Group #153389 — subgroups

Subgroups — #153389

Order 512.

186 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#13165	C₂	1	2	yes	maximal
4	#7471	C₂	1	2	yes	maximal
5	#7471	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×#366	C₂²	1	2	yes	—
9	C₂×#366	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1942	—	2	2	no	—
12	#1423	—	2	2	no	—
13	C₄⋊C₃₂	—	2	2	no	—
14	#1380	—	2	2	no	—
15	C₂²×C₁₆	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	C₂³	1	1	yes	characteristic
17	C₂×#107	—	2	2	no	—
18	C₂×#107	—	2	2	no	—
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	#389	—	4	2	no	—
21	C₂×C₃₂	—	2	2	no	—
22	C₂×C₃₂	—	2	1	no	—
23	C₂×C₃₂	—	2	2	no	—
24	C₂×C₃₂	D₄	1	2	yes	—
25	#366	—	2	2	no	—
26	#366	—	2	2	no	—
27	#366	—	2	2	no	—
28	#366	Q₈	1	4	yes	—
29	C₄×C₁₆	—	2	1	no	—
30	C₄×C₁₆	—	2	1	no	—
31	#340	—	4	2	no	—
32	C₂²×C₈	D₈	1	1	yes	characteristic
33	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄	—	4	2	no	—
35	C₂×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄⋊C₄	—	8	2	no	—
37	C₂×C₄²	D₈	1	1	yes	characteristic
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Q₈	1	2	yes	—
42	C₂×C₁₆	#41	1	2	yes	—
43	C₂×C₁₆	#41	1	1	yes	characteristic
44	#107	—	2	4	no	—
45	#107	—	4	2	no	—
46	#107	—	4	2	no	—
47	#107	—	4	2	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	#95	—	8	2	no	—
51	C₃₂	—	2	2	no	—
52	C₃₂	—	4	1	no	—
53	C₂²×C₄	—	8	2	no	—
54	C₂²×C₄	—	8	2	no	—
55	C₂²×C₄	D₁₆	1	1	yes	characteristic
56	C₂²×C₄	D₁₆	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	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	—	4	4	no	—
65	C₄⋊C₄	—	8	2	no	—
66	C₄⋊C₄	—	8	2	no	—
67	C₄⋊C₄	—	8	2	no	—
68	C₄⋊C₄	—	16	4	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	2	1	no	—
71	C₁₆	—	4	1	no	—
72	C₁₆	#123	1	1	yes	characteristic
73	C₁₆	#125	1	2	yes	—
74	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
75	C₂³	C₂×D₁₆	1	1	yes	characteristic
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	8	4	no	—
81	C₂×C₄	—	16	2	no	—
82	C₂×C₄	—	16	2	no	—
83	C₂×C₄	—	16	2	no	—
84	C₂×C₄	—	16	2	no	—
85	C₂×C₄	—	16	2	no	—
86	C₂×C₄	Q₆₄	1	2	yes	—
87	C₂×C₄	#466	1	1	yes	characteristic
88	C₂×C₄	#474	1	2	yes	—
89	C₂×C₄	#508	1	1	yes	characteristic
90	C₂×C₄	#510	1	2	yes	—
91	C₈	—	4	1	no	—
92	C₈	#398	1	1	yes	characteristic
93	C₈	#470	1	1	yes	characteristic
94	C₈	#479	1	2	yes	—
95	C₂²	#2215	1	1	yes	characteristic
96	C₂²	#2224	1	2	yes	—
97	C₂²	C₂×Q₆₄	1	1	yes	characteristic
98	C₂²	#2262	1	1	yes	characteristic
99	C₂²	#2263	1	2	yes	—
100	C₄	—	2	2	no	—
101	C₄	—	4	1	no	—
102	C₄	—	16	4	no	—
103	C₄	—	32	2	no	—
104	C₄	#2086	1	1	yes	characteristic
105	C₄	#2219	1	1	yes	characteristic
106	C₄	#2229	1	2	yes	—
107	C₂	#13514	1	1	yes	characteristic
108	C₂	#13695	1	1	yes	characteristic
109	C₂	#13697	1	1	yes	characteristic
110	C₂	#13706	1	2	yes	—
111	C₂	#13707	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1423

C₂

yes

maximal, characteristic

C₂×C₄⋊C₃₂

C₂

yes

maximal, characteristic

#13165

C₂

yes

maximal

#7471

C₂

yes

maximal

#7471

C₂

yes

maximal, characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1942

—

#1423

—

C₄⋊C₃₂

—

#1380

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×#107

—

C₂×#107

—

C₂×C₄×C₈

D₄

yes

characteristic

#389

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

—

—

—

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

#123

yes

characteristic

C₁₆

#125

yes

—

C₁₆

C₄⋊Q₈

yes

characteristic

C₂³

C₂×D₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Q₆₄

yes

—

C₂×C₄

#466

yes

characteristic

C₂×C₄

#474

yes

—

C₂×C₄

#508

yes

characteristic

C₂×C₄

#510

yes

—

C₈

—

C₈

yes

C₈

yes

C₈

yes

—

C₂²

#2215

yes

characteristic

C₂²

#2224

yes

—

C₂²

C₂×Q₆₄

yes

C₂²

yes

C₂²

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#2086

yes

characteristic

105

C₄

#2219

yes

characteristic

106

C₄

#2229

yes

—

107

C₂

#13514

yes

characteristic

108

C₂

#13695

yes

characteristic

109

C₂

#13697

yes

characteristic

110

C₂

#13706

yes

—

111

C₂

#13707

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
C₂×C₄⋊C₃₂	2
#13165	2
#7471	2

subgroup

index

C₂×#1423

C₂×C₄⋊C₃₂

#13165

#7471

Subgroups — #153389

Conjugacy classes of subgroups

Maximal subgroups