Group #92913 — subgroups

Subgroups — #92913

Order 512.

174 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 102 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×#1367	C₂	1	2	yes	maximal
2	C₂×#1311	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆	—	2	2	no	—
4	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈²	C₂²	1	1	yes	characteristic
7	#1367	—	2	2	no	—
8	#1367	C₄	1	4	yes	—
9	#1311	—	2	2	no	—
10	C₂²×C₁₆	—	2	1	no	—
11	C₂²×C₁₆	—	4	2	no	—
12	C₂²×C₁₆	C₈	1	2	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	D₄	1	1	yes	characteristic
15	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
16	C₄⋊C₁₆	—	2	4	no	—
17	C₄⋊C₁₆	—	4	2	no	—
18	C₄×C₁₆	—	2	1	no	—
19	C₄×C₁₆	—	2	1	no	—
20	C₄×C₁₆	—	2	1	no	—
21	C₄×C₁₆	C₈	1	2	yes	—
22	C₈²	C₂×C₄	1	2	yes	—
23	C₈²	D₄	1	2	yes	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₈	M₁₆	1	1	yes	characteristic
27	C₂²×C₈	SD₁₆	1	2	yes	—
28	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₁₆	—	2	4	no	—
30	C₂×C₁₆	—	2	2	no	—
31	C₂×C₁₆	—	2	2	no	—
32	C₂×C₁₆	—	2	2	no	—
33	C₂×C₁₆	—	4	4	no	—
34	C₂×C₁₆	—	8	4	no	—
35	C₂×C₁₆	C₁₆	1	4	yes	complemented
36	C₄×C₈	—	2	1	no	—
37	C₄×C₈	C₄²	1	1	yes	characteristic
38	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×C₈	C₂²⋊C₄	1	2	yes	—
40	C₄×C₈	C₄⋊C₄	1	2	yes	—
41	C₄×C₈	C₂×C₈	1	1	yes	characteristic
42	C₄×C₈	M₁₆	1	1	yes	characteristic
43	C₄×C₈	D₈	1	1	yes	characteristic
44	C₄×C₈	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
46	C₂²×C₄	#106	1	1	yes	characteristic
47	C₂²×C₄	#108	1	1	yes	characteristic
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	C₄×C₈	1	1	yes	characteristic
53	C₂×C₈	#97	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₈	#102	1	2	yes	—
57	C₂×C₈	#103	1	2	yes	—
58	C₂×C₈	C₂×C₁₆	1	2	yes	—
59	C₂×C₈	M₃₂	1	2	yes	—
60	C₄²	#95	1	1	yes	characteristic
61	C₄²	C₄⋊C₈	1	1	yes	characteristic
62	C₄²	#107	1	1	yes	characteristic
63	C₄²	#108	1	1	yes	characteristic
64	C₁₆	—	2	4	no	—
65	C₁₆	—	2	4	no	—
66	C₁₆	—	8	4	no	—
67	C₂³	#334	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	#334	1	1	yes	characteristic
70	C₂×C₄	#335	1	1	yes	characteristic
71	C₂×C₄	#335	1	1	yes	characteristic
72	C₂×C₄	#336	1	1	yes	characteristic
73	C₂×C₄	#340	1	1	yes	characteristic
74	C₂×C₄	#341	1	1	yes	characteristic
75	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
76	C₂×C₄	#364	1	2	yes	—
77	C₂×C₄	#365	1	4	yes	—
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	—	2	2	no	—
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	C₄×C₁₆	1	1	yes	characteristic
84	C₈	#346	1	1	yes	characteristic
85	C₈	C₂²⋊C₁₆	1	2	yes	—
86	C₂²	#1277	1	1	yes	characteristic
87	C₂²	#1367	1	2	yes	—
88	C₂²	#1368	1	2	yes	—
89	C₂²	#1369	1	2	yes	—
90	C₄	—	2	1	no	—
91	C₄	—	2	1	no	—
92	C₄	#1368	1	2	yes	—
93	C₄	#1370	1	2	yes	—
94	C₄	#1374	1	1	yes	characteristic
95	C₄	#1376	1	1	yes	characteristic
96	C₄	#1383	1	2	yes	—
97	C₂	#7099	1	4	yes	—
98	C₂	#7151	1	1	yes	characteristic
99	C₂	#7156	1	1	yes	characteristic
100	C₂	#7161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1367

C₂

yes

maximal

C₂×#1311

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

—

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1367

—

#1367

C₄

yes

—

#1311

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₁₆

yes

complemented

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

C₄⋊C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

#102

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₄²

#95

yes

characteristic

C₄²

C₄⋊C₈

yes

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#334

yes

characteristic

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊C₁₆

yes

—

C₂×C₄

#364

yes

—

C₂×C₄

#365

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₄×C₁₆

yes

characteristic

C₈

#346

yes

characteristic

C₈

C₂²⋊C₁₆

yes

—

C₂²

#1277

yes

characteristic

C₂²

#1367

yes

—

C₂²

#1368

yes

—

C₂²

#1369

yes

—

C₄

—

C₄

—

C₄

#1368

yes

—

C₄

#1370

yes

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

#7099

yes

—

C₂

yes

C₂

yes

100

C₂

#7161

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#106
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle, prime radical	C₂³	#334
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

#106

commutator

C₈

C₄×C₁₆

Frattini, cosocle

C₂×C₈²

C₂²

socle, prime radical

C₂³

#334

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
35. C₂×C₁₆	C₁₆	semidirect

H ≅ G/N

type

35. C₂×C₁₆

C₁₆

semidirect

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

subgroup	index
C₂×#1367	2
C₂×#1311	2

subgroup

index

C₂×#1367

C₂×#1311

Subgroups — #92913

Conjugacy classes of subgroups

Complements

Maximal subgroups