Group #93649 — subgroups

Subgroups — #93649

Order 512.

161 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 48 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	#12596	C₂	1	1	yes	maximal, characteristic
2	#12596	C₂	1	1	yes	maximal, characteristic
3	#8139	C₂	1	1	yes	maximal, characteristic
4	C₂×#347	—	2	1	no	—
5	C₂×#347	—	2	1	no	—
6	#1726	C₂²	1	1	yes	characteristic
7	#1452	—	2	1	no	—
8	#1377	—	2	1	no	—
9	#1377	C₄	1	2	yes	—
10	#1376	—	2	1	no	—
11	#1376	C₄	1	2	yes	—
12	#1275	—	2	1	no	—
13	#1274	C₄	1	2	yes	—
14	C₂×M₃₂	—	2	2	no	—
15	C₂×M₃₂	—	2	2	no	—
16	#431	—	2	2	no	—
17	C₂×#97	C₂×C₄	1	1	yes	characteristic
18	C₂×#97	D₄	1	1	yes	characteristic
19	C₂×#97	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
23	#376	C₂×C₄	1	1	yes	characteristic
24	#347	—	2	4	no	—
25	#347	—	2	4	no	—
26	#322	—	2	4	no	—
27	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄²	C₄²	1	1	yes	characteristic
32	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
34	M₃₂	—	2	4	no	—
35	M₃₂	—	2	4	no	—
36	M₃₂	—	4	2	no	—
37	M₃₂	—	4	2	no	—
38	C₂×C₁₆	—	4	2	no	—
39	C₂×C₁₆	—	4	2	no	—
40	C₄⋊C₈	—	4	2	no	—
41	C₂²⋊C₈	—	4	2	no	—
42	#97	—	2	2	no	—
43	#97	—	2	2	no	—
44	#97	—	4	1	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	—	2	4	no	—
47	C₄×C₈	—	4	1	no	—
48	C₄×C₈	—	4	1	no	—
49	#95	—	4	1	no	—
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	#95	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	4	4	no	—
65	C₂×C₈	#104	1	2	yes	—
66	C₂×C₈	#104	1	2	yes	—
67	C₄²	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₄²	#104	1	2	yes	—
73	C₁₆	—	4	4	no	—
74	C₁₆	—	4	4	no	—
75	C₂³	#344	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	8	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	—	8	1	no	—
82	C₂×C₄	#338	1	1	yes	characteristic
83	C₂×C₄	#339	1	1	yes	characteristic
84	C₂×C₄	#339	1	1	yes	characteristic
85	C₂×C₄	#339	1	1	yes	characteristic
86	C₂×C₄	#344	1	1	yes	characteristic
87	C₂×C₄	#344	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	4	1	no	—
92	C₈	—	8	4	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#1291	1	1	yes	characteristic
95	C₂²	#1300	1	1	yes	characteristic
96	C₂²	#1308	1	1	yes	characteristic
97	C₄	—	4	2	no	—
98	C₄	—	8	1	no	—
99	C₄	—	8	1	no	—
100	C₄	#1291	1	1	yes	characteristic
101	C₄	#1291	1	1	yes	characteristic
102	C₄	#1300	1	1	yes	characteristic
103	C₄	#1300	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	#7048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12596

C₂

yes

maximal, characteristic

#12596

C₂

yes

maximal, characteristic

#8139

C₂

yes

maximal, characteristic

C₂×#347

—

C₂×#347

—

#1726

C₂²

yes

—

—

C₄

yes

—

#1376

—

#1376

C₄

yes

—

#1275

—

#1274

C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

#431

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

#376

C₂×C₄

yes

—

—

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#95

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#104

yes

—

C₂×C₈

#104

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#104

yes

—

C₁₆

—

C₁₆

—

C₂³

#344

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

100

C₄

#1291

yes

characteristic

101

C₄

#1291

yes

characteristic

102

C₄

#1300

yes

characteristic

103

C₄

#1300

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

—

107

C₂

#7048

yes

characteristic

	subgroup	G/H
center	C₄	#1300
commutator	C₂×C₄²	C₄²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7048
prime radical	C₂³	#344
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1300

commutator

C₂×C₄²

C₄²

C₂²

C₂

C₂³

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G

LECS

G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12596	2
#8139	2

subgroup

index

#12596

#8139

Subgroups — #93649

Conjugacy classes of subgroups

Maximal subgroups