Group #34014484 — subgroups

Subgroups — #34014484

Order 18688.

198 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 96 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	#22626352	C₂	1	2	yes	maximal
2	#22625091	C₂	1	2	yes	maximal, complemented
3	#22625090	C₂	1	2	yes	maximal, complemented
4	#22611529	C₂	1	1	yes	maximal, characteristic
5	#17989520	—	2	2	no	—
6	#17989519	C₂²	1	4	yes	—
7	#17989518	—	2	2	no	—
8	C₂×#14556530	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
10	#17988541	C₄	1	1	yes	characteristic
11	#17988362	C₄	1	1	yes	characteristic
12	C₄×#11924003	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
14	#346×C₇₃	C₄	1	1	yes	characteristic
15	#14556641	—	2	4	no	—
16	C₂×#11924011	C₈	1	1	yes	characteristic
17	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
18	#14556548	C₈	1	1	yes	characteristic
19	#14556530	—	2	2	no	—
20	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924003	C₂³	1	1	yes	characteristic
22	C₁₆×D₇₃	D₄	1	2	yes	—
23	C₁₆×D₇₃	Q₈	1	2	yes	—
24	#14556502	C₈	1	1	yes	characteristic
25	#14556501	C₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
31	#11924011	C₁₆	1	4	yes	—
32	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
33	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
34	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
35	C₈×D₇₃	M₁₆	1	2	yes	—
36	#11924007	—	2	1	no	—
37	#11924005	—	2	1	no	—
38	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
39	#11924003	—	2	1	no	—
40	#11924003	C₂×D₄	1	1	yes	characteristic
41	#11924003	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₇₃	—	2	1	no	—
46	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×D₇₃	M₃₂	1	1	yes	characteristic
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	1	no	—
51	C₂×Dic₇₃	M₃₂	1	1	yes	characteristic
52	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
53	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
54	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
55	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
57	#10596831	C₄⋊C₈	1	2	yes	—
58	#10596831	C₂×C₁₆	1	2	yes	—
59	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
60	C₂×D₇₃	—	4	1	no	—
61	C₂×D₇₃	C₄⋊C₁₆	1	2	yes	—
62	C₄×C₇₃	—	4	1	no	—
63	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
65	Dic₇₃	—	4	1	no	—
66	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
67	#13630	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	—	2	1	no	—
69	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
70	D₇₃	—	2	2	no	—
71	C₂×M₆₄	—	73	2	no	—
72	#2108	—	73	1	no	—
73	#1422	—	73	2	no	—
74	#1422	—	73	2	no	—
75	C₇₃	#13630	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₃₂	—	73	1	no	—
77	C₂²×C₁₆	—	73	1	no	—
78	#405	—	73	1	no	—
79	M₆₄	—	73	4	no	—
80	M₆₄	—	146	2	no	—
81	C₂×C₃₂	—	146	2	no	—
82	#346	—	73	1	no	—
83	#346	#64408	1	1	yes	characteristic
84	C₄×C₁₆	—	73	1	no	—
85	C₄×C₁₆	—	73	1	no	—
86	C₂×M₁₆	—	73	1	no	—
87	C₂²×C₈	—	73	1	no	—
88	#117	—	73	1	no	—
89	M₃₂	—	146	2	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	2	no	—
92	C₂×C₁₆	—	73	2	no	—
93	C₂×C₁₆	—	73	1	no	—
94	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
95	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
96	#97	—	73	1	no	—
97	#97	—	73	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	C₂×#64408	1	1	yes	characteristic
100	C₃₂	—	146	4	no	—
101	C₂²×C₄	—	73	1	no	—
102	M₁₆	—	73	4	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	2	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	#11924036	1	1	yes	characteristic
108	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
109	C₄⋊C₄	—	146	1	no	—
110	C₂²⋊C₄	—	146	1	no	—
111	C₄²	—	73	1	no	—
112	C₄²	#11924036	1	1	yes	characteristic
113	C₁₆	—	2	1	no	—
114	C₁₆	—	73	1	no	—
115	C₁₆	—	73	1	no	—
116	C₁₆	—	146	1	no	—
117	C₁₆	C₄⋊#64408	1	2	yes	—
118	C₂³	—	73	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	73	2	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	146	1	no	—
124	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
125	C₈	—	73	2	no	—
126	C₈	—	73	2	no	—
127	C₈	#14556664	1	2	yes	—
128	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
129	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
130	C₂²	—	292	1	no	—
131	C₂²	—	73	2	no	—
132	C₂²	#17989643	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	292	1	no	—
135	C₄	—	73	2	no	—
136	C₄	#17989588	1	1	yes	characteristic
137	C₄	C₂×#14556664	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	146	2	no	—
140	C₂	#22626261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626352

C₂

yes

maximal

#22625091

C₂

yes

maximal, complemented

#22625090

C₂

yes

maximal, complemented

#22611529

C₂

yes

maximal, characteristic

#17989520

—

#17989519

C₂²

yes

—

#17989518

—

C₂×#14556530

C₄

yes

characteristic

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

#17988541

C₄

yes

characteristic

#17988362

C₄

yes

characteristic

C₄×#11924003

C₂²

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

characteristic

#346×C₇₃

C₄

yes

characteristic

#14556641

—

C₂×#11924011

C₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

C₈

yes

characteristic

#14556530

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂³

yes

characteristic

C₁₆×D₇₃

D₄

yes

—

C₁₆×D₇₃

Q₈

yes

—

#14556502

C₈

yes

characteristic

#14556501

C₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

#11924011

C₁₆

yes

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₈×D₇₃

C₄⋊C₄

yes

—

C₈×D₇₃

M₁₆

yes

—

#11924007

—

#11924005

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

#11924003

—

#11924003

C₂×D₄

yes

characteristic

#11924003

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₄⋊C₄

yes

—

C₂²×D₇₃

M₃₂

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

M₃₂

yes

characteristic

C₄×D₇₃

C₄⋊C₈

yes

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

C₄⋊C₈

yes

—

#10596831

C₂×C₁₆

yes

—

C₂²×C₇₃

C₂×M₃₂

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

C₄⋊C₁₆

yes

—

#13630

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

characteristic

D₇₃

—

C₂×M₆₄

—

#2108

—

#1422

—

#1422

—

C₇₃

#13630

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

146

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

100

C₃₂

—

146

—

101

C₂²×C₄

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

#11924036

yes

characteristic

108

C₂×C₈

C₂²×#64408

yes

characteristic

109

C₄⋊C₄

—

146

—

110

C₂²⋊C₄

—

146

—

111

C₄²

—

112

C₄²

#11924036

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

146

—

117

C₁₆

C₄⋊#64408

yes

—

118

C₂³

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₂×#11924036

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

#14556664

yes

—

128

C₈

C₂×C₄⋊#64408

yes

characteristic

129

C₈

D₇₃⋊M₁₆

yes

characteristic

130

C₂²

—

292

—

131

C₂²

—

132

C₂²

#17989643

yes

characteristic

133

C₄

—

134

C₄

—

292

—

135

C₄

—

136

C₄

#17989588

yes

characteristic

137

C₄

C₂×#14556664

yes

characteristic

138

C₂

—

139

C₂

—

146

—

140

C₂

#22626261

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	#346×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄⋊C₁₆
prime radical	C₂²×D₇₃	M₃₂
cosocle	C₂×#11924003	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#13630
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

subgroup

G/H

center

C₈

C₂×C₄⋊#64408

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#64408

Fitting

#346×C₇₃

C₄

socle

C₂×C₇₃

C₂×C₄⋊C₁₆

prime radical

C₂²×D₇₃

M₃₂

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

N	H ≅ G/N	type
2. #22625091	C₂	semidirect
3. #22625090	C₂	semidirect
75. C₇₃	#13630	semidirect

H ≅ G/N

type

2. #22625091

C₂

semidirect

3. #22625090

C₂

semidirect

75. C₇₃

#13630

semidirect

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

subgroup	index
#22626352	2
#22625091	2
#22625090	2
#22611529	2
#13630	73

subgroup

index

Subgroups — #34014484

Conjugacy classes of subgroups

Complements

Maximal subgroups