Group #22627124 — subgroups

Subgroups — C₂×#17989691 #22627124

Order 9344.

168 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 71 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×#14556688	C₂	1	2	yes	maximal
2	#17989691	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	#14556688	—	2	2	no	—
5	#14556688	C₂²	1	4	yes	—
6	C₂×#11924040	—	2	2	no	—
7	C₂×#11924036	C₂²	1	1	yes	characteristic
8	C₂×C₄×#64408	C₄	1	1	yes	characteristic
9	C₈×#64408	C₄	1	2	yes	—
10	C₈×#64408	C₂²	1	2	yes	—
11	C₂×C₈×D₇₃	C₄	1	1	yes	characteristic
12	#11924040	—	2	4	no	—
13	C₂²×#64408	—	2	1	no	—
14	#11924036	C₂×C₄	1	1	yes	characteristic
15	#11924036	D₄	1	1	yes	characteristic
16	#11924036	Q₈	1	1	yes	characteristic
17	#11924036	C₂³	1	1	yes	characteristic
18	C₂×#10596833	D₄	1	1	yes	characteristic
19	C₂×#10596833	Q₈	1	1	yes	characteristic
20	C₄×#64408	—	2	1	no	—
21	C₄×#64408	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#10596831	C₈	1	1	yes	characteristic
24	C₈×D₇₃	C₈	1	2	yes	—
25	C₈×D₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₇₃	C₈	1	1	yes	characteristic
27	C₂²×D₇₃	M₁₆	1	1	yes	characteristic
28	C₂×#64408	—	2	4	no	—
29	C₂×#64408	—	2	2	no	—
30	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
34	C₄×D₇₃	M₁₆	1	1	yes	characteristic
35	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
36	#10596833	C₄⋊C₄	1	1	yes	characteristic
37	#10596833	C₄⋊C₄	1	1	yes	characteristic
38	#10596833	C₂×D₄	1	1	yes	characteristic
39	#10596833	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₇₃	C₂×C₈	1	2	yes	—
41	#10596831	C₂×C₈	1	2	yes	—
42	C₂²×C₇₃	C₄⋊C₈	1	1	yes	characteristic
43	C₂×D₇₃	—	2	2	no	—
44	C₂×D₇₃	C₄⋊C₈	1	1	yes	characteristic
45	C₂×D₇₃	C₂×M₁₆	1	1	yes	characteristic
46	#64408	—	2	4	no	—
47	C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
48	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
50	Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇₃	#364	1	2	yes	complemented
52	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
53	D₇₃	—	2	1	no	—
54	D₇₃	—	2	1	no	—
55	C₂×#364	—	73	1	no	maximal, Sylow
56	C₇₃	C₂×#364	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	73	2	no	—
58	C₂×C₄×C₈	—	73	1	no	—
59	#364	—	73	4	no	—
60	C₂²×C₈	—	73	1	no	—
61	C₂²×C₈	—	73	1	no	—
62	C₂×C₄²	—	73	1	no	—
63	M₃₂	—	73	4	no	—
64	M₃₂	—	146	2	no	—
65	C₂×C₁₆	—	146	2	no	—
66	C₄×C₈	—	73	2	no	—
67	C₄×C₈	—	73	2	no	—
68	C₂²×C₄	—	73	1	no	—
69	C₂²×C₄	—	146	1	no	—
70	C₂×C₈	—	73	1	no	—
71	C₂×C₈	—	73	1	no	—
72	C₂×C₈	—	73	1	no	—
73	C₂×C₈	—	73	1	no	—
74	C₂×C₈	—	73	1	no	—
75	C₂×C₈	—	73	1	no	—
76	C₂×C₈	—	73	1	no	—
77	C₂×C₈	—	73	2	no	—
78	C₂×C₈	—	73	2	no	—
79	C₂×C₈	#10596842	1	1	yes	characteristic
80	C₄²	—	73	2	no	—
81	C₄²	—	146	1	no	—
82	C₁₆	—	146	4	no	—
83	C₂³	—	73	1	no	—
84	C₂×C₄	—	73	1	no	—
85	C₂×C₄	—	73	1	no	—
86	C₂×C₄	—	73	1	no	—
87	C₂×C₄	—	73	1	no	—
88	C₂×C₄	—	73	1	no	—
89	C₂×C₄	—	146	2	no	—
90	C₂×C₄	—	146	4	no	—
91	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
92	C₈	—	73	1	no	—
93	C₈	—	73	1	no	—
94	C₈	—	73	1	no	—
95	C₈	—	73	1	no	—
96	C₈	—	73	2	no	—
97	C₈	C₂×#10596842	1	2	yes	—
98	C₂²	—	73	1	no	—
99	C₂²	—	73	1	no	—
100	C₂²	—	146	2	no	—
101	C₂²	C₄⋊#10596842	1	1	yes	characteristic
102	C₄	—	73	1	no	—
103	C₄	—	73	1	no	—
104	C₄	—	146	4	no	—
105	C₄	C₄⋊#10596842	1	1	yes	characteristic
106	C₄	C₂²×#10596842	1	1	yes	characteristic
107	C₂	—	146	1	no	—
108	C₂	—	146	1	no	—
109	C₂	#17989691	1	2	yes	direct product factor
110	C₂	C₂×C₄⋊#10596842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556688

C₂

yes

maximal

#17989691

C₂

yes

maximal, direct product factor

C₂×C₈×#64408

C₂

yes

maximal, characteristic

#14556688

—

#14556688

C₂²

yes

—

C₂×#11924040

—

C₂×#11924036

C₂²

yes

characteristic

C₂×C₄×#64408

C₄

yes

characteristic

C₈×#64408

C₄

yes

—

C₈×#64408

C₂²

yes

—

C₂×C₈×D₇₃

C₄

yes

characteristic

#11924040

—

C₂²×#64408

—

#11924036

C₂×C₄

yes

characteristic

#11924036

D₄

yes

characteristic

#11924036

Q₈

yes

characteristic

#11924036

C₂³

yes

characteristic

C₂×#10596833

D₄

yes

characteristic

C₂×#10596833

Q₈

yes

characteristic

C₄×#64408

—

C₄×#64408

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×#10596831

C₈

yes

characteristic

C₈×D₇₃

C₈

yes

—

C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₈

yes

characteristic

C₂²×D₇₃

M₁₆

yes

characteristic

C₂×#64408

—

C₂×#64408

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

C₂×C₈

yes

characteristic

C₄×D₇₃

M₁₆

yes

characteristic

C₄×D₇₃

C₂²×C₄

yes

characteristic

#10596833

C₄⋊C₄

yes

characteristic

#10596833

C₄⋊C₄

yes

characteristic

#10596833

C₂×D₄

yes

characteristic

#10596833

C₂×Q₈

yes

characteristic

C₈×C₇₃

C₂×C₈

yes

—

#10596831

C₂×C₈

yes

—

C₂²×C₇₃

C₄⋊C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₈

yes

characteristic

C₂×D₇₃

C₂×M₁₆

yes

characteristic

#64408

—

C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂²×C₈

yes

characteristic

Dic₇₃

C₄⋊C₈

yes

characteristic

Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₇₃

#364

yes

complemented

C₂×C₇₃

C₂×C₄⋊C₈

yes

characteristic

D₇₃

—

D₇₃

—

C₂×#364

—

maximal, Sylow

C₇₃

C₂×#364

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

#364

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#10596842

yes

characteristic

C₄²

—

C₄²

—

146

—

C₁₆

—

146

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂×#10596842

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×#10596842

yes

—

C₂²

—

C₂²

—

100

C₂²

—

146

—

101

C₂²

C₄⋊#10596842

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

146

—

105

C₄

C₄⋊#10596842

yes

characteristic

106

C₄

C₂²×#10596842

yes

characteristic

107

C₂

—

146

—

108

C₂

—

146

—

109

C₂

#17989691

yes

direct product factor

110

C₂

C₂×C₄⋊#10596842

yes

characteristic

	subgroup	G/H
center	C₂²	C₄⋊#10596842
commutator	C₄×C₇₃	C₂²×C₈
Frattini	C₄	C₂²×#10596842
Fitting	C₂×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₄⋊C₈
hypercenter	C₂×C₈	#10596842
hypocenter	C₇₃	C₂×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

C₄⋊#10596842

commutator

C₄×C₇₃

C₂²×C₈

Frattini

C₄

C₂²×#10596842

Fitting

C₂×C₈×C₇₃

C₈

socle

C₂²×C₇₃

C₄⋊C₈

C₂×C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
2. #17989691	C₂	direct (N × H)
51. C₂×C₇₃	#364	semidirect
56. C₇₃	C₂×#364	semidirect
109. C₂	#17989691	direct (N × H)

H ≅ G/N

type

2. #17989691

C₂

direct (N × H)

51. C₂×C₇₃

#364

semidirect

56. C₇₃

C₂×#364

semidirect

109. C₂

#17989691

direct (N × H)

Subgroups — C₂×#17989691 #22627124

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17989691 #22627124

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989691 #22627124