Group #13123213 — subgroups

Subgroups — #13123213

Order 1664.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#88123	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	4	yes	maximal
4	#11718638	C₂	1	4	yes	maximal
5	#11718635	C₂	1	4	yes	maximal
6	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂²×Dic₂₆	C₂²	1	2	yes	—
9	C₁₃×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
18	#88123	—	2	4	no	—
19	#88071	C₂²	1	16	yes	—
20	C₂×Q₈×C₁₃	—	2	4	no	—
21	C₂×Q₈×C₁₃	—	2	8	no	—
22	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	C₂³	1	4	yes	—
27	C₂²×Dic₁₃	C₂³	1	4	yes	—
28	C₂×Dic₂₆	—	2	4	no	—
29	C₂×Dic₂₆	—	2	8	no	—
30	C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₄²×C₁₃	D₄	1	4	yes	—
33	C₄⋊Dic₁₃	—	2	4	no	—
34	C₄⋊Dic₁₃	—	2	4	no	—
35	C₄×Dic₁₃	—	2	4	no	—
36	C₄×Dic₁₃	—	2	4	no	—
37	C₄×Dic₁₃	Q₈	1	8	yes	complemented
38	#2862	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
40	Q₈×C₁₃	—	4	8	no	—
41	C₂×C₄×C₁₃	—	2	4	no	—
42	C₂×C₄×C₁₃	—	2	8	no	—
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₁₃	#41	1	4	yes	—
48	C₂×C₄×C₁₃	#41	1	4	yes	—
49	C₂×Dic₁₃	—	2	4	no	—
50	C₂×Dic₁₃	—	2	4	no	—
51	C₂×Dic₁₃	—	2	4	no	—
52	C₂×Dic₁₃	—	2	8	no	—
53	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
54	Dic₂₆	—	4	8	no	—
55	C₂×C₄⋊Q₈	—	13	1	no	—
56	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
57	#391	—	13	4	no	—
58	#391	—	13	4	no	—
59	#382	—	13	4	no	—
60	C₄³	—	13	1	no	—
61	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
63	C₂²×C₁₃	C₂×#41	1	2	yes	—
64	C₂²×C₁₃	C₂×#41	1	2	yes	—
65	C₄×C₁₃	—	4	4	no	—
66	C₄×C₁₃	#124	1	4	yes	—
67	C₄×C₁₃	#124	1	4	yes	—
68	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
69	Dic₁₃	—	2	8	no	—
70	Dic₁₃	—	4	4	no	—
71	C₂²×Q₈	—	13	2	no	—
72	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
73	C₄⋊Q₈	—	2	4	no	—
74	C₄⋊Q₈	—	26	4	no	—
75	C₂×C₄⋊C₄	—	13	2	no	—
76	C₂×C₄⋊C₄	—	13	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
78	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
79	C₂×C₄²	—	13	2	no	—
80	C₂×C₄²	—	13	2	no	—
81	C₂×C₄²	—	13	2	no	—
82	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
83	#95	—	13	16	no	—
84	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
85	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
86	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₁₃	#533	1	4	yes	—
88	C₂×Q₈	—	2	8	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	26	8	no	—
91	C₂×Q₈	—	26	4	no	—
92	C₂²×C₄	—	13	4	no	—
93	C₂²×C₄	—	13	4	no	—
94	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	26	4	no	—
101	C₄⋊C₄	—	26	4	no	—
102	C₄²	—	13	8	no	—
103	C₄²	—	26	4	no	—
104	C₄²	—	26	4	no	—
105	C₄²	C₂²⋊D₁₃	1	4	yes	—
106	C₁₃	#2862	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂³×D₁₃	1	1	yes	characteristic
108	Q₈	—	4	8	no	—
109	Q₈	—	52	8	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	13	8	no	—
113	C₂×C₄	—	26	4	no	—
114	C₂×C₄	—	26	4	no	—
115	C₂×C₄	—	26	8	no	—
116	C₂×C₄	—	26	4	no	—
117	C₂×C₄	#6129	1	4	yes	—
118	C₂×C₄	Q₈×D₁₃	1	4	yes	—
119	C₂×C₄	#6131	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
123	C₂²	C₂×#6129	1	2	yes	—
124	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
125	C₂²	C₂×#6131	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	26	8	no	—
129	C₄	—	52	4	no	—
130	C₄	#88190	1	4	yes	—
131	C₄	#88198	1	4	yes	—
132	C₄	#88201	1	4	yes	—
133	C₂	#11719433	1	4	yes	—
134	C₂	C₂×#88190	1	1	yes	characteristic
135	C₂	C₂×#88198	1	1	yes	characteristic
136	C₂	C₂×#88201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#88123

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal

#11718638

C₂

yes

maximal

#11718635

C₂

yes

maximal

C₄²×Dic₁₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88123

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

Q₈

yes

complemented

#2862

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₁₃

yes

—

—

—

C₄³

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

#124

yes

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

#533

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²⋊D₁₃

yes

—

106

C₁₃

#2862

yes

characteristic, complemented, Sylow

107

C₂³

C₂³×D₁₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#6129

yes

—

118

C₂×C₄

Q₈×D₁₃

yes

—

119

C₂×C₄

#6131

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

121

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

122

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

123

C₂²

C₂×#6129

yes

—

124

C₂²

C₂×Q₈×D₁₃

yes

—

125

C₂²

C₂×#6131

yes

—

126

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#88190

yes

—

131

C₄

#88198

yes

—

132

C₄

#88201

yes

—

133

C₂

#11719433

yes

—

134

C₂

C₂×#88190

yes

characteristic

135

C₂

C₂×#88198

yes

characteristic

136

C₂

C₂×#88201

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2862
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
37. C₄×Dic₁₃	Q₈	semidirect
106. C₁₃	#2862	semidirect

H ≅ G/N

type

37. C₄×Dic₁₃

Q₈

semidirect

106. C₁₃

#2862

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88123	2
#11718891	2
#11718638	2
#11718635	2
C₄²×Dic₁₃	2
#2862	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

C₄²×Dic₁₃

Subgroups — #13123213

Conjugacy classes of subgroups

Complements

Maximal subgroups