
4	#14519766	C₂	1	1	yes	maximal, characteristic
5	#14519765	C₂	1	1	yes	maximal, characteristic
6	#14519755	C₂	1	2	yes	maximal, complemented
7	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519742	C₂	1	1	yes	maximal, characteristic, complemented
9	#14519739	C₂	1	1	yes	maximal, characteristic
10	#14519734	C₂	1	2	yes	maximal, complemented
11	C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×D₂₄₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₆₁	C₂²	1	2	yes	—
15	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₄²×C₆₁	C₂²	1	2	yes	—
20	C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	#11758198	C₂²	1	2	yes	—
22	#11758198	C₂²	1	2	yes	—
23	#11758198	C₂²	1	2	yes	—
24	#11758198	C₂²	1	2	yes	—
25	#11758198	C₂²	1	2	yes	—
26	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
27	#11758196	C₂²	1	1	yes	characteristic
28	#11758196	C₂²	1	2	yes	—
29	#11758196	C₂²	1	2	yes	—
30	#11758196	C₂²	1	1	yes	characteristic
31	#11758196	C₂²	1	2	yes	—
32	C₄×Dic₆₁	C₂²	1	2	yes	—
33	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
35	C₂²×D₆₁	C₂³	1	2	yes	—
36	Q₈×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	C₂³	1	2	yes	—
38	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₆₁	C₂³	1	2	yes	—
45	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
47	D₂₄₄	—	2	2	no	—
48	C₄×D₆₁	—	2	4	no	—
49	Dic₁₂₂	—	2	2	no	—
50	Dic₁₂₂	—	2	2	no	—
51	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₆₁	—	2	2	no	—
53	C₂×D₆₁	—	2	2	no	—
54	C₂×D₆₁	—	2	2	no	—
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	2	2	no	—
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	#41	1	2	yes	—
60	Dic₆₁	—	2	1	no	—
61	Dic₆₁	—	2	2	no	—
62	Dic₆₁	—	2	1	no	—
63	Dic₆₁	—	2	1	no	—
64	Dic₆₁	#41	1	2	yes	—
65	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
66	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
67	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
68	D₆₁	—	4	2	no	—
69	#556	—	61	1	no	maximal, Sylow
70	C₆₁	#556	1	1	yes	characteristic, complemented, Sylow
71	C₄⋊Q₈	—	61	1	no	—
72	#126	—	61	2	no	—
73	#126	—	61	2	no	—
74	#124	—	61	1	no	—
75	#124	—	61	1	no	—
76	C₂²⋊Q₈	—	61	2	no	—
77	C₄×Q₈	—	61	1	no	—
78	C₄×Q₈	—	61	1	no	—
79	C₄×Q₈	D₆₁	1	1	yes	characteristic, complemented
80	C₄×D₄	—	61	1	no	—
81	#117	—	61	2	no	—
82	C₂×Q₈	—	61	1	no	—
83	C₂×Q₈	—	61	1	no	—
84	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
85	C₂×D₄	—	61	1	no	—
86	C₂²×C₄	—	61	2	no	—
87	C₄⋊C₄	—	61	1	no	—
88	C₄⋊C₄	—	61	2	no	—
89	C₄⋊C₄	—	61	2	no	—
90	C₄⋊C₄	—	61	1	no	—
91	C₄⋊C₄	—	61	2	no	—
92	C₄⋊C₄	—	61	1	no	—
93	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
95	C₂²⋊C₄	—	61	2	no	—
96	C₂²⋊C₄	—	61	2	no	—
97	C₂²⋊C₄	—	61	2	no	—
98	C₂²⋊C₄	—	61	2	no	—
99	C₂²⋊C₄	—	61	2	no	—
100	C₄²	—	61	2	no	—
101	C₄²	—	61	1	no	—
102	C₄²	—	61	1	no	—
103	C₄²	C₂×D₆₁	1	2	yes	—
104	C₄²	C₂×D₆₁	1	1	yes	characteristic
105	C₂³	—	61	2	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	122	2	no	—
108	Q₈	—	122	2	no	—
109	D₄	—	122	2	no	—
110	C₂×C₄	—	61	1	no	—
111	C₂×C₄	—	61	1	no	—
112	C₂×C₄	—	61	1	no	—
113	C₂×C₄	—	61	1	no	—
114	C₂×C₄	—	61	2	no	—
115	C₂×C₄	—	122	4	no	—
116	C₂×C₄	C₂²×D₆₁	1	2	yes	—
117	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₆₁	1	2	yes	—
119	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
121	C₂²	—	122	2	no	—
122	C₂²	—	122	2	no	—
123	C₂²	—	122	2	no	—
124	C₂²	C₂³×D₆₁	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	61	2	no	—
130	C₄	—	122	1	no	—
131	C₄	—	122	2	no	—
132	C₄	—	122	1	no	—
133	C₄	—	122	1	no	—
134	C₄	#11758222	1	2	yes	—
135	C₂	—	244	2	no	—
136	C₂	#41×D₆₁	1	1	yes	characteristic
137	C₂	C₂×#11758222	1	1	yes	characteristic
138	C₂	#14519825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14519775

C₂

yes

maximal, characteristic, complemented

#14519772

C₂

yes

maximal, complemented

#14519771

C₂

yes

maximal, complemented

#14519766

C₂

yes

maximal, characteristic

#14519765

C₂

yes

maximal, characteristic

#14519755

C₂

yes

maximal, complemented

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

#14519742

C₂

yes

maximal, characteristic, complemented

#14519739

C₂

yes

maximal, characteristic

#14519734

C₂

yes

maximal, complemented

C₄×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

C₂²

yes

characteristic

C₂×C₄×D₆₁

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

characteristic

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

characteristic

#11758196

C₂²

yes

characteristic

#11758196

C₂²

yes

—

#11758196

C₂²

yes

—

#11758196

C₂²

yes

characteristic

#11758196

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₆₁

C₂²

yes

characteristic

C₂²×D₆₁

C₂³

yes

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₂²×C₆₁

C₂⁴

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#41

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#41

yes

—

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

characteristic

D₆₁

—

#556

—

maximal, Sylow

C₆₁

#556

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₆₁

yes

characteristic, complemented

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₄⋊C₄

C₂×D₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×D₆₁

yes

—

104

C₄²

C₂×D₆₁

yes

characteristic

105

C₂³

—

106

Q₈

—

107

Q₈

—

122

—

108

Q₈

—

122

—

109

D₄

—

122

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

122

—

116

C₂×C₄

C₂²×D₆₁

yes

—

117

C₂×C₄

C₂²×D₆₁

yes

characteristic

118

C₂×C₄

C₂²×D₆₁

yes

—

119

C₂×C₄

C₂²×D₆₁

yes

characteristic

120

C₂×C₄

C₂²×D₆₁

yes

characteristic

121

C₂²

—

122

—

122

C₂²

—

122

—

123

C₂²

—

122

—

124

C₂²

C₂³×D₆₁

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

122

—

131

C₄

—

122

—

132

C₄

—

122

—

133

C₄

—

122

—

134

C₄

#11758222

yes

—

135

C₂

—

244

—

136

C₂

#41×D₆₁

yes

characteristic

137

C₂

C₂×#11758222

yes

characteristic

138

C₂

#14519825

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₄×Q₈×C₆₁	C₂
prime radical	C₂×D₂₄₄	C₂²
hypercenter	C₄×Q₈	D₆₁
hypocenter	C₆₁	#556
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, cosocle

C₂²×C₆₁

C₂⁴

Fitting

C₄×Q₈×C₆₁

C₂

prime radical

C₂×D₂₄₄

C₂²

hypercenter

C₄×Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #14519775	C₂	semidirect
2. #14519772	C₂	semidirect
3. #14519771	C₂	semidirect
6. #14519755	C₂	semidirect
7. C₄×Dic₁₂₂	C₂	semidirect
8. #14519742	C₂	semidirect
10. #14519734	C₂	semidirect
11. C₄×Q₈×C₆₁	C₂	semidirect
70. C₆₁	#556	semidirect
79. C₄×Q₈	D₆₁	semidirect

H ≅ G/N

type

1. #14519775

C₂

semidirect

2. #14519772

C₂

semidirect

3. #14519771

C₂

semidirect

6. #14519755

C₂

semidirect

7. C₄×Dic₁₂₂

C₂

semidirect

8. #14519742

C₂

semidirect

10. #14519734

C₂

semidirect

11. C₄×Q₈×C₆₁

C₂

semidirect

70. C₆₁

#556

semidirect

79. C₄×Q₈

D₆₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14519775	2
#14519772	2
#14519771	2
#14519766	2
#14519765	2
#14519755	2
C₄×Dic₁₂₂	2
#14519742	2
#14519739	2
#14519734	2
C₄×Q₈×C₆₁	2
#556	61

subgroup

index

C₄×Dic₁₂₂

C₄×Q₈×C₆₁

Subgroups — #16840484

Conjugacy classes of subgroups

Complements

Maximal subgroups