Group #22608379 — subgroups

Subgroups — #22608379

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×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	#17988644	C₂	1	1	yes	maximal, characteristic
3	#17988319	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	C₄	1	1	yes	characteristic
6	C₄×Dic₁₄₆	—	2	1	no	—
7	C₄×Dic₁₄₆	—	2	1	no	—
8	C₄×Dic₁₄₆	C₄	1	2	yes	—
9	C₄×Dic₁₄₆	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924007	—	2	1	no	—
12	C₂×C₄×Dic₇₃	—	2	1	no	—
13	C₄⋊#10596831	C₂²	1	2	yes	—
14	C₂²⋊#10596831	—	2	1	no	—
15	#14556502	—	2	1	no	—
16	C₂×M₁₆×C₇₃	—	2	1	no	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#97×C₇₃	—	2	1	no	—
19	C₄×C₈×C₇₃	C₂²	1	2	yes	—
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×Dic₁₄₆	—	2	4	no	—
22	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	—	2	1	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
28	#11924007	—	2	2	no	—
29	#11924007	—	2	2	no	—
30	C₄×Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	M₁₆×C₇₃	—	2	4	no	—
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	4	no	—
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	4	2	no	—
55	Dic₁₄₆	—	2	4	no	—
56	Dic₁₄₆	—	2	2	no	—
57	Dic₁₄₆	M₁₆	1	4	yes	complemented
58	C₈×C₇₃	—	2	4	no	—
59	#10596831	—	4	2	no	—
60	C₂²×C₇₃	—	2	1	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	#104	1	2	yes	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	C₂×M₁₆	1	2	yes	—
69	C₄×C₇₃	#137	1	2	yes	—
70	Dic₇₃	—	2	4	no	—
71	Dic₇₃	—	4	2	no	—
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	#407	1	1	yes	characteristic
74	C₂×C₇₃	#419	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
76	#1491	—	73	1	no	maximal, Sylow
77	C₇₃	#1491	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄×Q₈	—	73	1	no	—
79	#432	—	73	1	no	—
80	C₄×M₁₆	D₇₃	1	1	yes	characteristic
81	Q₈⋊C₈	—	73	4	no	—
82	C₂²×Q₈	—	73	1	no	—
83	C₂×M₁₆	—	2	1	no	—
84	C₄×Q₈	—	73	2	no	—
85	C₄×Q₈	—	73	2	no	—
86	C₄×Q₈	—	146	1	no	—
87	C₄×Q₈	—	146	1	no	—
88	C₂×C₄⋊C₄	—	73	1	no	—
89	C₂×C₄⋊C₄	—	146	1	no	—
90	C₂×C₄²	—	146	1	no	—
91	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
92	C₄⋊C₈	—	73	2	no	—
93	C₂²⋊C₈	—	146	1	no	—
94	#97	—	2	1	no	—
95	#97	—	146	1	no	—
96	C₄×C₈	C₂×D₇₃	1	2	yes	—
97	C₂×Q₈	—	73	2	no	—
98	C₂×Q₈	—	146	4	no	—
99	C₂×Q₈	—	146	1	no	—
100	C₂²×C₄	—	146	1	no	—
101	C₂²×C₄	—	146	1	no	—
102	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
103	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
105	M₁₆	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	146	2	no	—
108	C₄⋊C₄	—	73	2	no	—
109	C₄⋊C₄	—	146	2	no	—
110	C₄⋊C₄	—	146	1	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄²	—	146	2	no	—
113	C₄²	—	146	2	no	—
114	C₄²	C₄×D₇₃	1	1	yes	characteristic
115	C₄²	D₂₉₂	1	1	yes	characteristic
116	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
117	C₄²	C₂²×D₇₃	1	1	yes	characteristic
118	C₂³	#11924005	1	1	yes	characteristic
119	Q₈	—	73	4	no	—
120	Q₈	—	146	4	no	—
121	Q₈	—	146	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	146	2	no	—
127	C₂×C₄	—	146	4	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	#11924005	1	1	yes	characteristic
130	C₂×C₄	#11924005	1	1	yes	characteristic
131	C₂×C₄	#11924005	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
134	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	292	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#14556519	1	2	yes	—
140	C₂²	C₂×#11924005	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	292	2	no	—
146	C₄	—	146	4	no	—
147	C₄	M₁₆×D₇₃	1	2	yes	—
148	C₄	#14556624	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#17988550	1	1	yes	characteristic
151	C₂	#17988597	1	1	yes	characteristic
152	C₂	C₂×#14556519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

#17988644

C₂

yes

maximal, characteristic

#17988319

C₂

yes

maximal, complemented

C₄×M₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₄

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

—

C₄×Dic₁₄₆

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

C₄⋊#10596831

C₂²

yes

—

C₂²⋊#10596831

—

#14556502

—

C₂×M₁₆×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#97×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

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

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

M₁₆

yes

complemented

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#104

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

—

C₄×C₇₃

#137

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#432

—

C₄×M₁₆

D₇₃

yes

characteristic

Q₈⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

146

—

#97

—

#97

—

146

—

C₄×C₈

C₂×D₇₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

100

C₂²×C₄

—

146

—

101

C₂²×C₄

—

146

—

102

C₂²×C₄

C₄×D₇₃

yes

characteristic

103

C₂²×C₄

D₂₉₂

yes

characteristic

104

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

146

—

110

C₄⋊C₄

—

146

—

111

C₄⋊C₄

—

146

—

112

C₄²

—

146

—

113

C₄²

—

146

—

114

C₄²

C₄×D₇₃

yes

characteristic

115

C₄²

D₂₉₂

yes

characteristic

116

C₄²

C₂²⋊D₇₃

yes

characteristic

117

C₄²

C₂²×D₇₃

yes

characteristic

118

C₂³

#11924005

yes

characteristic

119

Q₈

—

120

Q₈

—

146

—

121

Q₈

—

146

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

—

146

—

129

C₂×C₄

#11924005

yes

characteristic

130

C₂×C₄

#11924005

yes

characteristic

131

C₂×C₄

#11924005

yes

characteristic

132

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

134

C₂×C₄

C₂×D₂₉₂

yes

characteristic

135

C₈

—

136

C₈

—

292

—

137

C₂²

—

138

C₂²

—

139

C₂²

#14556519

yes

—

140

C₂²

C₂×#11924005

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

292

—

146

C₄

—

146

—

147

C₄

M₁₆×D₇₃

yes

—

148

C₄

#14556624

yes

—

149

C₂

—

150

C₂

#17988550

yes

characteristic

151

C₂

#17988597

yes

characteristic

152

C₂

C₂×#14556519

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	C₄×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₇₃
hypocenter	C₇₃	#1491
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

C₄×M₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₇₃

hypocenter

C₇₃

#1491

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #17988319	C₂	semidirect
57. Dic₁₄₆	M₁₆	semidirect
77. C₇₃	#1491	semidirect

H ≅ G/N

type

3. #17988319

C₂

semidirect

57. Dic₁₄₆

M₁₆

semidirect

77. C₇₃

#1491

semidirect

Subgroups — #22608379

Conjugacy classes of subgroups

Complements