Group #22610058 — subgroups

Subgroups — #22610058

Order 9344.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
2	#17988457	C₂	1	1	yes	maximal, characteristic
3	#17988349	C₂	1	4	yes	maximal
4	#384×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10596831	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₃	—	2	1	no	—
7	C₂×C₄⋊Dic₇₃	—	2	1	no	—
8	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
9	C₄×#10596831	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
11	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
12	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇₃	—	2	2	no	—
15	C₂×#10596831	—	2	2	no	—
16	C₂×#10596831	—	2	2	no	—
17	C₂×#10596831	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₇₃	—	2	4	no	—
19	C₄⋊Dic₇₃	—	4	2	no	—
20	C₄⋊Dic₇₃	—	4	2	no	—
21	C₂²×C₄×C₇₃	—	2	2	no	—
22	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₇₃	—	2	4	no	—
26	C₄⋊C₄×C₇₃	—	4	2	no	—
27	C₄⋊C₄×C₇₃	—	4	2	no	—
28	C₄²×C₇₃	C₂×C₄	1	2	yes	—
29	C₄²×C₇₃	Q₈	1	2	yes	—
30	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₇₃	—	4	2	no	—
32	C₂×C₄×C₇₃	—	4	2	no	—
33	C₂×C₄×C₇₃	—	4	2	no	—
34	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₇₃	D₈	1	2	yes	—
37	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
38	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
39	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
40	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#41	1	2	yes	—
43	C₂×C₄×C₇₃	#41	1	2	yes	—
44	C₂×Dic₇₃	—	4	2	no	—
45	C₂×Dic₇₃	—	4	2	no	—
46	C₂×Dic₇₃	—	4	2	no	—
47	#10596831	—	2	4	no	—
48	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₇₃	#124	1	1	yes	characteristic
50	C₂²×C₇₃	#124	1	1	yes	characteristic
51	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₇₃	—	8	2	no	—
56	C₄×C₇₃	#106	1	4	yes	—
57	C₄×C₇₃	#107	1	4	yes	—
58	C₄×C₇₃	#117	1	2	yes	—
59	C₄×C₇₃	#125	1	2	yes	—
60	Dic₇₃	—	8	2	no	—
61	C₂×C₇₃	#382	1	1	yes	characteristic
62	C₂×C₇₃	C₂×#106	1	1	yes	characteristic
63	C₂×C₇₃	C₂×#107	1	1	yes	characteristic
64	C₂×C₇₃	#486	1	1	yes	characteristic
65	C₂×C₇₃	#486	1	1	yes	characteristic
66	C₂×C₇₃	#487	1	1	yes	characteristic
67	C₂×C₇₃	#487	1	1	yes	characteristic
68	#1834	—	73	1	no	maximal, Sylow
69	C₇₃	#1834	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄×C₈	—	73	1	no	—
71	#384	—	73	1	no	—
72	#384	D₇₃	1	1	yes	characteristic
73	#340	—	73	4	no	—
74	C₂²×C₈	—	73	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	73	2	no	—
78	C₂×C₄⋊C₄	—	146	1	no	—
79	C₂×C₄⋊C₄	—	146	1	no	—
80	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
81	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
82	C₄×C₈	—	73	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	146	2	no	—
85	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
88	C₂×C₈	—	73	4	no	—
89	C₂×C₈	—	146	2	no	—
90	C₂×C₈	—	146	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	292	2	no	—
95	C₄⋊C₄	—	292	2	no	—
96	C₄⋊C₄	—	146	4	no	—
97	C₄²	Dic₁₄₆	1	2	yes	—
98	C₄²	C₄×D₇₃	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	292	2	no	—
104	C₂×C₄	—	292	2	no	—
105	C₂×C₄	—	292	2	no	—
106	C₂×C₄	#11924007	1	2	yes	—
107	C₂×C₄	#11924007	1	2	yes	—
108	C₂×C₄	#11924013	1	2	yes	—
109	C₂×C₄	#11924015	1	2	yes	—
110	C₂×C₄	#11924016	1	2	yes	—
111	C₂×C₄	#11924018	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
114	C₂×C₄	#11924027	1	2	yes	—
115	C₂×C₄	#11924029	1	2	yes	—
116	C₈	—	146	4	no	—
117	C₂²	C₂×#11924007	1	1	yes	characteristic
118	C₂²	#14556578	1	1	yes	characteristic
119	C₂²	#14556580	1	1	yes	characteristic
120	C₂²	C₂×#11924013	1	1	yes	characteristic
121	C₂²	C₂×#11924015	1	1	yes	characteristic
122	C₂²	C₂×#11924016	1	1	yes	characteristic
123	C₂²	C₂×#11924018	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	584	2	no	—
126	C₄	#14556522	1	4	yes	—
127	C₄	#14556524	1	4	yes	—
128	C₄	#14556549	1	2	yes	—
129	C₄	#14556582	1	2	yes	—
130	C₂	#17988452	1	1	yes	characteristic
131	C₂	C₂×#14556522	1	1	yes	characteristic
132	C₂	C₂×#14556524	1	1	yes	characteristic
133	C₂	#17988933	1	1	yes	characteristic
134	C₂	#17988935	1	1	yes	characteristic
135	C₂	#17988937	1	1	yes	characteristic
136	C₂	#17988939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#10596831

C₂

yes

maximal, characteristic

#17988457

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal

#384×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₄×#10596831

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

Q₈

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#106

yes

—

C₄×C₇₃

#107

yes

—

C₄×C₇₃

#117

yes

—

C₄×C₇₃

#125

yes

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

#384

—

#384

D₇₃

yes

characteristic

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

C₂²×C₄

C₂²×D₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄²

Dic₁₄₆

yes

—

C₄²

C₄×D₇₃

yes

—

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

292

—

104

C₂×C₄

—

292

—

105

C₂×C₄

—

292

—

106

C₂×C₄

#11924007

yes

—

107

C₂×C₄

#11924007

yes

—

108

C₂×C₄

#11924013

yes

—

109

C₂×C₄

#11924015

yes

—

110

C₂×C₄

#11924016

yes

—

111

C₂×C₄

#11924018

yes

—

112

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

113

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

114

C₂×C₄

#11924027

yes

—

115

C₂×C₄

#11924029

yes

—

116

C₈

—

146

—

117

C₂²

C₂×#11924007

yes

characteristic

118

C₂²

#14556578

yes

characteristic

119

C₂²

#14556580

yes

characteristic

120

C₂²

C₂×#11924013

yes

characteristic

121

C₂²

C₂×#11924015

yes

characteristic

122

C₂²

C₂×#11924016

yes

characteristic

123

C₂²

C₂×#11924018

yes

characteristic

124

C₄

—

125

C₄

—

584

—

126

C₄

#14556522

yes

—

127

C₄

#14556524

yes

—

128

C₄

#14556549

yes

—

129

C₄

#14556582

yes

—

130

C₂

#17988452

yes

characteristic

131

C₂

C₂×#14556522

yes

characteristic

132

C₂

C₂×#14556524

yes

characteristic

133

C₂

#17988933

yes

characteristic

134

C₂

#17988935

yes

characteristic

135

C₂

#17988937

yes

characteristic

136

C₂

#17988939

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#384×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	#384	D₇₃
hypocenter	C₇₃	#1834
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#384×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#10596831	C₄	semidirect
69. C₇₃	#1834	semidirect

H ≅ G/N

type

9. C₄×#10596831

C₄

semidirect

69. C₇₃

#1834

semidirect

Subgroups — #22610058

Conjugacy classes of subgroups

Complements