Group #72286415 — subgroups

Subgroups — #72286415

Order 1024.

195 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 125 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×#7511	C₂	1	2	yes	maximal
2	C₂×#7330	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₂₈	C₄	1	4	yes	—
4	C₂×#1397	C₄	1	1	yes	characteristic
5	C₂×#1397	C₂²	1	1	yes	characteristic
6	C₂×#1311	C₄	1	1	yes	characteristic
7	#7511	—	2	2	no	—
8	#7511	—	2	2	no	—
9	#7330	—	2	2	no	—
10	C₂²×C₃₂	C₈	1	2	yes	—
11	C₂²×C₃₂	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₆	C₈	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈²	C₈	1	1	yes	characteristic
15	M₁₂₈	—	2	4	no	—
16	M₁₂₈	C₈	1	8	yes	complemented
17	C₂×C₆₄	—	2	4	no	—
18	#1397	—	2	1	no	—
19	#1397	—	2	1	no	—
20	#1397	D₄	1	2	yes	—
21	#1397	D₄	1	1	yes	characteristic
22	#1397	Q₈	1	1	yes	characteristic
23	#1311	—	2	1	no	—
24	#1311	—	2	1	no	—
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	C₄²	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₈	—	2	1	no	—
29	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
30	C₂×C₃₂	—	2	2	no	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	C₁₆	1	4	yes	—
33	C₂×C₃₂	C₂²⋊C₄	1	2	yes	—
34	C₂×C₃₂	C₄⋊C₄	1	2	yes	—
35	C₂×C₃₂	C₂×C₈	1	4	yes	—
36	C₂×C₃₂	M₁₆	1	4	yes	—
37	C₄×C₁₆	—	2	1	no	—
38	C₄×C₁₆	C₁₆	1	2	yes	—
39	C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₆	M₁₆	1	1	yes	characteristic
42	C₄×C₁₆	M₁₆	1	1	yes	characteristic
43	C₈²	—	2	1	no	—
44	C₈²	C₁₆	1	2	yes	—
45	C₆₄	—	2	8	no	—
46	C₂²×C₈	—	4	1	no	—
47	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
48	C₂²×C₈	M₃₂	1	1	yes	characteristic
49	C₂×C₄²	M₃₂	1	1	yes	characteristic
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	4	1	no	—
52	C₂×C₁₆	—	4	1	no	—
53	C₂×C₁₆	#95	1	1	yes	characteristic
54	C₂×C₁₆	C₄×C₈	1	2	yes	—
55	C₂×C₁₆	#97	1	1	yes	characteristic
56	C₂×C₁₆	#97	1	2	yes	—
57	C₂×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₁₆	C₂×C₁₆	1	2	yes	—
60	C₂×C₁₆	M₃₂	1	2	yes	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	4	1	no	—
63	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
64	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
65	C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
66	C₄×C₈	C₂×C₁₆	1	1	yes	characteristic
67	C₃₂	—	2	4	no	—
68	C₃₂	C₂²⋊C₈	1	4	yes	—
69	C₃₂	C₄⋊C₈	1	4	yes	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	#346	1	1	yes	characteristic
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	C₈²	1	1	yes	characteristic
77	C₂×C₈	#322	1	1	yes	characteristic
78	C₂×C₈	#336	1	1	yes	characteristic
79	C₂×C₈	C₄×C₁₆	1	1	yes	characteristic
80	C₂×C₈	C₄×C₁₆	1	1	yes	characteristic
81	C₂×C₈	#346	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊C₁₆	1	1	yes	characteristic
83	C₂×C₈	C₄⋊C₁₆	1	1	yes	characteristic
84	C₄²	—	2	1	no	—
85	C₄²	C₂²⋊C₁₆	1	1	yes	characteristic
86	C₄²	C₄⋊C₁₆	1	1	yes	characteristic
87	C₁₆	—	4	2	no	—
88	C₁₆	#322	1	2	yes	—
89	C₁₆	#336	1	2	yes	—
90	C₁₆	C₂²⋊C₁₆	1	2	yes	—
91	C₁₆	C₄⋊C₁₆	1	2	yes	—
92	C₂³	#1398	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	#1311	1	1	yes	characteristic
97	C₂×C₄	#1312	1	1	yes	characteristic
98	C₂×C₄	#1374	1	1	yes	characteristic
99	C₂×C₄	#1398	1	1	yes	characteristic
100	C₂×C₄	#1398	1	1	yes	characteristic
101	C₂×C₄	#1398	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	8	2	no	—
105	C₈	#1273	1	1	yes	characteristic
106	C₈	C₈×C₁₆	1	1	yes	characteristic
107	C₈	#1312	1	1	yes	characteristic
108	C₈	#1374	1	1	yes	characteristic
109	C₂²	—	2	2	no	—
110	C₂²	#7330	1	1	yes	characteristic
111	C₂²	#7332	1	1	yes	characteristic
112	C₂²	#7466	1	1	yes	characteristic
113	C₄	—	4	1	no	—
114	C₄	—	4	1	no	—
115	C₄	#7136	1	1	yes	characteristic
116	C₄	#7330	1	1	yes	characteristic
117	C₄	#7332	1	1	yes	characteristic
118	C₄	#7466	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#93990	1	1	yes	characteristic
122	C₂	#94509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#7511

C₂

yes

maximal

C₂×#7330

C₂

yes

maximal, characteristic

C₂×M₁₂₈

C₄

yes

—

C₂×#1397

C₄

yes

characteristic

C₂×#1397

C₂²

yes

characteristic

C₂×#1311

C₄

yes

—

—

—

C₂²×C₃₂

C₈

yes

—

C₂²×C₃₂

C₂×C₄

yes

—

C₂×C₄×C₁₆

C₈

yes

characteristic

C₂×C₄×C₁₆

C₂×C₄

yes

characteristic

C₂×C₈²

C₈

yes

characteristic

M₁₂₈

—

M₁₂₈

C₈

yes

complemented

C₂×C₆₄

—

#1397

—

#1397

—

#1397

D₄

yes

—

#1397

D₄

yes

characteristic

#1397

Q₈

yes

characteristic

#1311

—

#1311

—

C₂²×C₁₆

—

C₂²×C₁₆

C₄²

yes

characteristic

C₂²×C₁₆

C₂×C₈

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₈

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₁₆

yes

—

C₂×C₃₂

C₂²⋊C₄

yes

—

C₂×C₃₂

C₄⋊C₄

yes

—

C₂×C₃₂

C₂×C₈

yes

—

C₂×C₃₂

M₁₆

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₁₆

yes

—

C₄×C₁₆

C₂²⋊C₄

yes

characteristic

C₄×C₁₆

C₄⋊C₄

yes

characteristic

C₄×C₁₆

M₁₆

yes

characteristic

C₄×C₁₆

M₁₆

yes

characteristic

C₈²

—

C₈²

C₁₆

yes

—

C₆₄

—

C₂²×C₈

—

C₂²×C₈

C₄×C₈

yes

characteristic

C₂²×C₈

M₃₂

yes

characteristic

C₂×C₄²

M₃₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#95

yes

characteristic

C₂×C₁₆

C₄×C₈

yes

—

C₂×C₁₆

#97

yes

characteristic

C₂×C₁₆

#97

yes

—

C₂×C₁₆

C₂²⋊C₈

yes

characteristic

C₂×C₁₆

C₄⋊C₈

yes

characteristic

C₂×C₁₆

yes

—

C₂×C₁₆

M₃₂

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₈

yes

characteristic

C₄×C₈

C₄⋊C₈

yes

characteristic

C₄×C₈

C₂×C₁₆

yes

characteristic

C₄×C₈

C₂×C₁₆

yes

characteristic

C₃₂

—

C₃₂

C₂²⋊C₈

yes

—

C₃₂

C₄⋊C₈

yes

—

C₂²×C₄

—

C₂²×C₄

#346

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈²

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₄×C₁₆

yes

characteristic

C₂×C₈

C₄×C₁₆

yes

characteristic

C₂×C₈

#346

yes

characteristic

C₂×C₈

C₂²⋊C₁₆

yes

characteristic

C₂×C₈

C₄⋊C₁₆

yes

characteristic

C₄²

—

C₄²

C₂²⋊C₁₆

yes

characteristic

C₄²

C₄⋊C₁₆

yes

characteristic

C₁₆

—

C₁₆

#322

yes

—

C₁₆

#336

yes

—

C₁₆

C₂²⋊C₁₆

yes

—

C₁₆

C₄⋊C₁₆

yes

—

C₂³

#1398

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#1398

yes

characteristic

101

C₂×C₄

#1398

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#1273

yes

characteristic

106

C₈

C₈×C₁₆

yes

characteristic

107

C₈

#1312

yes

characteristic

108

C₈

#1374

yes

characteristic

109

C₂²

—

110

C₂²

#7330

yes

characteristic

111

C₂²

#7332

yes

characteristic

112

C₂²

#7466

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

#7136

yes

characteristic

116

C₄

#7330

yes

characteristic

117

C₄

#7332

yes

characteristic

118

C₄

#7466

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#93990

yes

characteristic

122

C₂

#94509

yes

—

	subgroup	G/H
center	C₂×C₈	#322
commutator	C₈	C₈×C₁₆
Frattini, cosocle	C₂×#1397	C₂²
socle	C₂²	#7332
prime radical	C₂³	#1398
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#322

commutator

C₈

C₈×C₁₆

C₂²

C₂²

C₂³

G ▷ C₈ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×#1397 ◁ G

LECS

G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. M₁₂₈	C₈	semidirect

H ≅ G/N

type

16. M₁₂₈

C₈

semidirect

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

subgroup	index
C₂×#7511	2
C₂×#7330	2

subgroup

index

C₂×#7511

C₂×#7330

Subgroups — #72286415

Conjugacy classes of subgroups

Complements

Maximal subgroups