Group #15778334 — subgroups

Subgroups — #15778334

Order 3328.

306 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 135 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×#11718085	C₂	1	2	yes	maximal
2	C₂×#11718094	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₁₃	C₂	1	2	yes	maximal
4	#13112930	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#108	C₂²	1	2	yes	—
8	M₁₆×Dic₁₃	—	2	2	no	—
9	C₂×#88059	C₂²	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6090	C₂²	1	1	yes	characteristic
12	#11718150	—	2	2	no	—
13	#11718094	C₄	1	4	yes	—
14	#11718085	C₄	1	8	yes	complemented
15	C₂×C₁₃×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
18	C₂×#6099	—	2	2	no	—
19	C₂²×#1055	D₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	—	2	2	no	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
22	C₁₃×#108	—	2	2	no	—
23	C₁₃×#108	C₂×C₄	1	4	yes	—
24	#88059	—	2	2	no	—
25	#88059	C₂×C₄	1	4	yes	—
26	#88055	—	4	2	no	—
27	C₈×Dic₁₃	—	4	2	no	—
28	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₁₃	Q₈	1	2	yes	—
30	C₂×#6090	C₂×C₄	1	4	yes	—
31	C₂×#6090	C₂×C₄	1	2	yes	—
32	#12676	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	Q₁₆	1	2	yes	—
35	C₁₃×M₁₆	—	2	4	no	—
36	C₁₃×M₁₆	—	4	2	no	—
37	C₂×C₈×C₁₃	—	4	2	no	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	#41	1	2	yes	—
42	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
43	C₄×Dic₁₃	D₈	1	2	yes	—
44	#6099	—	2	4	no	—
45	#6099	—	4	2	no	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	4	2	no	—
48	C₂×#1055	C₄⋊C₄	1	2	yes	—
49	C₁₃×C₁₆	—	2	2	no	—
50	#6090	C₄²	1	4	yes	—
51	C₂×#368	—	13	2	no	—
52	C₂×#346	—	13	1	no	—
53	C₄×#108	—	13	2	no	—
54	#1381	—	13	1	no	—
55	#1381	D₁₃	1	1	yes	characteristic
56	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#135	1	2	yes	—
61	C₂×Dic₁₃	—	4	2	no	—
62	C₂×Dic₁₃	#107	1	4	yes	—
63	C₈×C₁₃	—	4	4	no	—
64	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
65	C₈×C₁₃	#117	1	1	yes	characteristic
66	C₈×C₁₃	C₄×Q₈	1	2	yes	—
67	#1055	—	2	2	no	—
68	#1055	—	4	4	no	—
69	C₂²×C₁₆	—	13	1	no	—
70	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
71	C₂×#108	—	13	2	no	—
72	C₂×#108	C₂×D₁₃	1	2	yes	—
73	C₄×M₁₆	—	26	2	no	—
74	C₂×C₄×C₈	—	13	1	no	—
75	#368	—	13	8	no	—
76	#346	—	13	4	no	—
77	#341	—	26	2	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
81	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₁₃	#427	1	1	yes	characteristic
83	C₄×C₁₃	C₄×D₈	1	2	yes	—
84	Dic₁₃	—	2	4	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	26	2	no	—
87	C₂²×C₈	—	13	1	no	—
88	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
89	C₂×C₄²	—	13	1	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	13	2	no	—
92	C₂×C₁₆	—	13	4	no	—
93	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
94	#108	—	2	2	no	—
95	#108	—	13	4	no	—
96	#108	—	26	2	no	—
97	#108	C₄×D₁₃	1	4	yes	—
98	#97	—	52	2	no	—
99	C₄×C₈	—	13	2	no	—
100	C₄×C₈	—	13	2	no	—
101	C₄×C₈	—	52	2	no	—
102	C₂×C₁₃	—	2	2	no	—
103	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
104	C₂×C₁₃	#2161	1	2	yes	—
105	C₂²×C₄	—	13	2	no	—
106	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
107	M₁₆	—	2	4	no	—
108	M₁₆	—	4	2	no	—
109	M₁₆	—	26	4	no	—
110	M₁₆	—	52	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	13	2	no	—
113	C₂×C₈	—	26	2	no	—
114	C₂×C₈	—	52	2	no	—
115	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
116	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
117	C₂×C₈	#6129	1	2	yes	—
118	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
119	C₄²	—	13	2	no	—
120	C₄²	—	13	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	13	4	no	—
123	C₁₃	#12676	1	1	yes	characteristic, complemented, Sylow
124	C₂³	Q₁₆×D₁₃	1	1	yes	characteristic
125	C₂×C₄	—	13	4	no	—
126	C₂×C₄	—	52	2	no	—
127	C₂×C₄	#88135	1	2	yes	—
128	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
129	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
130	C₂×C₄	#88170	1	2	yes	—
131	C₈	—	4	4	no	—
132	C₈	—	26	2	no	—
133	C₈	—	52	4	no	—
134	C₈	C₄²×D₁₃	1	1	yes	characteristic
135	C₈	#88131	1	1	yes	characteristic
136	C₈	#88141	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#11718451	1	2	yes	—
139	C₂²	D₁₃×#107	1	1	yes	characteristic
140	C₄	—	26	4	no	—
141	C₄	#11718310	1	1	yes	characteristic
142	C₄	#11718417	1	2	yes	—
143	C₄	#11718521	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#13114728	1	1	yes	characteristic
146	C₂	#13116629	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718085

C₂

yes

maximal

C₂×#11718094

C₂

yes

maximal, characteristic

#108×Dic₁₃

C₂

yes

maximal

#13112930

C₂

yes

maximal, characteristic

C₁₃×#1381

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₂×C₁₃×#108

C₂²

yes

—

M₁₆×Dic₁₃

—

C₂×#88059

C₂²

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

#11718150

—

#11718094

C₄

yes

—

#11718085

C₄

yes

complemented

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂×#6099

—

C₂²×#1055

D₄

yes

characteristic

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₁₃×#108

—

C₁₃×#108

C₂×C₄

yes

—

#88059

—

#88059

C₂×C₄

yes

—

#88055

—

C₈×Dic₁₃

—

C₈×Dic₁₃

C₂×C₄

yes

—

C₈×Dic₁₃

Q₈

yes

—

C₂×#6090

C₂×C₄

yes

—

C₂×#6090

C₂×C₄

yes

—

#12676

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

Q₁₆

yes

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₄×Dic₁₃

C₄⋊C₄

yes

—

C₄×Dic₁₃

D₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₁₃×C₁₆

—

#6090

C₄²

yes

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#107

yes

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₄²

yes

characteristic

C₈×C₁₃

#117

yes

characteristic

C₈×C₁₃

C₄×Q₈

yes

—

#1055

—

#1055

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₃

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₁₃

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#368

—

#346

—

#341

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#107

yes

characteristic

C₂²×C₁₃

C₄×Q₁₆

yes

—

C₄×C₁₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₁₃

#427

yes

characteristic

C₄×C₁₃

C₄×D₈

yes

—

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₃

yes

—

—

—

—

C₄×D₁₃

yes

—

#97

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂×C₁₃

—

103

C₂×C₁₃

C₄×#107

yes

characteristic

104

C₂×C₁₃

#2161

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

D₄×D₁₃

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

116

C₂×C₈

C₂×C₄×D₁₃

yes

—

117

C₂×C₈

#6129

yes

—

118

C₂×C₈

Q₈×D₁₃

yes

characteristic

119

C₄²

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

123

C₁₃

#12676

yes

characteristic, complemented, Sylow

124

C₂³

Q₁₆×D₁₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#88135

yes

—

128

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

129

C₂×C₄

D₈×D₁₃

yes

characteristic

130

C₂×C₄

#88170

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₄²×D₁₃

yes

characteristic

135

C₈

#88131

yes

characteristic

136

C₈

#88141

yes

—

137

C₂²

—

138

C₂²

#11718451

yes

—

139

C₂²

D₁₃×#107

yes

characteristic

140

C₄

—

141

C₄

#11718310

yes

characteristic

142

C₄

#11718417

yes

—

143

C₄

#11718521

yes

characteristic

144

C₂

—

145

C₂

#13114728

yes

characteristic

146

C₂

#13116629

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1381	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1381	D₁₃
hypocenter	C₁₃	#12676
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₁₃

commutator

C₈×C₁₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#1381

C₂

socle

C₂²×C₁₃

C₂×#107

prime radical

C₂³×C₁₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
14. #11718085	C₄	semidirect
123. C₁₃	#12676	semidirect

H ≅ G/N

type

14. #11718085

C₄

semidirect

123. C₁₃

#12676

semidirect

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

subgroup	index
C₂×#11718085	2
C₂×#11718094	2
#108×Dic₁₃	2
#13112930	2
C₁₃×#1381	2
#12676	13

subgroup

index

#108×Dic₁₃

C₁₃×#1381

Subgroups — #15778334

Conjugacy classes of subgroups

Complements

Maximal subgroups