Group #22586993 — subgroups

Subgroups — C₂×#17985748 #22586993

Order 9088.

256 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
2	#17985748	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	—	2	2	no	—
5	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
6	C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
7	C₂²×Q₈×C₇₁	C₄	1	2	yes	—
8	C₄⋊Q₈×C₇₁	—	2	2	no	—
9	C₄⋊Q₈×C₇₁	C₄	1	2	yes	—
10	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₇₁⋊C₈	—	2	4	no	—
14	C₂×C₇₁⋊C₈	—	4	4	no	—
15	C₂×Q₈×C₇₁	—	2	2	no	—
16	C₂×Q₈×C₇₁	—	4	4	no	—
17	C₂×Q₈×C₇₁	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₇₁	—	2	2	no	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	D₄	1	2	yes	—
21	C₄⋊C₄×C₇₁	—	2	4	no	—
22	C₄⋊C₄×C₇₁	—	2	4	no	—
23	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₁	D₄	1	2	yes	—
25	C₄²×C₇₁	C₂³	1	1	yes	characteristic
26	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₇₁	—	4	4	no	—
28	Q₈×C₇₁	—	4	4	no	—
29	C₂×C₄×C₇₁	—	2	2	no	—
30	C₂×C₄×C₇₁	—	2	4	no	—
31	C₂×C₄×C₇₁	—	4	4	no	—
32	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₇₁	SD₁₆	1	4	yes	—
35	C₂×C₄×C₇₁	Q₁₆	1	4	yes	—
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
38	C₇₁⋊C₈	—	4	4	no	—
39	C₂²×C₇₁	#100	1	2	yes	—
40	C₂²×C₇₁	#103	1	4	yes	—
41	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₇₁	—	2	1	no	—
43	C₄×C₇₁	—	2	1	no	—
44	C₄×C₇₁	—	4	4	no	—
45	C₄×C₇₁	#103	1	4	yes	—
46	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
47	C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
48	C₂×C₇₁	#333	1	4	yes	complemented
49	C₂×C₇₁	C₂×#100	1	1	yes	characteristic
50	C₂×C₇₁	C₂×#103	1	2	yes	—
51	C₂×#333	—	71	1	no	maximal, Sylow
52	C₇₁	C₂×#333	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
54	C₂×C₄⋊C₈	—	71	2	no	—
55	#333	—	71	4	no	—
56	C₂²×Q₈	Dic₇₁	1	2	yes	—
57	C₂²×C₈	—	142	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	Dic₇₁	1	2	yes	—
60	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
63	C₄⋊C₈	—	71	4	no	—
64	C₄⋊C₈	—	142	2	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	4	4	no	—
67	C₂×Q₈	C₂×Dic₇₁	1	4	yes	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
70	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
71	C₂×C₈	—	284	4	no	—
72	C₂×C₈	—	142	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
76	C₄²	C₂²⋊D₇₁	1	2	yes	—
77	C₄²	C₂²×D₇₁	1	1	yes	characteristic
78	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
79	Q₈	—	4	4	no	—
80	Q₈	—	4	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
86	C₂×C₄	#11765852	1	4	yes	—
87	C₂×C₄	#11765855	1	4	yes	—
88	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
89	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
90	C₈	—	284	4	no	—
91	C₂²	#14556045	1	2	yes	—
92	C₂²	#14556052	1	4	yes	—
93	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	4	4	no	—
97	C₄	#14556052	1	4	yes	—
98	C₄	C₂×#11765852	1	2	yes	—
99	C₄	C₂×#11765855	1	2	yes	—
100	C₂	#17985748	1	4	yes	direct product factor
101	C₂	C₂×#14556045	1	1	yes	characteristic
102	C₂	C₂×#14556052	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal

#17985748

C₂

yes

maximal, direct product factor

C₂×C₄⋊Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²×Q₈×C₇₁

C₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₄

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

#100

yes

—

C₂²×C₇₁

#103

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#103

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

C₂×Q₁₆

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×#333

—

maximal, Sylow

C₇₁

C₂×#333

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₁

yes

characteristic

C₂×C₄⋊C₈

—

#333

—

C₂²×Q₈

Dic₇₁

yes

—

C₂²×C₈

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₇₁

yes

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

C₂²⋊Dic₇₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

—

C₂×C₄

#11765852

yes

—

C₂×C₄

#11765855

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

C₈

—

284

—

C₂²

#14556045

yes

—

C₂²

#14556052

yes

—

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#14556052

yes

—

C₄

C₂×#11765852

yes

—

C₄

C₂×#11765855

yes

—

100

C₂

#17985748

yes

direct product factor

101

C₂

C₂×#14556045

yes

characteristic

102

C₂

C₂×#14556052

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₂×C₄⋊Q₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	C₂×#333
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂²⋊Dic₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₂×C₄⋊Q₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₇₁

hypocenter

C₇₁

C₂×#333

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #17985748	C₂	direct (N × H)
48. C₂×C₇₁	#333	semidirect
52. C₇₁	C₂×#333	semidirect
100. C₂	#17985748	direct (N × H)

H ≅ G/N

type

2. #17985748

C₂

direct (N × H)

48. C₂×C₇₁

#333

semidirect

52. C₇₁

C₂×#333

semidirect

100. C₂

#17985748

direct (N × H)

Subgroups — C₂×#17985748 #22586993

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17985748 #22586993

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17985748 #22586993