Group #22609752 — subgroups

Subgroups — C₄×#14556516 #22609752

Order 9344.

332 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂×#14556516	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988350	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₄₆	—	2	2	no	—
8	C₄×Dic₁₄₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×#11924007	—	2	1	no	—
12	C₂×C₄×Dic₇₃	—	2	1	no	—
13	C₂×C₄×Dic₇₃	—	2	1	no	—
14	#14556516	C₄	1	8	yes	direct product factor
15	#14556498	—	2	1	no	—
16	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₇₃	—	2	2	no	—
19	C₂×Dic₁₄₆	—	2	1	no	—
20	C₂×Dic₁₄₆	—	2	2	no	—
21	C₂×Dic₁₄₆	C₂×C₄	1	4	yes	—
22	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	—	2	2	no	—
25	C₂²×Dic₇₃	—	2	1	no	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
29	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	#11924007	—	2	4	no	—
31	C₄×Dic₇₃	—	2	4	no	—
32	C₄×Dic₇₃	—	4	2	no	—
33	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	C₂×C₄	1	4	yes	—
39	C₄²×C₇₃	C₂×C₄	1	2	yes	—
40	C₄²×C₇₃	D₄	1	2	yes	—
41	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
47	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#41	1	2	yes	—
51	C₂×Dic₇₃	—	2	2	no	—
52	C₂×Dic₇₃	—	2	4	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	2	4	no	—
55	C₂×Dic₇₃	—	4	2	no	—
56	C₂×Dic₇₃	—	4	2	no	—
57	C₂×Dic₇₃	—	4	2	no	—
58	Dic₁₄₆	—	2	2	no	—
59	Dic₁₄₆	—	2	4	no	—
60	Dic₁₄₆	C₄²	1	4	yes	complemented
61	C₈×C₇₃	—	2	4	no	—
62	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₃	C₄×D₄	1	2	yes	—
64	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²×C₇₃	#135	1	2	yes	—
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	#103	1	4	yes	complemented
69	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
70	C₄×C₇₃	#117	1	1	yes	characteristic
71	C₄×C₇₃	C₄×D₄	1	2	yes	—
72	Dic₇₃	—	2	4	no	—
73	Dic₇₃	—	4	2	no	—
74	Dic₇₃	—	4	4	no	—
75	C₂×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
77	C₂×C₇₃	#416	1	1	yes	characteristic
78	C₂×C₇₃	C₄×SD₁₆	1	2	yes	—
79	C₂×C₇₃	C₄×Q₁₆	1	2	yes	—
80	C₄×#103	—	73	1	no	maximal, Sylow
81	C₇₃	C₄×#103	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄×Q₈	—	73	1	no	—
83	C₂×#103	—	73	2	no	—
84	C₂×C₄×C₈	D₇₃	1	1	yes	characteristic
85	C₄×C₄⋊C₄	—	73	1	no	—
86	#340	—	73	2	no	—
87	C₂²×Q₈	—	73	1	no	—
88	C₂²×C₈	C₂×D₇₃	1	2	yes	—
89	C₄×Q₈	—	73	4	no	—
90	C₄×Q₈	—	146	2	no	—
91	C₂×C₄⋊C₄	—	73	1	no	—
92	C₂×C₄⋊C₄	—	73	2	no	—
93	C₂×C₄⋊C₄	—	146	1	no	—
94	C₂×C₄²	—	146	1	no	—
95	C₂×C₄²	—	146	1	no	—
96	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
97	#103	—	73	8	no	—
98	C₄×C₈	—	2	2	no	—
99	#95	—	146	1	no	—
100	C₂×Q₈	—	73	4	no	—
101	C₂×Q₈	—	73	2	no	—
102	C₂×Q₈	—	146	1	no	—
103	C₂×Q₈	—	146	2	no	—
104	C₂²×C₄	—	146	2	no	—
105	C₂²×C₄	—	146	1	no	—
106	C₂²×C₄	—	146	1	no	—
107	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
108	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄×D₇₃	1	4	yes	—
113	C₄⋊C₄	—	73	4	no	—
114	C₄⋊C₄	—	73	2	no	—
115	C₄⋊C₄	—	146	4	no	—
116	C₄⋊C₄	—	146	1	no	—
117	C₄⋊C₄	—	146	2	no	—
118	C₄²	—	292	2	no	—
119	C₄²	—	146	4	no	—
120	C₄²	C₄×D₇₃	1	2	yes	—
121	C₄²	C₂²⋊D₇₃	1	2	yes	—
122	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
123	Q₈	—	73	4	no	—
124	Q₈	—	146	2	no	—
125	Q₈	—	146	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	—	292	2	no	—
129	C₂×C₄	—	292	2	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	—	146	4	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	—	146	4	no	—
134	C₂×C₄	#11924005	1	2	yes	—
135	C₂×C₄	#11924005	1	2	yes	—
136	C₂×C₄	#11924017	1	2	yes	—
137	C₂×C₄	Dic₂₉₂	1	2	yes	—
138	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
141	C₂×C₄	#11924028	1	2	yes	—
142	C₈	—	2	4	no	—
143	C₂²	C₂×#11924005	1	1	yes	characteristic
144	C₂²	C₄×D₂₉₂	1	2	yes	—
145	C₂²	C₂×#11924017	1	1	yes	characteristic
146	C₂²	C₂×Dic₂₉₂	1	1	yes	characteristic
147	C₂²	#14556613	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	292	4	no	—
150	C₄	—	292	2	no	—
151	C₄	—	146	4	no	—
152	C₄	#14556516	1	4	yes	direct product factor
153	C₄	C₄²×D₇₃	1	1	yes	characteristic
154	C₄	#14556548	1	1	yes	characteristic
155	C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
156	C₂	C₄×#11924005	1	1	yes	characteristic
157	C₂	C₂×#14556516	1	1	yes	characteristic
158	C₂	#17988584	1	1	yes	characteristic
159	C₂	C₄×#11924017	1	2	yes	—
160	C₂	C₄×Dic₂₉₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

C₂×#14556516

C₂

yes

maximal

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal

C₂×C₄×C₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

complemented

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

#14556516

C₄

yes

direct product factor

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

C₄²

yes

complemented

C₈×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

—

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×Q₁₆

yes

characteristic

C₂²×C₇₃

#135

yes

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

complemented

C₄×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

#117

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₄×C₂²⋊C₄

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

C₄×SD₁₆

yes

—

C₂×C₇₃

C₄×Q₁₆

yes

—

C₄×#103

—

maximal, Sylow

C₇₃

C₄×#103

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₇₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#103

—

C₄×C₈

—

#95

—

146

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

146

—

103

C₂×Q₈

—

146

—

104

C₂²×C₄

—

146

—

105

C₂²×C₄

—

146

—

106

C₂²×C₄

—

146

—

107

C₂²×C₄

D₂₉₂

yes

characteristic

108

C₂²×C₄

D₂₉₂

yes

characteristic

109

C₂²×C₄

C₂²×D₇₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₄×D₇₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

146

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄²

—

292

—

119

C₄²

—

146

—

120

C₄²

C₄×D₇₃

yes

—

121

C₄²

C₂²⋊D₇₃

yes

—

122

C₂³

C₂×D₂₉₂

yes

characteristic

123

Q₈

—

124

Q₈

—

146

—

125

Q₈

—

146

—

126

C₂×C₄

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

292

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

#11924005

yes

—

135

C₂×C₄

#11924005

yes

—

136

C₂×C₄

#11924017

yes

—

137

C₂×C₄

Dic₂₉₂

yes

—

138

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

139

C₂×C₄

C₂×C₄×D₇₃

yes

—

140

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

141

C₂×C₄

#11924028

yes

—

142

C₈

—

143

C₂²

C₂×#11924005

yes

characteristic

144

C₂²

C₄×D₂₉₂

yes

—

145

C₂²

C₂×#11924017

yes

characteristic

146

C₂²

C₂×Dic₂₉₂

yes

characteristic

147

C₂²

#14556613

yes

—

148

C₄

—

149

C₄

—

292

—

150

C₄

—

292

—

151

C₄

—

146

—

152

C₄

#14556516

yes

direct product factor

153

C₄

C₄²×D₇₃

yes

characteristic

154

C₄

#14556548

yes

characteristic

155

C₄

C₄×C₂²⋊D₇₃

yes

—

156

C₂

C₄×#11924005

yes

characteristic

157

C₂

C₂×#14556516

yes

characteristic

158

C₂

#17988584

yes

characteristic

159

C₂

C₄×#11924017

yes

—

160

C₂

C₄×Dic₂₉₂

yes

—

	subgroup	G/H
center	C₂²×C₄	D₂₉₂
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄×C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₇₃
hypocenter	C₇₃	C₄×#103
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₉₂

commutator

C₄×C₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄×C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₇₃

hypocenter

C₇₃

C₄×#103

LCS

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. C₄×Dic₁₄₆	C₄	semidirect
14. #14556516	C₄	direct (N × H)
60. Dic₁₄₆	C₄²	semidirect
68. C₄×C₇₃	#103	semidirect
81. C₇₃	C₄×#103	semidirect
152. C₄	#14556516	direct (N × H)

H ≅ G/N

type

8. C₄×Dic₁₄₆

C₄

semidirect

14. #14556516

C₄

direct (N × H)

60. Dic₁₄₆

C₄²

semidirect

68. C₄×C₇₃

#103

semidirect

81. C₇₃

C₄×#103

semidirect

152. C₄

#14556516

direct (N × H)

Subgroups — C₄×#14556516 #22609752

Conjugacy classes of subgroups

Complements

Subgroups — C4×#14556516 #22609752

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14556516 #22609752