Group #34003376 — subgroups

Subgroups — #34003376

Order 18688.

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

normal subgroups only

This group has 294 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	#22625649	C₂	1	1	yes	maximal, characteristic
2	#22625145	C₂	1	1	yes	maximal, characteristic
3	#22624756	C₂	1	2	yes	maximal
4	#22624754	C₂	1	2	yes	maximal
5	#22608213	C₂	1	1	yes	maximal, characteristic
6	#17989605	C₂²	1	2	yes	—
7	#17989546	C₂²	1	1	yes	characteristic
8	#17989523	C₂²	1	1	yes	characteristic
9	#17989452	C₄	1	4	yes	—
10	#17989429	C₄	1	4	yes	complemented
11	#17988638	C₂²	1	2	yes	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988309	C₄	1	2	yes	complemented
14	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#11924036	C₂×C₄	1	2	yes	—
17	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
18	#14556658	—	2	2	no	—
19	#14556648	—	2	2	no	—
20	#14556646	—	2	2	no	—
21	C₄×#10596833	C₂×C₄	1	4	yes	—
22	(C₂×#64408)⋊C₄	—	2	2	no	—
23	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	C₄²×D₇₃	C₂³	1	1	yes	characteristic
25	#14556520	—	2	2	no	—
26	#14556503	—	2	2	no	—
27	C₄×#10596831	C₂×C₄	1	2	yes	—
28	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#64408	—	2	2	no	—
33	#11924036	—	2	4	no	—
34	C₂×#10596833	—	2	2	no	—
35	C₂×#10596833	—	2	4	no	—
36	C₄×#64408	—	2	4	no	—
37	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	#41	1	2	yes	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	C₄²	1	2	yes	—
41	C₈×D₇₃	—	2	4	no	—
42	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₇₃	#41	1	2	yes	—
44	C₂×C₈×C₇₃	—	2	2	no	—
45	C₂×C₈×C₇₃	C₄²	1	2	yes	—
46	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₇₃	#117	1	1	yes	characteristic
48	C₂×#64408	—	4	4	no	—
49	C₂×#64408	—	4	2	no	—
50	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#117	1	2	yes	—
52	C₂×Dic₇₃	#117	1	2	yes	—
53	C₂×Dic₇₃	#117	1	1	yes	characteristic
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	#131	1	4	yes	—
56	#10596833	—	4	4	no	—
57	#10596833	—	4	2	no	—
58	C₈×C₇₃	—	2	2	no	—
59	C₈×C₇₃	—	4	2	no	—
60	#10596831	—	2	2	no	—
61	#10596831	—	4	2	no	—
62	C₂²×C₇₃	#376	1	1	yes	characteristic
63	C₂×D₇₃	—	2	2	no	—
64	C₂×D₇₃	#433	1	2	yes	—
65	#64408	—	8	2	no	—
66	C₄×C₇₃	—	2	2	no	—
67	C₄×C₇₃	#405	1	2	yes	—
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	#433	1	2	yes	—
70	#8142	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	#1746	1	1	yes	characteristic
72	C₂×C₇₃	#1747	1	2	yes	—
73	D₇₃	—	2	2	no	—
74	#1746	—	73	1	no	—
75	#1452	—	73	1	no	—
76	#1450	—	73	1	no	—
77	#1275	—	73	2	no	—
78	#1274	—	73	2	no	—
79	C₇₃	#8142	1	1	yes	characteristic, complemented, Sylow
80	#431	—	73	2	no	—
81	#431	—	73	2	no	—
82	C₂×#97	—	73	1	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	#376	—	73	1	no	—
85	#336	—	73	4	no	—
86	#322	—	73	4	no	—
87	#322	—	73	2	no	—
88	C₈²	—	73	1	no	—
89	C₈²	#64408	1	1	yes	characteristic, complemented
90	C₂²×C₈	—	73	2	no	—
91	C₂²×C₈	—	73	2	no	—
92	C₂×C₄²	—	73	1	no	—
93	C₂×C₄²	—	73	2	no	—
94	C₄⋊C₈	—	146	2	no	—
95	C₄⋊C₈	—	146	2	no	—
96	C₂²⋊C₈	—	146	2	no	—
97	C₂²⋊C₈	—	146	2	no	—
98	#97	—	146	2	no	—
99	C₄×C₈	—	73	2	no	—
100	C₄×C₈	—	73	4	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	2	no	—
103	C₄×C₈	C₂×#64408	1	2	yes	—
104	C₄×C₈	C₂×#64408	1	1	yes	characteristic
105	#95	—	146	2	no	—
106	C₂²×C₄	—	73	2	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	146	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	146	4	no	—
112	C₂×C₈	—	146	4	no	—
113	C₂×C₈	—	146	2	no	—
114	C₂×C₈	—	146	4	no	—
115	C₂×C₈	—	146	2	no	—
116	C₂×C₈	C₄×#64408	1	2	yes	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	2	no	—
119	C₄²	—	146	4	no	—
120	C₄²	C₂²×#64408	1	1	yes	characteristic
121	C₂³	—	73	1	no	—
122	C₂×C₄	—	292	2	no	—
123	C₂×C₄	—	292	2	no	—
124	C₂×C₄	—	292	4	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	73	4	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
129	C₂×C₄	#14556670	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	4	2	no	—
132	C₈	—	292	2	no	—
133	C₈	—	292	2	no	—
134	C₈	—	292	4	no	—
135	C₈	—	146	2	no	—
136	C₂²	—	73	2	no	—
137	C₂²	—	146	2	no	—
138	C₂²	#17989523	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	584	2	no	—
141	C₄	—	73	2	no	—
142	C₄	—	146	2	no	—
143	C₄	#17989550	1	2	yes	—
144	C₂	—	146	2	no	—
145	C₂	#22625649	1	1	yes	characteristic
146	C₂	#22625652	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625649

C₂

yes

maximal, characteristic

#22625145

C₂

yes

maximal, characteristic

#22624756

C₂

yes

maximal

#22624754

C₂

yes

maximal

#22608213

C₂

yes

maximal, characteristic

#17989605

C₂²

yes

—

#17989546

C₂²

yes

characteristic

#17989523

C₂²

yes

characteristic

#17989452

C₄

yes

—

#17989429

C₄

yes

complemented

#17988638

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₄

yes

complemented

C₈×#10596831

C₄

yes

characteristic, complemented

C₈²×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924036

C₂×C₄

yes

—

C₂×C₄×#64408

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

(C₂×#64408)⋊C₄

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₇₃

#41

yes

—

C₂×#10596831

—

C₂×#10596831

C₄²

yes

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂²×C₄

yes

C₂²×D₇₃

yes

—

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×Dic₇₃

#117

yes

—

C₂×Dic₇₃

#117

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#131

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#376

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#433

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#405

yes

—

Dic₇₃

—

Dic₇₃

#433

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

—

—

C₇₃

#8142

yes

characteristic, complemented, Sylow

#431

—

#431

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

C₈²

—

C₈²

#64408

yes

characteristic, complemented

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

#97

—

146

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#64408

yes

—

104

C₄×C₈

C₂×#64408

yes

characteristic

105

#95

—

146

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

146

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

146

—

112

C₂×C₈

—

146

—

113

C₂×C₈

—

146

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₂×C₈

C₄×#64408

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

146

—

120

C₄²

C₂²×#64408

yes

characteristic

121

C₂³

—

122

C₂×C₄

—

292

—

123

C₂×C₄

—

292

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₄×#64408

yes

characteristic

129

C₂×C₄

#14556670

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

292

—

133

C₈

—

292

—

134

C₈

—

292

—

135

C₈

—

146

—

136

C₂²

—

137

C₂²

—

146

—

138

C₂²

#17989523

yes

characteristic

139

C₄

—

140

C₄

—

584

—

141

C₄

—

142

C₄

—

146

—

143

C₄

#17989550

yes

—

144

C₂

—

146

—

145

C₂

#22625649

yes

characteristic

146

C₂

#22625652

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#376
prime radical	C₂²×D₇₃	#117
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8142
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
10. #17989429	C₄	semidirect
13. #17988309	C₄	semidirect
14. C₈×#10596831	C₄	semidirect
15. C₈²×C₇₃	C₄	semidirect
79. C₇₃	#8142	semidirect
89. C₈²	#64408	semidirect

H ≅ G/N

type

10. #17989429

C₄

semidirect

13. #17988309

C₄

semidirect

14. C₈×#10596831

C₄

semidirect

15. C₈²×C₇₃

C₄

semidirect

79. C₇₃

#8142

semidirect

89. C₈²

#64408

semidirect

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

subgroup	index
#22625649	2
#22625145	2
#22624756	2
#22624754	2
#22608213	2
#8142	73

subgroup

index

Subgroups — #34003376

Conjugacy classes of subgroups

Complements

Maximal subgroups