Group #22609980 — subgroups

Subgroups — #22609980

Order 9344.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#14556599	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556526	C₂	1	2	yes	maximal, complemented
3	#17988569	C₂	1	2	yes	maximal
4	#17988345	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924028	C₂²	1	1	yes	characteristic
7	#14556599	—	2	4	no	—
8	C₂×#11924012	C₂²	1	2	yes	—
9	C₂×#11924011	C₄	1	2	yes	—
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₇₃	C₄	1	2	yes	—
12	#14556526	—	2	4	no	—
13	#14556509	—	2	2	no	—
14	#14556507	—	2	2	no	—
15	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	#11924028	—	2	2	no	—
18	#11924028	—	2	2	no	—
19	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
20	C₂×D₂₉₂	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₇₃	Q₈	1	2	yes	—
22	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
23	#11924012	—	2	2	no	—
24	#11924012	—	2	2	no	—
25	#11924011	—	2	4	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×#10596831	D₄	1	4	yes	—
30	C₈×D₇₃	—	2	4	no	—
31	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₇₃	—	2	2	no	—
33	M₁₆×C₇₃	—	2	2	no	—
34	M₁₆×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
37	C₂³×C₇₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₇₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#41	1	2	yes	—
43	C₂²⋊D₇₃	—	2	4	no	—
44	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
45	D₂₉₂	—	2	2	no	—
46	C₄×D₇₃	—	2	2	no	—
47	C₄×D₇₃	—	2	2	no	—
48	Dic₁₄₆	—	2	2	no	—
49	C₈×C₇₃	—	2	2	no	—
50	C₈×C₇₃	—	4	2	no	—
51	#10596831	—	4	2	no	—
52	#10596831	C₂²⋊C₄	1	2	yes	—
53	#10596831	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₇₃	—	4	1	no	—
55	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₇₃	—	2	2	no	—
58	C₂×D₇₃	—	4	2	no	—
59	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₇₃	#117	1	1	yes	characteristic
61	C₄×C₇₃	#121	1	2	yes	—
62	Dic₇₃	—	2	2	no	—
63	C₂×C₇₃	—	2	1	no	—
64	C₂×C₇₃	—	2	2	no	—
65	C₂×C₇₃	#380	1	1	yes	characteristic
66	D₇₃	—	4	2	no	—
67	#1813	—	73	1	no	maximal, Sylow
68	C₇₃	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	73	1	no	—
70	C₂×#108	—	73	2	no	—
71	#413	—	73	2	no	—
72	#338	—	73	1	no	—
73	#338	D₇₃	1	1	yes	characteristic, complemented
74	C₂×#41	—	73	1	no	—
75	#131	—	146	4	no	—
76	C₂×M₁₆	—	73	2	no	—
77	C₂×M₁₆	—	73	2	no	—
78	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
79	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
80	C₂²×C₈	—	73	1	no	—
81	C₂²×C₈	—	73	2	no	—
82	#108	—	146	4	no	—
83	#101	—	146	2	no	—
84	#100	—	146	2	no	—
85	#41	—	146	2	no	—
86	#41	—	146	2	no	—
87	C₂×Q₈	—	73	1	no	—
88	C₂×D₄	—	73	2	no	—
89	C₂×D₄	—	73	1	no	—
90	C₂²×C₄	—	73	2	no	—
91	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	146	4	no	—
96	M₁₆	—	146	2	no	—
97	M₁₆	—	146	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	—	146	4	no	—
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	C₄×D₇₃	1	2	yes	—
105	C₂³	—	73	2	no	—
106	C₂³	Q₈×D₇₃	1	1	yes	characteristic
107	Q₈	—	146	2	no	—
108	D₄	—	146	4	no	—
109	D₄	—	146	2	no	—
110	C₂×C₄	—	73	2	no	—
111	C₂×C₄	—	146	2	no	—
112	C₂×C₄	—	146	2	no	—
113	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
114	C₂×C₄	D₄×D₇₃	1	2	yes	—
115	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
116	C₂×C₄	#11924028	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	292	2	no	—
120	C₈	—	73	2	no	—
121	C₈	—	73	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	292	2	no	—
124	C₂²	—	146	2	no	—
125	C₂²	C₄⋊C₄×D₇₃	1	1	yes	characteristic
126	C₂²	#14556569	1	2	yes	—
127	C₄	—	146	2	no	—
128	C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
129	C₄	#14556548	1	1	yes	characteristic
130	C₄	#14556564	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	292	2	no	—
134	C₂	#17988446	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556599

C₂

yes

maximal, characteristic

C₂×#14556526

C₂

yes

maximal, complemented

#17988569

C₂

yes

maximal

#17988345

C₂

yes

maximal, characteristic, complemented

#338×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

characteristic

#14556599

—

C₂×#11924012

C₂²

yes

—

C₂×#11924011

C₄

yes

—

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₄

yes

—

#14556526

—

#14556509

—

#14556507

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

#11924028

—

#11924028

—

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₂²⋊D₇₃

Q₈

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

D₄

yes

—

C₈×D₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂×Q₈

yes

characteristic

C₂²×D₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₄⋊C₄

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

C₂²⋊C₄

yes

—

#10596831

C₂²⋊C₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#380

yes

characteristic

D₇₃

—

#1813

—

maximal, Sylow

C₇₃

#1813

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×#41

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

146

—

#101

—

146

—

#100

—

146

—

#41

—

146

—

#41

—

146

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

146

—

M₁₆

—

146

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

146

—

103

C₂×C₈

—

146

—

104

C₂×C₈

C₄×D₇₃

yes

—

105

C₂³

—

106

C₂³

Q₈×D₇₃

yes

characteristic

107

Q₈

—

146

—

108

D₄

—

146

—

109

D₄

—

146

—

110

C₂×C₄

—

111

C₂×C₄

—

146

—

112

C₂×C₄

—

146

—

113

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

114

C₂×C₄

D₄×D₇₃

yes

—

115

C₂×C₄

D₄×D₇₃

yes

characteristic

116

C₂×C₄

#11924028

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

292

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

292

—

124

C₂²

—

146

—

125

C₂²

C₄⋊C₄×D₇₃

yes

characteristic

126

C₂²

#14556569

yes

—

127

C₄

—

146

—

128

C₄

C₂²⋊C₄×D₇₃

yes

characteristic

129

C₄

#14556548

yes

characteristic

130

C₄

#14556564

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

292

—

134

C₂

#17988446

yes

characteristic

	subgroup	G/H
center	C₄	#14556548
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#338×C₇₃	C₂
socle	C₂×C₇₃	#380
hypercenter	#338	D₇₃
hypocenter	C₇₃	#1813
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#14556548

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#338×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#14556526	C₂	semidirect
4. #17988345	C₂	semidirect
5. #338×C₇₃	C₂	semidirect
68. C₇₃	#1813	semidirect
73. #338	D₇₃	semidirect

H ≅ G/N

type

2. C₂×#14556526

C₂

semidirect

4. #17988345

C₂

semidirect

5. #338×C₇₃

C₂

semidirect

68. C₇₃

#1813

semidirect

73. #338

D₇₃

semidirect

Subgroups — #22609980

Conjugacy classes of subgroups

Complements