Group #135713 — subgroups

Subgroups — #135713

Order 512.

324 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 166 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	#12900	C₂	1	2	yes	maximal
2	#11996	C₂	1	4	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	4	no	—
5	#2150	C₂²	1	4	yes	—
6	C₄²×C₈	C₄	1	2	yes	—
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈²	C₄	1	1	yes	characteristic
9	C₂×C₈²	C₄	1	1	yes	characteristic
10	C₂×C₈²	C₂²	1	2	yes	—
11	#1367	—	2	8	no	—
12	C₂×M₃₂	—	2	4	no	—
13	C₂×C₄×C₈	—	2	4	no	—
14	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	D₄	1	2	yes	—
18	C₂×C₄×C₈	Q₈	1	2	yes	—
19	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
20	C₄³	C₂×C₄	1	1	yes	characteristic
21	#364	—	4	8	no	—
22	C₄⋊C₁₆	—	2	8	no	—
23	C₈²	—	2	2	no	—
24	C₈²	C₈	1	1	yes	characteristic
25	C₈²	C₈	1	2	yes	—
26	C₈²	C₈	1	1	yes	characteristic
27	C₈²	D₄	1	4	yes	—
28	C₈²	Q₈	1	4	yes	—
29	C₂²×C₈	—	2	2	no	—
30	C₂²×C₈	C₄⋊C₄	1	2	yes	—
31	C₂²×C₈	M₁₆	1	4	yes	—
32	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄²	—	2	2	no	—
35	C₂×C₄²	C₄⋊C₄	1	2	yes	—
36	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
37	M₃₂	—	4	8	no	—
38	C₂×C₁₆	—	4	4	no	—
39	C₄×C₈	—	2	16	no	—
40	C₄×C₈	—	2	4	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	C₄⋊C₄	1	4	yes	—
43	C₄×C₈	C₄⋊C₄	1	4	yes	—
44	C₄×C₈	C₂×C₈	1	2	yes	—
45	C₄×C₈	C₂×C₈	1	1	yes	characteristic
46	C₄×C₈	C₂×C₈	1	1	yes	characteristic
47	C₄×C₈	C₂×C₈	1	2	yes	—
48	C₄×C₈	M₁₆	1	2	yes	—
49	C₄×C₈	SD₁₆	1	8	yes	—
50	C₄×C₈	C₂×D₄	1	2	yes	—
51	C₄×C₈	C₂×Q₈	1	2	yes	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×M₁₆	1	2	yes	—
55	C₂×C₈	—	2	8	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	8	no	—
59	C₂×C₈	C₄⋊C₈	1	4	yes	—
60	C₂×C₈	C₄⋊C₈	1	2	yes	—
61	C₂×C₈	C₄⋊C₈	1	2	yes	—
62	C₂×C₈	C₄⋊C₈	1	4	yes	—
63	C₂×C₈	#106	1	4	yes	—
64	C₂×C₈	#108	1	4	yes	—
65	C₂×C₈	#127	1	1	yes	characteristic
66	C₂×C₈	C₄⋊Q₈	1	2	yes	—
67	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₈	C₂×SD₁₆	1	4	yes	—
69	C₄²	—	2	8	no	—
70	C₄²	—	2	2	no	—
71	C₄²	#106	1	4	yes	—
72	C₄²	C₂×C₄⋊C₄	1	2	yes	—
73	C₄²	C₂²×C₈	1	1	yes	characteristic
74	C₄²	C₂×M₁₆	1	1	yes	characteristic
75	C₁₆	—	8	4	no	—
76	C₂³	#423	1	1	yes	characteristic
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	#384	1	1	yes	characteristic
81	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
83	C₂×C₄	#423	1	2	yes	—
84	C₂×C₄	C₂×#106	1	2	yes	—
85	C₂×C₄	C₂×#108	1	2	yes	—
86	C₈	—	2	8	no	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	#334	1	4	yes	—
90	C₈	#334	1	4	yes	—
91	C₈	#364	1	8	yes	complemented
92	C₈	#492	1	2	yes	—
93	C₈	#498	1	2	yes	—
94	C₂²	—	2	2	no	—
95	C₂²	#1566	1	2	yes	—
96	C₂²	#1842	1	1	yes	characteristic
97	C₄	—	2	8	no	—
98	C₄	—	2	2	no	—
99	C₄	C₂×#334	1	2	yes	—
100	C₄	#1844	1	1	yes	characteristic
101	C₄	#1848	1	1	yes	characteristic
102	C₄	C₂×#364	1	4	yes	—
103	C₂	—	2	2	no	—
104	C₂	#8805	1	1	yes	characteristic
105	C₂	#12900	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12900

C₂

yes

maximal

#11996

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

C₂×#364

—

#2150

C₂²

yes

—

C₄²×C₈

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

—

#1367

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₈²

—

C₈²

C₈

yes

characteristic

C₈²

C₈

yes

—

C₈²

C₈

yes

characteristic

C₈²

D₄

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

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

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

—

C₄²

—

C₄²

—

C₄²

#106

yes

—

C₄²

C₂×C₄⋊C₄

yes

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#423

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#384

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

—

C₂×C₄

#423

yes

—

C₂×C₄

C₂×#106

yes

—

C₂×C₄

C₂×#108

yes

—

C₈

—

C₈

—

C₈

—

C₈

#334

yes

—

C₈

#334

yes

—

C₈

#364

yes

complemented

C₈

#492

yes

—

C₈

#498

yes

—

C₂²

—

C₂²

#1566

yes

—

C₂²

#1842

yes

characteristic

C₄

—

C₄

—

C₄

C₂×#334

yes

—

100

C₄

#1844

yes

characteristic

101

C₄

#1848

yes

characteristic

102

C₄

C₂×#364

yes

—

103

C₂

—

104

C₂

#8805

yes

characteristic

105

C₂

#12900

yes

—

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1842
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
91. C₈	#364	semidirect

H ≅ G/N

type

91. C₈

#364

semidirect

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

subgroup	index
#12900	2
#11996	2
C₄×C₈²	2

subgroup

index

#12900

#11996

C₄×C₈²

Subgroups — #135713

Conjugacy classes of subgroups

Complements

Maximal subgroups