Group #16788734 — subgroups

Subgroups — #16788734

Order 3328.

216 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13130397	C₂	1	2	yes	maximal
2	#13129861	C₂	1	2	yes	maximal, complemented
3	#13129860	C₂	1	2	yes	maximal, complemented
4	C₄×#88038	C₂	1	1	yes	maximal, characteristic
5	#11719054	—	2	2	no	—
6	C₂×#88038	C₄	1	1	yes	characteristic
7	C₂×#88038	C₄	1	1	yes	characteristic
8	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	#11718254	—	2	2	no	—
10	#11718253	C₂²	1	4	yes	—
11	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
12	#11718094	C₂²	1	1	yes	characteristic
13	#11718094	C₂²	1	1	yes	characteristic
14	C₄×#6090	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
17	C₄²×D₁₃	D₄	1	1	yes	characteristic
18	#88111	—	4	2	no	—
19	#88110	—	4	2	no	—
20	C₄⋊#1057	—	2	4	no	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₁₃	D₄	1	1	yes	characteristic
25	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂×#6090	C₂×C₄	1	1	yes	characteristic
27	C₂×#6090	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	Q₈	1	1	yes	characteristic
29	#88038	—	2	2	no	—
30	#88038	—	2	2	no	—
31	#12021	—	13	1	no	maximal, Sylow
32	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
34	C₂×#1057	—	4	4	no	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄×Dic₁₃	SD₁₆	1	2	yes	—
39	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
42	C₈×D₁₃	—	2	2	no	—
43	C₈×D₁₃	—	2	1	no	—
44	C₈×D₁₃	—	2	1	no	—
45	C₁₃×C₁₆	M₁₆	1	2	yes	—
46	C₁₃×C₁₆	M₁₆	1	2	yes	—
47	#6090	—	2	1	no	—
48	#6090	—	2	1	no	—
49	C₄×M₃₂	—	13	1	no	—
50	#1568	—	13	2	no	—
51	#1368	—	13	2	no	—
52	#1368	—	13	2	no	—
53	C₂²×D₁₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#106	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₃	#106	1	1	yes	characteristic
57	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
58	C₄×D₁₃	—	2	1	no	—
59	C₄×D₁₃	—	2	1	no	—
60	C₄×D₁₃	—	4	2	no	—
61	#1057	—	8	4	no	—
62	C₈×C₁₃	#108	1	2	yes	—
63	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
65	#1055	—	2	1	no	—
66	#1055	—	2	1	no	—
67	C₂×M₃₂	—	13	1	no	—
68	C₂×M₃₂	—	13	1	no	—
69	#432	—	26	2	no	—
70	C₂×C₄×C₈	—	13	1	no	—
71	#346	—	13	1	no	—
72	#346	—	13	1	no	—
73	C₄×C₁₆	—	13	1	no	—
74	C₄×C₁₆	#261	1	1	yes	characteristic
75	#335	—	26	2	no	—
76	#334	—	13	4	no	—
77	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
78	C₂×D₁₃	—	2	1	no	—
79	C₂×D₁₃	—	2	2	no	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	#423	1	1	yes	characteristic
82	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
83	Dic₁₃	—	2	1	no	—
84	Dic₁₃	—	2	2	no	—
85	C₂²×C₈	—	13	1	no	—
86	C₂²×C₈	—	13	1	no	—
87	C₂×C₄²	—	13	1	no	—
88	M₃₂	—	26	2	no	—
89	M₃₂	—	26	2	no	—
90	C₂×C₁₆	—	13	1	no	—
91	C₂×C₁₆	—	13	1	no	—
92	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
93	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
94	C₄⋊C₈	—	26	4	no	—
95	C₂²⋊C₈	—	52	2	no	—
96	#97	—	52	2	no	—
97	C₄×C₈	—	13	1	no	—
98	C₄×C₈	—	13	1	no	—
99	C₄×C₈	—	13	1	no	—
100	C₄×C₈	C₂×#261	1	1	yes	characteristic
101	C₂×C₁₃	#1566	1	1	yes	characteristic
102	C₂×C₁₃	#2155	1	2	yes	—
103	D₁₃	—	4	1	no	—
104	C₂²×C₄	—	13	1	no	—
105	C₂²×C₄	—	13	2	no	—
106	C₂×C₈	—	13	1	no	—
107	C₂×C₈	—	13	1	no	—
108	C₂×C₈	—	26	2	no	—
109	C₂×C₈	—	26	1	no	—
110	C₂×C₈	—	26	1	no	—
111	C₂×C₈	—	52	4	no	—
112	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
113	C₂×C₈	C₂²×#261	1	1	yes	characteristic
114	C₄²	—	13	2	no	—
115	C₄²	—	13	1	no	—
116	C₄²	C₄⋊#261	1	1	yes	characteristic
117	C₁₆	—	26	1	no	—
118	C₁₆	—	26	1	no	—
119	C₁₆	#6118	1	2	yes	—
120	C₁₆	#6118	1	2	yes	—
121	C₁₃	#12021	1	1	yes	characteristic, complemented, Sylow
122	C₂³	—	13	1	no	—
123	C₂×C₄	—	13	2	no	—
124	C₂×C₄	—	13	1	no	—
125	C₂×C₄	—	26	1	no	—
126	C₂×C₄	—	26	1	no	—
127	C₂×C₄	—	52	2	no	—
128	C₂×C₄	#88101	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
130	C₈	—	26	1	no	—
131	C₈	—	26	1	no	—
132	C₈	—	104	4	no	—
133	C₈	#88103	1	2	yes	—
134	C₈	C₂×#6118	1	1	yes	characteristic
135	C₈	C₂×#6118	1	1	yes	characteristic
136	C₂²	—	26	1	no	—
137	C₂²	—	26	2	no	—
138	C₂²	C₂×#88101	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	26	1	no	—
141	C₄	—	26	2	no	—
142	C₄	#11719052	1	1	yes	characteristic
143	C₄	C₂×#88103	1	1	yes	characteristic
144	C₂	—	52	1	no	—
145	C₂	#13130391	1	1	yes	characteristic
146	C₂	#13131183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130397

C₂

yes

maximal

#13129861

C₂

yes

maximal, complemented

#13129860

C₂

yes

maximal, complemented

C₄×#88038

C₂

yes

maximal, characteristic

#11719054

—

C₂×#88038

C₄

yes

characteristic

C₂×#88038

C₄

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718254

—

#11718253

C₂²

yes

—

C₄×C₁₃×C₁₆

C₄

yes

characteristic

#11718094

C₂²

yes

characteristic

#11718094

C₂²

yes

characteristic

C₄×#6090

C₄

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

Q₈

yes

characteristic

C₄²×D₁₃

D₄

yes

—

—

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₄×#1055

Q₈

yes

—

—

—

C₂×C₄×D₁₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

—

C₂×#1057

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₂×Q₈

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

M₁₆

yes

—

C₁₃×C₁₆

M₁₆

yes

—

#6090

—

#6090

—

C₄×M₃₂

—

#1568

—

#1368

—

#1368

—

C₂²×D₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₃

#106

yes

characteristic

C₂×Dic₁₃

C₂×SD₁₆

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

C₈×C₁₃

#108

yes

—

C₈×C₁₃

C₂×M₁₆

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

characteristic

#1055

—

#1055

—

C₂×M₃₂

—

C₂×M₃₂

—

#432

—

C₂×C₄×C₈

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

C₂²×C₁₃

C₂×#106

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×#261

yes

characteristic

101

C₂×C₁₃

#1566

yes

characteristic

102

C₂×C₁₃

#2155

yes

—

103

D₁₃

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊#261

yes

characteristic

113

C₂×C₈

C₂²×#261

yes

characteristic

114

C₄²

—

115

C₄²

—

116

C₄²

C₄⋊#261

yes

characteristic

117

C₁₆

—

118

C₁₆

—

119

C₁₆

#6118

yes

—

120

C₁₆

#6118

yes

—

121

C₁₃

#12021

yes

characteristic, complemented, Sylow

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#88101

yes

—

129

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

104

—

133

C₈

#88103

yes

—

134

C₈

C₂×#6118

yes

characteristic

135

C₈

C₂×#6118

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×#88101

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#11719052

yes

characteristic

143

C₄

C₂×#88103

yes

characteristic

144

C₂

—

145

C₂

#13130391

yes

characteristic

146

C₂

#13131183

yes

—

	subgroup	G/H
center	C₂²	C₂×#88101
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#261
Fitting	C₄×C₁₃×C₁₆	C₄
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂²×D₁₃	#106
cosocle	C₈×Dic₁₃	C₂³
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	#12021
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#88101

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#261

Fitting

C₄×C₁₃×C₁₆

C₄

C₂²×C₁₃

C₂²×D₁₃

C₈×Dic₁₃

C₂³

C₄×C₁₆

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #13129861	C₂	semidirect
3. #13129860	C₂	semidirect
121. C₁₃	#12021	semidirect

H ≅ G/N

type

2. #13129861

C₂

semidirect

3. #13129860

C₂

semidirect

121. C₁₃

#12021

semidirect

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

subgroup	index
#13130397	2
#13129861	2
#13129860	2
C₄×#88038	2
#12021	13

subgroup

index

Subgroups — #16788734

Conjugacy classes of subgroups

Complements

Maximal subgroups