Group #22626220 — subgroups

Subgroups — D₇₃⋊#356 #22626220

Order 9344.

240 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 54 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	D₇₃⋊#101	C₂	1	2	yes	maximal
2	#17989498	C₂	1	2	yes	maximal, complemented
3	#17989497	C₂	1	2	yes	maximal, complemented
4	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	D₇₃⋊M₁₆	—	2	2	no	—
6	#14556652	—	2	2	no	—
7	#14556652	C₂²	1	4	yes	—
8	C₂×Q₈×D₇₃	C₄	1	1	yes	characteristic
9	C₂×Q₈×D₇₃	C₂²	1	1	yes	characteristic
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556590	—	2	2	no	—
12	#14556589	C₄	1	1	yes	characteristic
13	#14556589	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×D₇₃	—	2	2	no	—
15	C₄²×D₇₃	C₄	1	1	yes	characteristic
16	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
17	#11924038	—	4	2	no	—
18	#11924037	—	2	4	no	—
19	#11924036	—	4	2	no	—
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₄₆	D₄	1	2	yes	—
23	C₂×Dic₁₄₆	C₂³	1	1	yes	characteristic
24	Q₈×D₇₃	—	4	1	no	—
25	Q₈×D₇₃	—	4	1	no	—
26	Q₈×D₇₃	—	4	1	no	—
27	Q₈×D₇₃	—	4	1	no	—
28	C₂×C₄×D₇₃	—	2	1	no	—
29	C₂×C₄×D₇₃	—	2	1	no	—
30	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	D₄	1	2	yes	—
32	C₄⋊Dic₇₃	—	2	2	no	—
33	#11924007	—	2	2	no	—
34	#11924007	—	2	2	no	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₇₃	—	4	1	no	—
43	Q₈×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	1	no	—
49	C₂×Dic₇₃	—	2	1	no	—
50	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
52	C₄×D₇₃	—	2	4	no	—
53	C₄×D₇₃	—	2	1	no	—
54	C₄×D₇₃	—	2	1	no	—
55	C₄×D₇₃	—	2	2	no	—
56	C₄×D₇₃	—	2	2	no	—
57	C₄×D₇₃	—	4	1	no	—
58	C₄×D₇₃	—	4	1	no	—
59	Dic₁₄₆	—	4	2	no	—
60	Dic₁₄₆	—	4	1	no	—
61	Dic₁₄₆	—	4	2	no	—
62	Dic₁₄₆	—	4	1	no	—
63	#10596833	—	4	4	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	#99	1	1	yes	characteristic
67	C₂×D₇₃	#99	1	1	yes	characteristic
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	—	4	1	no	—
72	Dic₇₃	—	2	2	no	—
73	Dic₇₃	—	2	2	no	—
74	Dic₇₃	—	2	1	no	—
75	Dic₇₃	—	4	1	no	—
76	C₂×C₇₃	—	2	1	no	—
77	C₂×C₇₃	C₂×#99	1	1	yes	characteristic
78	D₇₃	—	2	1	no	—
79	D₇₃	#356	1	2	yes	complemented
80	C₂×#356	—	73	1	no	maximal, Sylow
81	C₇₃	C₂×#356	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	73	1	no	—
83	C₂×#101	—	73	2	no	—
84	#356	—	73	2	no	—
85	#356	—	73	2	no	—
86	C₂²×Q₈	—	73	1	no	—
87	C₂²×Q₈	—	73	1	no	—
88	C₂×M₁₆	—	146	2	no	—
89	C₄⋊Q₈	—	73	1	no	—
90	C₄⋊Q₈	—	73	1	no	—
91	C₄⋊Q₈	—	73	1	no	—
92	C₄⋊Q₈	—	146	2	no	—
93	C₄⋊Q₈	#64408	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	146	2	no	—
95	C₂×C₄²	—	73	1	no	—
96	#101	—	73	4	no	—
97	#101	—	146	2	no	—
98	C₂×Q₈	—	292	1	no	—
99	C₂×Q₈	—	292	1	no	—
100	C₂×Q₈	—	292	1	no	—
101	C₂×Q₈	—	292	1	no	—
102	C₂×Q₈	—	73	2	no	—
103	C₂×Q₈	—	73	1	no	—
104	C₂×Q₈	—	73	1	no	—
105	C₂×Q₈	—	146	1	no	—
106	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
107	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
108	C₂²×C₄	—	73	2	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	146	1	no	—
111	C₂²×C₄	—	146	1	no	—
112	M₁₆	—	292	2	no	—
113	M₁₆	—	146	4	no	—
114	C₂×C₈	—	292	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	146	2	no	—
117	C₄⋊C₄	—	146	2	no	—
118	C₄⋊C₄	—	146	2	no	—
119	C₄²	—	73	1	no	—
120	C₄²	—	146	1	no	—
121	C₄²	C₂×#64408	1	1	yes	characteristic
122	C₂³	—	73	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	292	1	no	—
126	Q₈	—	292	2	no	—
127	Q₈	—	292	1	no	—
128	Q₈	—	292	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	292	1	no	—
132	C₂×C₄	—	292	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	2	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	1	no	—
138	C₂×C₄	—	146	1	no	—
139	C₂×C₄	—	146	4	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	C₂²⋊#64408	1	2	yes	—
143	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
144	C₈	—	292	4	no	—
145	C₂²	—	73	1	no	—
146	C₂²	—	73	1	no	—
147	C₂²	—	146	2	no	—
148	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	1	no	—
153	C₄	—	292	1	no	—
154	C₄	—	146	2	no	—
155	C₄	—	146	2	no	—
156	C₄	—	146	1	no	—
157	C₂	—	2	1	no	—
158	C₂	—	73	2	no	—
159	C₂	—	146	1	no	—
160	C₂	D₇₃⋊#99	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇₃⋊#101

C₂

yes

maximal

#17989498

C₂

yes

maximal, complemented

#17989497

C₂

yes

maximal, complemented

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

D₇₃⋊M₁₆

—

#14556652

—

#14556652

C₂²

yes

—

C₂×Q₈×D₇₃

C₄

yes

characteristic

C₂×Q₈×D₇₃

C₂²

yes

characteristic

#14556591

C₂²

yes

characteristic

#14556590

—

#14556589

C₄

yes

characteristic

#14556589

C₂²

yes

characteristic

C₄⋊C₄×D₇₃

—

C₄²×D₇₃

C₄

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

—

C₂×Dic₁₄₆

C₂³

yes

characteristic

Q₈×D₇₃

—

Q₈×D₇₃

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#99

yes

characteristic

D₇₃

—

D₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#101

—

#356

—

#356

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

#64408

yes

characteristic

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

#101

—

#101

—

146

—

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

100

C₂×Q₈

—

292

—

101

C₂×Q₈

—

292

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

146

—

106

C₂×Q₈

C₂×#64408

yes

characteristic

107

C₂×Q₈

C₂×#64408

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

—

146

—

112

M₁₆

—

292

—

113

M₁₆

—

146

—

114

C₂×C₈

—

292

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

—

146

—

119

C₄²

—

120

C₄²

—

146

—

121

C₄²

C₂×#64408

yes

characteristic

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

292

—

126

Q₈

—

292

—

127

Q₈

—

292

—

128

Q₈

—

292

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

292

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

C₂²⋊#64408

yes

—

143

C₂×C₄

C₂²×#64408

yes

characteristic

144

C₈

—

292

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

146

—

148

C₂²

C₂×C₂²⋊#64408

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

—

146

—

155

C₄

—

146

—

156

C₄

—

146

—

157

C₂

—

158

C₂

—

159

C₂

—

146

—

160

C₂

D₇₃⋊#99

yes

characteristic

	subgroup	G/H
center	C₂	D₇₃⋊#99
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂×C₇₃	C₂×#99
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	C₂×#356
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

subgroup

G/H

center

C₂

D₇₃⋊#99

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄⋊Q₈×C₇₃

C₄

socle

C₂×C₇₃

C₂×#99

C₄⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #17989498	C₂	semidirect
3. #17989497	C₂	semidirect
79. D₇₃	#356	semidirect
81. C₇₃	C₂×#356	semidirect

H ≅ G/N

type

2. #17989498

C₂

semidirect

3. #17989497

C₂

semidirect

79. D₇₃

#356

semidirect

81. C₇₃

C₂×#356

semidirect

Subgroups — D₇₃⋊#356 #22626220

Conjugacy classes of subgroups

Complements

Subgroups — D73⋊#356 #22626220

Conjugacy classes of subgroups

Complements

Subgroups — D₇₃⋊#356 #22626220