Group #34032894 — subgroups

Subgroups — #34032894

Order 18688.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic
3	#17989700	—	2	2	no	—
4	#17989677	C₄	1	4	yes	—
5	#17989676	—	2	2	no	—
6	C₂×#14556664	C₄	1	1	yes	characteristic
7	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
8	C₁₆×#64408	—	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	SD₁₆	1	2	yes	—
27	#11924036	—	2	1	no	—
28	#11924036	C₄⋊C₄	1	2	yes	—
29	C₂×#10596833	—	2	1	no	—
30	C₄×#64408	C₂²⋊C₄	1	2	yes	—
31	C₄×#64408	D₈	1	1	yes	characteristic
32	C₄×#64408	Q₁₆	1	1	yes	characteristic
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	#106	1	1	yes	characteristic
44	C₂×#64408	—	2	4	no	—
45	C₂×#64408	#102	1	2	yes	—
46	C₂×#64408	#103	1	2	yes	—
47	#10596842	—	8	4	no	—
48	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	#108	1	1	yes	characteristic
50	C₄×D₇₃	#95	1	1	yes	characteristic
51	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₄×D₇₃	#107	1	1	yes	characteristic
53	C₄×D₇₃	#108	1	1	yes	characteristic
54	#10596833	—	2	2	no	—
55	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
56	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
57	#10596831	#97	1	1	yes	characteristic
58	#10596831	C₂²⋊C₈	1	1	yes	characteristic
59	C₂²×C₇₃	#334	1	1	yes	characteristic
60	C₂×D₇₃	—	2	2	no	—
61	C₂×D₇₃	#334	1	1	yes	characteristic
62	C₂×D₇₃	#340	1	1	yes	characteristic
63	#64408	—	2	2	no	—
64	#64408	#359	1	4	yes	—
65	C₄×C₇₃	#335	1	1	yes	characteristic
66	C₄×C₇₃	#336	1	1	yes	characteristic
67	Dic₇₃	#335	1	1	yes	characteristic
68	Dic₇₃	#341	1	1	yes	characteristic
69	#7155	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	#1277	1	1	yes	characteristic
72	D₇₃	—	2	1	no	—
73	D₇₃	#1382	1	2	yes	—
74	C₂×C₄×C₁₆	—	73	1	no	—
75	#1564	—	73	2	no	—
76	C₇₃	#7155	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₁₆	—	73	1	no	—
78	C₂²×C₁₆	—	73	1	no	—
79	#431	—	146	2	no	—
80	C₂×C₄×C₈	—	73	1	no	—
81	C₄×C₁₆	—	146	2	no	—
82	#335	—	73	4	no	—
83	#334	—	146	2	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₂²×C₈	—	73	1	no	—
86	C₂×C₄²	—	73	1	no	—
87	C₂×C₁₆	—	73	2	no	—
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	—	146	1	no	—
90	C₂×C₁₆	—	146	1	no	—
91	C₂×C₁₆	—	146	1	no	—
92	C₂×C₁₆	—	146	1	no	—
93	C₂×C₁₆	#10596842	1	1	yes	characteristic, complemented
94	C₄⋊C₈	—	146	4	no	—
95	C₂²⋊C₈	—	292	2	no	—
96	C₄×C₈	—	292	2	no	—
97	C₄×C₈	—	73	2	no	—
98	C₄×C₈	—	73	2	no	—
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	73	2	no	—
101	C₂×C₈	—	292	4	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	2	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	—	146	1	no	—
110	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
111	C₄²	—	73	2	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	146	2	no	—
116	C₁₆	—	146	1	no	—
117	C₂³	—	73	1	no	—
118	C₂×C₄	—	73	2	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	73	2	no	—
125	C₂×C₄	—	146	4	no	—
126	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
127	C₈	—	584	4	no	—
128	C₈	—	73	1	no	—
129	C₈	—	73	1	no	—
130	C₈	—	146	2	no	—
131	C₈	C₄×#10596842	1	1	yes	characteristic
132	C₈	C₂²⋊#10596842	1	1	yes	characteristic
133	C₂²	—	73	1	no	—
134	C₂²	—	73	1	no	—
135	C₂²	—	146	2	no	—
136	C₂²	#17989676	1	1	yes	characteristic
137	C₄	—	73	4	no	—
138	C₄	—	73	1	no	—
139	C₄	—	73	1	no	—
140	C₄	—	146	2	no	—
141	C₄	#17989677	1	1	yes	characteristic
142	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	73	2	no	—
145	C₂	—	146	1	no	—
146	C₂	#22626937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627055

C₂

yes

maximal

C₂×C₁₆×#64408

C₂

yes

maximal, characteristic

#17989700

—

#17989677

C₄

yes

—

#17989676

—

C₂×#14556664

C₄

yes

characteristic

C₂×C₈×#64408

C₂²

yes

characteristic

C₁₆×#64408

—

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

SD₁₆

yes

—

#11924036

—

#11924036

C₄⋊C₄

yes

—

C₂×#10596833

—

C₄×#64408

C₂²⋊C₄

yes

—

C₄×#64408

D₈

yes

characteristic

C₄×#64408

Q₁₆

yes

characteristic

#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

#334

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1277

yes

characteristic

D₇₃

—

D₇₃

#1382

yes

—

C₂×C₄×C₁₆

—

#1564

—

C₇₃

#7155

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

146

—

C₂×C₄×C₈

—

C₄×C₁₆

—

146

—

#335

—

#334

—

146

—

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

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂×C₈

—

292

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₂×C₈

C₂×#10596842

yes

characteristic

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₁₆

—

115

C₁₆

—

146

—

116

C₁₆

—

146

—

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

146

—

126

C₂×C₄

C₄⋊#10596842

yes

characteristic

127

C₈

—

584

—

128

C₈

—

129

C₈

—

130

C₈

—

146

—

131

C₈

C₄×#10596842

yes

characteristic

132

C₈

C₂²⋊#10596842

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

146

—

136

C₂²

#17989676

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

146

—

141

C₄

#17989677

yes

characteristic

142

C₄

(C₂×#10596842)⋊C₄

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

—

146

C₂

#22626937

yes

characteristic

	subgroup	G/H
center	C₄	#17989677
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₈	C₂×#10596842
Fitting	C₂×C₁₆×C₇₃	C₈
socle	C₂×C₇₃	#1277
prime radical	C₂²×D₇₃	#106
cosocle	C₂×C₈×#64408	C₂²
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	#7155
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#17989677

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
76. C₇₃	#7155	semidirect
93. C₂×C₁₆	#10596842	semidirect

H ≅ G/N

type

16. #14556664

C₈

semidirect

21. C₂×#11924003

C₈

semidirect

24. C₂×C₁₆×C₇₃

C₈

semidirect

76. C₇₃

#7155

semidirect

93. 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×C₁₆×#64408	2
#7155	73

subgroup

index

#22627055

C₂×C₁₆×#64408

#7155

Subgroups — #34032894

Conjugacy classes of subgroups

Complements

Maximal subgroups