Group #13268519 — subgroups

Subgroups — #13268519

Order 1792.

330 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 99 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	#11726820	C₂	1	1	yes	maximal, characteristic, complemented
2	#11726815	C₂	1	2	yes	maximal, complemented
3	#11726814	C₂	1	1	yes	maximal, characteristic
4	#11725814	C₂	1	2	yes	maximal, complemented
5	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₆	—	2	2	no	—
7	C₂×#6374	—	2	4	no	—
8	C₂×D₁₁₂	—	2	2	no	—
9	#89471	C₂²	1	2	yes	—
10	#89467	C₂²	1	2	yes	—
11	#89466	—	2	4	no	—
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	#12518	—	7	1	no	maximal, Sylow
16	C₂×Dic₂₈	—	2	2	no	—
17	C₂×Dic₂₈	—	2	2	no	—
18	C₂×D₅₆	—	2	2	no	—
19	C₂×D₅₆	—	2	2	no	—
20	#6437	—	2	2	no	—
21	#6432	—	2	2	no	—
22	C₂×C₇×C₁₆	D₄	1	2	yes	—
23	C₂×C₇×C₁₆	D₄	1	2	yes	—
24	C₄×C₇×C₈	D₄	1	2	yes	—
25	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
26	#6395	—	4	8	no	—
27	Dic₅₆	—	4	4	no	—
28	#6374	—	4	8	no	—
29	D₁₁₂	—	4	4	no	—
30	#2248	—	7	2	no	—
31	#2247	—	7	1	no	—
32	#2246	—	7	1	no	—
33	#1709	—	7	2	no	—
34	C₈×C₁₆	D₇	1	1	yes	characteristic, complemented
35	C₂×D₂₈	—	4	2	no	—
36	C₂×D₂₈	—	4	4	no	—
37	C₂×Dic₁₄	—	4	2	no	—
38	C₂×C₇×C₈	D₈	1	4	yes	—
39	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₇	—	4	4	no	—
43	Dic₂₈	—	4	4	no	—
44	Dic₂₈	—	4	4	no	—
45	D₅₆	—	4	4	no	—
46	D₅₆	—	4	4	no	—
47	C₇×C₁₆	D₈	1	4	yes	—
48	C₇×C₁₆	D₈	1	4	yes	—
49	C₂×Q₃₂	—	14	2	no	—
50	C₂×SD₃₂	—	14	4	no	—
51	C₂×D₁₆	—	14	2	no	—
52	#494	—	7	2	no	—
53	#493	—	7	2	no	—
54	#486	—	14	4	no	—
55	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
56	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
57	C₈²	C₂×D₇	1	1	yes	characteristic
58	C₂²×D₇	—	8	2	no	—
59	C₂×C₄×C₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
61	C₂×Dic₇	—	8	2	no	—
62	D₂₈	—	8	2	no	—
63	D₂₈	—	8	2	no	—
64	D₂₈	—	8	8	no	—
65	Dic₁₄	—	8	2	no	—
66	Dic₁₄	—	8	2	no	—
67	C₇×C₈	—	2	4	no	—
68	C₇×C₈	C₂×D₈	1	2	yes	—
69	C₇×C₈	C₂×D₈	1	2	yes	—
70	C₂×Q₁₆	—	14	2	no	—
71	C₂×Q₁₆	—	14	2	no	—
72	C₂×D₈	—	14	2	no	—
73	C₂×D₈	—	14	2	no	—
74	C₄⋊Q₈	—	14	2	no	—
75	#127	—	14	2	no	—
76	Q₃₂	—	28	4	no	—
77	SD₃₂	—	28	8	no	—
78	D₁₆	—	28	4	no	—
79	C₂×C₁₆	D₂₈	1	2	yes	—
80	C₂×C₁₆	D₂₈	1	2	yes	—
81	#102	—	28	8	no	—
82	C₄×C₈	D₂₈	1	2	yes	—
83	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
84	C₂²×C₇	#493	1	1	yes	characteristic
85	C₂×D₇	—	16	2	no	—
86	C₂×D₇	—	16	4	no	—
87	C₄×C₇	#493	1	1	yes	characteristic
88	C₄×C₇	#493	1	1	yes	characteristic
89	C₄×C₇	#508	1	4	yes	—
90	Dic₇	—	16	2	no	—
91	C₂×Q₈	—	28	2	no	—
92	C₂×D₄	—	28	2	no	—
93	C₂×D₄	—	28	4	no	—
94	Q₁₆	—	28	4	no	—
95	Q₁₆	—	28	4	no	—
96	D₈	—	28	4	no	—
97	D₈	—	28	4	no	—
98	C₂×C₈	D₅₆	1	4	yes	—
99	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
100	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
101	C₄⋊C₄	—	28	4	no	—
102	C₄²	C₂×D₂₈	1	1	yes	characteristic
103	C₁₆	D₅₆	1	4	yes	—
104	C₁₆	D₅₆	1	4	yes	—
105	C₂×C₇	#1712	1	1	yes	characteristic
106	C₂×C₇	#2249	1	2	yes	—
107	D₇	—	32	2	no	—
108	C₂³	—	56	2	no	—
109	Q₈	—	56	2	no	—
110	Q₈	—	56	2	no	—
111	D₄	—	56	8	no	—
112	D₄	—	56	2	no	—
113	D₄	—	56	2	no	—
114	C₂×C₄	—	56	2	no	—
115	C₂×C₄	#6437	1	1	yes	characteristic
116	C₂×C₄	C₂×D₅₆	1	2	yes	—
117	C₈	—	2	4	no	—
118	C₈	C₂×D₅₆	1	2	yes	—
119	C₈	C₂×D₅₆	1	2	yes	—
120	C₇	#12518	1	1	yes	characteristic, complemented, Sylow
121	C₂²	—	112	4	no	—
122	C₂²	—	112	2	no	—
123	C₂²	#89467	1	1	yes	characteristic
124	C₄	—	112	2	no	—
125	C₄	#89467	1	1	yes	characteristic
126	C₄	#89467	1	1	yes	characteristic
127	C₄	#89676	1	4	yes	—
128	C₂	—	224	2	no	—
129	C₂	#11725816	1	1	yes	characteristic
130	C₂	#11726816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11726820

C₂

yes

maximal, characteristic, complemented

#11726815

C₂

yes

maximal, complemented

#11726814

C₂

yes

maximal, characteristic

#11725814

C₂

yes

maximal, complemented

C₇×C₈×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₆

—

C₂×#6374

—

C₂×D₁₁₂

—

#89471

C₂²

yes

—

#89467

C₂²

yes

—

#89466

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₇×C₈²

C₂²

yes

characteristic

#12518

—

maximal, Sylow

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6395

—

Dic₅₆

—

#6374

—

D₁₁₂

—

—

—

—

—

C₈×C₁₆

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

Dic₂₈

—

Dic₂₈

—

D₅₆

—

D₅₆

—

C₇×C₁₆

D₈

yes

—

C₇×C₁₆

D₈

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

—

#493

—

#486

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₄×C₁₆

C₂×D₇

yes

characteristic

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

D₂₈

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₂×D₈

yes

—

C₇×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₂₈

yes

—

C₂×C₁₆

D₂₈

yes

—

#102

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

D₅₆

yes

—

C₂×C₈

C₂×D₂₈

yes

characteristic

100

C₂×C₈

C₂×D₂₈

yes

characteristic

101

C₄⋊C₄

—

102

C₄²

C₂×D₂₈

yes

characteristic

103

C₁₆

D₅₆

yes

—

104

C₁₆

D₅₆

yes

—

105

C₂×C₇

#1712

yes

characteristic

106

C₂×C₇

#2249

yes

—

107

D₇

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

#6437

yes

characteristic

116

C₂×C₄

C₂×D₅₆

yes

—

117

C₈

—

118

C₈

C₂×D₅₆

yes

—

119

C₈

C₂×D₅₆

yes

—

120

C₇

#12518

yes

characteristic, complemented, Sylow

121

C₂²

—

112

—

122

C₂²

—

112

—

123

C₂²

#89467

yes

characteristic

124

C₄

—

112

—

125

C₄

#89467

yes

characteristic

126

C₄

#89467

yes

characteristic

127

C₄

#89676

yes

—

128

C₂

—

224

—

129

C₂

#11725816

yes

characteristic

130

C₂

#11726816

yes

—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×C₁₆	C₂
socle	C₂²×C₇	#493
prime radical	#11726814	C₂
hypercenter	C₈×C₁₆	D₇
hypocenter	C₇	#12518
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

Fitting

C₇×C₈×C₁₆

C₂

C₂²×C₇

C₂

C₈×C₁₆

D₇

hypocenter

C₇

#12518

LCS

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #11726820	C₂	semidirect
2. #11726815	C₂	semidirect
4. #11725814	C₂	semidirect
5. C₇×C₈×C₁₆	C₂	semidirect
34. C₈×C₁₆	D₇	semidirect
120. C₇	#12518	semidirect

H ≅ G/N

type

1. #11726820

C₂

semidirect

2. #11726815

C₂

semidirect

4. #11725814

C₂

semidirect

5. C₇×C₈×C₁₆

C₂

semidirect

34. C₈×C₁₆

D₇

semidirect

120. C₇

#12518

semidirect

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

subgroup	index
#11726820	2
#11726815	2
#11726814	2
#11725814	2
C₇×C₈×C₁₆	2
#12518	7

subgroup

index

C₇×C₈×C₁₆

Subgroups — #13268519

Conjugacy classes of subgroups

Complements

Maximal subgroups