Group #34000447 — subgroups

Subgroups — #34000447

Order 18688.

180 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 44 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	#22626303	C₂	1	1	yes	maximal, characteristic
2	#22625861	C₂	1	1	yes	maximal, characteristic
3	#22608225	C₂	1	1	yes	maximal, characteristic
4	#17989510	—	2	1	no	—
5	#17989484	—	2	1	no	—
6	#17989480	C₄	1	2	yes	complemented
7	#17989469	—	2	1	no	—
8	#17989458	—	2	1	no	—
9	#17989456	C₄	1	1	yes	characteristic
10	#17989455	C₄	1	1	yes	characteristic
11	#17988638	—	2	1	no	—
12	C₄×#11924011	C₂²	1	1	yes	characteristic
13	#17988309	—	2	1	no	—
14	#17988308	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
16	D₇₃⋊M₁₆	—	2	1	no	—
17	#14556670	—	2	1	no	—
18	#14556665	—	2	2	no	—
19	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924011	D₄	1	1	yes	characteristic
21	C₄²×D₇₃	D₄	1	1	yes	characteristic
22	#14556520	—	2	2	no	—
23	#14556503	—	2	2	no	—
24	#14556501	C₂×C₄	1	1	yes	characteristic
25	#14556501	Q₈	1	1	yes	characteristic
26	C₄×#10596831	—	2	1	no	—
27	C₄×#10596831	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	—	2	1	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#11924038	—	4	1	no	—
31	#11924037	—	4	1	no	—
32	#11924036	—	4	1	no	—
33	C₄⋊#64408	—	4	1	no	—
34	C₂²⋊#64408	—	4	1	no	—
35	C₄×#64408	—	4	1	no	—
36	#11924031	—	4	2	no	—
37	C₂×C₄×D₇₃	—	2	1	no	—
38	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	#11924011	—	4	1	no	—
40	#11924011	—	4	1	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	—	2	1	no	—
45	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×D₇₃	#104	1	1	yes	characteristic
51	C₂×#64408	—	8	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	1	no	—
55	C₂×Dic₇₃	#104	1	1	yes	characteristic
56	C₄×D₇₃	—	2	1	no	—
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	4	2	no	—
59	#10596833	—	8	1	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	—	4	2	no	—
63	#10596831	—	2	1	no	—
64	#10596831	—	2	1	no	—
65	#10596831	—	4	2	no	—
66	C₂²×C₇₃	#339	1	1	yes	characteristic
67	C₂×D₇₃	—	2	1	no	—
68	C₂×D₇₃	—	4	1	no	—
69	#64408	—	16	1	no	—
70	C₄×C₇₃	—	2	2	no	—
71	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₇₃	#344	1	1	yes	characteristic
73	Dic₇₃	—	2	1	no	—
74	Dic₇₃	—	4	1	no	—
75	#7226	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	2	1	no	—
77	C₂×C₇₃	#1288	1	1	yes	characteristic
78	D₇₃	—	4	1	no	—
79	#2165	—	73	1	no	—
80	#1841	—	73	1	no	—
81	#1453	—	73	1	no	—
82	C₇₃	#7226	1	1	yes	characteristic, complemented, Sylow
83	#431	—	146	1	no	—
84	C₄×M₁₆	—	73	1	no	—
85	#364	—	146	1	no	—
86	#349	—	146	1	no	—
87	#347	—	73	2	no	—
88	#344	—	146	1	no	—
89	#339	—	146	1	no	—
90	#338	—	73	1	no	—
91	H(3,ℤ/4)	—	73	1	no	—
92	#322	—	73	1	no	—
93	#322	—	146	1	no	—
94	C₈²	#64408	1	1	yes	characteristic, complemented
95	C₂×M₁₆	—	73	1	no	—
96	C₂×M₁₆	—	73	1	no	—
97	C₂×M₁₆	—	146	1	no	—
98	#117	—	146	1	no	—
99	C₂×C₄²	—	73	1	no	—
100	M₃₂	—	146	2	no	—
101	C₄⋊C₈	—	146	2	no	—
102	C₂²⋊C₈	—	146	2	no	—
103	#97	—	73	1	no	—
104	#97	—	73	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	146	1	no	—
108	C₄×C₈	C₂×#64408	1	1	yes	characteristic
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	146	1	no	—
111	M₁₆	—	292	1	no	—
112	M₁₆	—	292	1	no	—
113	M₁₆	—	292	1	no	—
114	M₁₆	—	292	1	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	292	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	C₄×#64408	1	1	yes	characteristic
121	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
122	C₄⋊C₄	—	292	1	no	—
123	C₂²⋊C₄	—	292	1	no	—
124	C₄²	—	292	1	no	—
125	C₄²	—	73	1	no	—
126	C₄²	—	146	1	no	—
127	C₄²	C₂²⋊#64408	1	1	yes	characteristic
128	C₁₆	—	292	2	no	—
129	C₂³	—	73	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	292	2	no	—
132	C₂×C₄	—	584	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	146	1	no	—
135	C₂×C₄	—	146	1	no	—
136	C₂×C₄	—	146	1	no	—
137	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	292	2	no	—
142	C₈	—	584	1	no	—
143	C₈	—	146	1	no	—
144	C₈	—	146	1	no	—
145	C₂²	—	292	1	no	—
146	C₂²	—	146	1	no	—
147	C₂²	#17989458	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	292	1	no	—
150	C₄	—	1168	1	no	—
151	C₄	—	146	1	no	—
152	C₄	#17989455	1	1	yes	characteristic
153	C₄	#17989469	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	292	1	no	—
156	C₂	#22624791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626303

C₂

yes

maximal, characteristic

#22625861

C₂

yes

maximal, characteristic

#22608225

C₂

yes

maximal, characteristic

#17989510

—

#17989484

—

#17989480

C₄

yes

—

—

C₄

yes

characteristic

#17989455

C₄

yes

characteristic

#17988638

—

C₄×#11924011

C₂²

yes

characteristic

#17988309

—

#17988308

C₄

yes

characteristic, complemented

C₈²×C₇₃

C₄

yes

characteristic, complemented

D₇₃⋊M₁₆

—

#14556670

—

#14556665

—

C₂×#11924011

C₂×C₄

yes

characteristic

C₂×#11924011

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

C₂×C₄

yes

characteristic

#14556501

Q₈

yes

characteristic

C₄×#10596831

—

C₄×#10596831

D₄

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

C₂²×D₇₃

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#104

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

#339

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7226

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1288

yes

characteristic

D₇₃

—

#2165

—

#1841

—

#1453

—

C₇₃

#7226

yes

characteristic, complemented, Sylow

#431

—

146

—

C₄×M₁₆

—

#364

—

146

—

#349

—

146

—

#347

—

#344

—

146

—

#339

—

146

—

—

—

—

—

146

—

C₈²

#64408

yes

characteristic, complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

#117

—

146

—

C₂×C₄²

—

100

M₃₂

—

146

—

101

C₄⋊C₈

—

146

—

102

C₂²⋊C₈

—

146

—

103

#97

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

146

—

108

C₄×C₈

C₂×#64408

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

146

—

111

M₁₆

—

292

—

112

M₁₆

—

292

—

113

M₁₆

—

292

—

114

M₁₆

—

292

—

115

C₂×C₈

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

146

—

120

C₂×C₈

C₄×#64408

yes

characteristic

121

C₂×C₈

C₄⋊#64408

yes

characteristic

122

C₄⋊C₄

—

292

—

123

C₂²⋊C₄

—

292

—

124

C₄²

—

292

—

125

C₄²

—

126

C₄²

—

146

—

127

C₄²

C₂²⋊#64408

yes

characteristic

128

C₁₆

—

292

—

129

C₂³

—

130

C₂×C₄

—

131

C₂×C₄

—

292

—

132

C₂×C₄

—

584

—

133

C₂×C₄

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

292

—

142

C₈

—

584

—

143

C₈

—

146

—

144

C₈

—

146

—

145

C₂²

—

292

—

146

C₂²

—

146

—

147

C₂²

#17989458

yes

characteristic

148

C₄

—

149

C₄

—

292

—

150

C₄

—

1168

—

151

C₄

—

146

—

152

C₄

#17989455

yes

characteristic

153

C₄

#17989469

yes

characteristic

154

C₂

—

155

C₂

—

292

—

156

C₂

#22624791

yes

characteristic

	subgroup	G/H
center	C₄	#17989455
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂×C₇₃	#1288
prime radical	C₂²×D₇₃	#104
cosocle	C₄×#11924011	C₂²
hypercenter	C₈²	#64408
hypocenter	C₇₃	#7226
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₈²

subgroup

G/H

center

C₄

#17989455

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

C₈²×C₇₃

C₄

C₂×C₇₃

C₂²×D₇₃

C₂²

C₈²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₈²

N	H ≅ G/N	type
6. #17989480	C₄	semidirect
14. #17988308	C₄	semidirect
15. C₈²×C₇₃	C₄	semidirect
82. C₇₃	#7226	semidirect
94. C₈²	#64408	semidirect

H ≅ G/N

type

6. #17989480

C₄

semidirect

14. #17988308

C₄

semidirect

15. C₈²×C₇₃

C₄

semidirect

82. C₇₃

#7226

semidirect

94. C₈²

#64408

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22626303	2
#22625861	2
#22608225	2
#7226	73

subgroup

index

Subgroups — #34000447

Conjugacy classes of subgroups

Complements

Maximal subgroups