Group #93993 — subgroups

Subgroups — #93993

Order 512.

170 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 114 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×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#1397	C₂	1	1	yes	maximal, characteristic
3	#11399	C₂	1	1	yes	maximal, characteristic
4	C₂×M₆₄	C₄	1	2	yes	—
5	C₂²×C₃₂	C₄	1	2	yes	—
6	C₄×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#1450	C₄	1	1	yes	characteristic
9	#1397	—	2	1	no	—
10	#1397	—	2	1	no	—
11	#1397	—	2	1	no	—
12	C₄×C₃₂	—	2	1	no	—
13	#1311	—	2	1	no	—
14	C₈×C₁₆	—	2	1	no	—
15	C₂×M₃₂	C₈	1	2	yes	—
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	#431	C₈	1	2	yes	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	M₆₄	—	2	2	no	—
21	M₆₄	C₈	1	4	yes	complemented
22	C₂×C₃₂	—	2	2	no	—
23	C₂×C₃₂	—	2	2	no	—
24	C₂×C₃₂	—	2	2	no	—
25	C₂×C₃₂	C₈	1	4	yes	complemented
26	#346	—	2	1	no	—
27	C₄×C₁₆	—	2	1	no	—
28	C₄×C₁₆	D₄	1	1	yes	characteristic
29	C₄×C₁₆	D₄	1	1	yes	characteristic
30	C₄×C₁₆	D₄	1	1	yes	characteristic
31	C₄×C₁₆	Q₈	1	1	yes	characteristic
32	#322	—	2	1	no	—
33	C₈²	—	2	1	no	—
34	C₂²×C₈	C₄²	1	1	yes	characteristic
35	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	M₃₂	C₁₆	1	4	yes	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₆	C₂×C₈	1	2	yes	—
45	C₂×C₁₆	C₂×C₈	1	2	yes	—
46	C₂×C₁₆	M₁₆	1	2	yes	—
47	C₂×C₁₆	M₁₆	1	2	yes	—
48	C₄⋊C₈	—	2	2	no	—
49	C₂²⋊C₈	C₁₆	1	4	yes	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈	M₁₆	1	1	yes	characteristic
54	C₄×C₈	M₁₆	1	1	yes	characteristic
55	C₃₂	—	2	4	no	—
56	C₃₂	—	2	4	no	—
57	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×C₁₆	1	2	yes	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	#95	1	1	yes	characteristic
61	C₂×C₈	C₄×C₈	1	1	yes	characteristic
62	C₂×C₈	C₄×C₈	1	1	yes	characteristic
63	C₂×C₈	#97	1	1	yes	characteristic
64	C₂×C₈	#97	1	1	yes	characteristic
65	C₂×C₈	#97	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₈	C₂×C₁₆	1	2	yes	—
69	C₂×C₈	M₃₂	1	2	yes	—
70	C₄²	C₂²⋊C₈	1	1	yes	characteristic
71	C₄²	C₄⋊C₈	1	1	yes	characteristic
72	C₄²	M₃₂	1	2	yes	—
73	C₁₆	—	2	4	no	—
74	C₁₆	C₂²⋊C₈	1	2	yes	—
75	C₁₆	C₂²⋊C₈	1	2	yes	—
76	C₁₆	C₄⋊C₈	1	2	yes	—
77	C₁₆	C₄⋊C₈	1	2	yes	—
78	C₂³	C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	C₈²	1	1	yes	characteristic
81	C₂×C₄	#322	1	1	yes	characteristic
82	C₂×C₄	#336	1	1	yes	characteristic
83	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
84	C₂×C₄	#346	1	1	yes	characteristic
85	C₂×C₄	#346	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
87	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
88	C₈	—	4	4	no	—
89	C₈	#322	1	1	yes	characteristic
90	C₈	#322	1	1	yes	characteristic
91	C₈	#336	1	1	yes	characteristic
92	C₈	#336	1	1	yes	characteristic
93	C₈	C₂²⋊C₁₆	1	2	yes	—
94	C₈	C₄⋊C₁₆	1	2	yes	—
95	C₂²	—	2	2	no	—
96	C₂²	C₈×C₁₆	1	1	yes	characteristic
97	C₂²	#1312	1	1	yes	characteristic
98	C₂²	#1374	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	#1273	1	1	yes	characteristic
102	C₄	#1311	1	1	yes	characteristic
103	C₄	#1312	1	1	yes	characteristic
104	C₄	#1374	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	#7136	1	1	yes	characteristic
108	C₂	#7469	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₆₄

C₂

yes

maximal, characteristic

C₂×#1397

C₂

yes

maximal, characteristic

#11399

C₂

yes

maximal, characteristic

C₂×M₆₄

C₄

yes

—

C₂²×C₃₂

C₄

yes

—

C₄×M₃₂

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1450

C₄

yes

—

—

—

C₄×C₃₂

—

#1311

—

C₈×C₁₆

—

C₂×M₃₂

C₈

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

M₆₄

C₈

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₈

yes

complemented

#346

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

Q₈

yes

characteristic

#322

—

C₈²

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

M₃₂

C₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₄⋊C₄

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₂²⋊C₈

C₁₆

yes

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₃₂

—

C₃₂

—

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

C₂×C₁₆

yes

—

C₂×C₈

—

C₂×C₈

#95

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₄²

C₂²⋊C₈

yes

characteristic

C₄²

C₄⋊C₈

yes

characteristic

C₄²

M₃₂

yes

—

C₁₆

—

C₁₆

C₂²⋊C₈

yes

—

C₁₆

C₂²⋊C₈

yes

—

C₁₆

C₄⋊C₈

yes

—

C₁₆

C₄⋊C₈

yes

—

C₂³

C₄×C₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

C₈²

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×C₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊C₁₆

yes

—

C₂×C₄

C₄⋊C₁₆

yes

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

C₈

yes

C₈

C₂²⋊C₁₆

yes

—

C₈

C₄⋊C₁₆

yes

—

C₂²

—

C₂²

C₈×C₁₆

yes

C₂²

yes

C₂²

yes

C₄

—

100

C₄

—

101

C₄

#1273

yes

characteristic

102

C₄

#1311

yes

characteristic

103

C₄

#1312

yes

characteristic

104

C₄

#1374

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

#7136

yes

characteristic

108

C₂

#7469

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₈×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	C₄×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

C₂²⋊C₄

commutator

C₂²

C₈×C₁₆

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

socle

C₂²

#1374

prime radical

C₂³

C₄×C₁₆

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
21. M₆₄	C₈	semidirect
25. C₂×C₃₂	C₈	semidirect

H ≅ G/N

type

21. M₆₄

C₈

semidirect

25. C₂×C₃₂

C₈

semidirect

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

subgroup	index
C₄×M₆₄	2
C₂×#1397	2
#11399	2

subgroup

index

C₄×M₆₄

C₂×#1397

#11399

Subgroups — #93993

Conjugacy classes of subgroups

Complements

Maximal subgroups