Group #153391 — subgroups

Subgroups — #153391

Order 512.

162 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 68 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×#1423	C₂	1	1	yes	maximal, characteristic
2	#13628	C₂	1	1	yes	maximal, characteristic
3	#13171	C₂	1	2	yes	maximal
4	#7478	C₂	1	2	yes	maximal
5	#7472	C₂	1	1	yes	maximal, characteristic
6	C₂×M₆₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	2	yes	—
9	C₂×#366	C₂²	1	2	yes	—
10	#2108	C₂²	1	1	yes	characteristic
11	#1952	—	2	2	no	—
12	#1423	—	2	2	no	—
13	C₄⋊C₃₂	—	2	2	no	—
14	#1387	—	2	2	no	—
15	C₂×M₃₂	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂³	1	1	yes	characteristic
17	C₂×#108	—	2	2	no	—
18	#428	—	4	2	no	—
19	C₂×#107	—	2	2	no	—
20	#405	D₄	1	1	yes	characteristic
21	M₆₄	—	2	2	no	—
22	C₂×C₃₂	—	2	2	no	—
23	C₂×C₃₂	—	2	1	no	—
24	C₂×C₃₂	D₄	1	2	yes	—
25	#368	—	2	2	no	—
26	#368	Q₈	1	4	yes	—
27	#366	—	2	2	no	—
28	#366	—	2	2	no	—
29	#346	—	2	1	no	—
30	C₄×C₁₆	—	2	1	no	—
31	#343	—	4	2	no	—
32	C₂×M₁₆	—	4	2	no	—
33	C₂×M₁₆	D₈	1	1	yes	characteristic
34	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
35	#117	D₈	1	1	yes	characteristic
36	C₂×C₄⋊C₄	—	4	2	no	—
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
40	C₂×C₁₆	C₂×Q₈	1	2	yes	—
41	C₂×C₁₆	#41	1	2	yes	—
42	C₂×C₁₆	#41	1	1	yes	characteristic
43	#108	—	2	4	no	—
44	#108	—	4	2	no	—
45	#107	—	4	2	no	—
46	#107	—	4	2	no	—
47	#107	—	8	2	no	—
48	#106	—	8	2	no	—
49	#97	—	2	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₃₂	—	2	2	no	—
52	C₃₂	—	4	1	no	—
53	C₂²×C₄	—	8	2	no	—
54	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	M₁₆	—	4	4	no	—
57	M₁₆	—	8	2	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	8	2	no	—
60	C₂×C₈	#121	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₈	#135	1	1	yes	characteristic
63	C₂×C₈	#137	1	2	yes	—
64	C₄⋊C₄	—	8	2	no	—
65	C₄⋊C₄	—	8	2	no	—
66	C₄⋊C₄	D₁₆	1	1	yes	characteristic
67	C₄⋊C₄	D₁₆	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	—	2	1	no	—
70	C₁₆	—	4	1	no	—
71	C₁₆	#123	1	1	yes	characteristic
72	C₁₆	#125	1	2	yes	—
73	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
74	C₂³	#508	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	16	2	no	—
78	C₂×C₄	—	32	2	no	—
79	C₂×C₄	#466	1	1	yes	characteristic
80	C₂×C₄	#474	1	2	yes	—
81	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
82	C₂×C₄	#510	1	2	yes	—
83	C₈	—	4	1	no	—
84	C₈	—	8	4	no	—
85	C₈	#398	1	1	yes	characteristic
86	C₈	#470	1	1	yes	characteristic
87	C₈	#479	1	2	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	#2215	1	1	yes	characteristic
91	C₂²	#2229	1	2	yes	—
92	C₄	—	4	1	no	—
93	C₄	—	4	1	no	—
94	C₄	—	32	2	no	—
95	C₄	#2086	1	1	yes	characteristic
96	C₄	#2219	1	1	yes	characteristic
97	C₄	#2224	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#13514	1	1	yes	characteristic
100	C₂	#13700	1	1	yes	characteristic
101	C₂	#13701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1423

C₂

yes

maximal, characteristic

#13628

C₂

yes

maximal, characteristic

#13171

C₂

yes

maximal

#7478

C₂

yes

maximal

#7472

C₂

yes

maximal, characteristic

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#366

C₂²

yes

—

#2108

C₂²

yes

characteristic

#1952

—

#1423

—

C₄⋊C₃₂

—

#1387

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

#368

—

#368

Q₈

yes

—

#366

—

#366

—

#346

—

C₄×C₁₆

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#135

yes

characteristic

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

characteristic

C₄⋊C₄

D₁₆

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

#123

yes

characteristic

C₁₆

#125

yes

—

C₁₆

C₄⋊Q₈

yes

characteristic

C₂³

#508

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

#474

yes

—

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

#510

yes

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

—

C₂²

—

C₂²

—

C₂²

#2215

yes

characteristic

C₂²

#2229

yes

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#13514

yes

characteristic

100

C₂

#13700

yes

characteristic

101

C₂

#13701

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#508

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#1423	2
#13628	2
#13171	2
#7478	2
#7472	2

subgroup

index

Subgroups — #153391

Conjugacy classes of subgroups

Maximal subgroups