Group #22612017 — subgroups

Subgroups — #22612017

Order 9344.

174 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 57 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×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988823	C₂	1	2	yes	maximal, complemented
3	#17988413	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988389	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₈₄	C₂²	1	2	yes	—
7	#14556565	—	2	2	no	—
8	D₁₁₆₈	—	2	2	no	—
9	#14556525	C₂²	1	2	yes	—
10	#14556513	—	2	2	no	—
11	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	—	2	2	no	—
13	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₉₂	—	2	2	no	—
16	C₂×C₂²⋊D₇₃	—	4	2	no	—
17	D₅₈₄	—	2	2	no	—
18	D₅₈₄	—	2	2	no	—
19	C₄⋊Dic₇₃	—	2	2	no	—
20	#11924005	—	4	2	no	—
21	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₇₃	—	2	1	no	—
23	C₂×C₈×C₇₃	—	2	1	no	—
24	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₁₆×C₇₃	—	2	1	no	—
27	C₁₆×C₇₃	D₄	1	2	yes	complemented
28	C₂³×C₇₃	D₈	1	1	yes	characteristic
29	C₂²×D₇₃	—	4	2	no	—
30	C₂×C₄×C₇₃	—	2	1	no	—
31	C₂×C₄×C₇₃	—	2	1	no	—
32	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₇₃	—	8	4	no	—
35	C₂×Dic₇₃	—	4	2	no	—
36	D₂₉₂	—	4	2	no	—
37	D₂₉₂	—	4	2	no	—
38	C₈×C₇₃	—	2	1	no	—
39	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₇₃	#41	1	1	yes	characteristic
41	C₂²×C₇₃	—	2	1	no	—
42	C₂²×C₇₃	—	2	1	no	—
43	C₂²×C₇₃	D₁₆	1	2	yes	complemented
44	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
45	C₂×D₇₃	—	8	2	no	—
46	C₂×D₇₃	—	8	2	no	—
47	C₂×D₇₃	—	8	2	no	—
48	C₄×C₇₃	—	2	1	no	—
49	C₄×C₇₃	#121	1	1	yes	characteristic
50	C₄×C₇₃	#135	1	1	yes	characteristic
51	Dic₇₃	—	8	2	no	—
52	C₂×C₇₃	—	2	2	no	—
53	C₂×C₇₃	#466	1	1	yes	characteristic
54	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
55	C₂×C₇₃	#508	1	1	yes	characteristic
56	D₇₃	—	16	2	no	—
57	#2215	—	73	1	no	maximal, Sylow
58	C₇₃	#2215	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₁₆	—	73	1	no	—
60	C₂²×C₁₆	D₇₃	1	1	yes	characteristic, complemented
61	#466	—	73	2	no	—
62	#366	—	73	1	no	—
63	#357	—	73	2	no	—
64	C₂×D₈	—	73	2	no	—
65	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
66	#121	—	146	2	no	—
67	D₁₆	—	146	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
70	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
71	#107	—	73	2	no	—
72	#102	—	146	2	no	—
73	C₂×D₄	—	292	2	no	—
74	C₂×D₄	—	146	2	no	—
75	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
76	D₈	—	146	2	no	—
77	D₈	—	146	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	D₂₉₂	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
82	C₄⋊C₄	—	146	2	no	—
83	C₂²⋊C₄	—	292	2	no	—
84	C₁₆	—	2	1	no	—
85	C₁₆	C₂²⋊D₇₃	1	2	yes	complemented
86	C₂³	—	292	2	no	—
87	C₂³	D₅₈₄	1	1	yes	characteristic
88	D₄	—	292	2	no	—
89	D₄	—	292	2	no	—
90	D₄	—	584	4	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	292	2	no	—
94	C₂×C₄	D₅₈₄	1	1	yes	characteristic
95	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
98	C₈	#11924028	1	1	yes	characteristic
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	584	2	no	—
102	C₂²	—	584	2	no	—
103	C₂²	—	584	2	no	—
104	C₂²	D₁₁₆₈	1	2	yes	complemented
105	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	584	2	no	—
108	C₄	#14556565	1	1	yes	characteristic
109	C₄	#14556613	1	1	yes	characteristic
110	C₂	—	2	2	no	—
111	C₂	—	1168	2	no	—
112	C₂	#17988823	1	1	yes	characteristic
113	C₂	C₂×D₁₁₆₈	1	1	yes	characteristic
114	C₂	#17989019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₁₆₈

C₂

yes

maximal, characteristic, complemented

#17988823

C₂

yes

maximal, complemented

#17988413

C₂

yes

maximal, characteristic, complemented

#17988389

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₅₈₄

C₂²

yes

—

#14556565

—

D₁₁₆₈

—

#14556525

C₂²

yes

—

#14556513

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

D₄

yes

complemented

C₂³×C₇₃

D₈

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

D₁₆

yes

complemented

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

characteristic

C₂×C₇₃

#508

yes

characteristic

D₇₃

—

#2215

—

maximal, Sylow

C₇₃

#2215

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂²×C₁₆

D₇₃

yes

characteristic, complemented

#466

—

#366

—

#357

—

C₂×D₈

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

#121

—

146

—

D₁₆

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#102

—

146

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

C₂²×C₄

D₂₉₂

yes

characteristic

D₈

—

146

—

D₈

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₉₂

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₁₆

—

C₁₆

C₂²⋊D₇₃

yes

complemented

C₂³

—

292

—

C₂³

D₅₈₄

yes

characteristic

D₄

—

292

—

D₄

—

292

—

D₄

—

584

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

D₅₈₄

yes

characteristic

C₂×C₄

C₂×D₂₉₂

yes

characteristic

C₈

—

C₈

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

#11924028

yes

characteristic

C₂²

—

100

C₂²

—

101

C₂²

—

584

—

102

C₂²

—

584

—

103

C₂²

—

584

—

104

C₂²

D₁₁₆₈

yes

complemented

105

C₂²

C₂×D₅₈₄

yes

characteristic

106

C₄

—

107

C₄

—

584

—

108

C₄

#14556565

yes

characteristic

109

C₄

#14556613

yes

characteristic

110

C₂

—

111

C₂

—

1168

—

112

C₂

#17988823

yes

characteristic

113

C₂

C₂×D₁₁₆₈

yes

characteristic

114

C₂

#17989019

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂²×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂²×C₁₆	D₇₃
hypocenter	C₇₃	#2215
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂²×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₂²×C₁₆

D₇₃

hypocenter

C₇₃

#2215

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×D₁₁₆₈	C₂	semidirect
2. #17988823	C₂	semidirect
3. #17988413	C₂	semidirect
4. #17988389	C₂	semidirect
5. C₂²×C₁₆×C₇₃	C₂	semidirect
27. C₁₆×C₇₃	D₄	semidirect
43. C₂²×C₇₃	D₁₆	semidirect
58. C₇₃	#2215	semidirect
60. C₂²×C₁₆	D₇₃	semidirect
85. C₁₆	C₂²⋊D₇₃	semidirect
104. C₂²	D₁₁₆₈	semidirect

H ≅ G/N

type

1. C₂×D₁₁₆₈

C₂

semidirect

2. #17988823

C₂

semidirect

3. #17988413

C₂

semidirect

4. #17988389

C₂

semidirect

5. C₂²×C₁₆×C₇₃

C₂

semidirect

27. C₁₆×C₇₃

D₄

semidirect

43. C₂²×C₇₃

D₁₆

semidirect

58. C₇₃

#2215

semidirect

60. C₂²×C₁₆

D₇₃

semidirect

85. C₁₆

C₂²⋊D₇₃

semidirect

104. C₂²

D₁₁₆₈

semidirect

Subgroups — #22612017

Conjugacy classes of subgroups

Complements