Group #22609835 — subgroups

Subgroups — #108×Dic₇₃ #22609835

Order 9344.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14556527	C₂	1	1	yes	maximal, characteristic
2	M₁₆×Dic₇₃	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988351	C₂	1	2	yes	maximal
5	C₂×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	C₂²	1	2	yes	—
7	C₂²×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
9	#14556527	C₄	1	4	yes	complemented
10	#14556501	—	2	2	no	—
11	C₈×Dic₇₃	—	2	2	no	—
12	C₈×Dic₇₃	C₄	1	4	yes	—
13	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#108×C₇₃	C₄	1	4	yes	complemented
16	C₂²×Dic₇₃	Q₈	1	2	yes	—
17	#11924012	—	2	2	no	—
18	#11924012	C₂×C₄	1	4	yes	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	2	1	no	—
21	C₂×#10596831	—	2	1	no	—
22	C₂×#10596831	C₂×C₄	1	2	yes	—
23	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₄×Dic₇₃	D₄	1	1	yes	characteristic
25	C₄×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₇₃	—	2	2	no	—
28	M₁₆×C₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₇₃	—	2	2	no	—
30	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₂³×C₇₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
39	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
40	C₂×Dic₇₃	—	2	4	no	—
41	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
42	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
43	C₈×C₇₃	—	2	4	no	—
44	C₈×C₇₃	C₄²	1	4	yes	—
45	#10596831	—	2	2	no	—
46	#10596831	—	2	4	no	—
47	C₂²×C₇₃	—	2	2	no	—
48	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
50	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₇₃	#117	1	1	yes	characteristic
53	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
54	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
55	Dic₇₃	—	2	2	no	—
56	Dic₇₃	#108	1	4	yes	direct product factor
57	C₂×C₇₃	—	2	1	no	—
58	C₂×C₇₃	—	2	1	no	—
59	C₂×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₇₃	C₂×#108	1	2	yes	—
61	C₄×#108	—	73	1	no	maximal, Sylow
62	C₇₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	73	1	no	—
64	C₂×#108	D₇₃	1	1	yes	characteristic
65	C₄×M₁₆	—	73	2	no	—
66	C₂×C₄×C₈	—	73	1	no	—
67	#341	—	73	2	no	—
68	C₂×M₁₆	—	73	2	no	—
69	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
70	C₂²×C₈	—	73	1	no	—
71	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
72	C₂×C₄²	—	73	1	no	—
73	#108	—	73	4	no	—
74	#108	Dic₇₃	1	4	yes	direct product factor
75	#97	—	146	2	no	—
76	C₄×C₈	—	73	4	no	—
77	C₄×C₈	—	146	2	no	—
78	C₂²×C₄	—	73	2	no	—
79	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	73	4	no	—
82	M₁₆	—	146	2	no	—
83	M₁₆	C₂×Dic₇₃	1	4	yes	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	73	2	no	—
86	C₂×C₈	—	146	2	no	—
87	C₂×C₈	—	146	1	no	—
88	C₂×C₈	—	146	1	no	—
89	C₂×C₈	C₄×D₇₃	1	2	yes	—
90	C₂×C₈	C₄×D₇₃	1	2	yes	—
91	C₂×C₈	C₂×Dic₇₃	1	2	yes	—
92	C₄²	—	73	2	no	—
93	C₄²	—	73	1	no	—
94	C₄²	—	73	1	no	—
95	C₂³	Q₈×D₇₃	1	1	yes	characteristic
96	C₂×C₄	—	73	2	no	—
97	C₂×C₄	—	73	2	no	—
98	C₂×C₄	—	146	4	no	—
99	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
101	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
102	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
103	C₂×C₄	#11924027	1	1	yes	characteristic
104	C₂×C₄	#11924029	1	1	yes	characteristic
105	C₈	—	2	4	no	—
106	C₈	—	146	2	no	—
107	C₈	—	146	4	no	—
108	C₈	C₄×Dic₇₃	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄⋊C₄×D₇₃	1	1	yes	characteristic
111	C₂²	#14556556	1	1	yes	characteristic
112	C₂²	Q₈×Dic₇₃	1	1	yes	characteristic
113	C₄	—	73	4	no	—
114	C₄	—	146	2	no	—
115	C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
116	C₄	#14556549	1	1	yes	characteristic
117	C₄	#14556553	1	1	yes	characteristic
118	C₄	D₄×Dic₇₃	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
122	C₂	#108×D₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556527

C₂

yes

maximal, characteristic

M₁₆×Dic₇₃

C₂

yes

maximal

C₂×C₈×Dic₇₃

C₂

yes

maximal, characteristic

#17988351

C₂

yes

maximal

C₂×#108×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

—

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556527

C₄

yes

complemented

#14556501

—

C₈×Dic₇₃

—

C₈×Dic₇₃

C₄

yes

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#108×C₇₃

C₄

yes

complemented

C₂²×Dic₇₃

Q₈

yes

—

#11924012

—

#11924012

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₄

yes

—

C₂×Dic₇₃

C₄⋊C₄

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄²

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₄×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

Dic₇₃

—

Dic₇₃

#108

yes

direct product factor

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₃

C₂×#108

yes

—

C₄×#108

—

maximal, Sylow

C₇₃

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

D₇₃

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂×C₄²

—

#108

—

#108

Dic₇₃

yes

direct product factor

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

146

—

M₁₆

C₂×Dic₇₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₄×D₇₃

yes

—

C₂×C₈

C₄×D₇₃

yes

—

C₂×C₈

C₂×Dic₇₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

100

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

101

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

102

C₂×C₄

D₄×D₇₃

yes

characteristic

103

C₂×C₄

#11924027

yes

characteristic

104

C₂×C₄

#11924029

yes

characteristic

105

C₈

—

106

C₈

—

146

—

107

C₈

—

146

—

108

C₈

C₄×Dic₇₃

yes

—

109

C₂²

—

110

C₂²

C₄⋊C₄×D₇₃

yes

characteristic

111

C₂²

#14556556

yes

characteristic

112

C₂²

Q₈×Dic₇₃

yes

characteristic

113

C₄

—

114

C₄

—

146

—

115

C₄

C₂×C₄×Dic₇₃

yes

characteristic

116

C₄

#14556549

yes

characteristic

117

C₄

#14556553

yes

characteristic

118

C₄

D₄×Dic₇₃

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

C₄⋊C₄×Dic₇₃

yes

characteristic

122

C₂

#108×D₇₃

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	C₂×#108	D₇₃
hypocenter	C₇₃	C₄×#108
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₇₃

commutator

C₄×C₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#108×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₄⋊C₄

D₇₃

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #14556527	C₄	semidirect
15. #108×C₇₃	C₄	semidirect
56. Dic₇₃	#108	direct (N × H)
62. C₇₃	C₄×#108	semidirect
74. #108	Dic₇₃	direct (N × H)

H ≅ G/N

type

9. #14556527

C₄

semidirect

15. #108×C₇₃

C₄

semidirect

56. Dic₇₃

#108

direct (N × H)

62. C₇₃

C₄×#108

semidirect

74. #108

Dic₇₃

direct (N × H)

Subgroups — #108×Dic₇₃ #22609835

Conjugacy classes of subgroups

Complements

Subgroups — #108×Dic73 #22609835

Conjugacy classes of subgroups

Complements

Subgroups — #108×Dic₇₃ #22609835