Group #22612100 — subgroups

Subgroups — #22612100

Order 9344.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#17988984	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988704	C₂	1	1	yes	maximal, characteristic
3	#17988410	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988409	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988407	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988397	C₂	1	1	yes	maximal, characteristic
7	#360×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556598	—	2	1	no	—
9	#14556590	—	2	1	no	—
10	#14556531	C₂²	1	1	yes	characteristic
11	#14556527	C₂²	1	1	yes	characteristic
12	#14556523	C₂²	1	1	yes	characteristic
13	#14556522	—	2	1	no	—
14	#14556518	—	2	1	no	—
15	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	#135×C₇₃	C₂²	1	1	yes	characteristic
17	M₃₂×C₇₃	C₂²	1	1	yes	characteristic
18	#106×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₆	—	4	1	no	—
20	#11924012	—	2	1	no	—
21	#11924012	—	4	1	no	—
22	C₂×#10596831	—	4	1	no	—
23	C₂×#10596831	D₄	1	1	yes	characteristic
24	C₄⋊Dic₇₃	—	2	1	no	—
25	#11924007	—	4	1	no	—
26	C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	#11924003	—	2	1	no	—
28	#41×C₇₃	—	2	1	no	—
29	Q₁₆×C₇₃	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₇₃	—	2	1	no	—
31	D₈×C₇₃	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	2	1	no	—
34	C₁₆×C₇₃	—	2	1	no	—
35	Q₈×C₇₃	—	2	1	no	—
36	D₄×C₇₃	—	2	1	no	—
37	D₄×C₇₃	—	4	1	no	—
38	C₂×C₄×C₇₃	—	4	1	no	—
39	C₂×C₄×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₇₃	—	2	1	no	—
42	C₂×Dic₇₃	—	4	1	no	—
43	Dic₁₄₆	—	8	2	no	—
44	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₃	#41	1	1	yes	characteristic
46	#10596831	—	4	1	no	—
47	#10596831	SD₁₆	1	1	yes	characteristic
48	#10596831	SD₁₆	1	1	yes	characteristic
49	C₂²×C₇₃	—	4	1	no	—
50	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₇₃	—	4	1	no	—
52	C₄×C₇₃	—	8	1	no	—
53	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₇₃	—	2	2	no	—
56	Dic₇₃	—	8	1	no	—
57	C₂×C₇₃	—	2	1	no	—
58	C₂×C₇₃	—	8	1	no	—
59	C₂×C₇₃	#476	1	1	yes	characteristic
60	#2230	—	73	1	no	maximal, Sylow
61	C₇₃	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	73	1	no	—
63	#443	—	73	1	no	—
64	#365	—	73	1	no	—
65	#365	—	73	1	no	—
66	#364	—	73	1	no	—
67	#360	—	73	1	no	—
68	#360	D₇₃	1	1	yes	characteristic
69	#135	C₂×D₇₃	1	1	yes	characteristic
70	#131	—	146	1	no	—
71	C₄⋊Q₈	—	146	1	no	—
72	M₃₂	—	73	1	no	—
73	M₃₂	C₂×D₇₃	1	1	yes	characteristic
74	#108	—	73	1	no	—
75	#106	—	73	1	no	—
76	#106	—	146	1	no	—
77	#106	C₂×D₇₃	1	1	yes	characteristic
78	#104	—	146	1	no	—
79	C₄×C₈	—	73	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	292	1	no	—
82	Q₁₆	Dic₁₄₆	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₁₄₆	1	1	yes	characteristic, complemented
85	M₁₆	—	292	1	no	—
86	M₁₆	—	146	1	no	—
87	C₂×C₈	—	292	1	no	—
88	C₂×C₈	—	73	1	no	—
89	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	292	1	no	—
92	C₄⋊C₄	—	146	1	no	—
93	C₄²	—	73	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	146	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	584	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	292	1	no	—
103	C₂×C₄	—	146	1	no	—
104	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
105	C₈	—	292	1	no	—
106	C₈	—	73	1	no	—
107	C₈	—	73	1	no	—
108	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
109	C₈	#11924027	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#14556615	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	584	1	no	—
115	C₄	—	146	2	no	—
116	C₄	#14556568	1	1	yes	characteristic
117	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#17988875	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988984

C₂

yes

maximal, characteristic, complemented

#17988704

C₂

yes

maximal, characteristic

#17988410

C₂

yes

maximal, characteristic, complemented

#17988409

C₂

yes

maximal, characteristic, complemented

#17988407

C₂

yes

maximal, characteristic, complemented

#17988397

C₂

yes

maximal, characteristic

#360×C₇₃

C₂

yes

maximal, characteristic

#14556598

—

#14556590

—

#14556531

C₂²

yes

characteristic

#14556527

C₂²

yes

characteristic

#14556523

C₂²

yes

characteristic

#14556522

—

#14556518

—

C₈×Dic₇₃

C₂²

yes

characteristic

#135×C₇₃

C₂²

yes

characteristic

M₃₂×C₇₃

C₂²

yes

characteristic

#106×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

—

—

—

D₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

—

#41×C₇₃

—

Q₁₆×C₇₃

Q₈

yes

characteristic, complemented

SD₁₆×C₇₃

—

D₈×C₇₃

Q₈

yes

characteristic, complemented

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

C₈×C₇₃

yes

—

SD₁₆

yes

characteristic

#10596831

SD₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₃

yes

characteristic

#135

C₂×D₇₃

yes

characteristic

#131

—

146

—

C₄⋊Q₈

—

146

—

M₃₂

—

M₃₂

C₂×D₇₃

yes

—

—

—

146

—

#106

C₂×D₇₃

yes

characteristic

#104

—

146

—

C₄×C₈

—

#41

—

C₂×Q₈

—

292

—

Q₁₆

Dic₁₄₆

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₄₆

yes

characteristic, complemented

M₁₆

—

292

—

M₁₆

—

146

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄²

—

C₁₆

—

C₁₆

—

146

—

Q₈

—

Q₈

—

584

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

292

—

103

C₂×C₄

—

146

—

104

C₂×C₄

D₄×D₇₃

yes

characteristic

105

C₈

—

292

—

106

C₈

—

107

C₈

—

108

C₈

C₂×Dic₁₄₆

yes

characteristic

109

C₈

#11924027

yes

characteristic

110

C₂²

—

111

C₂²

#14556615

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

584

—

115

C₄

—

146

—

116

C₄

#14556568

yes

characteristic

117

C₄

SD₁₆×D₇₃

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#17988875

yes

characteristic

	subgroup	G/H
center	C₂	#17988875
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#360×C₇₃	C₂
socle	C₂×C₇₃	#476
hypercenter	#360	D₇₃
hypocenter	C₇₃	#2230
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#17988875

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#360×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #17988984	C₂	semidirect
3. #17988410	C₂	semidirect
4. #17988409	C₂	semidirect
5. #17988407	C₂	semidirect
29. Q₁₆×C₇₃	Q₈	semidirect
31. D₈×C₇₃	Q₈	semidirect
61. C₇₃	#2230	semidirect
82. Q₁₆	Dic₁₄₆	semidirect
84. D₈	Dic₁₄₆	semidirect

H ≅ G/N

type

1. #17988984

C₂

semidirect

3. #17988410

C₂

semidirect

4. #17988409

C₂

semidirect

5. #17988407

C₂

semidirect

29. Q₁₆×C₇₃

Q₈

semidirect

31. D₈×C₇₃

Q₈

semidirect

61. C₇₃

#2230

semidirect

82. Q₁₆

Dic₁₄₆

semidirect

84. D₈

Dic₁₄₆

semidirect

Subgroups — #22612100

Conjugacy classes of subgroups

Complements