Group #34009650 — subgroups

Subgroups — #34009650

Order 18688.

330 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 176 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	C₄×#14556666	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556664	C₂	1	1	yes	maximal, characteristic
3	#22624843	C₂	1	2	yes	maximal, complemented
4	C₈×#11924031	C₂	1	2	yes	maximal, complemented
5	#22608216	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556666	C₄	1	2	yes	—
7	C₂×#14556664	C₄	1	2	yes	—
8	#17989476	C₄	1	2	yes	—
9	#17989473	C₄	1	2	yes	—
10	C₄×#11924031	C₂²	1	1	yes	characteristic
11	C₄×#11924031	C₂²	1	2	yes	—
12	C₄×#11924031	C₂²	1	1	yes	characteristic
13	#17988638	C₄	1	2	yes	—
14	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	#17988308	C₄	1	1	yes	characteristic
16	C₈×#10596831	C₂²	1	2	yes	—
17	#322×C₇₃	C₄	1	1	yes	characteristic
18	#14556666	C₈	1	8	yes	complemented
19	#14556664	C₈	1	8	yes	complemented
20	C₂×#11924031	C₂×C₄	1	2	yes	—
21	C₂×#11924031	C₂×C₄	1	2	yes	—
22	C₂×#11924031	C₂×C₄	1	4	yes	—
23	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
26	#14556520	C₈	1	4	yes	—
27	#14556503	C₈	1	4	yes	—
28	C₄×#10596831	C₂×C₄	1	2	yes	—
29	C₄×#10596831	C₂³	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	#11924031	C₂×C₈	1	4	yes	—
35	#11924031	C₂×C₈	1	4	yes	—
36	#11924031	C₂×C₈	1	8	yes	—
37	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₂×C₈	1	4	yes	—
40	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
42	C₈×D₇₃	C₂×C₈	1	4	yes	—
43	C₈×D₇₃	C₂×C₈	1	4	yes	—
44	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
49	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
53	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
54	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
55	C₄×D₇₃	—	2	4	no	—
56	C₄×D₇₃	C₄×C₈	1	2	yes	—
57	C₄×D₇₃	C₄×C₈	1	2	yes	—
58	C₈×C₇₃	—	2	4	no	—
59	C₈×C₇₃	C₄×C₈	1	2	yes	—
60	C₈×C₇₃	C₄×C₈	1	2	yes	—
61	#10596831	—	2	4	no	—
62	#10596831	C₂²×C₈	1	2	yes	—
63	#10596831	C₂²×C₈	1	2	yes	—
64	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	C₈²	1	2	yes	—
67	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₇₃	#504	1	4	yes	—
70	Dic₇₃	—	2	2	no	—
71	Dic₇₃	C₈²	1	2	yes	—
72	#11399	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
74	C₂×C₇₃	#2108	1	2	yes	—
75	D₇₃	—	2	2	no	—
76	C₄×M₃₂	—	73	1	no	—
77	C₂×C₄×C₁₆	—	73	1	no	—
78	#1450	—	73	1	no	—
79	#1311	—	73	2	no	—
80	C₈×C₁₆	—	73	2	no	—
81	C₇₃	#11399	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₃₂	—	73	2	no	—
83	C₂²×C₁₆	—	73	2	no	—
84	#431	—	73	2	no	—
85	C₂×C₄×C₈	—	73	1	no	—
86	#346	—	73	2	no	—
87	C₄×C₁₆	—	73	1	no	—
88	C₄×C₁₆	—	73	2	no	—
89	C₄×C₁₆	—	73	1	no	—
90	C₄×C₁₆	—	73	2	no	—
91	#322	—	73	1	no	—
92	#322	#64408	1	1	yes	characteristic
93	C₈²	—	73	2	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	1	no	—
96	C₂×C₄²	—	73	1	no	—
97	M₃₂	—	73	8	no	—
98	C₂×C₁₆	—	73	2	no	—
99	C₂×C₁₆	—	73	2	no	—
100	C₂×C₁₆	—	73	4	no	—
101	C₂×C₁₆	—	73	8	no	—
102	C₄⋊C₈	—	73	4	no	—
103	C₂²⋊C₈	—	73	4	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	C₂×#64408	1	1	yes	characteristic
109	C₄×C₈	C₂×#64408	1	2	yes	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	73	2	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	73	4	no	—
117	C₂×C₈	C₄×#64408	1	1	yes	characteristic
118	C₂×C₈	C₄×#64408	1	1	yes	characteristic
119	C₂×C₈	#11924036	1	4	yes	—
120	C₄²	—	73	2	no	—
121	C₄²	—	73	1	no	—
122	C₄²	C₂²×#64408	1	1	yes	characteristic
123	C₁₆	—	73	4	no	—
124	C₁₆	—	73	4	no	—
125	C₁₆	—	73	8	no	—
126	C₂³	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	73	2	no	—
131	C₂×C₄	—	146	4	no	—
132	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
133	C₂×C₄	C₂×#11924036	1	2	yes	—
134	C₈	—	2	4	no	—
135	C₈	—	73	2	no	—
136	C₈	—	73	2	no	—
137	C₈	—	146	4	no	—
138	C₈	C₈×#64408	1	2	yes	—
139	C₈	C₈×#64408	1	2	yes	—
140	C₂²	—	73	2	no	—
141	C₂²	—	146	2	no	—
142	C₂²	C₄×#11924036	1	1	yes	characteristic
143	C₄	—	73	2	no	—
144	C₄	—	146	2	no	—
145	C₄	C₂×C₈×#64408	1	1	yes	characteristic
146	C₄	C₂×C₈×#64408	1	1	yes	characteristic
147	C₄	#17989646	1	4	yes	—
148	C₂	—	146	2	no	—
149	C₂	C₈×#11924036	1	1	yes	characteristic
150	C₂	#22626141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556666

C₂

yes

maximal, characteristic

C₄×#14556664

C₂

yes

maximal, characteristic

#22624843

C₂

yes

maximal, complemented

C₈×#11924031

C₂

yes

maximal, complemented

#22608216

C₂

yes

maximal, characteristic

C₂×#14556666

C₄

yes

—

C₂×#14556664

C₄

yes

—

#17989476

C₄

yes

—

#17989473

C₄

yes

—

C₄×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

—

C₄×#11924031

C₂²

yes

characteristic

#17988638

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988308

C₄

yes

characteristic

C₈×#10596831

C₂²

yes

—

#322×C₇₃

C₄

yes

characteristic

#14556666

C₈

yes

complemented

#14556664

C₈

yes

complemented

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

#14556520

C₈

yes

—

#14556503

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

#11924031

C₂×C₈

yes

—

#11924031

C₂×C₈

yes

—

#11924031

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₄×C₈

yes

—

C₄×D₇₃

C₄×C₈

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

—

C₈×C₇₃

C₄×C₈

yes

—

#10596831

—

#10596831

C₂²×C₈

yes

—

#10596831

C₂²×C₈

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₈²

yes

—

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

#504

yes

—

Dic₇₃

—

Dic₇₃

C₈²

yes

—

#11399

—

maximal, Sylow

C₂×C₇₃

C₂×C₈²

yes

characteristic

C₂×C₇₃

#2108

yes

—

D₇₃

—

C₄×M₃₂

—

C₂×C₄×C₁₆

—

#1450

—

#1311

—

C₈×C₁₆

—

C₇₃

#11399

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

—

yes

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#64408

yes

characteristic

109

C₄×C₈

C₂×#64408

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₄×#64408

yes

characteristic

118

C₂×C₈

C₄×#64408

yes

characteristic

119

C₂×C₈

#11924036

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂²×#64408

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

146

—

132

C₂×C₄

C₂×C₄×#64408

yes

characteristic

133

C₂×C₄

C₂×#11924036

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

146

—

138

C₈

C₈×#64408

yes

—

139

C₈

C₈×#64408

yes

—

140

C₂²

—

141

C₂²

—

146

—

142

C₂²

C₄×#11924036

yes

characteristic

143

C₄

—

144

C₄

—

146

—

145

C₄

C₂×C₈×#64408

yes

characteristic

146

C₄

C₂×C₈×#64408

yes

characteristic

147

C₄

#17989646

yes

—

148

C₂

—

146

—

149

C₂

C₈×#11924036

yes

characteristic

150

C₂

#22626141

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂²×#64408
commutator	C₂×C₇₃	C₂×C₈²
Fitting	#322×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	#322	#64408
hypocenter	C₇₃	#11399
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×#64408

commutator

C₂×C₇₃

C₂×C₈²

Fitting

#322×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄×C₈

prime radical

C₂²×D₇₃

C₄×C₈

C₂³

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
3. #22624843	C₂	semidirect
4. C₈×#11924031	C₂	semidirect
18. #14556666	C₈	semidirect
19. #14556664	C₈	semidirect
81. C₇₃	#11399	semidirect

H ≅ G/N

type

3. #22624843

C₂

semidirect

4. C₈×#11924031

C₂

semidirect

18. #14556666

C₈

semidirect

19. #14556664

C₈

semidirect

81. C₇₃

#11399

semidirect

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

subgroup	index
C₄×#14556666	2
C₄×#14556664	2
#22624843	2
C₈×#11924031	2
#22608216	2
#11399	73

subgroup

index

Subgroups — #34009650

Conjugacy classes of subgroups

Complements

Maximal subgroups