Group #22611688 — subgroups

Subgroups — #22611688

Order 9344.

160 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 79 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×#14556530	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988541	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988405	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988405	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924011	C₄	1	1	yes	characteristic
9	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#14556548	C₄	1	1	yes	characteristic
11	#14556530	—	2	2	no	—
12	C₂×#11924003	C₂²	1	1	yes	characteristic
13	C₂×#11924003	C₂²	1	1	yes	characteristic
14	C₁₆×D₇₃	—	2	2	no	—
15	#14556502	C₄	1	1	yes	characteristic
16	#14556501	C₄	1	1	yes	characteristic
17	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
22	#11924011	—	2	2	no	—
23	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₈×D₇₃	D₄	1	2	yes	—
26	C₈×D₇₃	Q₈	1	2	yes	—
27	#11924007	C₈	1	1	yes	characteristic
28	#11924007	C₈	1	1	yes	characteristic
29	#11924005	C₈	1	1	yes	characteristic
30	#11924005	C₈	1	1	yes	characteristic
31	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	#11924003	—	2	1	no	—
33	#11924003	—	2	1	no	—
34	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₁₆×C₇₃	—	2	1	no	—
38	C₁₆×C₇₃	—	2	1	no	—
39	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
46	C₄×D₇₃	C₄⋊C₄	1	2	yes	—
47	C₄×D₇₃	M₁₆	1	2	yes	—
48	C₈×C₇₃	—	2	1	no	—
49	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	#10596831	—	2	1	no	—
52	#10596831	C₄⋊C₄	1	2	yes	—
53	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₇₃	—	2	1	no	—
55	C₂×D₇₃	—	2	1	no	—
56	C₂×D₇₃	C₄⋊C₈	1	2	yes	—
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
61	Dic₇₃	—	2	1	no	—
62	Dic₇₃	—	2	1	no	—
63	Dic₇₃	C₄⋊C₈	1	2	yes	—
64	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₇₃	#504	1	1	yes	characteristic
66	C₂×C₇₃	#504	1	1	yes	characteristic
67	D₇₃	—	2	2	no	—
68	#2151	—	73	1	no	maximal, Sylow
69	C₇₃	#2151	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₃₂	—	73	1	no	—
71	C₂²×C₁₆	—	73	1	no	—
72	#405	—	73	1	no	—
73	C₄⋊C₁₆	—	73	1	no	—
74	C₄⋊C₁₆	—	73	1	no	—
75	C₄⋊C₁₆	—	73	1	no	—
76	C₄⋊C₁₆	D₇₃	1	1	yes	characteristic, complemented
77	C₂×M₁₆	—	73	1	no	—
78	C₂²×C₈	—	73	1	no	—
79	#117	—	73	1	no	—
80	M₃₂	—	146	2	no	—
81	C₂×C₁₆	—	73	1	no	—
82	C₂×C₁₆	—	73	1	no	—
83	C₂×C₁₆	—	146	2	no	—
84	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
85	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
86	#97	—	73	1	no	—
87	#97	—	73	1	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
90	C₂²×C₄	—	73	1	no	—
91	M₁₆	—	146	2	no	—
92	C₂×C₈	—	73	1	no	—
93	C₂×C₈	—	73	2	no	—
94	C₂×C₈	—	73	2	no	—
95	C₂×C₈	—	73	1	no	—
96	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
97	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
98	C₄⋊C₄	—	73	1	no	—
99	C₄⋊C₄	—	73	1	no	—
100	C₂²⋊C₄	—	73	1	no	—
101	C₂²⋊C₄	—	73	1	no	—
102	C₄²	—	73	1	no	—
103	C₄²	C₄×D₇₃	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	146	1	no	—
107	C₁₆	—	146	1	no	—
108	C₂³	—	73	1	no	—
109	C₂×C₄	—	73	1	no	—
110	C₂×C₄	—	73	2	no	—
111	C₂×C₄	—	73	2	no	—
112	C₂×C₄	—	73	1	no	—
113	C₂×C₄	—	73	1	no	—
114	C₂×C₄	C₈×D₇₃	1	1	yes	characteristic
115	C₂×C₄	C₈×D₇₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	73	2	no	—
119	C₈	—	146	1	no	—
120	C₈	D₄×D₇₃	1	1	yes	characteristic
121	C₈	Q₈×D₇₃	1	1	yes	characteristic
122	C₂²	—	73	2	no	—
123	C₂²	—	146	1	no	—
124	C₂²	—	146	1	no	—
125	C₂²	C₂×C₈×D₇₃	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	73	2	no	—
129	C₄	—	146	1	no	—
130	C₄	—	146	1	no	—
131	C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
132	C₄	M₁₆×D₇₃	1	1	yes	characteristic
133	C₂	—	146	2	no	—
134	C₂	C₄⋊C₈×D₇₃	1	1	yes	characteristic
135	C₂	#17989003	1	1	yes	characteristic
136	C₂	#17989005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556530

C₂

yes

maximal, characteristic

C₂×C₁₆×D₇₃

C₂

yes

maximal, characteristic

#17988541

C₂

yes

maximal, characteristic

C₄⋊#11924003

C₂

yes

maximal, characteristic, complemented

#17988405

C₂

yes

maximal, characteristic, complemented

#17988405

C₂

yes

maximal, characteristic, complemented

C₄⋊C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924011

C₄

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556548

C₄

yes

characteristic

#14556530

—

C₂×#11924003

C₂²

yes

characteristic

C₂×#11924003

C₂²

yes

characteristic

C₁₆×D₇₃

—

#14556502

C₄

yes

characteristic

#14556501

C₄

yes

characteristic

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

#11924011

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

—

C₈×D₇₃

Q₈

yes

—

#11924007

C₈

yes

characteristic

#11924007

C₈

yes

characteristic

#11924005

C₈

yes

characteristic

#11924005

C₈

yes

characteristic

C₄×Dic₇₃

C₂×C₄

yes

characteristic

#11924003

—

#11924003

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂²×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

—

C₄×D₇₃

M₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

C₂×Q₈

yes

characteristic

#10596831

—

#10596831

C₄⋊C₄

yes

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₄⋊C₈

yes

—

C₂×C₇₃

C₂×C₄⋊C₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2151

—

maximal, Sylow

C₇₃

#2151

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₇₃

yes

characteristic, complemented

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂²×C₄

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

C₄×D₇₃

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

146

—

107

C₁₆

—

146

—

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₈×D₇₃

yes

characteristic

115

C₂×C₄

C₈×D₇₃

yes

characteristic

116

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

146

—

120

C₈

D₄×D₇₃

yes

characteristic

121

C₈

Q₈×D₇₃

yes

characteristic

122

C₂²

—

123

C₂²

—

146

—

124

C₂²

—

146

—

125

C₂²

C₂×C₈×D₇₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

146

—

130

C₄

—

146

—

131

C₄

C₄⋊C₄×D₇₃

yes

characteristic

132

C₄

M₁₆×D₇₃

yes

characteristic

133

C₂

—

146

—

134

C₂

C₄⋊C₈×D₇₃

yes

characteristic

135

C₂

#17989003

yes

characteristic

136

C₂

#17989005

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₃
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₄⋊C₁₆×C₇₃	C₂
hypercenter	C₄⋊C₁₆	D₇₃
hypocenter	C₇₃	#2151
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₃

commutator, socle

C₂²×C₇₃

C₂²×C₈

Fitting

C₄⋊C₁₆×C₇₃

C₂

hypercenter

C₄⋊C₁₆

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
4. C₄⋊#11924003	C₂	semidirect
5. #17988405	C₂	semidirect
6. #17988405	C₂	semidirect
7. C₄⋊C₁₆×C₇₃	C₂	semidirect
69. C₇₃	#2151	semidirect
76. C₄⋊C₁₆	D₇₃	semidirect

H ≅ G/N

type

4. C₄⋊#11924003

C₂

semidirect

5. #17988405

C₂

semidirect

6. #17988405

C₂

semidirect

7. C₄⋊C₁₆×C₇₃

C₂

semidirect

69. C₇₃

#2151

semidirect

76. C₄⋊C₁₆

D₇₃

semidirect

Subgroups — #22611688

Conjugacy classes of subgroups

Complements