Group #15774921 — subgroups

Subgroups — #15774921

Order 3328.

206 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13113732	C₂	1	2	yes	maximal
3	#13112799	C₂	1	2	yes	maximal
4	#13112752	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1372	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic
7	#11718492	—	2	2	no	—
8	C₄×Dic₅₂	C₂²	1	2	yes	—
9	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#335	C₂²	1	2	yes	—
11	#11718096	C₄	1	1	yes	characteristic
12	#11718095	C₄	1	2	yes	—
13	#11718076	—	2	2	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₂₆	—	2	2	no	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₁₃×C₄⋊C₈	—	2	2	no	—
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88058	C₂×C₄	1	2	yes	—
23	#88056	—	2	2	no	—
24	#88055	—	4	2	no	—
25	#88054	—	4	2	no	—
26	C₂×#6090	—	2	1	no	—
27	C₄×#1055	D₄	1	1	yes	characteristic
28	C₄×#1055	D₄	1	1	yes	characteristic
29	Dic₁₀₄	—	2	2	no	—
30	#12085	—	13	1	no	maximal, Sylow
31	C₂×Dic₂₆	—	2	2	no	—
32	C₂×C₈×C₁₃	—	4	2	no	—
33	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₃	—	2	2	no	—
37	#6101	—	4	2	no	—
38	C₄×Dic₁₃	—	4	2	no	—
39	C₂×#1055	—	2	1	no	—
40	C₂×#1055	D₈	1	2	yes	—
41	Dic₅₂	—	2	2	no	—
42	Dic₅₂	—	2	2	no	—
43	C₁₃×C₁₆	—	2	1	no	—
44	C₁₃×C₁₆	M₁₆	1	2	yes	—
45	#6090	—	2	2	no	—
46	C₄×Q₃₂	—	13	1	no	—
47	#1591	—	13	2	no	—
48	#1372	D₁₃	1	1	yes	characteristic
49	#1332	—	13	2	no	—
50	#1311	—	13	1	no	—
51	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
54	C₂×Dic₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	4	2	no	—
56	Dic₂₆	—	4	2	no	—
57	Dic₂₆	—	4	2	no	—
58	C₈×C₁₃	—	2	1	no	—
59	C₈×C₁₃	—	8	2	no	—
60	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₁₃	#131	1	1	yes	characteristic
62	#1055	—	2	2	no	—
63	#1055	—	4	1	no	—
64	C₂×Q₃₂	—	13	1	no	—
65	#446	—	26	2	no	—
66	C₄×Q₁₆	—	13	2	no	—
67	#366	—	13	1	no	—
68	#358	—	13	2	no	—
69	C₄×C₁₆	—	13	1	no	—
70	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
71	#335	C₂×D₁₃	1	2	yes	—
72	Q₈⋊C₈	—	26	2	no	—
73	C₈²	—	13	1	no	—
74	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
75	C₄×C₁₃	—	2	1	no	—
76	C₄×C₁₃	#436	1	1	yes	characteristic
77	C₄×C₁₃	#443	1	1	yes	characteristic
78	C₄×C₁₃	#510	1	2	yes	—
79	Dic₁₃	—	8	2	no	—
80	Dic₁₃	—	8	2	no	—
81	C₂×Q₁₆	—	13	2	no	—
82	C₄×Q₈	—	26	2	no	—
83	Q₃₂	—	26	2	no	—
84	C₂×C₁₆	—	26	1	no	—
85	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
86	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
87	#107	—	13	2	no	—
88	C₄⋊C₈	—	2	2	no	—
89	C₄⋊C₈	—	52	2	no	—
90	#103	—	26	2	no	—
91	#97	—	52	2	no	—
92	C₄×C₈	—	13	1	no	—
93	C₄×C₈	—	13	1	no	—
94	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
95	C₂×C₁₃	#1589	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	—	26	2	no	—
100	Q₁₆	—	26	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	13	2	no	—
103	C₂×C₈	—	26	1	no	—
104	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
105	C₂×C₈	#6131	1	1	yes	characteristic
106	C₄⋊C₄	—	26	2	no	—
107	C₄⋊C₄	—	52	2	no	—
108	C₄²	—	52	2	no	—
109	C₄²	D₄×D₁₃	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	26	2	no	—
112	C₁₆	#6094	1	2	yes	—
113	C₁₃	#12085	1	1	yes	characteristic, complemented, Sylow
114	Q₈	—	52	2	no	—
115	Q₈	—	52	2	no	—
116	C₂×C₄	—	52	2	no	—
117	C₂×C₄	—	52	2	no	—
118	C₂×C₄	#88147	1	1	yes	characteristic
119	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
120	C₂×C₄	#88173	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	26	2	no	—
124	C₈	—	52	1	no	—
125	C₈	C₂×#6094	1	1	yes	characteristic
126	C₈	#88161	1	1	yes	characteristic
127	C₂²	#11718533	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	104	2	no	—
130	C₄	—	104	2	no	—
131	C₄	#11718499	1	1	yes	characteristic
132	C₄	#11718556	1	1	yes	characteristic
133	C₄	#11718576	1	2	yes	—
134	C₂	#13113720	1	1	yes	characteristic
135	C₂	#13116706	1	1	yes	characteristic
136	C₂	#13116721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₁₀₄

C₂

yes

maximal, characteristic

#13113732

C₂

yes

maximal

#13112799

C₂

yes

maximal

#13112752

C₂

yes

maximal, characteristic

C₁₃×#1372

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

C₄

yes

characteristic

#11718492

—

C₄×Dic₅₂

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#335

C₂²

yes

—

#11718096

C₄

yes

characteristic

#11718095

C₄

yes

—

#11718076

—

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₂×Dic₅₂

C₂×C₄

yes

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

Dic₁₀₄

—

#12085

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

D₈

yes

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

M₁₆

yes

—

#6090

—

C₄×Q₃₂

—

#1591

—

#1372

D₁₃

yes

characteristic

#1332

—

#1311

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×M₁₆

yes

C₈×C₁₃

yes

—

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

C₂×D₁₃

yes

—

Q₈⋊C₈

—

C₈²

—

C₂²×C₁₃

C₄×D₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₃

yes

characteristic

C₂×C₁₆

C₄×D₁₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

105

C₂×C₈

#6131

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

D₄×D₁₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

#6094

yes

—

113

C₁₃

#12085

yes

characteristic, complemented, Sylow

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#88147

yes

characteristic

119

C₂×C₄

D₈×D₁₃

yes

characteristic

120

C₂×C₄

#88173

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂×#6094

yes

characteristic

126

C₈

#88161

yes

characteristic

127

C₂²

#11718533

yes

characteristic

128

C₄

—

129

C₄

—

104

—

130

C₄

—

104

—

131

C₄

#11718499

yes

characteristic

132

C₄

#11718556

yes

characteristic

133

C₄

#11718576

yes

—

134

C₂

#13113720

yes

characteristic

135

C₂

#13116706

yes

characteristic

136

C₂

#13116721

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1372	C₂
socle, prime radical	C₂²×C₁₃	C₄×D₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1372	D₁₃
hypocenter	C₁₃	#12085
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂×C₄

D₈×D₁₃

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1372

C₂

socle, prime radical

C₂²×C₁₃

C₄×D₈

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
113. C₁₃	#12085	semidirect

H ≅ G/N

type

113. C₁₃

#12085

semidirect

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

subgroup	index
C₄×Dic₁₀₄	2
#13113732	2
#13112799	2
#13112752	2
C₁₃×#1372	2
#12085	13

subgroup

index

C₄×Dic₁₀₄

C₁₃×#1372

Subgroups — #15774921

Conjugacy classes of subgroups

Complements

Maximal subgroups