Group #22612109 — subgroups

Subgroups — #22612109

Order 9344.

150 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 53 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×#14556525	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988412	C₂	1	2	yes	maximal, complemented
3	#17988388	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₃	—	2	1	no	—
7	C₂×#11924003	C₂²	1	2	yes	—
8	#14556525	—	2	1	no	—
9	#14556525	C₂²	1	2	yes	—
10	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#121×C₇₃	—	2	1	no	—
13	#107×C₇₃	C₂²	1	1	yes	characteristic
14	#102×C₇₃	—	2	1	no	—
15	C₂²×Dic₇₃	—	4	1	no	—
16	C₄⋊Dic₇₃	—	2	2	no	—
17	C₄⋊Dic₇₃	—	4	1	no	—
18	#11924003	—	2	2	no	—
19	C₂×D₄×C₇₃	—	2	1	no	—
20	C₂×D₄×C₇₃	—	4	1	no	—
21	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
22	D₈×C₇₃	—	2	2	no	—
23	C₂×C₈×C₇₃	—	2	2	no	—
24	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₇₃	—	2	1	no	—
27	C₂²⋊C₄×C₇₃	—	4	1	no	—
28	C₂³×C₇₃	—	4	1	no	—
29	C₂³×C₇₃	D₈	1	1	yes	characteristic
30	D₄×C₇₃	—	4	1	no	—
31	D₄×C₇₃	—	4	1	no	—
32	D₄×C₇₃	—	8	2	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	4	1	no	—
36	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₇₃	—	4	2	no	—
39	C₂×Dic₇₃	—	8	2	no	—
40	C₈×C₇₃	—	2	1	no	—
41	C₈×C₇₃	#41	1	2	yes	—
42	C₂²×C₇₃	—	2	1	no	—
43	C₂²×C₇₃	—	2	1	no	—
44	C₂²×C₇₃	—	8	1	no	—
45	C₂²×C₇₃	—	8	1	no	—
46	C₂²×C₇₃	—	8	1	no	—
47	C₂²×C₇₃	D₁₆	1	2	yes	—
48	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	—	8	1	no	—
51	C₄×C₇₃	#123	1	1	yes	characteristic
52	C₄×C₇₃	#137	1	1	yes	characteristic
53	Dic₇₃	—	8	2	no	—
54	C₂×C₇₃	—	2	2	no	—
55	C₂×C₇₃	—	16	1	no	—
56	C₂×C₇₃	#480	1	1	yes	characteristic
57	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
58	C₂×C₇₃	#510	1	1	yes	characteristic
59	#2232	—	73	1	no	maximal, Sylow
60	C₇₃	#2232	1	1	yes	characteristic, complemented, Sylow
61	#466	D₇₃	1	1	yes	characteristic
62	C₂×#107	—	73	1	no	—
63	#366	—	73	2	no	—
64	#357	—	73	2	no	—
65	C₂²⋊C₁₆	—	73	1	no	—
66	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
68	#121	—	2	1	no	—
69	C₂×C₄⋊C₄	—	146	1	no	—
70	C₂×C₁₆	—	73	2	no	—
71	#107	—	73	2	no	—
72	#107	—	146	1	no	—
73	#107	C₂×D₇₃	1	1	yes	characteristic
74	#102	—	2	1	no	—
75	C₂×D₄	—	2	1	no	—
76	C₂×D₄	—	4	1	no	—
77	C₂²×C₄	—	292	1	no	—
78	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
79	D₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
82	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	292	1	no	—
85	C₄⋊C₄	—	146	2	no	—
86	C₂²⋊C₄	—	4	1	no	—
87	C₁₆	—	146	2	no	—
88	C₂³	—	4	1	no	—
89	C₂³	#11924013	1	1	yes	characteristic
90	D₄	—	4	1	no	—
91	D₄	—	4	1	no	—
92	D₄	—	8	2	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	292	2	no	—
97	C₂×C₄	—	584	2	no	—
98	C₂×C₄	#11924013	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	#11924027	1	2	yes	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	8	1	no	—
105	C₂²	—	8	1	no	—
106	C₂²	—	8	1	no	—
107	C₂²	#14556532	1	2	yes	—
108	C₂²	C₂×#11924013	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	8	1	no	—
111	C₄	—	584	2	no	—
112	C₄	#14556572	1	1	yes	characteristic
113	C₄	#14556621	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	—	16	1	no	—
116	C₂	#17988898	1	1	yes	characteristic
117	C₂	C₂×#14556532	1	1	yes	characteristic
118	C₂	#17989027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556525

C₂

yes

maximal, characteristic, complemented

#17988412

C₂

yes

maximal, complemented

#17988388

C₂

yes

maximal, complemented

C₂²⋊#11924003

C₂

yes

maximal, characteristic, complemented

#466×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₃

—

C₂×#11924003

C₂²

yes

—

#14556525

—

#14556525

C₂²

yes

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#121×C₇₃

—

#107×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

D₈

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#41

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

D₁₆

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#480

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#466

D₇₃

yes

—

—

—

C₂²⋊C₁₆

—

C₂×D₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

#121

—

C₂×C₄⋊C₄

—

146

—

C₂×C₁₆

—

#107

—

#107

—

146

—

#107

C₂×D₇₃

yes

characteristic

#102

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

292

—

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

C₁₆

—

146

—

C₂³

—

C₂³

#11924013

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

584

—

C₂×C₄

#11924013

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

100

C₈

—

101

C₈

#11924027

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#14556532

yes

—

108

C₂²

C₂×#11924013

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

584

—

112

C₄

#14556572

yes

characteristic

113

C₄

#14556621

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#17988898

yes

characteristic

117

C₂

C₂×#14556532

yes

characteristic

118

C₂

#17989027

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#466×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#466	D₇₃
hypocenter	C₇₃	#2232
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#466×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

N	H ≅ G/N	type
1. C₂×#14556525	C₂	semidirect
2. #17988412	C₂	semidirect
3. #17988388	C₂	semidirect
4. C₂²⋊#11924003	C₂	semidirect
60. C₇₃	#2232	semidirect

H ≅ G/N

type

1. C₂×#14556525

C₂

semidirect

2. #17988412

C₂

semidirect

3. #17988388

C₂

semidirect

4. C₂²⋊#11924003

C₂

semidirect

60. C₇₃

#2232

semidirect

Subgroups — #22612109

Conjugacy classes of subgroups

Complements