Group #16782378 — subgroups

Subgroups — #16782378

Order 3328.

228 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 80 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	#13129693	C₂	1	4	yes	maximal
3	#13113118	C₂	1	1	yes	maximal, characteristic
4	#11719054	C₂²	1	4	yes	—
5	#11718346	C₂²	1	2	yes	—
6	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	#11718254	—	2	2	no	—
8	#11718253	—	2	2	no	—
9	C₈²×C₁₃	C₄	1	1	yes	characteristic
10	#11718046	C₄	1	2	yes	—
11	C₈×#1055	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
13	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
14	C₄²×D₁₃	C₂³	1	1	yes	characteristic
15	#88111	—	2	4	no	—
16	#88110	—	2	4	no	—
17	C₄⋊#1057	—	2	4	no	—
18	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	#88060	—	2	2	no	—
21	#88054	—	2	2	no	—
22	C₈×Dic₁₃	D₄	1	1	yes	characteristic
23	C₈×Dic₁₃	D₄	1	1	yes	characteristic
24	C₄×#1055	C₂×C₄	1	2	yes	—
25	C₄×#1055	C₂×C₄	1	1	yes	characteristic
26	#8796	—	13	1	no	maximal, Sylow
27	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×D₁₃	#41	1	2	yes	—
29	C₂×#1057	—	4	4	no	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
35	C₄×Dic₁₃	#41	1	2	yes	—
36	C₂×#1055	—	2	2	no	—
37	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
39	C₈×D₁₃	—	2	2	no	—
40	C₈×D₁₃	—	2	2	no	—
41	#1568	—	13	2	no	—
42	#1450	—	13	1	no	—
43	#1287	—	13	4	no	—
44	C₂²×D₁₃	#125	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#117	1	2	yes	—
47	C₂×Dic₁₃	#124	1	2	yes	—
48	C₂×Dic₁₃	#125	1	1	yes	characteristic
49	C₄×D₁₃	—	2	1	no	—
50	C₄×D₁₃	—	2	1	no	—
51	C₄×D₁₃	—	4	2	no	—
52	#1057	—	8	4	no	—
53	C₈×C₁₃	—	4	2	no	—
54	C₈×C₁₃	#108	1	2	yes	—
55	C₈×C₁₃	#108	1	2	yes	—
56	#1055	—	2	1	no	—
57	#1055	—	2	1	no	—
58	#1055	—	4	2	no	—
59	#432	—	13	4	no	—
60	#431	—	13	2	no	—
61	C₂×C₄×C₈	—	13	1	no	—
62	#335	—	26	2	no	—
63	#334	—	26	2	no	—
64	#322	—	13	2	no	—
65	C₈²	—	13	1	no	—
66	C₈²	#261	1	1	yes	characteristic
67	C₂²×C₁₃	#382	1	1	yes	characteristic
68	C₂×D₁₃	—	2	2	no	—
69	C₂×D₁₃	—	2	1	no	—
70	C₄×C₁₃	—	2	2	no	—
71	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
72	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
73	Dic₁₃	—	2	2	no	—
74	Dic₁₃	—	2	1	no	—
75	C₂²×C₈	—	13	1	no	—
76	C₂²×C₈	—	13	1	no	—
77	C₂×C₄²	—	13	1	no	—
78	C₄⋊C₈	—	26	2	no	—
79	C₄⋊C₈	—	26	4	no	—
80	C₂²⋊C₈	—	26	2	no	—
81	C₂²⋊C₈	—	26	4	no	—
82	#97	—	26	4	no	—
83	C₄×C₈	—	13	1	no	—
84	C₄×C₈	—	13	1	no	—
85	C₄×C₈	—	13	1	no	—
86	C₄×C₈	—	13	2	no	—
87	C₄×C₈	C₂×#261	1	2	yes	—
88	C₄×C₈	C₂×#261	1	1	yes	characteristic
89	C₂×C₁₃	#1833	1	2	yes	—
90	C₂×C₁₃	#1837	1	1	yes	characteristic
91	D₁₃	—	4	1	no	—
92	C₂²×C₄	—	13	1	no	—
93	C₂²×C₄	—	13	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	13	1	no	—
96	C₂×C₈	—	13	1	no	—
97	C₂×C₈	—	26	2	no	—
98	C₂×C₈	—	26	2	no	—
99	C₂×C₈	—	26	2	no	—
100	C₂×C₈	—	52	4	no	—
101	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
102	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
103	C₄²	—	13	1	no	—
104	C₄²	—	13	2	no	—
105	C₄²	C₂²×#261	1	1	yes	characteristic
106	C₁₃	#8796	1	1	yes	characteristic, complemented, Sylow
107	C₂³	—	13	1	no	—
108	C₂×C₄	—	13	2	no	—
109	C₂×C₄	—	13	1	no	—
110	C₂×C₄	—	26	1	no	—
111	C₂×C₄	—	26	1	no	—
112	C₂×C₄	—	52	2	no	—
113	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
114	C₂×C₄	#88237	1	2	yes	—
115	C₈	—	4	2	no	—
116	C₈	—	26	1	no	—
117	C₈	—	26	1	no	—
118	C₈	—	52	2	no	—
119	C₈	—	104	4	no	—
120	C₈	#88103	1	2	yes	—
121	C₈	#88104	1	2	yes	—
122	C₂²	—	26	2	no	—
123	C₂²	—	26	1	no	—
124	C₂²	#11719060	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	26	2	no	—
127	C₄	—	26	1	no	—
128	C₄	C₂×#88103	1	1	yes	characteristic
129	C₄	C₂×#88104	1	1	yes	characteristic
130	C₂	—	52	1	no	—
131	C₂	#13130755	1	2	yes	—
132	C₂	#13130760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130397

C₂

yes

maximal

#13129693

C₂

yes

maximal

#13113118

C₂

yes

maximal, characteristic

#11719054

C₂²

yes

—

#11718346

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718254

—

#11718253

—

C₈²×C₁₃

C₄

yes

characteristic

#11718046

C₄

yes

—

C₈×#1055

C₄

yes

characteristic

C₂×C₈×D₁₃

Q₈

yes

characteristic

C₂×C₈×D₁₃

Q₈

yes

characteristic

C₄²×D₁₃

C₂³

yes

—

—

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88060

—

#88054

—

C₈×Dic₁₃

D₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#8796

—

maximal, Sylow

C₂×C₄×D₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₁₃

#41

yes

—

C₂×#1057

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₄×Dic₁₃

#41

yes

—

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

#1568

—

#1450

—

#1287

—

C₂²×D₁₃

#125

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×Dic₁₃

#124

yes

—

C₂×Dic₁₃

#125

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

#108

yes

—

C₈×C₁₃

#108

yes

—

—

—

—

—

—

C₂×C₄×C₈

—

#335

—

#334

—

#322

—

C₈²

—

C₈²

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

—

C₄×C₈

C₂×#261

yes

characteristic

C₂×C₁₃

#1833

yes

—

C₂×C₁₃

#1837

yes

characteristic

D₁₃

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄⋊#261

yes

characteristic

102

C₂×C₈

C₄⋊#261

yes

characteristic

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×#261

yes

characteristic

106

C₁₃

#8796

yes

characteristic, complemented, Sylow

107

C₂³

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

114

C₂×C₄

#88237

yes

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

104

—

120

C₈

#88103

yes

—

121

C₈

#88104

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

#11719060

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₂×#88103

yes

characteristic

129

C₄

C₂×#88104

yes

characteristic

130

C₂

—

131

C₂

#13130755

yes

—

132

C₂

#13130760

yes

characteristic

	subgroup	G/H
center	C₂²	#11719060
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₈²×C₁₃	C₄
socle	C₂²×C₁₃	#382
prime radical	C₂²×D₁₃	#125
cosocle	C₄²×D₁₃	C₂³
hypercenter	C₈²	#261
hypocenter	C₁₃	#8796
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11719060

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₈²×C₁₃

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₈²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
106. C₁₃	#8796	semidirect

H ≅ G/N

type

106. C₁₃

#8796

semidirect

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

subgroup	index
#13130397	2
#13129693	2
#13113118	2
#8796	13

subgroup

index

Subgroups — #16782378

Conjugacy classes of subgroups

Complements

Maximal subgroups