Group #15774515 — subgroups

Subgroups — #15774515

Order 3328.

220 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 143 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×#11718055	C₂	1	2	yes	maximal
2	#13112912	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
5	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88052	—	2	2	no	—
8	C₁₃×#346	C₄	1	2	yes	—
9	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
10	#11718055	—	2	2	no	—
11	#11718055	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
15	C₁₃×M₃₂	—	2	2	no	—
16	C₁₃×M₃₂	C₈	1	4	yes	—
17	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	#88052	—	2	4	no	—
24	#88052	—	4	2	no	—
25	C₂×#6090	—	4	2	no	—
26	#12011	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	—	2	1	no	—
28	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₁₃	D₈	1	2	yes	—
34	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₁₃	—	2	1	no	—
38	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₁₃×C₁₆	—	2	2	no	—
41	C₁₃×C₁₆	C₂×C₈	1	4	yes	—
42	#6090	—	4	4	no	—
43	C₂×#364	—	13	2	no	—
44	C₄×M₃₂	D₁₃	1	1	yes	characteristic
45	#1373	—	13	4	no	—
46	C₂³×C₁₃	#108	1	1	yes	characteristic
47	C₂×C₄×C₁₃	—	2	1	no	—
48	C₂×C₄×C₁₃	—	2	1	no	—
49	C₂×C₄×C₁₃	—	4	2	no	—
50	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
51	C₂×C₄×C₁₃	#107	1	2	yes	—
52	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
55	C₈×C₁₃	#107	1	2	yes	—
56	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×M₃₂	—	26	2	no	—
61	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic
62	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
64	#364	—	13	4	no	—
65	#364	—	26	2	no	—
66	#346	Dic₁₃	1	2	yes	—
67	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
68	C₂²×C₁₃	—	4	1	no	—
69	C₂²×C₁₃	#335	1	2	yes	—
70	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	—	4	1	no	—
73	C₄×C₁₃	#335	1	2	yes	—
74	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
76	C₂²×C₈	D₅₂	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
78	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	M₃₂	—	26	4	no	—
81	M₃₂	—	52	2	no	—
82	M₃₂	#1055	1	4	yes	—
83	C₂×C₁₆	—	52	2	no	—
84	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
86	C₄×C₈	Dic₂₆	1	1	yes	characteristic
87	C₄×C₈	D₅₂	1	1	yes	characteristic
88	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
89	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
90	C₂×C₁₃	—	2	1	no	—
91	C₂×C₁₃	—	2	2	no	—
92	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
95	C₂×C₈	D₁₀₄	1	2	yes	—
96	C₂×C₈	Dic₅₂	1	2	yes	—
97	C₂×C₈	C₂×#1055	1	2	yes	—
98	C₂×C₈	#6099	1	2	yes	—
99	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
100	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
101	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
102	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
103	C₄²	—	2	1	no	—
104	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
105	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	52	4	no	—
108	C₁₆	C₂×#1055	1	4	yes	—
109	C₁₃	#12011	1	1	yes	characteristic, complemented, Sylow
110	C₂³	#88059	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₄⋊#1055	1	2	yes	—
115	C₂×C₄	#88058	1	2	yes	—
116	C₂×C₄	#88059	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
118	C₈	C₄⋊#1055	1	2	yes	—
119	C₈	#88058	1	2	yes	—
120	C₈	C₂×D₁₀₄	1	1	yes	characteristic
121	C₈	C₂×Dic₅₂	1	1	yes	characteristic
122	C₈	C₂²×#1055	1	1	yes	characteristic
123	C₈	C₂×#6099	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	#11718050	1	2	yes	—
126	C₂²	C₂×#88059	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	#11718050	1	2	yes	—
130	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
131	C₄	C₂×#88058	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×#11718050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718055

C₂

yes

maximal

#13112912

C₂

yes

maximal, complemented

C₄×C₁₃×M₃₂

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₄

yes

characteristic

C₂×C₁₃×M₃₂

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88052

—

C₁₃×#346

C₄

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

—

#11718055

—

#11718055

C₂²

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

Q₈

yes

characteristic

C₁₃×M₃₂

—

C₁₃×M₃₂

C₈

yes

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

Q₈

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

#6090

—

C₂×#364

—

C₄×M₃₂

D₁₃

yes

characteristic

#1373

—

C₂³×C₁₃

#108

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

#107

yes

—

C₂×C₄×C₁₃

#108

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₃

C₄⋊C₈

yes

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

characteristic

C₈×C₁₃

C₂×D₈

yes

characteristic

C₈×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

Dic₁₃

yes

characteristic

C₂×M₃₂

Dic₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

Dic₁₃

yes

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

#335

yes

—

C₂²×C₁₃

C₂×#108

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#335

yes

—

C₄×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₃

C₂×#107

yes

characteristic

C₂²×C₈

D₅₂

yes

characteristic

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

Dic₂₆

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

#1055

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₄×C₈

Dic₂₆

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#335

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

D₁₀₄

yes

—

C₂×C₈

Dic₅₂

yes

—

C₂×C₈

C₂×#1055

yes

—

C₂×C₈

#6099

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

100

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

101

C₂×C₈

C₂×D₅₂

yes

characteristic

102

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

103

C₄²

—

104

C₄²

C₄⋊Dic₁₃

yes

characteristic

105

C₄²

C₂×Dic₂₆

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

C₂×#1055

yes

—

109

C₁₃

#12011

yes

characteristic, complemented, Sylow

110

C₂³

#88059

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄⋊#1055

yes

—

115

C₂×C₄

#88058

yes

—

116

C₂×C₄

#88059

yes

characteristic

117

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

118

C₈

C₄⋊#1055

yes

—

119

C₈

#88058

yes

—

120

C₈

C₂×D₁₀₄

yes

characteristic

121

C₈

C₂×Dic₅₂

yes

characteristic

122

C₈

C₂²×#1055

yes

characteristic

123

C₈

C₂×#6099

yes

characteristic

124

C₂²

—

125

C₂²

#11718050

yes

—

126

C₂²

C₂×#88059

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

#11718050

yes

—

130

C₄

C₂×C₄⋊#1055

yes

characteristic

131

C₄

C₂×#88058

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

C₂×#11718050

yes

characteristic

	subgroup	G/H
center	C₈	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂×C₁₃	C₂×#335
prime radical	C₂³×C₁₃	#108
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12011
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₈

C₂×D₁₀₄

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₄×C₁₃×M₃₂

C₂

C₂×C₁₃

C₂³×C₁₃

C₄×C₈×C₁₃

C₂³

hypercenter

C₄×M₃₂

D₁₃

hypocenter

C₁₃

#12011

LCS

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
2. #13112912	C₂	semidirect
109. C₁₃	#12011	semidirect

H ≅ G/N

type

2. #13112912

C₂

semidirect

109. C₁₃

#12011

semidirect

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

subgroup	index
C₂×#11718055	2
#13112912	2
C₄×C₁₃×M₃₂	2
#12011	13

subgroup

index

C₂×#11718055

#13112912

C₄×C₁₃×M₃₂

#12011

Subgroups — #15774515

Conjugacy classes of subgroups

Complements

Maximal subgroups