Group #16840481 — subgroups

Subgroups — #16840481

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	#14519770	C₂	1	2	yes	maximal, complemented
3	#14519770	C₂	1	2	yes	maximal, complemented
4	#14519764	C₂	1	1	yes	maximal, characteristic
5	#14519754	C₂	1	2	yes	maximal, complemented
6	Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic
10	#14519733	C₂	1	2	yes	maximal, complemented
11	#124×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₆₁	C₂²	1	2	yes	—
15	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₆₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₆₁	C₂²	1	2	yes	—
22	C₂²⋊Dic₆₁	C₂²	1	2	yes	—
23	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₆₁	C₂²	1	2	yes	—
26	#11758196	C₂²	1	1	yes	characteristic
27	#11758196	C₂²	1	2	yes	—
28	#11758196	C₂²	1	2	yes	—
29	#11758196	C₂²	1	1	yes	characteristic
30	#11758196	C₂²	1	2	yes	—
31	C₄×Dic₆₁	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	C₂³×C₆₁	C₂³	1	2	yes	—
36	Q₈×C₆₁	—	2	2	no	—
37	D₄×C₆₁	—	2	2	no	—
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	—	2	4	no	—
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	2	yes	—
47	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
49	Dic₁₂₂	—	2	2	no	—
50	Dic₁₂₂	—	2	2	no	—
51	C₂²×C₆₁	—	2	2	no	—
52	C₂²×C₆₁	—	2	2	no	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	2	1	no	—
58	C₄×C₆₁	#41	1	2	yes	—
59	Dic₆₁	—	2	1	no	—
60	Dic₆₁	—	2	2	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	—	4	2	no	—
66	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
67	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
68	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
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	D₆₁	1	1	yes	characteristic
76	C₂²⋊Q₈	—	61	2	no	—
77	C₄×Q₈	—	61	1	no	—
78	C₄×Q₈	—	61	1	no	—
79	C₄×Q₈	—	61	1	no	—
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	C₂×D₆₁	1	1	yes	characteristic
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	—	61	2	no	—
94	C₄⋊C₄	—	61	1	no	—
95	C₂²⋊C₄	—	61	2	no	—
96	C₂²⋊C₄	—	61	2	no	—
97	C₂²⋊C₄	—	61	2	no	—
98	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
99	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
100	C₄²	—	61	2	no	—
101	C₄²	—	61	2	no	—
102	C₄²	—	61	1	no	—
103	C₄²	—	61	1	no	—
104	C₄²	C₂×D₆₁	1	1	yes	characteristic
105	C₂³	C₂²×D₆₁	1	2	yes	—
106	Q₈	—	2	2	no	—
107	Q₈	—	122	2	no	—
108	Q₈	—	122	2	no	—
109	D₄	—	2	2	no	—
110	C₂×C₄	—	61	1	no	—
111	C₂×C₄	—	61	1	no	—
112	C₂×C₄	—	61	2	no	—
113	C₂×C₄	—	61	1	no	—
114	C₂×C₄	—	61	1	no	—
115	C₂×C₄	—	61	2	no	—
116	C₂×C₄	—	122	4	no	—
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	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂³×D₆₁	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	61	2	no	—
129	C₄	—	122	2	no	—
130	C₄	—	122	1	no	—
131	C₄	—	122	2	no	—
132	C₄	—	122	1	no	—
133	C₄	—	122	1	no	—
134	C₄	#11758224	1	2	yes	—
135	C₂	—	4	2	no	—
136	C₂	#41×D₆₁	1	1	yes	characteristic
137	C₂	C₂×#11758224	1	1	yes	characteristic
138	C₂	#14519824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14519775

C₂

yes

maximal, characteristic, complemented

#14519770

C₂

yes

maximal, complemented

#14519770

C₂

yes

maximal, complemented

#14519764

C₂

yes

maximal, characteristic

#14519754

C₂

yes

maximal, complemented

Q₈×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

C₄×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

D₄×Dic₆₁

C₂

yes

maximal, characteristic

#14519733

C₂

yes

maximal, complemented

#124×C₆₁

C₂

yes

maximal, characteristic

C₂×Q₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

C₂²

yes

characteristic

C₂²×Dic₆₁

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₂×Dic₁₂₂

C₂²

yes

characteristic

C₂²⋊C₄×C₆₁

C₂²

yes

—

C₂²⋊C₄×C₆₁

C₂²

yes

—

C₄²×C₆₁

C₂²

yes

characteristic

C₂²⋊Dic₆₁

C₂²

yes

—

C₂²⋊Dic₆₁

C₂²

yes

—

C₂²⋊Dic₆₁

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

—

#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

—

C₄×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₆₁

C₂²

yes

characteristic

C₂³×C₆₁

C₂³

yes

—

Q₈×C₆₁

—

D₄×C₆₁

—

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

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

Dic₁₂₂

—

Dic₁₂₂

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂⁴

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#41

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#41

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

—

C₆₁

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

D₆₁

yes

characteristic

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₆₁

yes

characteristic

C₂×D₄

C₂×D₆₁

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₆₁

yes

—

C₂²⋊C₄

C₂×D₆₁

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×D₆₁

yes

characteristic

105

C₂³

C₂²×D₆₁

yes

—

106

Q₈

—

107

Q₈

—

122

—

108

Q₈

—

122

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

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

C₂²

—

123

C₂²

—

124

C₂²

C₂³×D₆₁

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

122

—

130

C₄

—

122

—

131

C₄

—

122

—

132

C₄

—

122

—

133

C₄

—

122

—

134

C₄

#11758224

yes

—

135

C₂

—

136

C₂

#41×D₆₁

yes

characteristic

137

C₂

C₂×#11758224

yes

characteristic

138

C₂

#14519824

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	#124×C₆₁	C₂
prime radical	C₂×D₄×C₆₁	C₂²
hypercenter	#124	D₆₁
hypocenter	C₆₁	#556
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, cosocle

C₂²×C₆₁

C₂⁴

Fitting

#124×C₆₁

C₂

prime radical

C₂×D₄×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #14519775	C₂	semidirect
2. #14519770	C₂	semidirect
3. #14519770	C₂	semidirect
5. #14519754	C₂	semidirect
6. Q₈×Dic₆₁	C₂	semidirect
7. C₄×Dic₁₂₂	C₂	semidirect
8. C₄×Dic₁₂₂	C₂	semidirect
10. #14519733	C₂	semidirect
70. C₆₁	#556	semidirect

H ≅ G/N

type

1. #14519775

C₂

semidirect

2. #14519770

C₂

semidirect

3. #14519770

C₂

semidirect

5. #14519754

C₂

semidirect

6. Q₈×Dic₆₁

C₂

semidirect

7. C₄×Dic₁₂₂

C₂

semidirect

8. C₄×Dic₁₂₂

C₂

semidirect

10. #14519733

C₂

semidirect

70. C₆₁

#556

semidirect

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

subgroup	index
#14519775	2
#14519770	2
#14519764	2
#14519754	2
Q₈×Dic₆₁	2
C₄×Dic₁₂₂	2
D₄×Dic₆₁	2
#14519733	2
#124×C₆₁	2
#556	61

subgroup

index

Q₈×Dic₆₁

C₄×Dic₁₂₂

D₄×Dic₆₁

#14519733

#124×C₆₁

#556

Subgroups — #16840481

Conjugacy classes of subgroups

Complements

Maximal subgroups