Group #15774758 — subgroups

Subgroups — #15774758

Order 3328.

212 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 117 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	Q₈×#6090	C₂	1	2	yes	maximal
2	#13112900	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#6090	C₂	1	2	yes	maximal
4	#13112754	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
8	C₁₃×#335	C₄	1	1	yes	characteristic
9	C₁₃×Q₈⋊C₈	C₄	1	2	yes	—
10	C₈²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄⋊#6090	—	2	2	no	—
12	C₄⋊#6090	C₂²	1	2	yes	—
13	C₄×#6090	—	2	2	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₄×#6090	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₁₃×#107	C₈	1	1	yes	characteristic
19	C₁₃×C₄⋊C₈	—	2	2	no	—
20	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
21	C₁₃×#103	C₈	1	2	yes	—
22	C₄×C₈×C₁₃	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×#6090	—	2	2	no	—
27	C₂×#6090	—	2	1	no	—
28	C₂×#6090	D₄	1	2	yes	—
29	#12059	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
31	C₁₃×Q₁₆	C₁₆	1	4	yes	complemented
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	—	2	1	no	—
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
38	C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
40	C₄²×C₁₃	—	2	2	no	—
41	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	#6090	—	2	2	no	—
43	#6090	—	4	1	no	—
44	#6090	—	4	2	no	—
45	Q₈×C₁₆	—	13	2	no	—
46	C₈×Q₁₆	D₁₃	1	1	yes	characteristic
47	#1367	—	13	1	no	—
48	Q₈⋊C₁₆	—	13	2	no	—
49	#1312	—	13	1	no	—
50	Q₈×C₁₃	—	2	2	no	—
51	Q₈×C₁₃	C₂×C₁₆	1	4	yes	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
57	C₈×C₁₃	—	2	1	no	—
58	C₈×C₁₃	—	2	1	no	—
59	C₈×C₁₃	—	4	1	no	—
60	C₈×C₁₃	—	4	2	no	—
61	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
62	C₈×C₁₃	#137	1	2	yes	—
63	C₈×Q₈	C₂×D₁₃	1	2	yes	—
64	C₄×Q₁₆	Dic₁₃	1	1	yes	characteristic
65	C₄⋊C₁₆	—	13	2	no	—
66	C₄⋊C₁₆	—	26	2	no	—
67	C₄×C₁₆	—	13	1	no	—
68	C₄×C₁₆	—	13	1	no	—
69	C₄×C₁₆	—	26	2	no	—
70	#335	Dic₁₃	1	1	yes	characteristic
71	Q₈⋊C₈	Dic₁₃	1	2	yes	—
72	C₈²	C₂×D₁₃	1	1	yes	characteristic
73	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
74	C₄×C₁₃	—	2	4	no	—
75	C₄×C₁₃	—	2	1	no	—
76	C₄×C₁₃	—	4	2	no	—
77	C₄×C₁₃	#441	1	1	yes	characteristic
78	C₄×C₁₃	#444	1	1	yes	characteristic
79	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
80	C₄×C₁₃	#504	1	1	yes	characteristic
81	C₂×Q₁₆	#1055	1	1	yes	characteristic
82	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
83	C₂×C₁₆	—	13	2	no	—
84	C₂×C₁₆	—	26	2	no	—
85	C₂×C₁₆	—	26	1	no	—
86	#107	#1055	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
89	#103	#1055	1	2	yes	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
92	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
94	C₂×C₁₃	#1579	1	1	yes	characteristic
95	C₂×C₁₃	D₄×C₁₆	1	1	yes	characteristic
96	C₂×C₁₃	#2171	1	1	yes	characteristic
97	C₂×Q₈	C₂×#1055	1	2	yes	—
98	Q₁₆	#6090	1	4	yes	complemented
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	C₂×#1055	1	1	yes	characteristic
102	C₂×C₈	C₂×#1055	1	1	yes	characteristic
103	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
104	C₂×C₈	#6129	1	1	yes	characteristic
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×#1055	1	2	yes	—
107	C₄²	—	2	2	no	—
108	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
109	C₁₆	—	26	2	no	—
110	C₁₆	—	52	1	no	—
111	C₁₆	—	52	2	no	—
112	C₁₃	#12059	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	2	2	no	—
114	Q₈	C₂×#6090	1	4	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
118	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
119	C₂×C₄	#88202	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	C₂×#6090	1	2	yes	—
125	C₈	#88172	1	2	yes	—
126	C₂²	D₄×#1055	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	4	no	—
129	C₄	—	4	2	no	—
130	C₄	C₂²×#6090	1	1	yes	characteristic
131	C₄	#11718801	1	1	yes	characteristic
132	C₄	#11718845	1	1	yes	characteristic
133	C₄	#11718858	1	1	yes	characteristic
134	C₂	#13113655	1	1	yes	characteristic
135	C₂	D₄×#6090	1	1	yes	characteristic
136	C₂	#13116678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#6090

C₂

yes

maximal

#13112900

C₂

yes

maximal, characteristic

Q₈⋊#6090

C₂

yes

maximal

#13112754

C₂

yes

maximal, characteristic

C₈×C₁₃×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

C₂²

yes

—

C₄×C₁₃×Q₁₆

C₄

yes

characteristic

C₁₃×#335

C₄

yes

characteristic

C₁₃×Q₈⋊C₈

C₄

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₄⋊#6090

—

C₄⋊#6090

C₂²

yes

—

C₄×#6090

—

C₄×#6090

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

C₈

yes

characteristic

C₄×Q₈×C₁₃

C₂×C₄

yes

—

C₁₃×#107

C₈

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₁₃×#103

C₈

yes

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

D₄

yes

—

#12059

—

maximal, Sylow

C₂×Q₈×C₁₃

C₂×C₈

yes

—

C₁₃×Q₁₆

C₁₆

yes

complemented

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

—

—

—

Q₈×C₁₆

—

C₈×Q₁₆

D₁₃

yes

characteristic

#1367

—

Q₈⋊C₁₆

—

#1312

—

Q₈×C₁₃

—

Q₈×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₁₆

yes

—

C₈×C₁₃

#137

yes

—

C₈×Q₈

C₂×D₁₃

yes

—

C₄×Q₁₆

Dic₁₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₁₃

yes

characteristic

Q₈⋊C₈

Dic₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂²×C₁₆

yes

C₄×C₁₃

yes

C₂×Q₁₆

yes

C₄×Q₈

C₂×Dic₁₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#1055

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₁₃

yes

—

#103

#1055

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

#1579

yes

characteristic

C₂×C₁₃

D₄×C₁₆

yes

C₂×C₁₃

yes

C₂×Q₈

yes

—

Q₁₆

#6090

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×#1055

yes

characteristic

102

C₂×C₈

C₂×#1055

yes

characteristic

103

C₂×C₈

D₄×D₁₃

yes

characteristic

104

C₂×C₈

#6129

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×#1055

yes

—

107

C₄²

—

108

C₄²

C₂²×Dic₁₃

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₃

#12059

yes

characteristic, complemented, Sylow

113

Q₈

—

114

Q₈

C₂×#6090

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×#1055

yes

characteristic

118

C₂×C₄

D₄×Dic₁₃

yes

characteristic

119

C₂×C₄

#88202

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#6090

yes

—

125

C₈

#88172

yes

—

126

C₂²

D₄×#1055

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₂²×#6090

yes

characteristic

131

C₄

#11718801

yes

characteristic

132

C₄

#11718845

yes

characteristic

133

C₄

#11718858

yes

characteristic

134

C₂

#13113655

yes

characteristic

135

C₂

D₄×#6090

yes

characteristic

136

C₂

#13116678

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×Q₁₆	C₂
socle, prime radical	C₂²×C₁₃	C₈×D₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#12059
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₁₃

commutator

C₄×C₁₃

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₈×C₁₃×Q₁₆

C₂

socle, prime radical

C₂²×C₁₃

C₈×D₄

cosocle

C₄×C₈×C₁₃

C₂³

hypercenter

C₈×Q₁₆

D₁₃

hypocenter

C₁₃

#12059

LCS

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
31. C₁₃×Q₁₆	C₁₆	semidirect
98. Q₁₆	#6090	semidirect
112. C₁₃	#12059	semidirect

H ≅ G/N

type

31. C₁₃×Q₁₆

C₁₆

semidirect

98. Q₁₆

#6090

semidirect

112. C₁₃

#12059

semidirect

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

subgroup	index
Q₈×#6090	2
#13112900	2
Q₈⋊#6090	2
#13112754	2
C₈×C₁₃×Q₁₆	2
#12059	13

subgroup

index

C₈×C₁₃×Q₁₆

Subgroups — #15774758

Conjugacy classes of subgroups

Complements

Maximal subgroups