Group #16838012 — subgroups

Subgroups — #16838012

Order 3776.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14273725	C₂	1	1	yes	maximal, characteristic, complemented
2	#14273720	C₂	1	2	yes	maximal
3	#14273720	C₂	1	2	yes	maximal
4	#14273718	C₂	1	1	yes	maximal, characteristic, complemented
5	#14273718	C₂	1	2	yes	maximal, complemented
6	#14273717	C₂	1	2	yes	maximal, complemented
7	#14273706	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#14273692	C₂	1	1	yes	maximal, characteristic, complemented
10	#14273686	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₁₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
18	C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
19	C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₅₉	C₂²	1	2	yes	—
21	C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₅₉	C₂²	1	2	yes	—
25	C₄⋊Dic₅₉	C₂²	1	2	yes	—
26	Dic₅₉⋊C₄	C₂²	1	2	yes	—
27	Dic₅₉⋊C₄	C₂²	1	2	yes	—
28	Dic₅₉⋊C₄	C₂²	1	2	yes	—
29	Dic₅₉⋊C₄	C₂²	1	2	yes	—
30	Dic₅₉⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₅₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₅₉	C₂²	1	1	yes	characteristic
33	C₄×Dic₅₉	C₂²	1	2	yes	—
34	C₄×Dic₅₉	C₂²	1	2	yes	—
35	C₂³×C₅₉	C₂³	1	1	yes	characteristic
36	Q₈×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	C₂³	1	2	yes	—
39	C₂×C₄×C₅₉	C₂³	1	2	yes	—
40	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
43	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₉	C₂³	1	2	yes	—
46	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
47	C₂×Dic₅₉	C₂³	1	2	yes	—
48	Dic₁₁₈	—	2	2	no	—
49	C₂²×C₅₉	—	2	1	no	—
50	C₂²×C₅₉	—	2	1	no	—
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	C₂⁴	1	1	yes	characteristic
53	C₄×C₅₉	—	2	1	no	—
54	C₄×C₅₉	—	2	2	no	—
55	C₄×C₅₉	—	2	2	no	—
56	C₄×C₅₉	#41	1	2	yes	—
57	Dic₅₉	—	2	2	no	—
58	Dic₅₉	—	2	1	no	—
59	Dic₅₉	—	2	1	no	—
60	Dic₅₉	—	2	1	no	—
61	Dic₅₉	—	2	1	no	—
62	Dic₅₉	—	2	2	no	—
63	C₂×C₅₉	—	4	1	no	—
64	C₂×C₅₉	C₂×#41	1	1	yes	characteristic
65	C₂×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
67	#541	—	59	1	no	maximal, Sylow
68	C₅₉	#541	1	1	yes	characteristic, complemented, Sylow
69	C₄⋊Q₈	—	59	1	no	—
70	#126	—	59	2	no	—
71	#126	—	59	2	no	—
72	#125	—	59	2	no	—
73	#125	—	59	1	no	—
74	#125	—	59	2	no	—
75	C₂²⋊Q₈	—	59	1	no	—
76	C₂²⋊Q₈	D₅₉	1	1	yes	characteristic
77	C₄×Q₈	—	59	1	no	—
78	C₄×Q₈	—	59	1	no	—
79	#117	—	59	1	no	—
80	C₂×Q₈	—	59	1	no	—
81	C₂×Q₈	C₂×D₅₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₅₉	1	1	yes	characteristic
83	C₄⋊C₄	—	59	1	no	—
84	C₄⋊C₄	—	59	1	no	—
85	C₄⋊C₄	—	59	1	no	—
86	C₄⋊C₄	—	59	2	no	—
87	C₄⋊C₄	—	59	2	no	—
88	C₄⋊C₄	—	59	2	no	—
89	C₄⋊C₄	—	59	2	no	—
90	C₄⋊C₄	—	59	2	no	—
91	C₄⋊C₄	—	59	2	no	—
92	C₄⋊C₄	—	59	2	no	—
93	C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₅₉	1	2	yes	—
95	C₂²⋊C₄	—	59	1	no	—
96	C₂²⋊C₄	—	59	1	no	—
97	C₂²⋊C₄	—	59	2	no	—
98	C₂²⋊C₄	C₂×D₅₉	1	2	yes	—
99	C₄²	—	59	1	no	—
100	C₄²	—	59	1	no	—
101	C₄²	—	59	2	no	—
102	C₄²	—	59	2	no	—
103	C₂³	C₂²×D₅₉	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	118	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	59	1	no	—
108	C₂×C₄	—	59	1	no	—
109	C₂×C₄	—	59	1	no	—
110	C₂×C₄	—	59	1	no	—
111	C₂×C₄	—	59	2	no	—
112	C₂×C₄	—	59	2	no	—
113	C₂×C₄	C₂²×D₅₉	1	2	yes	—
114	C₂×C₄	C₂²×D₅₉	1	2	yes	—
115	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₅₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	118	2	no	—
125	C₄	—	118	1	no	—
126	C₄	—	118	1	no	—
127	C₄	—	118	1	no	—
128	C₄	—	118	1	no	—
129	C₄	—	118	2	no	—
130	C₄	#11745687	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#11745687	1	1	yes	characteristic
133	C₂	#14273774	1	1	yes	characteristic
134	C₂	#14273775	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14273725

C₂

yes

maximal, characteristic, complemented

#14273720

C₂

yes

maximal

#14273720

C₂

yes

maximal

#14273718

C₂

yes

maximal, characteristic, complemented

#14273718

C₂

yes

maximal, complemented

#14273717

C₂

yes

maximal, complemented

#14273706

C₂

yes

maximal, characteristic

Q₈×Dic₅₉

C₂

yes

maximal, characteristic, complemented

#14273692

C₂

yes

maximal, characteristic, complemented

#14273686

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×Q₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₄⋊C₄×C₅₉

C₂²

yes

—

C₂²⋊C₄×C₅₉

C₂²

yes

—

C₂²⋊Dic₅₉

C₂²

yes

characteristic

C₂²⋊Dic₅₉

C₂²

yes

characteristic

C₂²⋊Dic₅₉

C₂²

yes

—

C₄⋊Dic₅₉

C₂²

yes

characteristic

C₄⋊Dic₅₉

C₂²

yes

characteristic

C₄⋊Dic₅₉

C₂²

yes

characteristic

C₄⋊Dic₅₉

C₂²

yes

—

C₄⋊Dic₅₉

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

Dic₅₉⋊C₄

C₂²

yes

—

C₄×Dic₅₉

C₂²

yes

characteristic

C₄×Dic₅₉

C₂²

yes

characteristic

C₄×Dic₅₉

C₂²

yes

—

C₄×Dic₅₉

C₂²

yes

—

C₂³×C₅₉

C₂³

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

C₂³

yes

—

C₂×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₅₉

C₂³

yes

characteristic

C₂×Dic₅₉

C₂³

yes

characteristic

C₂×Dic₅₉

C₂³

yes

characteristic

C₂×Dic₅₉

C₂³

yes

—

C₂×Dic₅₉

C₂³

yes

characteristic

C₂×Dic₅₉

C₂³

yes

—

Dic₁₁₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂⁴

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#41

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#41

yes

characteristic

C₂×C₅₉

2₋¹⁺⁴

yes

characteristic

C₂×C₅₉

2₋¹⁺⁴

yes

—

C₅₉

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₅₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₉

yes

characteristic

C₂²×C₄

C₂×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₉

yes

characteristic

C₄⋊C₄

C₂×D₅₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅₉

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂³

C₂²×D₅₉

yes

characteristic

104

Q₈

—

105

Q₈

—

118

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₅₉

yes

—

114

C₂×C₄

C₂²×D₅₉

yes

—

115

C₂×C₄

C₂²×D₅₉

yes

characteristic

116

C₂×C₄

C₂²×D₅₉

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₅₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

118

—

125

C₄

—

118

—

126

C₄

—

118

—

127

C₄

—

118

—

128

C₄

—

118

—

129

C₄

—

118

—

130

C₄

#11745687

yes

—

131

C₂

—

132

C₂

C₂×#11745687

yes

characteristic

133

C₂

#14273774

yes

characteristic

134

C₂

#14273775

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₉
commutator, socle, cosocle	C₂²×C₅₉	C₂⁴
Fitting	C₂²⋊Q₈×C₅₉	C₂
prime radical	C₂³×C₅₉	C₂³
hypercenter	C₂²⋊Q₈	D₅₉
hypocenter	C₅₉	#541
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₉

commutator, socle, cosocle

C₂²×C₅₉

C₂⁴

Fitting

C₂²⋊Q₈×C₅₉

C₂

prime radical

C₂³×C₅₉

C₂³

hypercenter

C₂²⋊Q₈

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #14273725	C₂	semidirect
4. #14273718	C₂	semidirect
5. #14273718	C₂	semidirect
6. #14273717	C₂	semidirect
8. Q₈×Dic₅₉	C₂	semidirect
9. #14273692	C₂	semidirect
68. C₅₉	#541	semidirect

H ≅ G/N

type

1. #14273725

C₂

semidirect

4. #14273718

C₂

semidirect

5. #14273718

C₂

semidirect

6. #14273717

C₂

semidirect

8. Q₈×Dic₅₉

C₂

semidirect

9. #14273692

C₂

semidirect

68. C₅₉

#541

semidirect

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

subgroup	index
#14273725	2
#14273720	2
#14273718	2
#14273717	2
#14273706	2
Q₈×Dic₅₉	2
#14273692	2
#14273686	2
C₂²⋊Q₈×C₅₉	2
#541	59

subgroup

index

Q₈×Dic₅₉

C₂²⋊Q₈×C₅₉

Subgroups — #16838012

Conjugacy classes of subgroups

Complements

Maximal subgroups