Group #16788807 — subgroups

Subgroups — #16788807

Order 3328.

222 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 74 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	#13130424	C₂	1	2	yes	maximal
2	#13129869	C₂	1	1	yes	maximal, characteristic
3	#13129793	C₂	1	2	yes	maximal
4	#13129747	C₂	1	1	yes	maximal, characteristic
5	#13116694	C₂	1	1	yes	maximal, characteristic
6	#11719074	—	2	2	no	—
7	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic
8	#11718534	C₂²	1	2	yes	—
9	#11718257	C₂²	1	1	yes	characteristic
10	#11718254	C₂²	1	2	yes	—
11	C₈×#1057	C₂²	1	1	yes	characteristic
12	#11718244	—	2	2	no	—
13	C₁₃×#366	C₄	1	1	yes	characteristic
14	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718084	C₄	1	2	yes	—
16	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
17	#88141	—	2	2	no	—
18	#88112	—	4	2	no	—
19	C₄⋊#1057	—	2	2	no	—
20	C₄×#1057	—	4	2	no	—
21	C₄×#1057	D₄	1	1	yes	characteristic
22	C₄×#1057	D₄	1	1	yes	characteristic
23	C₂×#6092	—	2	1	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#107	C₂×C₄	1	2	yes	—
26	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂×#6090	C₂×C₄	1	1	yes	characteristic
28	#88050	—	2	2	no	—
29	Dic₁₀₄	C₈	1	4	yes	complemented
30	#12044	—	13	1	no	maximal, Sylow
31	C₂×Dic₂₆	—	2	2	no	—
32	C₂×#1057	—	2	1	no	—
33	C₂×#1057	—	4	2	no	—
34	C₂×#1057	D₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₁₃×C₄⋊C₄	—	2	2	no	—
37	#6101	—	4	2	no	—
38	C₄×Dic₁₃	—	4	2	no	—
39	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×#1055	#41	1	1	yes	characteristic
41	Dic₅₂	—	2	2	no	—
42	Dic₅₂	C₂×C₈	1	4	yes	—
43	#6092	—	2	2	no	—
44	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
45	#6090	—	2	1	no	—
46	C₄×Q₃₂	—	13	1	no	—
47	C₈×Q₁₆	—	13	2	no	—
48	#1372	—	13	1	no	—
49	#1332	—	13	2	no	—
50	#1311	—	13	1	no	—
51	C₂×C₄×C₁₃	—	4	2	no	—
52	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
53	C₂×Dic₁₃	—	4	2	no	—
54	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₁₃	#135	1	1	yes	characteristic
56	Dic₂₆	—	2	4	no	—
57	Dic₂₆	—	4	2	no	—
58	#1057	—	2	2	no	—
59	#1057	—	4	1	no	—
60	#1057	—	8	2	no	—
61	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₁₃	#131	1	1	yes	characteristic
63	#1055	—	2	1	no	—
64	C₂×Q₃₂	—	13	1	no	—
65	C₈×Q₈	—	26	2	no	—
66	C₄×Q₁₆	—	13	2	no	—
67	#366	#261	1	1	yes	characteristic
68	#358	—	13	2	no	—
69	C₄×C₁₆	—	13	1	no	—
70	C₄×C₁₆	—	13	1	no	—
71	#335	—	13	2	no	—
72	Q₈⋊C₈	—	26	2	no	—
73	C₈²	—	13	1	no	—
74	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
75	C₄×C₁₃	—	8	2	no	—
76	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
77	C₄×C₁₃	#443	1	1	yes	characteristic
78	Dic₁₃	—	2	1	no	—
79	Dic₁₃	—	4	4	no	—
80	Dic₁₃	#510	1	2	yes	—
81	C₂×Q₁₆	—	13	2	no	—
82	C₄×Q₈	—	26	2	no	—
83	Q₃₂	—	13	4	no	—
84	C₂×C₁₆	—	13	1	no	—
85	C₂×C₁₆	—	26	1	no	—
86	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
87	#107	C₂×#261	1	2	yes	—
88	C₄⋊C₈	—	26	2	no	—
89	C₄⋊C₈	—	52	2	no	—
90	#103	—	26	2	no	—
91	C₄×C₈	—	13	1	no	—
92	C₄×C₈	—	13	1	no	—
93	C₄×C₈	—	13	1	no	—
94	C₄×C₈	—	52	2	no	—
95	C₂×C₁₃	C₈×D₈	1	1	yes	characteristic
96	C₂×C₁₃	#2176	1	1	yes	characteristic
97	C₂×C₁₃	#2179	1	1	yes	characteristic
98	C₂×Q₈	—	26	2	no	—
99	Q₁₆	—	13	4	no	—
100	Q₁₆	—	26	2	no	—
101	C₂×C₈	—	13	2	no	—
102	C₂×C₈	—	13	1	no	—
103	C₂×C₈	—	26	1	no	—
104	C₂×C₈	—	52	2	no	—
105	C₂×C₈	C₂²×#261	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	52	2	no	—
108	C₄²	—	13	1	no	—
109	C₄²	—	52	2	no	—
110	C₁₆	—	26	2	no	—
111	C₁₆	—	26	1	no	—
112	C₁₆	D₁₃⋊C₈	1	2	yes	—
113	C₁₃	#12044	1	1	yes	characteristic, complemented, Sylow
114	Q₈	—	26	4	no	—
115	Q₈	—	52	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	13	1	no	—
118	C₂×C₄	—	13	1	no	—
119	C₂×C₄	—	52	2	no	—
120	C₂×C₄	D₄×#261	1	1	yes	characteristic
121	C₈	—	26	2	no	—
122	C₈	—	26	1	no	—
123	C₈	—	52	1	no	—
124	C₈	—	104	2	no	—
125	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
126	C₈	#88240	1	1	yes	characteristic
127	C₂²	D₈×#261	1	1	yes	characteristic
128	C₄	—	8	2	no	—
129	C₄	—	13	2	no	—
130	C₄	—	26	1	no	—
131	C₄	—	52	4	no	—
132	C₄	#11719073	1	1	yes	characteristic
133	C₄	#11719111	1	1	yes	characteristic
134	C₂	#13130417	1	1	yes	characteristic
135	C₂	#13131233	1	1	yes	characteristic
136	C₂	#13131240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130424

C₂

yes

maximal

#13129869

C₂

yes

maximal, characteristic

#13129793

C₂

yes

maximal

#13129747

C₂

yes

maximal, characteristic

#13116694

C₂

yes

maximal, characteristic

#11719074

—

C₂×Dic₁₀₄

C₄

yes

characteristic

#11718534

C₂²

yes

—

#11718257

C₂²

yes

characteristic

#11718254

C₂²

yes

—

C₈×#1057

C₂²

yes

characteristic

#11718244

—

C₁₃×#366

C₄

yes

characteristic

C₁₆×Dic₁₃

C₂²

yes

characteristic

#11718084

C₄

yes

—

C₂×Dic₅₂

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#1057

D₄

yes

characteristic

C₂×#6092

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×#107

C₂×C₄

yes

—

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

#88050

—

Dic₁₀₄

C₈

yes

complemented

#12044

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×#1057

—

C₂×#1057

—

C₂×#1057

D₈

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×C₄⋊C₄

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

yes

Dic₅₂

—

Dic₅₂

C₂×C₈

yes

—

#6092

—

C₁₃×C₁₆

C₂×C₈

yes

—

#6090

—

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

#135

yes

characteristic

Dic₂₆

—

Dic₂₆

—

#1057

—

#1057

—

#1057

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂²×C₁₃

C₄×D₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#443

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#510

yes

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

C₈×D₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×#261

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

D₁₃⋊C₈

yes

—

113

C₁₃

#12044

yes

characteristic, complemented, Sylow

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×#261

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

104

—

125

C₈

C₂×D₁₃⋊C₈

yes

characteristic

126

C₈

#88240

yes

characteristic

127

C₂²

D₈×#261

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#11719073

yes

characteristic

133

C₄

#11719111

yes

characteristic

134

C₂

#13130417

yes

characteristic

135

C₂

#13131233

yes

characteristic

136

C₂

#13131240

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#261
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#366	C₄
socle, prime radical	C₂²×C₁₃	C₄×D₈
cosocle	C₈×Dic₁₃	C₂³
hypercenter	#366	#261
hypocenter	C₁₃	#12044
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×#261

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#261

Fitting

C₁₃×#366

C₄

socle, prime radical

C₂²×C₁₃

C₄×D₈

cosocle

C₈×Dic₁₃

C₂³

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
29. Dic₁₀₄	C₈	semidirect
113. C₁₃	#12044	semidirect

H ≅ G/N

type

29. Dic₁₀₄

C₈

semidirect

113. C₁₃

#12044

semidirect

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

subgroup	index
#13130424	2
#13129869	2
#13129793	2
#13129747	2
#13116694	2
#12044	13

subgroup

index

Subgroups — #16788807

Conjugacy classes of subgroups

Complements

Maximal subgroups