Group #22570713 — subgroups

Subgroups — C₂×#16863798 #22570713

Order 8576.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14524035	C₂	1	2	yes	maximal
2	#16863798	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14523977	C₂	1	2	yes	maximal
4	C₂×#14523975	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524035	—	2	4	no	—
7	#14524035	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
9	C₂×Dic₆₇⋊C₄	—	2	4	no	—
10	#14523977	C₂²	1	8	yes	—
11	#14523975	C₂²	1	8	yes	—
12	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₇	C₂²	1	4	yes	—
16	C₂²×Dic₆₇	—	2	4	no	—
17	C₄⋊Dic₆₇	—	2	4	no	—
18	C₄⋊Dic₆₇	C₂³	1	8	yes	—
19	Dic₆₇⋊C₄	—	2	16	no	—
20	C₂²×C₄×C₆₇	D₄	1	2	yes	—
21	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₆₇	Q₈	1	4	yes	—
23	C₂×C₈×C₆₇	Q₈	1	4	yes	—
24	C₂×C₈×C₆₇	C₂³	1	2	yes	—
25	C₂×C₈×C₆₇	C₂³	1	2	yes	—
26	C₄²×C₆₇	D₄	1	2	yes	—
27	C₄²×C₆₇	C₂³	1	2	yes	—
28	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	4	no	—
30	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₆₇	—	2	8	no	—
36	C₂×Dic₆₇	—	4	8	no	—
37	C₈×C₆₇	C₂×Q₈	1	4	yes	—
38	C₈×C₆₇	C₂×Q₈	1	4	yes	—
39	C₂²×C₆₇	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₆₇	#135	1	4	yes	—
41	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₆₇	—	2	4	no	—
43	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
44	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
46	Dic₆₇	—	4	8	no	—
47	C₂×C₆₇	#499	1	4	yes	complemented
48	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₆₇	C₂×#135	1	2	yes	—
50	C₂×#499	—	67	1	no	maximal, Sylow
51	C₆₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
52	C₂×#125	—	67	2	no	—
53	#499	—	67	8	no	—
54	C₂×#107	—	67	2	no	—
55	C₂×#106	—	67	2	no	—
56	C₂×C₄×C₈	D₆₇	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
59	#125	—	67	8	no	—
60	#125	—	134	4	no	—
61	C₂×C₄⋊C₄	—	67	4	no	—
62	C₂×C₄⋊C₄	—	134	4	no	—
63	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
64	#107	—	67	8	no	—
65	#106	—	67	8	no	—
66	C₄×C₈	C₂×D₆₇	1	4	yes	—
67	C₂²×C₄	—	134	4	no	—
68	C₂²×C₄	D₂₆₈	1	2	yes	—
69	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
70	C₂×C₈	Dic₁₃₄	1	4	yes	—
71	C₂×C₈	Dic₁₃₄	1	4	yes	—
72	C₂×C₈	C₂²×D₆₇	1	2	yes	—
73	C₂×C₈	C₂²×D₆₇	1	2	yes	—
74	C₄⋊C₄	—	67	8	no	—
75	C₄⋊C₄	—	134	4	no	—
76	C₄⋊C₄	—	134	16	no	—
77	C₄²	D₂₆₈	1	2	yes	—
78	C₄²	C₂²×D₆₇	1	2	yes	—
79	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	268	8	no	—
82	C₂×C₄	—	134	8	no	—
83	C₂×C₄	C₂×D₂₆₈	1	4	yes	—
84	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
86	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
87	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
88	C₈	C₂×Dic₁₃₄	1	4	yes	—
89	C₈	C₂×Dic₁₃₄	1	4	yes	—
90	C₂²	#14524043	1	2	yes	—
91	C₂²	#14524065	1	4	yes	—
92	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	268	8	no	—
95	C₄	#14524043	1	2	yes	—
96	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
97	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
98	C₂	#16863798	1	4	yes	direct product factor
99	C₂	C₂×#14524043	1	1	yes	characteristic
100	C₂	C₂×#14524065	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal

#16863798

C₂

yes

maximal, direct product factor

C₂×#14523977

C₂

yes

maximal

C₂×#14523975

C₂

yes

maximal

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

#14524035

—

#14524035

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

#14523977

C₂²

yes

—

#14523975

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

C₂×Q₈

yes

—

C₈×C₆₇

C₂×Q₈

yes

—

C₂²×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

#135

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂²×Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

#499

yes

complemented

C₂×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

C₂×#135

yes

—

C₂×#499

—

maximal, Sylow

C₆₇

C₂×#499

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄×C₈

D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#125

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₆₇

yes

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

C₂²×D₆₇

yes

—

C₂×C₈

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

D₂₆₈

yes

—

C₄²

C₂²×D₆₇

yes

—

C₂³

C₂×D₂₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

C₂×D₂₆₈

yes

—

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₂×C₄

C₂×Dic₁₃₄

yes

—

C₂×C₄

C₂×Dic₁₃₄

yes

—

C₂×C₄

C₂³×D₆₇

yes

characteristic

C₈

C₂×Dic₁₃₄

yes

—

C₈

C₂×Dic₁₃₄

yes

—

C₂²

#14524043

yes

—

C₂²

#14524065

yes

—

C₂²

C₂²×D₂₆₈

yes

characteristic

C₄

—

C₄

—

268

—

C₄

#14524043

yes

—

C₄

C₂²×Dic₁₃₄

yes

characteristic

C₄

C₂²×Dic₁₃₄

yes

characteristic

C₂

#16863798

yes

direct product factor

C₂

C₂×#14524043

yes

characteristic

100

C₂

C₂×#14524065

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#499
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄×C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₆₇

hypocenter

C₆₇

C₂×#499

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16863798	C₂	direct (N × H)
47. C₂×C₆₇	#499	semidirect
51. C₆₇	C₂×#499	semidirect
98. C₂	#16863798	direct (N × H)

H ≅ G/N

type

2. #16863798

C₂

direct (N × H)

47. C₂×C₆₇

#499

semidirect

51. C₆₇

C₂×#499

semidirect

98. C₂

#16863798

direct (N × H)

Subgroups — C₂×#16863798 #22570713

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863798 #22570713

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863798 #22570713