Group #22627123 — subgroups

Subgroups — #22627123

Order 9344.

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

normal subgroups only

This group has 160 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	1	yes	maximal, characteristic
2	C₂×#14556686	C₂	1	1	yes	maximal, characteristic
3	#17989690	C₂	1	2	yes	maximal, complemented
4	C₄⋊#11924040	C₂	1	2	yes	maximal, complemented
5	#17989552	C₂	1	1	yes	maximal, characteristic
6	#14556688	—	2	2	no	—
7	#14556686	—	2	2	no	—
8	C₂×#11924040	C₂²	1	1	yes	characteristic
9	C₂×#11924040	C₂²	1	2	yes	—
10	C₂×#11924040	C₂²	1	1	yes	characteristic
11	C₂×#11924038	C₄	1	1	yes	characteristic
12	C₂×#11924036	C₂²	1	1	yes	characteristic
13	#14556646	C₄	1	2	yes	—
14	C₄×#10596833	C₂²	1	2	yes	—
15	#14556549	C₄	1	1	yes	characteristic
16	#11924040	—	2	1	no	—
17	#11924040	—	2	1	no	—
18	#11924040	—	2	2	no	—
19	#11924038	—	2	2	no	—
20	#11924036	D₄	1	2	yes	—
21	#11924036	Q₈	1	2	yes	—
22	C₂×#10596833	C₂×C₄	1	2	yes	—
23	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
24	C₂×#10596833	C₂³	1	1	yes	characteristic
25	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
27	#11924005	C₈	1	2	yes	—
28	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
30	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₇₃	C₂×C₈	1	2	yes	—
34	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×D₇₃	C₄⋊C₄	1	2	yes	—
36	C₄×D₇₃	M₁₆	1	2	yes	—
37	#10596833	—	2	2	no	—
38	#10596833	C₄⋊C₄	1	2	yes	—
39	#10596833	C₂×D₄	1	1	yes	characteristic
40	#10596833	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
42	C₂×D₇₃	—	2	1	no	—
43	C₂×D₇₃	—	2	1	no	—
44	C₂×D₇₃	C₄⋊C₈	1	2	yes	—
45	C₄×C₇₃	—	2	2	no	—
46	C₄×C₇₃	C₄⋊C₈	1	2	yes	—
47	Dic₇₃	—	2	2	no	—
48	Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₇₃	#504	1	1	yes	characteristic
52	C₂×C₇₃	#504	1	1	yes	characteristic
53	D₇₃	—	2	2	no	—
54	#2151	—	73	1	no	maximal, Sylow
55	C₇₃	#2151	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	73	1	no	—
57	C₂²×C₁₆	—	73	1	no	—
58	#405	—	73	1	no	—
59	C₄⋊C₁₆	—	73	2	no	—
60	C₄⋊C₁₆	—	73	2	no	—
61	C₂×M₁₆	—	73	1	no	—
62	C₂²×C₈	—	73	1	no	—
63	#117	—	73	1	no	—
64	M₃₂	—	146	2	no	—
65	C₂×C₁₆	—	73	1	no	—
66	C₂×C₁₆	—	73	1	no	—
67	C₂×C₁₆	—	73	2	no	—
68	C₂×C₁₆	—	146	2	no	—
69	#97	—	73	2	no	—
70	C₄×C₈	—	73	2	no	—
71	C₂²×C₄	—	73	1	no	—
72	M₁₆	—	146	2	no	—
73	C₂×C₈	—	73	1	no	—
74	C₂×C₈	—	73	1	no	—
75	C₂×C₈	—	73	2	no	—
76	C₂×C₈	—	73	2	no	—
77	C₂×C₈	—	73	2	no	—
78	C₄⋊C₄	—	73	1	no	—
79	C₄⋊C₄	#10596842	1	1	yes	characteristic
80	C₂²⋊C₄	—	73	2	no	—
81	C₄²	—	73	2	no	—
82	C₁₆	—	146	1	no	—
83	C₁₆	—	146	1	no	—
84	C₁₆	—	146	2	no	—
85	C₂³	—	73	1	no	—
86	C₂×C₄	—	73	1	no	—
87	C₂×C₄	—	73	2	no	—
88	C₂×C₄	—	73	2	no	—
89	C₂×C₄	—	73	2	no	—
90	C₂×C₄	C₂×#10596842	1	2	yes	—
91	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
92	C₈	—	73	1	no	—
93	C₈	—	73	2	no	—
94	C₈	—	73	1	no	—
95	C₈	—	146	2	no	—
96	C₂²	—	73	2	no	—
97	C₂²	—	146	1	no	—
98	C₂²	—	146	1	no	—
99	C₂²	C₂²×#10596842	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	73	1	no	—
102	C₄	—	73	1	no	—
103	C₄	—	146	2	no	—
104	C₄	C₄⋊#10596842	1	2	yes	—
105	C₂	—	146	2	no	—
106	C₂	C₂×C₄⋊#10596842	1	1	yes	characteristic
107	C₂	#17989708	1	1	yes	characteristic
108	C₂	#17989709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556688

C₂

yes

maximal, characteristic

C₂×#14556686

C₂

yes

maximal, characteristic

#17989690

C₂

yes

maximal, complemented

C₄⋊#11924040

C₂

yes

maximal, complemented

#17989552

C₂

yes

maximal, characteristic

#14556688

—

#14556686

—

C₂×#11924040

C₂²

yes

characteristic

C₂×#11924040

C₂²

yes

—

C₂×#11924040

C₂²

yes

characteristic

C₂×#11924038

C₄

yes

characteristic

C₂×#11924036

C₂²

yes

characteristic

#14556646

C₄

yes

—

C₄×#10596833

C₂²

yes

—

#14556549

C₄

yes

—

—

—

—

D₄

yes

—

#11924036

Q₈

yes

—

C₂×#10596833

C₂×C₄

yes

—

C₂×#10596833

C₂×C₄

yes

characteristic

C₂×#10596833

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₄⋊Dic₇₃

C₈

yes

characteristic

#11924005

C₈

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

C₈

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

—

C₄×D₇₃

M₁₆

yes

—

#10596833

—

#10596833

C₄⋊C₄

yes

—

#10596833

C₂×D₄

yes

characteristic

#10596833

C₂×Q₈

yes

characteristic

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊C₈

yes

—

Dic₇₃

—

Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

Dic₇₃

C₂×M₁₆

yes

characteristic

C₂×C₇₃

C₂×C₄⋊C₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2151

—

maximal, Sylow

C₇₃

#2151

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#97

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

#10596842

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₁₆

—

146

—

C₁₆

—

146

—

C₁₆

—

146

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#10596842

yes

—

C₂×C₄

C₂×#10596842

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

146

—

C₂²

—

C₂²

—

146

—

C₂²

—

146

—

C₂²

C₂²×#10596842

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

146

—

104

C₄

C₄⋊#10596842

yes

—

105

C₂

—

146

—

106

C₂

C₂×C₄⋊#10596842

yes

characteristic

107

C₂

#17989708

yes

characteristic

108

C₂

#17989709

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₄⋊C₄×C₇₃	C₈
hypercenter	C₄⋊C₄	#10596842
hypocenter	C₇₃	#2151
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₈

Fitting

C₄⋊C₄×C₇₃

C₈

C₄⋊C₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₄⋊C₄

N	H ≅ G/N	type
3. #17989690	C₂	semidirect
4. C₄⋊#11924040	C₂	semidirect
55. C₇₃	#2151	semidirect

H ≅ G/N

type

3. #17989690

C₂

semidirect

4. C₄⋊#11924040

C₂

semidirect

55. C₇₃

#2151

semidirect

Subgroups — #22627123

Conjugacy classes of subgroups

Complements