Group #22608576 — subgroups

Subgroups — C₂×#17988334 #22608576

Order 9344.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	#17988334	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	—	2	1	no	—
6	C₂²×#10596831	—	2	1	no	—
7	#14556591	—	2	1	no	—
8	#14556591	—	2	1	no	—
9	#14556591	C₄	1	2	yes	—
10	#14556591	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₃	—	2	1	no	—
12	C₂×C₄⋊Dic₇₃	C₄	1	2	yes	—
13	C₄⋊#10596831	—	2	1	no	—
14	C₄⋊#10596831	C₂²	1	2	yes	—
15	C₂²×C₈×C₇₃	—	2	1	no	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₃	—	2	1	no	—
18	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
19	C₂×Dic₁₄₆	—	2	2	no	—
20	C₂×Dic₁₄₆	—	2	2	no	—
21	C₂×Dic₁₄₆	—	4	2	no	—
22	C₂×Dic₁₄₆	—	4	2	no	—
23	C₂²×Dic₇₃	—	2	2	no	—
24	C₂×#10596831	—	2	2	no	—
25	C₂×#10596831	—	4	2	no	—
26	C₄⋊Dic₇₃	—	2	2	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	4	2	no	—
35	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₇₃	D₄	1	1	yes	characteristic
37	C₄²×C₇₃	D₄	1	1	yes	characteristic
38	C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	D₈	1	2	yes	—
45	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
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	—	4	4	no	—
53	Dic₁₄₆	—	4	4	no	—
54	Dic₁₄₆	—	4	2	no	—
55	Dic₁₄₆	—	4	2	no	—
56	C₈×C₇₃	—	4	2	no	—
57	#10596831	—	4	2	no	—
58	C₂²×C₇₃	#101	1	2	yes	—
59	C₂²×C₇₃	#102	1	2	yes	—
60	C₂²×C₇₃	#103	1	2	yes	—
61	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	#102	1	2	yes	—
65	C₄×C₇₃	#103	1	2	yes	—
66	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
67	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
70	Dic₇₃	—	4	4	no	—
71	C₂×C₇₃	#332	1	4	yes	complemented
72	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
73	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
74	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
75	C₂×#332	—	73	1	no	maximal, Sylow
76	C₇₃	C₂×#332	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄⋊Q₈	—	73	1	no	—
78	C₂×C₄⋊C₈	—	73	1	no	—
79	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
80	#332	—	73	4	no	—
81	C₂²×Q₈	—	146	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	146	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	2	no	—
89	C₂×C₄⋊C₄	—	146	1	no	—
90	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	73	2	no	—
93	C₄⋊C₈	—	146	1	no	—
94	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
95	C₂×Q₈	—	292	2	no	—
96	C₂×Q₈	—	292	2	no	—
97	C₂×Q₈	—	146	2	no	—
98	C₂×Q₈	—	146	2	no	—
99	C₂²×C₄	—	146	2	no	—
100	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
101	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	292	2	no	—
106	C₂×C₈	—	146	2	no	—
107	C₄⋊C₄	—	73	4	no	—
108	C₄⋊C₄	—	146	2	no	—
109	C₄⋊C₄	—	146	2	no	—
110	C₄⋊C₄	—	146	2	no	—
111	C₄²	C₄×D₇₃	1	1	yes	characteristic
112	C₄²	D₂₉₂	1	1	yes	characteristic
113	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
114	C₄²	C₂²×D₇₃	1	1	yes	characteristic
115	C₂³	#11924005	1	1	yes	characteristic
116	Q₈	—	292	4	no	—
117	Q₈	—	292	2	no	—
118	Q₈	—	292	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	292	4	no	—
121	C₂×C₄	—	146	4	no	—
122	C₂×C₄	#11924005	1	1	yes	characteristic
123	C₂×C₄	#11924005	1	1	yes	characteristic
124	C₂×C₄	#11924005	1	1	yes	characteristic
125	C₂×C₄	#11924013	1	2	yes	—
126	C₂×C₄	#11924015	1	2	yes	—
127	C₂×C₄	#11924017	1	2	yes	—
128	C₂×C₄	Dic₂₉₂	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	292	2	no	—
134	C₂²	#14556509	1	2	yes	—
135	C₂²	#14556511	1	2	yes	—
136	C₂²	#14556516	1	2	yes	—
137	C₂²	C₂×#11924005	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	292	4	no	—
141	C₄	#14556511	1	2	yes	—
142	C₄	#14556516	1	2	yes	—
143	C₄	C₂×#11924013	1	1	yes	characteristic
144	C₄	C₂×#11924015	1	1	yes	characteristic
145	C₄	C₂×#11924017	1	1	yes	characteristic
146	C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
147	C₂	#17988334	1	4	yes	direct product factor
148	C₂	C₂×#14556509	1	1	yes	characteristic
149	C₂	C₂×#14556511	1	1	yes	characteristic
150	C₂	C₂×#14556516	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596831

C₂

yes

maximal, characteristic

#17988334

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

—

—

—

—

C₄

yes

—

#14556591

C₂²

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₄

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

C₂²

yes

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#101

yes

—

C₂²×C₇₃

#102

yes

—

C₂²×C₇₃

#103

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

—

C₄×C₇₃

#103

yes

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

characteristic

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₇₃

yes

characteristic

#332

—

C₂²×Q₈

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

—

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂²×C₄

—

146

—

100

C₂²×C₄

C₄×D₇₃

yes

characteristic

101

C₂²×C₄

D₂₉₂

yes

characteristic

102

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

146

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

146

—

109

C₄⋊C₄

—

146

—

110

C₄⋊C₄

—

146

—

111

C₄²

C₄×D₇₃

yes

characteristic

112

C₄²

D₂₉₂

yes

characteristic

113

C₄²

C₂²⋊D₇₃

yes

characteristic

114

C₄²

C₂²×D₇₃

yes

characteristic

115

C₂³

#11924005

yes

characteristic

116

Q₈

—

292

—

117

Q₈

—

292

—

118

Q₈

—

292

—

119

C₂×C₄

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

146

—

122

C₂×C₄

#11924005

yes

characteristic

123

C₂×C₄

#11924005

yes

characteristic

124

C₂×C₄

#11924005

yes

characteristic

125

C₂×C₄

#11924013

yes

—

126

C₂×C₄

#11924015

yes

—

127

C₂×C₄

#11924017

yes

—

128

C₂×C₄

Dic₂₉₂

yes

—

129

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

131

C₂×C₄

C₂×D₂₉₂

yes

characteristic

132

C₈

—

133

C₈

—

292

—

134

C₂²

#14556509

yes

—

135

C₂²

#14556511

yes

—

136

C₂²

#14556516

yes

—

137

C₂²

C₂×#11924005

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

292

—

141

C₄

#14556511

yes

—

142

C₄

#14556516

yes

—

143

C₄

C₂×#11924013

yes

characteristic

144

C₄

C₂×#11924015

yes

characteristic

145

C₄

C₂×#11924017

yes

characteristic

146

C₄

C₂×Dic₂₉₂

yes

characteristic

147

C₂

#17988334

yes

direct product factor

148

C₂

C₂×#14556509

yes

characteristic

149

C₂

C₂×#14556511

yes

characteristic

150

C₂

C₂×#14556516

yes

characteristic

	subgroup	G/H
center	C₂³	#11924005
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	C₂×#332
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

#11924005

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

C₂×#332

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #17988334	C₂	direct (N × H)
71. C₂×C₇₃	#332	semidirect
76. C₇₃	C₂×#332	semidirect
147. C₂	#17988334	direct (N × H)

H ≅ G/N

type

3. #17988334

C₂

direct (N × H)

71. C₂×C₇₃

#332

semidirect

76. C₇₃

C₂×#332

semidirect

147. C₂

#17988334

direct (N × H)

Subgroups — C₂×#17988334 #22608576

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17988334 #22608576

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17988334 #22608576