Group #22611702 — subgroups

Subgroups — #22611702

Order 9344.

144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#108×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988622	C₂	1	1	yes	maximal, characteristic
4	#17988411	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988410	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988409	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×D₇₃	—	2	1	no	—
9	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#14556549	—	2	1	no	—
11	#14556531	C₂²	1	1	yes	characteristic
12	#14556530	C₄	1	2	yes	complemented
13	C₁₆×D₇₃	C₄	1	2	yes	complemented
14	#14556527	C₂²	1	1	yes	characteristic
15	#14556526	—	2	1	no	—
16	#14556524	—	2	1	no	—
17	#14556523	C₂²	1	1	yes	characteristic
18	M₃₂×C₇₃	C₂²	1	1	yes	characteristic
19	#108×C₇₃	C₂²	1	1	yes	characteristic
20	#106×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
22	#11924012	—	2	1	no	—
23	#11924011	—	4	1	no	—
24	C₂×#10596831	D₄	1	1	yes	characteristic
25	C₈×D₇₃	—	4	1	no	—
26	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×D₇₃	Q₈	1	1	yes	characteristic
29	C₈×D₇₃	Q₈	1	1	yes	characteristic
30	C₄⋊Dic₇₃	—	2	1	no	—
31	#11924005	—	4	1	no	—
32	C₄×Dic₇₃	—	4	1	no	—
33	#11924003	C₂×C₄	1	2	yes	—
34	M₁₆×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₁₆×C₇₃	C₂×C₄	1	2	yes	—
38	C₂²×D₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₇₃	—	4	1	no	—
42	C₂×Dic₇₃	SD₁₆	1	1	yes	characteristic
43	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
46	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
47	C₈×C₇₃	—	4	1	no	—
48	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	#10596831	—	4	1	no	—
51	#10596831	C₄⋊C₄	1	1	yes	characteristic
52	#10596831	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×D₇₃	—	4	1	no	—
55	C₂×D₇₃	#106	1	1	yes	characteristic
56	C₂×D₇₃	#106	1	1	yes	characteristic
57	C₄×C₇₃	—	8	1	no	—
58	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₇₃	—	8	1	no	—
61	Dic₇₃	#106	1	1	yes	characteristic
62	Dic₇₃	#106	1	1	yes	characteristic
63	C₂×C₇₃	—	2	1	no	—
64	C₂×C₇₃	C₂×#106	1	1	yes	characteristic
65	D₇₃	—	2	1	no	—
66	D₇₃	—	2	1	no	—
67	#2155	—	73	1	no	maximal, Sylow
68	C₇₃	#2155	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	—	73	1	no	—
70	C₂×#108	—	73	1	no	—
71	#427	—	73	1	no	—
72	#365	—	73	1	no	—
73	#365	—	73	1	no	—
74	#365	—	73	1	no	—
75	#365	D₇₃	1	1	yes	characteristic, complemented
76	C₂×M₁₆	—	146	1	no	—
77	C₂²×C₈	—	73	1	no	—
78	#117	—	146	1	no	—
79	M₃₂	—	73	1	no	—
80	M₃₂	—	73	2	no	—
81	M₃₂	C₂×D₇₃	1	1	yes	characteristic
82	C₂×C₁₆	—	73	2	no	—
83	#108	—	73	1	no	—
84	#108	—	146	1	no	—
85	#108	C₂×D₇₃	1	1	yes	characteristic
86	#107	—	146	1	no	—
87	#106	—	73	1	no	—
88	#106	C₂×D₇₃	1	1	yes	characteristic
89	C₂²×C₄	—	73	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	292	1	no	—
92	M₁₆	—	146	1	no	—
93	C₂×C₈	—	292	1	no	—
94	C₂×C₈	—	73	1	no	—
95	C₂×C₈	—	73	1	no	—
96	C₂×C₈	—	73	1	no	—
97	C₂×C₈	—	73	1	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	146	1	no	—
102	C₂²⋊C₄	—	292	1	no	—
103	C₄²	—	292	1	no	—
104	C₁₆	—	73	2	no	—
105	C₁₆	C₄×D₇₃	1	2	yes	—
106	C₂³	—	73	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	292	1	no	—
109	C₂×C₄	—	73	1	no	—
110	C₂×C₄	—	73	1	no	—
111	C₂×C₄	—	73	1	no	—
112	C₂×C₄	—	73	1	no	—
113	C₂×C₄	—	73	1	no	—
114	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
115	C₈	—	4	1	no	—
116	C₈	—	292	1	no	—
117	C₈	—	73	1	no	—
118	C₈	—	73	1	no	—
119	C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
120	C₈	Q₈×D₇₃	1	1	yes	characteristic
121	C₂²	—	292	1	no	—
122	C₂²	—	73	1	no	—
123	C₂²	—	73	1	no	—
124	C₂²	SD₁₆×D₇₃	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	584	1	no	—
127	C₄	—	73	1	no	—
128	C₄	—	73	1	no	—
129	C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
130	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	146	1	no	—
133	C₂	—	146	1	no	—
134	C₂	#106×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₇₃

C₂

yes

maximal, characteristic

#108×D₇₃

C₂

yes

maximal, characteristic

#17988622

C₂

yes

maximal, characteristic

#17988411

C₂

yes

maximal, characteristic, complemented

#17988410

C₂

yes

maximal, characteristic, complemented

#17988409

C₂

yes

maximal, characteristic, complemented

#365×C₇₃

C₂

yes

maximal, characteristic, complemented

M₁₆×D₇₃

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556549

—

#14556531

C₂²

yes

characteristic

#14556530

C₄

yes

complemented

C₁₆×D₇₃

C₄

yes

complemented

#14556527

C₂²

yes

—

—

C₂²

yes

characteristic

M₃₂×C₇₃

C₂²

yes

characteristic

#108×C₇₃

C₂²

yes

characteristic

#106×C₇₃

C₂²

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

C₄⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

#11924003

C₂×C₄

yes

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

C₂×C₄

yes

—

C₂²×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂²×C₄

yes

characteristic

C₈×C₇₃

C₂×Q₈

yes

characteristic

#10596831

—

#10596831

C₄⋊C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

C₂×#106

yes

characteristic

D₇₃

—

D₇₃

—

#2155

—

maximal, Sylow

C₇₃

#2155

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×M₁₆

—

146

—

C₂²×C₈

—

#117

—

146

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₇₃

yes

characteristic

C₂×C₁₆

—

#108

—

#108

—

146

—

#108

C₂×D₇₃

yes

characteristic

#107

—

146

—

#106

—

#106

C₂×D₇₃

yes

characteristic

C₂²×C₄

—

M₁₆

—

M₁₆

—

292

—

M₁₆

—

146

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

146

—

102

C₂²⋊C₄

—

292

—

103

C₄²

—

292

—

104

C₁₆

—

105

C₁₆

C₄×D₇₃

yes

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

292

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₄×D₇₃

yes

characteristic

115

C₈

—

116

C₈

—

292

—

117

C₈

—

118

C₈

—

119

C₈

C₂×C₄×D₇₃

yes

characteristic

120

C₈

Q₈×D₇₃

yes

characteristic

121

C₂²

—

292

—

122

C₂²

—

123

C₂²

—

124

C₂²

SD₁₆×D₇₃

yes

characteristic

125

C₄

—

126

C₄

—

584

—

127

C₄

—

128

C₄

—

129

C₄

C₄⋊C₄×D₇₃

yes

characteristic

130

C₄

SD₁₆×D₇₃

yes

characteristic

131

C₂

—

132

C₂

—

146

—

133

C₂

—

146

—

134

C₂

#106×D₇₃

yes

characteristic

	subgroup	G/H
center	C₂	#106×D₇₃
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#365×C₇₃	C₂
socle	C₂×C₇₃	C₂×#106
hypercenter	#365	D₇₃
hypocenter	C₇₃	#2155
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#106×D₇₃

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#365×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
4. #17988411	C₂	semidirect
5. #17988410	C₂	semidirect
6. #17988409	C₂	semidirect
7. #365×C₇₃	C₂	semidirect
12. #14556530	C₄	semidirect
13. C₁₆×D₇₃	C₄	semidirect
68. C₇₃	#2155	semidirect
75. #365	D₇₃	semidirect

H ≅ G/N

type

4. #17988411

C₂

semidirect

5. #17988410

C₂

semidirect

6. #17988409

C₂

semidirect

7. #365×C₇₃

C₂

semidirect

12. #14556530

C₄

semidirect

13. C₁₆×D₇₃

C₄

semidirect

68. C₇₃

#2155

semidirect

75. #365

D₇₃

semidirect

Subgroups — #22611702

Conjugacy classes of subgroups

Complements