Group #34000441 — subgroups

Subgroups — #34000441

Order 18688.

192 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22626301	C₂	1	1	yes	maximal, characteristic
2	#22625154	C₂	1	1	yes	maximal, characteristic
3	#22610078	C₂	1	1	yes	maximal, characteristic
4	C₄×#11924037	—	2	1	no	—
5	#17989509	—	2	1	no	—
6	#17989483	—	2	1	no	—
7	#17989480	C₄	1	2	yes	complemented
8	#17989429	—	2	1	no	—
9	#17989428	C₄	1	1	yes	characteristic, complemented
10	C₈×#10596833	C₄	1	1	yes	characteristic, complemented
11	M₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
12	#17988357	—	2	1	no	—
13	#17988346	—	2	1	no	—
14	#17988345	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924037	—	2	2	no	—
17	#14556665	—	2	2	no	—
18	#14556646	—	2	2	no	—
19	C₄×#10596833	—	2	1	no	—
20	C₄×#10596833	—	2	1	no	—
21	C₂×#11924012	—	2	1	no	—
22	C₂×#11924012	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
24	#14556501	C₂×C₄	1	1	yes	characteristic
25	#14556501	Q₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#117×C₇₃	—	2	1	no	—
30	#11924037	—	4	4	no	—
31	C₂×#10596833	—	2	2	no	—
32	C₂×#10596833	—	2	2	no	—
33	#11924031	—	4	2	no	—
34	C₂²×Dic₇₃	—	2	1	no	—
35	#11924012	—	4	1	no	—
36	#11924012	—	4	1	no	—
37	#11924012	—	4	1	no	—
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₇₃	—	2	1	no	—
42	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	M₁₆×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₇₃	—	4	1	no	—
49	C₂²⋊C₄×C₇₃	—	4	1	no	—
50	C₄²×C₇₃	—	4	1	no	—
51	C₂³×C₇₃	#104	1	1	yes	characteristic
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	8	1	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	4	2	no	—
60	#10596833	—	2	4	no	—
61	#10596833	—	4	2	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	—	2	1	no	—
64	#10596831	—	2	1	no	—
65	#10596831	—	2	1	no	—
66	#10596831	—	8	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	#339	1	1	yes	characteristic
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	16	1	no	—
72	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₇₃	#344	1	1	yes	characteristic
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	—	4	1	no	—
76	#7225	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	—	4	1	no	—
79	C₂×C₇₃	#1288	1	1	yes	characteristic
80	#2165	—	73	1	no	—
81	#1841	—	73	1	no	—
82	#1455	—	73	1	no	—
83	C₇₃	#7225	1	1	yes	characteristic, complemented, Sylow
84	C₄×M₁₆	—	73	1	no	—
85	C₄×M₁₆	—	146	1	no	—
86	#364	—	146	1	no	—
87	#349	—	146	1	no	—
88	#347	—	73	2	no	—
89	#344	—	146	1	no	—
90	#339	—	146	1	no	—
91	#338	—	73	1	no	—
92	H(3,ℤ/4)	#64408	1	1	yes	characteristic
93	#322	—	73	1	no	—
94	#322	—	146	1	no	—
95	C₈²	—	73	1	no	—
96	C₂×M₁₆	—	73	1	no	—
97	C₂×M₁₆	—	146	1	no	—
98	C₂×M₁₆	—	146	2	no	—
99	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	C₂×C₄²	—	73	1	no	—
102	M₃₂	—	146	2	no	—
103	#97	—	73	1	no	—
104	#97	—	73	1	no	—
105	#97	—	146	2	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₄×C₈	—	146	1	no	—
109	C₄×C₈	—	146	1	no	—
110	C₂²×C₄	—	146	1	no	—
111	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
112	M₁₆	—	4	1	no	—
113	M₁₆	—	292	4	no	—
114	M₁₆	—	292	1	no	—
115	M₁₆	—	292	1	no	—
116	M₁₆	—	292	1	no	—
117	C₂×C₈	—	292	1	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	146	2	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	C₄×#64408	1	1	yes	characteristic
123	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
124	C₄⋊C₄	—	4	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	146	1	no	—
130	C₁₆	—	292	2	no	—
131	C₂³	#14556655	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	292	2	no	—
136	C₂×C₄	—	146	1	no	—
137	C₂×C₄	—	146	1	no	—
138	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	#14556656	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	292	2	no	—
143	C₈	—	584	1	no	—
144	C₈	—	146	4	no	—
145	C₈	—	146	1	no	—
146	C₈	—	146	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#17989457	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	16	1	no	—
152	C₄	—	292	1	no	—
153	C₄	—	146	2	no	—
154	C₄	#17989454	1	1	yes	characteristic
155	C₄	#17989468	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	4	1	no	—
158	C₂	#22624790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626301

C₂

yes

maximal, characteristic

#22625154

C₂

yes

maximal, characteristic

#22610078

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#17989429

—

#17989428

C₄

yes

characteristic, complemented

C₈×#10596833

C₄

yes

characteristic, complemented

M₁₆×Dic₇₃

C₂²

yes

—

—

C₄

yes

characteristic

H(3,ℤ/4)×C₇₃

C₄

yes

—

—

—

—

—

—

D₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

#14556501

C₂×C₄

yes

characteristic

#14556501

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂×M₁₆×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

—

—

—

—

—

C₂²×Dic₇₃

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#104

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#339

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7225

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

—

C₇₃

#7225

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

C₄×M₁₆

—

146

—

#364

—

146

—

#349

—

146

—

#347

—

#344

—

146

—

#339

—

146

—

—

yes

—

—

146

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₂×M₁₆

—

146

—

C₂×M₁₆

C₂×#64408

yes

characteristic

100

#117

—

101

C₂×C₄²

—

102

M₃₂

—

146

—

103

#97

—

104

#97

—

105

#97

—

146

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

146

—

109

C₄×C₈

—

146

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

C₂²⋊#64408

yes

characteristic

112

M₁₆

—

113

M₁₆

—

292

—

114

M₁₆

—

292

—

115

M₁₆

—

292

—

116

M₁₆

—

292

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

C₄×#64408

yes

characteristic

123

C₂×C₈

C₄⋊#64408

yes

characteristic

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

146

—

130

C₁₆

—

292

—

131

C₂³

#14556655

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

292

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

139

C₂×C₄

#14556656

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

292

—

143

C₈

—

584

—

144

C₈

—

146

—

145

C₈

—

146

—

146

C₈

—

146

—

147

C₂²

—

148

C₂²

—

149

C₂²

#17989457

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

292

—

153

C₄

—

146

—

154

C₄

#17989454

yes

characteristic

155

C₄

#17989468

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

#22624790

yes

characteristic

	subgroup	G/H
center	C₄	#17989454
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×M₁₆	C₂×#64408
Fitting	H(3,ℤ/4)×C₇₃	C₄
socle	C₂×C₇₃	#1288
prime radical	C₂³×C₇₃	#104
cosocle	M₁₆×Dic₇₃	C₂²
hypercenter	H(3,ℤ/4)	#64408
hypocenter	C₇₃	#7225
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#17989454

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂×M₁₆

C₂×#64408

Fitting

H(3,ℤ/4)×C₇₃

C₄

C₂×C₇₃

C₂³×C₇₃

M₁₆×Dic₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
7. #17989480	C₄	semidirect
9. #17989428	C₄	semidirect
10. C₈×#10596833	C₄	semidirect
83. C₇₃	#7225	semidirect

H ≅ G/N

type

7. #17989480

C₄

semidirect

9. #17989428

C₄

semidirect

10. C₈×#10596833

C₄

semidirect

83. C₇₃

#7225

semidirect

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

subgroup	index
#22626301	2
#22625154	2
#22610078	2
#7225	73

subgroup

index

Subgroups — #34000441

Conjugacy classes of subgroups

Complements

Maximal subgroups