Group #22608705 — subgroups

Subgroups — #22608705

Order 9344.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#17988640	C₂	1	1	yes	maximal, characteristic
2	#17988611	C₂	1	1	yes	maximal, characteristic
3	#17988339	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924012	—	2	1	no	—
6	C₄⋊#10596831	—	2	1	no	—
7	C₄⋊#10596831	C₄	1	2	yes	—
8	C₄⋊#10596831	C₂²	1	2	yes	—
9	C₂²⋊#10596831	C₄	1	2	yes	—
10	C₄×#10596831	C₂²	1	2	yes	—
11	C₂²×C₈×C₇₃	—	2	1	no	—
12	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₃	—	2	1	no	—
14	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
15	#11924012	—	4	2	no	—
16	C₂×#10596831	—	2	2	no	—
17	C₂×#10596831	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₇₃	—	2	2	no	—
22	C₂×C₈×C₇₃	—	4	2	no	—
23	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₃	D₄	1	1	yes	characteristic
25	C₄²×C₇₃	Q₈	1	1	yes	characteristic
26	C₄²×C₇₃	C₂³	1	1	yes	characteristic
27	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₃	—	2	2	no	—
29	C₂×C₄×C₇₃	—	4	1	no	—
30	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₃	D₈	1	2	yes	—
34	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
35	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₇₃	—	4	2	no	—
39	#10596831	—	2	4	no	—
40	#10596831	—	4	2	no	—
41	C₂²×C₇₃	—	2	1	no	—
42	C₂²×C₇₃	—	2	1	no	—
43	C₂²×C₇₃	#107	1	2	yes	—
44	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₇₃	—	2	1	no	—
46	C₄×C₇₃	—	2	1	no	—
47	C₄×C₇₃	—	2	1	no	—
48	C₄×C₇₃	#107	1	2	yes	—
49	C₄×C₇₃	#131	1	2	yes	—
50	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₇₃	—	2	2	no	—
53	C₂×C₇₃	#424	1	1	yes	characteristic
54	C₂×C₇₃	C₂×#107	1	1	yes	characteristic
55	C₂×C₇₃	#430	1	1	yes	characteristic
56	#1571	—	73	1	no	maximal, Sylow
57	C₇₃	#1571	1	1	yes	characteristic, complemented, Sylow
58	#431	—	73	1	no	—
59	#423	—	73	1	no	—
60	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
61	#335	—	73	4	no	—
62	C₂×M₁₆	—	146	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	73	2	no	—
67	C₄⋊C₈	—	73	2	no	—
68	C₄⋊C₈	—	146	1	no	—
69	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
70	C₂²⋊C₈	—	73	2	no	—
71	C₄×C₈	—	73	2	no	—
72	C₂²×C₄	Dic₁₄₆	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
75	M₁₆	—	292	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	73	4	no	—
79	C₂×C₈	—	146	2	no	—
80	C₄²	Dic₁₄₆	1	1	yes	characteristic
81	C₄²	C₄×D₇₃	1	1	yes	characteristic
82	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
83	C₄²	C₂²×D₇₃	1	1	yes	characteristic
84	C₂³	#11924007	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	#11924007	1	1	yes	characteristic
88	C₂×C₄	#11924007	1	1	yes	characteristic
89	C₂×C₄	#11924007	1	1	yes	characteristic
90	C₂×C₄	#11924013	1	2	yes	—
91	C₂×C₄	#11924018	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	292	2	no	—
97	C₈	—	146	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#14556524	1	2	yes	—
101	C₂²	C₂×#11924007	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#14556524	1	2	yes	—
106	C₄	#14556597	1	2	yes	—
107	C₄	C₂×#11924013	1	1	yes	characteristic
108	C₄	C₂×#11924018	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#17988613	1	1	yes	characteristic
111	C₂	C₂×#14556524	1	1	yes	characteristic
112	C₂	#17988635	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988640

C₂

yes

maximal, characteristic

#17988611

C₂

yes

maximal, characteristic

#17988339

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

—

C₄⋊#10596831

—

C₄⋊#10596831

C₄

yes

—

C₄⋊#10596831

C₂²

yes

—

C₂²⋊#10596831

C₄

yes

—

C₄×#10596831

C₂²

yes

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

—

#11924012

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

Q₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

Q₈

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#107

yes

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#107

yes

—

C₄×C₇₃

#131

yes

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄⋊C₈

D₇₃

yes

characteristic

#335

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₄₆

yes

characteristic

C₂²×C₄

C₄×D₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₄²

Dic₁₄₆

yes

characteristic

C₄²

C₄×D₇₃

yes

characteristic

C₄²

C₂²⋊D₇₃

yes

characteristic

C₄²

C₂²×D₇₃

yes

characteristic

C₂³

#11924007

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#11924018

yes

—

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

C₈

—

C₈

—

292

—

C₈

—

146

—

C₂²

—

C₂²

—

100

C₂²

#14556524

yes

—

101

C₂²

C₂×#11924007

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#14556524

yes

—

106

C₄

#14556597

yes

—

107

C₄

C₂×#11924013

yes

characteristic

108

C₄

C₂×#11924018

yes

characteristic

109

C₂

—

110

C₂

#17988613

yes

characteristic

111

C₂

C₂×#14556524

yes

characteristic

112

C₂

#17988635

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#1571
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₄⋊C₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#1571

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #17988339	C₂	semidirect
57. C₇₃	#1571	semidirect

H ≅ G/N

type

3. #17988339

C₂

semidirect

57. C₇₃

#1571

semidirect

Subgroups — #22608705

Conjugacy classes of subgroups

Complements