Group #22608776 — subgroups

Subgroups — #22608776

Order 9344.

166 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11924017	C₂	1	1	yes	maximal, characteristic
3	#17988666	C₂	1	1	yes	maximal, characteristic
4	#17988338	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988319	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988316	C₂	1	1	yes	maximal, characteristic
7	#322×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924017	C₄	1	1	yes	characteristic
9	C₂×#11924011	—	2	1	no	—
10	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
12	#14556523	C₄	1	1	yes	characteristic
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	#14556520	—	2	1	no	—
16	#14556516	C₄	1	1	yes	characteristic
17	#14556513	C₄	1	1	yes	characteristic
18	#14556503	—	2	1	no	—
19	C₈×Dic₇₃	—	2	1	no	—
20	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₇₃	—	2	1	no	—
26	#11924017	—	2	2	no	—
27	#11924011	—	4	2	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	#11924007	—	2	1	no	—
33	#11924005	—	2	1	no	—
34	C₄×Dic₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	1	no	—
42	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
45	C₂×Dic₇₃	—	2	1	no	—
46	C₂×Dic₇₃	—	2	1	no	—
47	C₂×Dic₇₃	—	2	1	no	—
48	D₂₉₂	—	2	1	no	—
49	D₂₉₂	—	2	1	no	—
50	C₄×D₇₃	—	4	1	no	—
51	C₄×D₇₃	—	4	1	no	—
52	Dic₁₄₆	—	2	1	no	—
53	Dic₁₄₆	M₁₆	1	2	yes	complemented
54	C₈×C₇₃	—	2	1	no	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	4	1	no	—
58	C₈×C₇₃	SD₁₆	1	2	yes	complemented
59	#10596831	—	4	1	no	—
60	#10596831	—	4	1	no	—
61	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₇₃	—	4	1	no	—
63	C₂×D₇₃	—	4	1	no	—
64	C₂×D₇₃	—	4	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₇₃	#131	1	1	yes	characteristic
68	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₃	#135	1	1	yes	characteristic
70	Dic₇₃	—	2	2	no	—
71	Dic₇₃	—	4	1	no	—
72	Dic₇₃	—	4	1	no	—
73	C₂×C₇₃	#435	1	1	yes	characteristic
74	C₂×C₇₃	C₄×SD₁₆	1	1	yes	characteristic
75	C₂×C₇₃	#444	1	1	yes	characteristic
76	D₇₃	—	8	1	no	—
77	#1583	—	73	1	no	maximal, Sylow
78	C₇₃	#1583	1	1	yes	characteristic, complemented, Sylow
79	C₈×Q₈	—	73	1	no	—
80	C₄×SD₁₆	—	73	1	no	—
81	#436	—	73	1	no	—
82	#334	—	73	1	no	—
83	Q₈⋊C₈	—	73	1	no	—
84	D₄⋊C₈	—	73	1	no	—
85	#322	D₇₃	1	1	yes	characteristic, complemented
86	C₂×SD₁₆	—	73	1	no	—
87	C₂×M₁₆	—	146	1	no	—
88	C₄×Q₈	—	73	1	no	—
89	C₄×D₄	—	73	1	no	—
90	#106	—	73	1	no	—
91	C₄⋊C₈	—	73	1	no	—
92	C₄⋊C₈	—	73	1	no	—
93	C₄⋊C₈	—	146	1	no	—
94	#103	—	73	1	no	—
95	#102	—	73	1	no	—
96	C₂²⋊C₈	—	146	1	no	—
97	C₄×C₈	—	146	1	no	—
98	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
100	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
101	C₂×Q₈	—	73	1	no	—
102	C₂×D₄	—	73	1	no	—
103	C₂²×C₄	—	146	1	no	—
104	SD₁₆	—	146	2	no	—
105	M₁₆	—	292	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
110	C₂×C₈	C₄×D₇₃	1	1	yes	characteristic
111	C₂×C₈	D₂₉₂	1	1	yes	characteristic
112	C₂×C₈	D₂₉₂	1	1	yes	characteristic
113	C₄⋊C₄	—	73	1	no	—
114	C₄⋊C₄	—	73	1	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₂²⋊C₄	—	146	1	no	—
117	C₄²	—	146	1	no	—
118	C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	C₂³	—	146	1	no	—
120	Q₈	—	73	2	no	—
121	Q₈	—	146	1	no	—
122	D₄	—	146	1	no	—
123	D₄	—	146	1	no	—
124	C₂×C₄	—	292	1	no	—
125	C₂×C₄	—	292	1	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	—	146	1	no	—
128	C₂×C₄	—	146	1	no	—
129	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
131	C₂×C₄	#11924028	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	292	1	no	—
137	C₈	—	292	1	no	—
138	C₈	#11924017	1	2	yes	complemented
139	C₂²	—	292	1	no	—
140	C₂²	—	292	1	no	—
141	C₂²	—	292	1	no	—
142	C₂²	C₄×D₂₉₂	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	292	1	no	—
145	C₄	—	292	1	no	—
146	C₄	—	146	2	no	—
147	C₄	M₁₆×D₇₃	1	1	yes	characteristic
148	C₄	#14556599	1	1	yes	characteristic
149	C₄	C₂×#11924017	1	1	yes	characteristic
150	C₄	#14556613	1	1	yes	characteristic
151	C₂	—	584	1	no	—
152	C₂	#17988660	1	1	yes	characteristic
153	C₂	C₄×#11924017	1	1	yes	characteristic
154	C₂	#17988710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₄₆

C₂

yes

maximal, characteristic, complemented

C₄×#11924017

C₂

yes

maximal, characteristic

#17988666

C₂

yes

maximal, characteristic

#17988338

C₂

yes

maximal, characteristic, complemented

#17988319

C₂

yes

maximal, characteristic, complemented

#17988316

C₂

yes

maximal, characteristic

#322×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924017

C₄

yes

characteristic

C₂×#11924011

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄×D₂₉₂

C₂²

yes

characteristic

#14556523

C₄

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

#14556520

—

#14556516

C₄

yes

characteristic

#14556513

C₄

yes

characteristic

#14556503

—

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

C₂×D₂₉₂

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

—

—

—

—

—

C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

characteristic

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

M₁₆

yes

complemented

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₁₆

yes

complemented

#10596831

—

#10596831

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

#131

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

#135

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#435

yes

characteristic

C₂×C₇₃

C₄×SD₁₆

yes

characteristic

C₂×C₇₃

#444

yes

characteristic

D₇₃

—

#1583

—

maximal, Sylow

C₇₃

#1583

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

D₇₃

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×M₁₆

—

146

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

—

#102

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₄×C₈

C₂×D₇₃

yes

characteristic

100

C₄×C₈

C₂×D₇₃

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

146

—

104

SD₁₆

—

146

—

105

M₁₆

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₂×C₈

C₄×D₇₃

yes

characteristic

110

C₂×C₈

C₄×D₇₃

yes

characteristic

111

C₂×C₈

D₂₉₂

yes

characteristic

112

C₂×C₈

D₂₉₂

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

146

—

116

C₂²⋊C₄

—

146

—

117

C₄²

—

146

—

118

C₄²

C₂²×D₇₃

yes

characteristic

119

C₂³

—

146

—

120

Q₈

—

121

Q₈

—

146

—

122

D₄

—

146

—

123

D₄

—

146

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

—

146

—

129

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

130

C₂×C₄

C₂×D₂₉₂

yes

characteristic

131

C₂×C₄

#11924028

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

292

—

137

C₈

—

292

—

138

C₈

#11924017

yes

complemented

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

292

—

142

C₂²

C₄×D₂₉₂

yes

characteristic

143

C₄

—

144

C₄

—

292

—

145

C₄

—

292

—

146

C₄

—

146

—

147

C₄

M₁₆×D₇₃

yes

characteristic

148

C₄

#14556599

yes

characteristic

149

C₄

C₂×#11924017

yes

characteristic

150

C₄

#14556613

yes

characteristic

151

C₂

—

584

—

152

C₂

#17988660

yes

characteristic

153

C₂

C₄×#11924017

yes

characteristic

154

C₂

#17988710

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#322×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₄
hypercenter	#322	D₇₃
hypocenter	C₇₃	#1583
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#322×C₇₃

C₂

socle

C₂²×C₇₃

C₄×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
1. C₈×Dic₁₄₆	C₂	semidirect
4. #17988338	C₂	semidirect
5. #17988319	C₂	semidirect
7. #322×C₇₃	C₂	semidirect
53. Dic₁₄₆	M₁₆	semidirect
58. C₈×C₇₃	SD₁₆	semidirect
78. C₇₃	#1583	semidirect
85. #322	D₇₃	semidirect
138. C₈	#11924017	semidirect

H ≅ G/N

type

1. C₈×Dic₁₄₆

C₂

semidirect

4. #17988338

C₂

semidirect

5. #17988319

C₂

semidirect

7. #322×C₇₃

C₂

semidirect

53. Dic₁₄₆

M₁₆

semidirect

58. C₈×C₇₃

SD₁₆

semidirect

78. C₇₃

#1583

semidirect

85. #322

D₇₃

semidirect

138. C₈

#11924017

semidirect

Subgroups — #22608776

Conjugacy classes of subgroups

Complements