Group #13125334 — subgroups

Subgroups — C₂×#11718516 #13125334

Order 1664.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#88142	C₂	1	2	yes	maximal
2	C₂×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×#88057	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88048	C₂	1	2	yes	maximal
6	#11718516	C₂	1	8	yes	maximal, direct product factor
7	C₂²×Dic₂₆	—	2	2	no	—
8	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×#6101	—	2	2	no	—
12	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#1055	C₂²	1	1	yes	characteristic
14	#88142	—	2	4	no	—
15	#88142	C₂²	1	8	yes	—
16	C₁₃×#106	C₂²	1	4	yes	—
17	#88057	C₂²	1	4	yes	—
18	C₈×Dic₁₃	C₂²	1	4	yes	—
19	#88048	C₂²	1	8	yes	—
20	C₂²×C₄×C₁₃	—	2	2	no	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₃	—	2	2	no	—
23	C₂²×Dic₁₃	D₄	1	2	yes	—
24	C₂×Dic₂₆	—	2	8	no	—
25	C₂×Dic₂₆	—	4	8	no	—
26	C₂×C₈×C₁₃	Q₈	1	4	yes	complemented
27	C₂×C₈×C₁₃	C₂³	1	2	yes	—
28	C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
30	C₄⋊Dic₁₃	—	2	2	no	—
31	C₄⋊Dic₁₃	C₂³	1	4	yes	—
32	#6101	—	2	8	no	—
33	C₄×Dic₁₃	D₄	1	2	yes	—
34	C₄×Dic₁₃	C₂³	1	2	yes	—
35	C₂×#1055	Q₈	1	4	yes	complemented
36	C₂×#1055	C₂³	1	2	yes	—
37	C₂×#498	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	—	2	4	no	—
40	C₂×C₄×C₁₃	—	4	4	no	—
41	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₁₃	—	2	4	no	—
46	C₂×Dic₁₃	—	4	4	no	—
47	C₂×Dic₁₃	SD₁₆	1	8	yes	—
48	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
49	Dic₂₆	—	4	16	no	—
50	C₈×C₁₃	C₂×Q₈	1	4	yes	complemented
51	#1055	C₂×Q₈	1	4	yes	complemented
52	C₂×C₄⋊Q₈	—	13	2	no	—
53	#498	—	13	8	no	—
54	C₂×#106	—	13	1	no	—
55	C₂×#106	—	13	2	no	—
56	C₂×#106	D₁₃	1	1	yes	characteristic
57	C₂×C₄×C₈	—	13	1	no	—
58	C₂²×C₁₃	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₁₃	C₂×SD₁₆	1	4	yes	—
60	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₁₃	—	4	4	no	—
62	C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
63	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
65	Dic₁₃	—	4	4	no	—
66	Dic₁₃	C₂×SD₁₆	1	8	yes	—
67	C₂²×Q₈	—	26	2	no	—
68	C₂²×C₈	—	13	1	no	—
69	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
70	C₄⋊Q₈	—	13	8	no	—
71	C₄⋊Q₈	—	26	4	no	—
72	C₂×C₄⋊C₄	—	13	2	no	—
73	C₂×C₄⋊C₄	—	26	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
75	C₂×C₄²	—	13	1	no	—
76	#106	—	13	4	no	—
77	#106	—	13	8	no	—
78	#106	C₂×D₁₃	1	4	yes	—
79	C₄×C₈	—	13	4	no	—
80	C₂×C₁₃	#498	1	4	yes	complemented
81	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₁₃	C₂²×SD₁₆	1	2	yes	—
83	C₂×Q₈	—	26	8	no	—
84	C₂×Q₈	—	52	8	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	13	2	no	—
87	C₂²×C₄	—	26	2	no	—
88	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
89	C₂×C₈	—	13	4	no	—
90	C₂×C₈	—	13	2	no	—
91	C₂×C₈	Dic₂₆	1	4	yes	complemented
92	C₂×C₈	C₂²×D₁₃	1	2	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	13	4	no	—
95	C₄⋊C₄	—	26	8	no	—
96	C₄⋊C₄	—	26	2	no	—
97	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
98	C₄²	—	13	2	no	—
99	C₄²	—	13	2	no	—
100	C₁₃	C₂×#498	1	1	yes	characteristic, complemented, Sylow
101	C₂³	D₄×D₁₃	1	1	yes	characteristic
102	Q₈	—	52	16	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	13	8	no	—
106	C₂×C₄	—	13	4	no	—
107	C₂×C₄	—	26	4	no	—
108	C₂×C₄	—	52	4	no	—
109	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
110	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₃	1	2	yes	—
112	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
113	C₈	—	13	4	no	—
114	C₈	C₂×Dic₂₆	1	4	yes	complemented
115	C₂²	#88142	1	2	yes	—
116	C₂²	SD₁₆×D₁₃	1	4	yes	—
117	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	13	8	no	—
120	C₄	—	52	4	no	—
121	C₄	#88142	1	2	yes	—
122	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
123	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
124	C₂	#11718516	1	4	yes	direct product factor
125	C₂	C₂×#88142	1	1	yes	characteristic
126	C₂	C₂×SD₁₆×D₁₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88142

C₂

yes

maximal

C₂×C₁₃×#106

C₂

yes

maximal, characteristic

C₂×#88057

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₃

C₂

yes

maximal, characteristic

C₂×#88048

C₂

yes

maximal

#11718516

C₂

yes

maximal, direct product factor

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

characteristic

#88142

—

#88142

C₂²

yes

—

C₁₃×#106

C₂²

yes

—

#88057

C₂²

yes

—

C₈×Dic₁₃

C₂²

yes

—

#88048

C₂²

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

D₄

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

Q₈

yes

complemented

C₂×C₈×C₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

C₂³

yes

—

C₂×#1055

Q₈

yes

complemented

C₂×#1055

C₂³

yes

—

C₂×#498

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

SD₁₆

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

Dic₂₆

—

C₈×C₁₃

C₂×Q₈

yes

complemented

#1055

C₂×Q₈

yes

complemented

C₂×C₄⋊Q₈

—

—

—

—

D₁₃

yes

characteristic

C₂×C₄×C₈

—

C₂²×C₁₃

C₄⋊Q₈

yes

—

C₂²×C₁₃

C₂×SD₁₆

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

C₄×C₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₃

C₂²×Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

C₂×SD₁₆

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₁₃

yes

—

C₄×C₈

—

C₂×C₁₃

#498

yes

complemented

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃

C₂²×SD₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₆

yes

complemented

C₂×C₈

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₃

yes

—

C₄²

—

C₄²

—

100

C₁₃

C₂×#498

yes

characteristic, complemented, Sylow

101

C₂³

D₄×D₁₃

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Dic₂₆

yes

—

110

C₂×C₄

D₄×D₁₃

yes

characteristic

111

C₂×C₄

Q₈×D₁₃

yes

—

112

C₂×C₄

C₂³×D₁₃

yes

characteristic

113

C₈

—

114

C₈

C₂×Dic₂₆

yes

complemented

115

C₂²

#88142

yes

—

116

C₂²

SD₁₆×D₁₃

yes

—

117

C₂²

C₂×D₄×D₁₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#88142

yes

—

122

C₄

C₂²×Dic₂₆

yes

characteristic

123

C₄

C₂×Q₈×D₁₃

yes

characteristic

124

C₂

#11718516

yes

direct product factor

125

C₂

C₂×#88142

yes

characteristic

126

C₂

C₂×SD₁₆×D₁₃

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#106	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	C₂×#106	D₁₃
hypocenter	C₁₃	C₂×#498
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₂×C₁₃×#106

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
6. #11718516	C₂	direct (N × H)
26. C₂×C₈×C₁₃	Q₈	semidirect
35. C₂×#1055	Q₈	semidirect
50. C₈×C₁₃	C₂×Q₈	semidirect
51. #1055	C₂×Q₈	semidirect
80. C₂×C₁₃	#498	semidirect
91. C₂×C₈	Dic₂₆	semidirect
100. C₁₃	C₂×#498	semidirect
114. C₈	C₂×Dic₂₆	semidirect
124. C₂	#11718516	direct (N × H)

H ≅ G/N

type

6. #11718516

C₂

direct (N × H)

26. C₂×C₈×C₁₃

Q₈

semidirect

35. C₂×#1055

Q₈

semidirect

50. C₈×C₁₃

C₂×Q₈

semidirect

51. #1055

C₂×Q₈

semidirect

80. C₂×C₁₃

#498

semidirect

91. C₂×C₈

Dic₂₆

semidirect

100. C₁₃

C₂×#498

semidirect

114. C₈

C₂×Dic₂₆

semidirect

124. C₂

#11718516

direct (N × H)

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

subgroup	index
C₂×#88142	2
C₂×C₁₃×#106	2
C₂×#88057	2
C₂×C₈×Dic₁₃	2
C₂×#88048	2
#11718516	2
C₂×#498	13

subgroup

index

C₂×C₁₃×#106

C₂×C₈×Dic₁₃

Subgroups — C₂×#11718516 #13125334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#11718516 #13125334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718516 #13125334