Group #34032896 — subgroups

Subgroups — #34032896

Order 18688.

214 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 73 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	#22627055	C₂	1	2	yes	maximal
2	C₂×#17989474	C₂	1	1	yes	maximal, characteristic
3	#17989700	—	2	2	no	—
4	#17989677	—	2	2	no	—
5	#17989676	C₄	1	4	yes	—
6	C₂×#14556664	C₄	1	1	yes	characteristic
7	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
8	#17989474	—	2	2	no	—
9	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
10	C₄⋊#10596842	—	2	4	no	—
11	C₂²⋊#10596842	—	4	2	no	—
12	C₄×#10596842	—	4	2	no	—
13	C₂×#11924036	Q₈	1	1	yes	characteristic
14	C₂×C₄×#64408	D₄	1	1	yes	characteristic
15	#14556664	—	2	1	no	—
16	#14556664	C₈	1	2	yes	complemented
17	C₂×#11924031	—	2	1	no	—
18	C₈×#64408	C₂×C₄	1	2	yes	—
19	C₈×#64408	D₄	1	2	yes	—
20	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924003	C₈	1	1	yes	characteristic, complemented
22	C₁₆×D₇₃	—	2	1	no	—
23	C₁₆×D₇₃	—	2	1	no	—
24	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic, complemented
25	C₂×#10596842	—	4	4	no	—
26	C₂²×#64408	D₈	1	1	yes	characteristic
27	C₂²×#64408	Q₁₆	1	1	yes	characteristic
28	#11924036	—	2	1	no	—
29	#11924036	C₄⋊C₄	1	2	yes	—
30	C₂×#10596833	—	2	1	no	—
31	C₄×#64408	C₂²⋊C₄	1	2	yes	—
32	C₄×#64408	SD₁₆	1	2	yes	—
33	#11924031	—	2	2	no	—
34	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#10596831	M₁₆	1	1	yes	characteristic
36	C₈×D₇₃	C₄²	1	1	yes	characteristic
37	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
39	C₈×D₇₃	M₁₆	1	1	yes	characteristic
40	#11924003	—	2	1	no	—
41	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₁₆×C₇₃	—	2	1	no	—
43	C₂²×D₇₃	#107	1	1	yes	characteristic
44	C₂×#64408	—	4	1	no	—
45	C₂×#64408	—	4	1	no	—
46	C₂×#64408	#102	1	2	yes	—
47	C₂×#64408	#103	1	2	yes	—
48	#10596842	—	8	4	no	—
49	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	#108	1	1	yes	characteristic
51	C₄×D₇₃	#95	1	1	yes	characteristic
52	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
53	C₄×D₇₃	#106	1	1	yes	characteristic
54	C₄×D₇₃	#108	1	1	yes	characteristic
55	#10596833	—	2	2	no	—
56	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
57	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	#10596831	#97	1	1	yes	characteristic
59	#10596831	C₂²⋊C₈	1	1	yes	characteristic
60	C₂²×C₇₃	#335	1	1	yes	characteristic
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	#335	1	1	yes	characteristic
64	C₂×D₇₃	#340	1	1	yes	characteristic
65	#64408	—	2	2	no	—
66	#64408	—	2	2	no	—
67	C₄×C₇₃	#334	1	1	yes	characteristic
68	C₄×C₇₃	#336	1	1	yes	characteristic
69	Dic₇₃	#334	1	1	yes	characteristic
70	Dic₇₃	#341	1	1	yes	characteristic
71	#7157	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	—	2	1	no	—
73	C₂×C₇₃	#1277	1	1	yes	characteristic
74	D₇₃	—	2	1	no	—
75	D₇₃	#1384	1	1	yes	characteristic
76	D₇₃	#1385	1	1	yes	characteristic
77	C₂×#346	—	73	1	no	—
78	#1564	—	73	2	no	—
79	C₇₃	#7157	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₁₆	—	73	1	no	—
81	C₂²×C₁₆	—	73	1	no	—
82	#431	—	146	2	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	#346	—	146	2	no	—
85	#335	—	146	2	no	—
86	#334	—	73	4	no	—
87	C₂²×C₈	—	73	1	no	—
88	C₂²×C₈	—	73	1	no	—
89	C₂×C₄²	—	73	1	no	—
90	C₂×C₁₆	—	73	2	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	1	no	—
93	C₂×C₁₆	—	146	1	no	—
94	C₂×C₁₆	—	146	1	no	—
95	C₂×C₁₆	—	146	1	no	—
96	C₂×C₁₆	#10596842	1	1	yes	characteristic, complemented
97	C₄⋊C₈	—	146	4	no	—
98	C₂²⋊C₈	—	292	2	no	—
99	C₄×C₈	—	292	2	no	—
100	C₄×C₈	—	73	2	no	—
101	C₄×C₈	—	73	2	no	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	73	1	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂×C₈	—	292	4	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	146	1	no	—
113	C₂×C₈	—	146	1	no	—
114	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
115	C₄²	—	73	2	no	—
116	C₄²	—	73	2	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	146	2	no	—
119	C₁₆	—	146	1	no	—
120	C₂³	—	73	1	no	—
121	C₂×C₄	—	292	1	no	—
122	C₂×C₄	—	292	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
131	C₈	—	584	4	no	—
132	C₈	—	73	1	no	—
133	C₈	—	73	1	no	—
134	C₈	—	146	2	no	—
135	C₈	C₄×#10596842	1	1	yes	characteristic
136	C₈	C₂²⋊#10596842	1	1	yes	characteristic
137	C₂²	—	73	1	no	—
138	C₂²	—	73	1	no	—
139	C₂²	—	146	1	no	—
140	C₂²	—	146	1	no	—
141	C₂²	#17989677	1	1	yes	characteristic
142	C₄	—	73	1	no	—
143	C₄	—	73	1	no	—
144	C₄	—	146	2	no	—
145	C₄	—	146	2	no	—
146	C₄	#17989676	1	1	yes	characteristic
147	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	73	1	no	—
150	C₂	—	73	1	no	—
151	C₂	—	146	1	no	—
152	C₂	#22626937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627055

C₂

yes

maximal

C₂×#17989474

C₂

yes

maximal, characteristic

#17989700

—

#17989677

—

#17989676

C₄

yes

—

C₂×#14556664

C₄

yes

characteristic

C₂×C₈×#64408

C₂²

yes

characteristic

#17989474

—

C₂×C₁₆×D₇₃

C₄

yes

—

—

—

Q₈

yes

characteristic

C₂×C₄×#64408

D₄

yes

characteristic

#14556664

—

#14556664

C₈

yes

complemented

C₂×#11924031

—

C₈×#64408

C₂×C₄

yes

—

C₈×#64408

D₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×#11924003

C₈

yes

characteristic, complemented

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₂×C₁₆×C₇₃

C₈

yes

characteristic, complemented

C₂×#10596842

—

C₂²×#64408

D₈

yes

characteristic

C₂²×#64408

Q₁₆

yes

characteristic

#11924036

—

#11924036

C₄⋊C₄

yes

—

C₂×#10596833

—

C₄×#64408

C₂²⋊C₄

yes

—

C₄×#64408

SD₁₆

yes

—

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×#10596831

M₁₆

yes

characteristic

C₈×D₇₃

C₄²

yes

characteristic

C₈×D₇₃

C₂²⋊C₄

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

—

—

yes

—

C₂×#64408

#103

yes

—

#10596842

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

C₂×Dic₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

C₄⋊C₈

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

—

C₈×C₇₃

C₄×C₈

yes

characteristic

C₈×C₇₃

C₂²⋊C₈

yes

yes

C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#335

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1277

yes

characteristic

D₇₃

—

D₇₃

yes

D₇₃

yes

—

—

C₇₃

#7157

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

146

—

C₂×C₄×C₈

—

#346

—

146

—

#335

—

146

—

#334

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

#10596842

yes

characteristic, complemented

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

292

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

—

146

—

114

C₂×C₈

C₂×#10596842

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₁₆

—

118

C₁₆

—

146

—

119

C₁₆

—

146

—

120

C₂³

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

292

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₄⋊#10596842

yes

characteristic

131

C₈

—

584

—

132

C₈

—

133

C₈

—

134

C₈

—

146

—

135

C₈

C₄×#10596842

yes

characteristic

136

C₈

C₂²⋊#10596842

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

146

—

140

C₂²

—

146

—

141

C₂²

#17989677

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

146

—

145

C₄

—

146

—

146

C₄

#17989676

yes

characteristic

147

C₄

(C₂×#10596842)⋊C₄

yes

characteristic

148

C₂

—

149

C₂

—

150

C₂

—

151

C₂

—

146

—

152

C₂

#22626937

yes

characteristic

	subgroup	G/H
center	C₄	#17989676
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₈	C₂×#10596842
Fitting	C₂×C₁₆×C₇₃	C₈
socle	C₂×C₇₃	#1277
prime radical	C₂²×D₇₃	#107
cosocle	C₂×C₈×#64408	C₂²
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	#7157
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#17989676

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₈

C₂×#10596842

Fitting

C₂×C₁₆×C₇₃

C₈

C₂×C₇₃

C₂²×D₇₃

C₂×C₈×#64408

C₂²

C₂×C₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
16. #14556664	C₈	semidirect
21. C₂×#11924003	C₈	semidirect
24. C₂×C₁₆×C₇₃	C₈	semidirect
79. C₇₃	#7157	semidirect
96. C₂×C₁₆	#10596842	semidirect

H ≅ G/N

type

16. #14556664

C₈

semidirect

21. C₂×#11924003

C₈

semidirect

24. C₂×C₁₆×C₇₃

C₈

semidirect

79. C₇₃

#7157

semidirect

96. C₂×C₁₆

#10596842

semidirect

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

subgroup	index
#22627055	2
C₂×#17989474	2
#7157	73

subgroup

index

#22627055

C₂×#17989474

#7157

Subgroups — #34032896

Conjugacy classes of subgroups

Complements

Maximal subgroups