Group #16788903 — subgroups

Subgroups — #16788903

Order 3328.

204 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13131228	C₂	1	2	yes	maximal, complemented
2	#13129863	C₂	1	1	yes	maximal, characteristic, complemented
3	#13129801	C₂	1	2	yes	maximal, complemented
4	#13129752	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116721	C₂	1	1	yes	maximal, characteristic
6	#11719096	—	2	2	no	—
7	#11718565	C₄	1	1	yes	characteristic
8	#11718556	C₂²	1	2	yes	—
9	#11718270	—	2	2	no	—
10	#11718261	C₂²	1	2	yes	—
11	#11718258	C₂²	1	1	yes	characteristic
12	#11718251	C₂²	1	1	yes	characteristic
13	C₁₃×#368	C₄	1	1	yes	characteristic
14	#11718094	C₂²	1	1	yes	characteristic
15	#11718089	C₄	1	2	yes	—
16	#88161	—	2	2	no	—
17	#88158	C₂×C₄	1	2	yes	—
18	C₄×#1057	—	2	1	no	—
19	#88106	—	2	2	no	—
20	C₂×#6092	D₄	1	2	yes	—
21	#88098	—	4	2	no	—
22	D₁₃⋊C₁₆	—	4	2	no	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#108	C₂×C₄	1	2	yes	—
25	#88065	—	2	2	no	—
26	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂×#6090	C₂×C₄	1	1	yes	characteristic
28	Dic₁₀₄	C₈	1	1	yes	characteristic, complemented
29	#88040	C₈	1	2	yes	complemented
30	D₂₀₈	C₈	1	1	yes	characteristic, complemented
31	#12068	—	13	1	no	maximal, Sylow
32	#6127	—	2	2	no	—
33	C₂×#1057	—	4	1	no	—
34	C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₁₃	#41	1	1	yes	characteristic
37	C₂×#1055	C₂×D₄	1	1	yes	characteristic
38	Dic₅₂	C₂×C₈	1	2	yes	—
39	D₁₀₄	C₂×C₈	1	2	yes	—
40	#6095	—	2	2	no	—
41	#6094	—	4	2	no	—
42	C₈×D₁₃	—	4	2	no	—
43	#6092	—	2	2	no	—
44	#6092	—	4	2	no	—
45	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
46	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
47	#6090	—	2	1	no	—
48	#2179	—	13	1	no	—
49	#2171	—	13	2	no	—
50	#1369	—	13	1	no	—
51	#1336	—	13	2	no	—
52	#1313	—	13	1	no	—
53	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
54	C₂²⋊D₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	2	1	no	—
56	D₅₂	—	2	2	no	—
57	C₄×D₁₃	—	4	2	no	—
58	Dic₂₆	—	2	2	no	—
59	#1057	—	8	1	no	—
60	C₈×C₁₃	—	4	2	no	—
61	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₁₃	#131	1	1	yes	characteristic
63	#1055	AGL(1,ℤ/8)	1	2	yes	—
64	#508	—	13	1	no	—
65	#504	—	26	2	no	—
66	#443	—	13	2	no	—
67	#368	#261	1	1	yes	characteristic
68	#364	—	13	2	no	—
69	#359	—	13	2	no	—
70	#350	—	26	2	no	—
71	#346	—	13	1	no	—
72	#346	—	13	1	no	—
73	#322	—	13	1	no	—
74	C₂²×C₁₃	#444	1	1	yes	characteristic
75	C₂×D₁₃	—	4	2	no	—
76	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
77	C₄×C₁₃	#442	1	1	yes	characteristic
78	Dic₁₃	—	4	1	no	—
79	Dic₁₃	—	4	2	no	—
80	#135	—	13	2	no	—
81	#131	—	26	2	no	—
82	Q₃₂	—	13	1	no	—
83	SD₃₂	—	13	2	no	—
84	D₁₆	—	13	1	no	—
85	M₃₂	—	26	2	no	—
86	M₃₂	—	52	2	no	—
87	C₂×C₁₆	—	13	2	no	—
88	C₂×C₁₆	—	13	1	no	—
89	C₂×C₁₆	—	52	2	no	—
90	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
91	#108	C₂×#261	1	2	yes	—
92	#104	—	26	2	no	—
93	C₄×C₈	—	13	1	no	—
94	C₄×C₈	—	26	1	no	—
95	C₂×C₁₃	—	2	1	no	—
96	C₂×C₁₃	#1580	1	1	yes	characteristic
97	D₁₃	—	8	2	no	—
98	#41	—	26	2	no	—
99	Q₁₆	—	13	2	no	—
100	SD₁₆	—	26	2	no	—
101	D₈	—	13	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	52	2	no	—
104	C₂×C₈	—	13	1	no	—
105	C₂×C₈	—	52	2	no	—
106	C₂×C₈	—	52	1	no	—
107	C₂×C₈	C₂²×#261	1	1	yes	characteristic
108	C₄²	—	13	1	no	—
109	C₁₆	—	26	2	no	—
110	C₁₆	—	26	1	no	—
111	C₁₆	—	52	2	no	—
112	C₁₆	D₁₃⋊C₈	1	1	yes	characteristic
113	C₁₆	D₁₃⋊C₈	1	1	yes	characteristic
114	C₁₃	#12068	1	1	yes	characteristic, complemented, Sylow
115	Q₈	—	26	2	no	—
116	D₄	—	26	2	no	—
117	D₄	—	52	2	no	—
118	C₂×C₄	—	26	1	no	—
119	C₂×C₄	—	52	2	no	—
120	C₂×C₄	D₄×#261	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	13	2	no	—
123	C₈	—	104	1	no	—
124	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
125	C₈	#88240	1	1	yes	characteristic
126	C₂²	—	52	2	no	—
127	C₂²	#11719112	1	1	yes	characteristic
128	C₄	—	52	1	no	—
129	C₄	—	52	2	no	—
130	C₄	#11719073	1	1	yes	characteristic
131	C₄	#11719102	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	104	2	no	—
134	C₂	#13130434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131228

C₂

yes

maximal, complemented

#13129863

C₂

yes

maximal, characteristic, complemented

#13129801

C₂

yes

maximal, complemented

#13129752

C₂

yes

maximal, characteristic, complemented

#13116721

C₂

yes

maximal, characteristic

#11719096

—

#11718565

C₄

yes

characteristic

#11718556

C₂²

yes

—

#11718270

—

#11718261

C₂²

yes

—

#11718258

C₂²

yes

characteristic

#11718251

C₂²

yes

characteristic

C₁₃×#368

C₄

yes

characteristic

#11718094

C₂²

yes

characteristic

#11718089

C₄

yes

—

#88161

—

#88158

C₂×C₄

yes

—

C₄×#1057

—

#88106

—

C₂×#6092

D₄

yes

—

#88098

—

D₁₃⋊C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×#108

C₂×C₄

yes

—

#88065

—

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

Dic₁₀₄

C₈

yes

characteristic, complemented

#88040

C₈

yes

complemented

D₂₀₈

C₈

yes

characteristic, complemented

—

—

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂²×C₄

yes

C₄×Dic₁₃

yes

C₂×D₄

yes

characteristic

Dic₅₂

C₂×C₈

yes

—

D₁₀₄

C₂×C₈

yes

—

#6095

—

#6094

—

C₈×D₁₃

—

#6092

—

#6092

—

C₁₃×C₁₆

C₂×C₈

yes

characteristic

C₁₃×C₁₆

C₂×C₈

yes

—

—

—

—

—

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

yes

—

—

—

—

yes

—

—

—

—

—

—

C₂²×C₁₃

#444

yes

characteristic

C₂×D₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#442

yes

characteristic

Dic₁₃

—

Dic₁₃

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

#104

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#1580

yes

characteristic

D₁₃

—

#41

—

Q₁₆

—

100

SD₁₆

—

101

D₈

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×#261

yes

characteristic

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

D₁₃⋊C₈

yes

characteristic

113

C₁₆

D₁₃⋊C₈

yes

characteristic

114

C₁₃

#12068

yes

characteristic, complemented, Sylow

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×#261

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

104

—

124

C₈

C₂×D₁₃⋊C₈

yes

characteristic

125

C₈

#88240

yes

characteristic

126

C₂²

—

127

C₂²

#11719112

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#11719073

yes

characteristic

131

C₄

#11719102

yes

characteristic

132

C₂

—

133

C₂

—

104

—

134

C₂

#13130434

yes

characteristic

	subgroup	G/H
center	C₄	#11719102
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#368	C₄
socle	C₂×C₁₃	#1580
prime radical	#11718565	C₄
cosocle	C₈×Dic₁₃	C₂³
hypercenter	#368	#261
hypocenter	C₁₃	#12068
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

#11719102

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#261

Fitting

C₁₃×#368

C₄

C₂×C₁₃

C₄

C₈×Dic₁₃

C₂³

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #13131228	C₂	semidirect
2. #13129863	C₂	semidirect
3. #13129801	C₂	semidirect
4. #13129752	C₂	semidirect
28. Dic₁₀₄	C₈	semidirect
29. #88040	C₈	semidirect
30. D₂₀₈	C₈	semidirect
114. C₁₃	#12068	semidirect

H ≅ G/N

type

1. #13131228

C₂

semidirect

2. #13129863

C₂

semidirect

3. #13129801

C₂

semidirect

4. #13129752

C₂

semidirect

28. Dic₁₀₄

C₈

semidirect

29. #88040

C₈

semidirect

30. D₂₀₈

C₈

semidirect

114. C₁₃

#12068

semidirect

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

subgroup	index
#13131228	2
#13129863	2
#13129801	2
#13129752	2
#13116721	2
#12068	13

subgroup

index

Subgroups — #16788903

Conjugacy classes of subgroups

Complements

Maximal subgroups