Group #22612081 — subgroups

Subgroups — #22612081

Order 9344.

130 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17988991	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988413	C₂	1	1	yes	maximal, characteristic
4	#17988412	C₂	1	1	yes	maximal, characteristic
5	#17988405	C₂	1	1	yes	maximal, characteristic
6	#17988391	C₂	1	1	yes	maximal, characteristic
7	#358×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556590	—	2	1	no	—
9	Q₈×Dic₇₃	—	2	1	no	—
10	C₂×#11924003	C₂²	1	1	yes	characteristic
11	#14556525	C₂²	1	1	yes	characteristic
12	#14556525	C₂²	1	1	yes	characteristic
13	#14556524	—	2	1	no	—
14	#14556515	—	2	1	no	—
15	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	#107×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₆	—	4	1	no	—
20	C₂×#10596831	D₄	1	1	yes	characteristic
21	C₄⋊Dic₇₃	—	2	1	no	—
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	4	1	no	—
24	#11924007	—	4	1	no	—
25	C₄×Dic₇₃	—	4	1	no	—
26	C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	#11924003	—	2	1	no	—
28	C₂×Q₈×C₇₃	—	2	1	no	—
29	Q₁₆×C₇₃	—	2	1	no	—
30	Q₁₆×C₇₃	Q₈	1	2	yes	complemented
31	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₃	—	2	1	no	—
33	C₁₆×C₇₃	—	2	1	no	—
34	Q₈×C₇₃	—	2	2	no	—
35	Q₈×C₇₃	—	4	1	no	—
36	C₂×C₄×C₇₃	—	4	1	no	—
37	C₂×C₄×C₇₃	—	4	1	no	—
38	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₃	—	4	1	no	—
40	C₂×Dic₇₃	—	4	1	no	—
41	C₂×Dic₇₃	D₈	1	1	yes	characteristic
42	C₂×Dic₇₃	D₈	1	1	yes	characteristic
43	Dic₁₄₆	—	8	2	no	—
44	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₃	#41	1	1	yes	characteristic
46	#10596831	—	2	1	no	—
47	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
48	C₄×C₇₃	—	4	2	no	—
49	C₄×C₇₃	—	8	1	no	—
50	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₇₃	#137	1	1	yes	characteristic
52	Dic₇₃	—	2	1	no	—
53	Dic₇₃	—	8	1	no	—
54	Dic₇₃	—	8	1	no	—
55	Dic₇₃	Q₃₂	1	2	yes	—
56	C₂×C₇₃	#474	1	1	yes	characteristic
57	C₂×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
58	C₂×C₇₃	#509	1	1	yes	characteristic
59	#2227	—	73	1	no	maximal, Sylow
60	C₇₃	#2227	1	1	yes	characteristic, complemented, Sylow
61	#500	—	73	1	no	—
62	C₄×Q₁₆	—	73	1	no	—
63	#366	—	73	1	no	—
64	#366	—	73	1	no	—
65	C₄⋊C₁₆	—	73	1	no	—
66	#358	—	73	1	no	—
67	#358	D₇₃	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
69	C₄⋊Q₈	—	146	1	no	—
70	C₄×Q₈	—	146	1	no	—
71	C₂×C₁₆	—	73	1	no	—
72	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
73	#107	—	73	1	no	—
74	#107	—	73	1	no	—
75	#107	—	146	1	no	—
76	#107	C₂×D₇₃	1	1	yes	characteristic
77	#103	—	146	1	no	—
78	C₄×C₈	—	73	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	292	1	no	—
81	Q₁₆	—	2	1	no	—
82	Q₁₆	Dic₁₄₆	1	2	yes	complemented
83	C₂×C₈	—	73	1	no	—
84	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	292	1	no	—
87	C₄⋊C₄	—	292	1	no	—
88	C₄⋊C₄	—	146	1	no	—
89	C₄⋊C₄	—	146	1	no	—
90	C₄²	—	292	1	no	—
91	C₄²	—	73	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	146	1	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	1	no	—
96	Q₈	—	584	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	292	1	no	—
100	C₂×C₄	—	292	1	no	—
101	C₂×C₄	—	73	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
104	C₈	—	146	1	no	—
105	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
106	C₈	#11924027	1	1	yes	characteristic
107	C₂²	D₈×D₇₃	1	1	yes	characteristic
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	—	584	1	no	—
111	C₄	—	584	1	no	—
112	C₄	—	73	2	no	—
113	C₄	—	146	1	no	—
114	C₄	#14556568	1	1	yes	characteristic
115	C₄	#14556620	1	1	yes	characteristic
116	C₂	#17988865	1	1	yes	characteristic
117	C₂	Q₃₂×D₇₃	1	1	yes	characteristic
118	C₂	#17989024	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988991

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₃

C₂

yes

maximal, characteristic

#17988413

C₂

yes

maximal, characteristic

#17988412

C₂

yes

maximal, characteristic

#17988405

C₂

yes

maximal, characteristic

#17988391

C₂

yes

maximal, characteristic

#358×C₇₃

C₂

yes

maximal, characteristic

#14556590

—

Q₈×Dic₇₃

—

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556524

—

#14556515

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

Q₈

yes

complemented

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₂×Dic₇₃

D₈

yes

characteristic

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#137

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

Q₃₂

yes

—

C₂×C₇₃

#474

yes

characteristic

C₂×C₇₃

C₂×Q₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#500

—

C₄×Q₁₆

—

#366

—

#366

—

C₄⋊C₁₆

—

#358

—

#358

D₇₃

yes

characteristic

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

—

—

—

146

—

#107

C₂×D₇₃

yes

characteristic

#103

—

146

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

Dic₁₄₆

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄²

—

292

—

C₄²

—

C₁₆

—

C₁₆

—

146

—

Q₈

—

Q₈

—

Q₈

—

584

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

100

C₂×C₄

—

292

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

D₄×D₇₃

yes

characteristic

104

C₈

—

146

—

105

C₈

C₂×Dic₁₄₆

yes

characteristic

106

C₈

#11924027

yes

characteristic

107

C₂²

D₈×D₇₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

584

—

111

C₄

—

584

—

112

C₄

—

113

C₄

—

146

—

114

C₄

#14556568

yes

characteristic

115

C₄

#14556620

yes

characteristic

116

C₂

#17988865

yes

characteristic

117

C₂

Q₃₂×D₇₃

yes

characteristic

118

C₂

#17989024

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#358×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#358	D₇₃
hypocenter	C₇₃	#2227
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#358×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
30. Q₁₆×C₇₃	Q₈	semidirect
60. C₇₃	#2227	semidirect
82. Q₁₆	Dic₁₄₆	semidirect

H ≅ G/N

type

30. Q₁₆×C₇₃

Q₈

semidirect

60. C₇₃

#2227

semidirect

82. Q₁₆

Dic₁₄₆

semidirect

Subgroups — #22612081

Conjugacy classes of subgroups

Complements