Group #34034442 — subgroups

Subgroups — #34034442

Order 18688.

194 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22627030	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14556681	C₂	1	2	yes	maximal
3	#22627017	C₂	1	2	yes	maximal, complemented
4	C₂×#17989473	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556681	C₂²	1	4	yes	—
6	C₂²×#11924031	C₄	1	1	yes	characteristic
7	C₂²×#11924031	C₂²	1	1	yes	characteristic
8	#17989473	C₄	1	1	yes	characteristic
9	#17989473	C₄	1	1	yes	characteristic
10	#17989473	C₂²	1	1	yes	characteristic
11	#17989473	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
13	#14556681	—	2	4	no	—
14	C₂×#11924031	—	2	2	no	—
15	C₂×#11924031	—	2	2	no	—
16	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
17	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
18	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924031	C₂³	1	1	yes	characteristic
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
22	C₈×Dic₇₃	C₈	1	1	yes	characteristic
23	C₈×Dic₇₃	C₈	1	1	yes	characteristic
24	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
27	#11924031	—	2	1	no	—
28	#11924031	—	2	1	no	—
29	#11924031	M₁₆	1	2	yes	—
30	#11924031	#41	1	2	yes	—
31	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
32	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	—	2	1	no	—
35	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
36	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₇₃	C₁₆	1	1	yes	characteristic
38	C₄×Dic₇₃	C₁₆	1	1	yes	characteristic
39	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
40	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	2	no	—
51	C₂×Dic₇₃	—	2	2	no	—
52	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	M₃₂	1	2	yes	—
58	#10596831	—	2	1	no	—
59	#10596831	#117	1	1	yes	characteristic
60	#10596831	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	M₆₄	1	2	yes	—
64	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	#431	1	1	yes	characteristic
67	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	—	2	1	no	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	M₆₄	1	2	yes	—
72	#13640	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	2	2	no	—
74	C₂×C₇₃	#2162	1	1	yes	characteristic
75	C₂×C₇₃	C₂×M₆₄	1	1	yes	characteristic
76	C₂×C₇₃	C₂×M₆₄	1	1	yes	characteristic
77	C₂×C₄×C₁₆	—	73	1	no	—
78	C₄⋊C₃₂	—	73	2	no	—
79	C₂²⋊C₃₂	—	73	2	no	—
80	#1397	—	73	2	no	—
81	C₇₃	#13640	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₁₆	—	73	1	no	—
83	C₂²×C₁₆	—	73	1	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	C₂×C₃₂	—	73	4	no	—
86	C₄×C₁₆	—	73	1	no	—
87	C₄×C₁₆	—	73	1	no	—
88	C₄×C₁₆	—	73	1	no	—
89	C₄×C₁₆	—	73	1	no	—
90	C₂²×C₈	—	73	1	no	—
91	C₂²×C₈	#10596842	1	1	yes	characteristic
92	C₂×C₄²	—	73	1	no	—
93	C₂×C₁₆	—	73	1	no	—
94	C₂×C₁₆	—	73	1	no	—
95	C₂×C₁₆	—	73	1	no	—
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	146	2	no	—
98	C₂×C₁₆	—	146	2	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₃₂	—	146	4	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	73	1	no	—
106	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	146	1	no	—
111	C₂×C₈	—	146	1	no	—
112	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
113	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
114	C₄²	—	73	1	no	—
115	C₄²	—	73	1	no	—
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₁₆	—	73	2	no	—
119	C₁₆	—	73	2	no	—
120	C₁₆	—	146	1	no	—
121	C₁₆	—	146	1	no	—
122	C₂³	#14556686	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	#14556686	1	1	yes	characteristic
132	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	73	1	no	—
135	C₈	—	73	1	no	—
136	C₈	—	146	1	no	—
137	C₈	#14556687	1	2	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#17989696	1	2	yes	—
141	C₂²	C₂×#14556686	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	73	2	no	—
144	C₄	—	146	1	no	—
145	C₄	—	146	1	no	—
146	C₄	—	146	1	no	—
147	C₄	#17989700	1	1	yes	characteristic
148	C₄	C₂×#14556687	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#22627129	1	1	yes	characteristic
151	C₂	#22627162	1	1	yes	characteristic
152	C₂	C₂×#17989696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627030

C₂

yes

maximal, complemented

C₂²⋊#14556681

C₂

yes

maximal

#22627017

C₂

yes

maximal, complemented

C₂×#17989473

C₂

yes

maximal, characteristic

C₂×#14556681

C₂²

yes

—

C₂²×#11924031

C₄

yes

characteristic

C₂²×#11924031

C₂²

yes

characteristic

#17989473

C₄

yes

characteristic

#17989473

C₄

yes

characteristic

#17989473

C₂²

yes

characteristic

#17989473

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₈

yes

—

—

M₁₆

yes

—

#11924031

#41

yes

—

C₂²×Dic₇₃

C₁₆

yes

characteristic

C₂²×Dic₇₃

C₁₆

yes

—

—

C₂×C₈

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₁₆

yes

characteristic

C₄×Dic₇₃

C₁₆

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

M₃₂

yes

—

—

yes

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

M₆₄

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#431

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

M₆₄

yes

—

#13640

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2162

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

—

C₇₃

#13640

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₃₂

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

#10596842

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₃₂

—

146

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×#10596842

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

146

—

111

C₂×C₈

—

146

—

112

C₂×C₈

C₂×#10596842

yes

characteristic

113

C₂×C₈

C₂×#10596842

yes

characteristic

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

146

—

121

C₁₆

—

146

—

122

C₂³

#14556686

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

146

—

130

C₂×C₄

—

146

—

131

C₂×C₄

#14556686

yes

characteristic

132

C₂×C₄

C₂²×#10596842

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

146

—

137

C₈

#14556687

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

#17989696

yes

—

141

C₂²

C₂×#14556686

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

146

—

145

C₄

—

146

—

146

C₄

—

146

—

147

C₄

#17989700

yes

characteristic

148

C₄

C₂×#14556687

yes

characteristic

149

C₂

—

150

C₂

#22627129

yes

characteristic

151

C₂

#22627162

yes

characteristic

152

C₂

C₂×#17989696

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂²×C₈×C₇₃	C₈
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	#13640
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂²×C₈×C₇₃

C₈

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂³

C₂²×C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. #22627030	C₂	semidirect
3. #22627017	C₂	semidirect
81. C₇₃	#13640	semidirect

H ≅ G/N

type

1. #22627030

C₂

semidirect

3. #22627017

C₂

semidirect

81. C₇₃

#13640

semidirect

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

subgroup	index
#22627030	2
C₂²⋊#14556681	2
#22627017	2
C₂×#17989473	2
#13640	73

subgroup

index

Subgroups — #34034442

Conjugacy classes of subgroups

Complements

Maximal subgroups