Group #135639 — subgroups

Subgroups — #135639

Order 512.

200 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 90 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	#12013	C₂	1	2	yes	maximal
2	#7845	C₂	1	1	yes	maximal, characteristic
3	#7196	C₂	1	2	yes	maximal
4	#7143	C₂	1	2	yes	maximal
5	#2163	C₂²	1	4	yes	—
6	#1744	C₄	1	1	yes	characteristic
7	#1744	C₄	1	1	yes	characteristic
8	#1744	C₂²	1	1	yes	characteristic
9	C₂×#322	C₄	1	1	yes	characteristic
10	C₂×#322	C₄	1	1	yes	characteristic
11	C₂×#322	C₂²	1	1	yes	characteristic
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1370	—	2	2	no	—
14	#1367	—	2	2	no	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	C₄³	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₁₆	—	2	4	no	—
23	C₂²⋊C₁₆	—	2	4	no	—
24	#346	—	2	4	no	—
25	#336	—	2	2	no	—
26	#336	—	2	2	no	—
27	#336	—	2	2	no	—
28	#322	—	2	2	no	—
29	#322	—	2	2	no	—
30	#322	C₈	1	4	yes	—
31	C₈²	D₄	1	2	yes	—
32	C₈²	Q₈	1	2	yes	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	M₁₆	1	2	yes	—
38	C₂²×C₈	#41	1	2	yes	—
39	C₂×C₄²	—	2	1	no	—
40	C₂×C₄²	—	2	1	no	—
41	C₂×C₄²	M₁₆	1	2	yes	—
42	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₁₆	—	4	4	no	—
44	C₄×C₈	—	2	1	no	—
45	C₄×C₈	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₄⋊C₄	1	2	yes	—
49	C₄×C₈	C₄⋊C₄	1	2	yes	—
50	C₄×C₈	C₂×C₈	1	2	yes	—
51	C₄×C₈	C₂×C₈	1	2	yes	—
52	C₄×C₈	C₂×C₈	1	2	yes	—
53	C₄×C₈	M₁₆	1	2	yes	—
54	C₄×C₈	C₂×D₄	1	1	yes	characteristic
55	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₄×C₈	#41	1	2	yes	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	#117	1	1	yes	characteristic
59	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
60	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	C₄⋊C₈	1	4	yes	—
72	C₂×C₈	#124	1	2	yes	—
73	C₂×C₈	#125	1	2	yes	—
74	C₄²	—	4	2	no	—
75	C₄²	—	4	2	no	—
76	C₄²	—	4	2	no	—
77	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄²	#117	1	1	yes	characteristic
79	C₄²	C₂²×C₈	1	1	yes	characteristic
80	C₄²	C₂×M₁₆	1	1	yes	characteristic
81	C₁₆	—	8	4	no	—
82	C₂³	#432	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	#382	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₄	#423	1	1	yes	characteristic
92	C₂×C₄	#431	1	1	yes	characteristic
93	C₂×C₄	#431	1	1	yes	characteristic
94	C₂×C₄	#432	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	2	4	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	2	no	—
100	C₈	—	4	2	no	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#1831	1	1	yes	characteristic
105	C₂²	#2166	1	2	yes	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	8	2	no	—
111	C₄	#1833	1	2	yes	—
112	C₄	#2153	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	#8781	1	1	yes	characteristic
115	C₂	#12881	1	1	yes	characteristic
116	C₂	#12884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12013

C₂

yes

maximal

#7845

C₂

yes

maximal, characteristic

#7196

C₂

yes

maximal

#7143

C₂

yes

maximal

#2163

C₂²

yes

—

#1744

C₄

yes

characteristic

#1744

C₄

yes

characteristic

#1744

C₂²

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1370

—

#1367

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

—

C₈

yes

—

C₈²

D₄

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

#41

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

M₁₆

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#124

yes

—

C₂×C₈

#125

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#117

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#432

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#382

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₂²

—

103

C₂²

—

104

C₂²

#1831

yes

characteristic

105

C₂²

#2166

yes

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#1833

yes

—

112

C₄

#2153

yes

—

113

C₂

—

114

C₂

#8781

yes

characteristic

115

C₂

#12881

yes

characteristic

116

C₂

#12884

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#382

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12013	2
#7845	2
#7196	2
#7143	2

subgroup

index

Subgroups — #135639

Conjugacy classes of subgroups

Maximal subgroups