Group #22620439 — subgroups

Subgroups — C₂×#17988992 #22620439

Order 9344.

424 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 213 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×#14556591	C₂	1	2	yes	maximal
2	#17988992	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14556525	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	—	2	2	no	—
6	#14556591	—	2	4	no	—
7	#14556591	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₇₃	—	2	2	no	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	4	yes	—
10	#14556525	C₂²	1	16	yes	—
11	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₇₃	C₂²	1	4	yes	—
14	C₂×Dic₁₄₆	—	2	8	no	—
15	C₂×Dic₁₄₆	—	4	8	no	—
16	C₂²×Dic₇₃	—	2	4	no	—
17	C₄⋊Dic₇₃	—	2	8	no	—
18	C₄⋊Dic₇₃	—	2	4	no	—
19	C₄⋊Dic₇₃	C₂³	1	8	yes	—
20	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₇₃	Q₈	1	8	yes	complemented
24	C₂×C₈×C₇₃	C₂³	1	4	yes	—
25	C₄²×C₇₃	D₄	1	2	yes	—
26	C₄²×C₇₃	C₂³	1	2	yes	—
27	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₇₃	D₈	1	4	yes	—
29	C₂×C₄×C₇₃	Q₁₆	1	4	yes	—
30	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₇₃	—	2	8	no	—
36	C₂×Dic₇₃	—	4	8	no	—
37	Dic₁₄₆	—	4	16	no	—
38	C₈×C₇₃	C₂×Q₈	1	8	yes	complemented
39	C₂²×C₇₃	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₇₃	C₂×D₈	1	2	yes	—
41	C₂²×C₇₃	C₂×Q₁₆	1	2	yes	—
42	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
43	C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
44	C₄×C₇₃	C₂×D₈	1	4	yes	—
45	C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
46	C₄×C₇₃	C₂²×Q₈	1	2	yes	—
47	Dic₇₃	—	4	8	no	—
48	C₂×C₇₃	#500	1	4	yes	complemented
49	C₂×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₇₃	C₂²×D₈	1	1	yes	characteristic
51	C₂×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
52	C₂×#500	—	73	1	no	maximal, Sylow
53	C₇₃	C₂×#500	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊Q₈	—	73	2	no	—
55	#500	—	73	8	no	—
56	C₂×#107	—	73	4	no	—
57	C₂×C₄×C₈	D₇₃	1	1	yes	characteristic
58	C₂²×Q₈	—	146	2	no	—
59	C₂²×C₈	C₂×D₇₃	1	2	yes	—
60	C₄⋊Q₈	—	73	8	no	—
61	C₄⋊Q₈	—	146	4	no	—
62	C₂×C₄⋊C₄	—	73	4	no	—
63	C₂×C₄⋊C₄	—	146	2	no	—
64	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
65	#107	—	73	16	no	—
66	C₄×C₈	C₂×D₇₃	1	4	yes	—
67	C₂×Q₈	—	292	8	no	—
68	C₂×Q₈	—	146	8	no	—
69	C₂²×C₄	—	146	4	no	—
70	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
71	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
73	C₂×C₈	Dic₁₄₆	1	8	yes	complemented
74	C₂×C₈	C₂²×D₇₃	1	4	yes	—
75	C₄⋊C₄	—	73	8	no	—
76	C₄⋊C₄	—	146	4	no	—
77	C₄⋊C₄	—	146	8	no	—
78	C₄²	D₂₉₂	1	2	yes	—
79	C₄²	C₂²×D₇₃	1	2	yes	—
80	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
81	Q₈	—	292	16	no	—
82	C₂×C₄	—	292	8	no	—
83	C₂×C₄	—	146	8	no	—
84	C₂×C₄	D₅₈₄	1	4	yes	—
85	C₂×C₄	Dic₂₉₂	1	4	yes	—
86	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
88	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
89	C₂×C₄	C₂×Dic₁₄₆	1	4	yes	—
90	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
91	C₈	C₂×Dic₁₄₆	1	8	yes	complemented
92	C₂²	#14556591	1	2	yes	—
93	C₂²	C₂×D₅₈₄	1	2	yes	—
94	C₂²	C₂×Dic₂₉₂	1	2	yes	—
95	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
96	C₄	—	292	8	no	—
97	C₄	#14556591	1	2	yes	—
98	C₄	C₂×D₅₈₄	1	4	yes	—
99	C₄	C₂×Dic₂₉₂	1	4	yes	—
100	C₄	C₂²×Dic₁₄₆	1	2	yes	—
101	C₂	#17988992	1	4	yes	direct product factor
102	C₂	C₂×#14556591	1	1	yes	characteristic
103	C₂	C₂²×D₅₈₄	1	1	yes	characteristic
104	C₂	C₂²×Dic₂₉₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal

#17988992

C₂

yes

maximal, direct product factor

C₂×#14556525

C₂

yes

maximal

C₂×C₄×C₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

C₂²

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

#14556525

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

Q₈

yes

complemented

C₂×C₈×C₇₃

C₂³

yes

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

complemented

C₂²×C₇₃

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂×D₈

yes

—

C₂²×C₇₃

C₂×Q₁₆

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

—

C₄×C₇₃

C₂×D₈

yes

—

C₄×C₇₃

C₂×Q₁₆

yes

—

C₄×C₇₃

C₂²×Q₈

yes

—

Dic₇₃

—

C₂×C₇₃

#500

yes

complemented

C₂×C₇₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₃

C₂²×D₈

yes

characteristic

C₂×C₇₃

C₂²×Q₁₆

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#500

—

C₂×#107

—

C₂×C₄×C₈

D₇₃

yes

characteristic

C₂²×Q₈

—

146

—

C₂²×C₈

C₂×D₇₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#107

—

C₄×C₈

C₂×D₇₃

yes

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

C₂²×D₇₃

yes

characteristic

C₂×C₈

Dic₁₄₆

yes

complemented

C₂×C₈

C₂²×D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄²

D₂₉₂

yes

—

C₄²

C₂²×D₇₃

yes

—

C₂³

C₂×D₂₉₂

yes

characteristic

Q₈

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

146

—

C₂×C₄

D₅₈₄

yes

—

C₂×C₄

Dic₂₉₂

yes

—

C₂×C₄

C₂×D₂₉₂

yes

characteristic

C₂×C₄

C₂×D₂₉₂

yes

—

C₂×C₄

C₂×D₂₉₂

yes

—

C₂×C₄

C₂×Dic₁₄₆

yes

—

C₂×C₄

C₂³×D₇₃

yes

characteristic

C₈

C₂×Dic₁₄₆

yes

complemented

C₂²

#14556591

yes

—

C₂²

C₂×D₅₈₄

yes

—

C₂²

C₂×Dic₂₉₂

yes

—

C₂²

C₂²×D₂₉₂

yes

characteristic

C₄

—

292

—

C₄

#14556591

yes

—

C₄

C₂×D₅₈₄

yes

—

C₄

C₂×Dic₂₉₂

yes

—

100

C₄

C₂²×Dic₁₄₆

yes

—

101

C₂

#17988992

yes

direct product factor

102

C₂

C₂×#14556591

yes

characteristic

103

C₂

C₂²×D₅₈₄

yes

characteristic

104

C₂

C₂²×Dic₂₉₂

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	C₂×C₄×C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₇₃
hypocenter	C₇₃	C₂×#500
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

C₂×C₄×C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₇₃

hypocenter

C₇₃

C₂×#500

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #17988992	C₂	direct (N × H)
23. C₂×C₈×C₇₃	Q₈	semidirect
38. C₈×C₇₃	C₂×Q₈	semidirect
48. C₂×C₇₃	#500	semidirect
53. C₇₃	C₂×#500	semidirect
73. C₂×C₈	Dic₁₄₆	semidirect
91. C₈	C₂×Dic₁₄₆	semidirect
101. C₂	#17988992	direct (N × H)

H ≅ G/N

type

2. #17988992

C₂

direct (N × H)

23. C₂×C₈×C₇₃

Q₈

semidirect

38. C₈×C₇₃

C₂×Q₈

semidirect

48. C₂×C₇₃

#500

semidirect

53. C₇₃

C₂×#500

semidirect

73. C₂×C₈

Dic₁₄₆

semidirect

91. C₈

C₂×Dic₁₄₆

semidirect

101. C₂

#17988992

direct (N × H)

Subgroups — C₂×#17988992 #22620439

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17988992 #22620439

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17988992 #22620439