Group #34012313 — subgroups

Subgroups — #34012313

Order 18688.

414 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 216 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	C₂×C₄×#11924031	C₂	1	2	yes	maximal
2	#22624970	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₂²	1	4	yes	—
5	C₄×#11924031	C₄	1	8	yes	complemented
6	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
9	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	8	no	—
12	C₂×#11924031	C₂×C₄	1	8	yes	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
17	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
18	C₄×#10596831	C₂×C₄	1	4	yes	—
19	C₈×Dic₇₃	—	2	4	no	—
20	C₈×Dic₇₃	—	2	4	no	—
21	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	C₈	1	4	yes	—
24	#11924031	—	2	8	no	—
25	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
26	C₂×#10596831	C₄²	1	2	yes	—
27	C₂×#10596831	M₁₆	1	4	yes	—
28	C₂×#10596831	C₂²×C₄	1	2	yes	—
29	C₂×#10596831	#41	1	4	yes	—
30	C₄×Dic₇₃	—	2	4	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	—	2	2	no	—
33	C₄×Dic₇₃	C₁₆	1	8	yes	complemented
34	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
40	C₄²×C₇₃	C₄²	1	2	yes	—
41	C₄²×C₇₃	C₂×C₈	1	2	yes	—
42	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
44	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
45	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
47	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#117	1	2	yes	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
51	C₂×Dic₇₃	—	2	4	no	—
52	C₂×Dic₇₃	—	2	4	no	—
53	C₂×Dic₇₃	C₂×C₁₆	1	8	yes	—
54	C₈×C₇₃	—	2	4	no	—
55	#10596831	#97	1	4	yes	—
56	#10596831	#117	1	4	yes	—
57	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
58	C₂²×C₇₃	#431	1	2	yes	—
59	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
60	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
61	C₄×C₇₃	C₄×C₁₆	1	4	yes	—
62	C₄×C₇₃	#346	1	4	yes	—
63	C₄×C₇₃	#376	1	1	yes	characteristic
64	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
65	C₄×C₇₃	#431	1	2	yes	—
66	Dic₇₃	—	2	8	no	—
67	#12590	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	#1744	1	1	yes	characteristic
69	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
70	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
71	C₂×C₇₃	#2162	1	4	yes	—
72	C₂×C₄×C₁₆	—	73	2	no	—
73	C₄²×C₈	—	73	1	no	—
74	#1374	—	73	4	no	—
75	C₇₃	#12590	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₁₆	—	73	4	no	—
77	C₂×C₄×C₈	—	73	1	no	—
78	C₂×C₄×C₈	—	73	2	no	—
79	C₂×C₄×C₈	—	73	2	no	—
80	C₂×C₄×C₈	#64408	1	1	yes	characteristic
81	C₄³	—	73	1	no	—
82	C₄×C₁₆	—	73	8	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₂²×C₈	C₂×#64408	1	2	yes	—
86	C₂×C₄²	—	73	2	no	—
87	C₂×C₄²	—	73	2	no	—
88	C₂×C₄²	—	73	2	no	—
89	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
90	C₂×C₁₆	—	73	8	no	—
91	C₂×C₁₆	—	146	8	no	—
92	C₄×C₈	—	73	4	no	—
93	C₄×C₈	—	146	4	no	—
94	C₄×C₈	—	146	4	no	—
95	C₄×C₈	#10596833	1	4	yes	—
96	C₂²×C₄	—	73	4	no	—
97	C₂²×C₄	#11924036	1	1	yes	characteristic
98	C₂²×C₄	#11924036	1	1	yes	characteristic
99	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	73	4	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	73	2	no	—
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	C₂×#10596833	1	4	yes	—
107	C₄²	—	73	8	no	—
108	C₄²	—	146	2	no	—
109	C₄²	—	146	4	no	—
110	C₄²	—	146	2	no	—
111	C₄²	C₄×#64408	1	2	yes	—
112	C₄²	C₂×#10596833	1	2	yes	—
113	C₁₆	—	146	8	no	—
114	C₂³	C₂×#11924036	1	1	yes	characteristic
115	C₂×C₄	—	73	8	no	—
116	C₂×C₄	—	146	4	no	—
117	C₂×C₄	—	146	4	no	—
118	C₂×C₄	C₄×#10596833	1	2	yes	—
119	C₂×C₄	C₄×#10596833	1	2	yes	—
120	C₂×C₄	#14556664	1	4	yes	—
121	C₂×C₄	#14556666	1	4	yes	—
122	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
123	C₂×C₄	#14556670	1	2	yes	—
124	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
125	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
126	C₈	—	2	4	no	—
127	C₈	—	73	4	no	—
128	C₈	—	73	4	no	—
129	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
130	C₂²	#17989605	1	2	yes	—
131	C₂²	C₂×#14556664	1	2	yes	—
132	C₂²	C₂×#14556666	1	2	yes	—
133	C₄	—	146	8	no	—
134	C₄	#17989473	1	4	yes	—
135	C₄	#17989476	1	4	yes	—
136	C₄	#17989523	1	1	yes	characteristic
137	C₄	#17989546	1	1	yes	characteristic
138	C₄	#17989603	1	2	yes	—
139	C₂	#22625645	1	1	yes	characteristic
140	C₂	C₂×#17989473	1	1	yes	characteristic
141	C₂	C₂×#17989476	1	1	yes	characteristic
142	C₂	#22626290	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11924031

C₂

yes

maximal

#22624970

C₂

yes

maximal

C₄×C₈×Dic₇₃

C₂

yes

maximal, characteristic

C₂²×#11924031

C₂²

yes

—

C₄×#11924031

C₄

yes

complemented

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₈

yes

—

#11924031

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

C₄²

yes

—

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₁₆

yes

complemented

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

—

#10596831

#97

yes

—

#10596831

#117

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

—

C₂²×C₇₃

C₂×M₃₂

yes

—

C₄×C₇₃

C₄×C₁₆

yes

—

C₄×C₇₃

#346

yes

—

C₄×C₇₃

#376

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

#431

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₄×C₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₄×C₁₆

—

C₄²×C₈

—

#1374

—

C₇₃

#12590

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄³

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₄×C₈

#10596833

yes

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²×#64408

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#10596833

yes

—

107

C₄²

—

108

C₄²

—

146

—

109

C₄²

—

146

—

110

C₄²

—

146

—

111

C₄²

C₄×#64408

yes

—

112

C₄²

C₂×#10596833

yes

—

113

C₁₆

—

146

—

114

C₂³

C₂×#11924036

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

146

—

117

C₂×C₄

—

146

—

118

C₂×C₄

C₄×#10596833

yes

—

119

C₂×C₄

C₄×#10596833

yes

—

120

C₂×C₄

#14556664

yes

—

121

C₂×C₄

#14556666

yes

—

122

C₂×C₄

C₂×C₄×#64408

yes

characteristic

123

C₂×C₄

#14556670

yes

—

124

C₂×C₄

C₂²×#10596833

yes

characteristic

125

C₂×C₄

D₇₃⋊M₁₆

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₂²

C₂×C₄×#10596833

yes

characteristic

130

C₂²

#17989605

yes

—

131

C₂²

C₂×#14556664

yes

—

132

C₂²

C₂×#14556666

yes

—

133

C₄

—

146

—

134

C₄

#17989473

yes

—

135

C₄

#17989476

yes

—

136

C₄

#17989523

yes

characteristic

137

C₄

#17989546

yes

characteristic

138

C₄

#17989603

yes

—

139

C₂

#22625645

yes

characteristic

140

C₂

C₂×#17989473

yes

characteristic

141

C₂

C₂×#17989476

yes

characteristic

142

C₂

#22626290

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12590
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×#64408

commutator

C₂×C₇₃

C₂×C₄×C₁₆

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂²×C₈

C₂³

C₂×C₄×C₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. C₄×#11924031	C₄	semidirect
33. C₄×Dic₇₃	C₁₆	semidirect
75. C₇₃	#12590	semidirect

H ≅ G/N

type

5. C₄×#11924031

C₄

semidirect

33. C₄×Dic₇₃

C₁₆

semidirect

75. C₇₃

#12590

semidirect

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

subgroup	index
C₂×C₄×#11924031	2
#22624970	2
C₄×C₈×Dic₇₃	2
#12590	73

subgroup

index

C₂×C₄×#11924031

#22624970

C₄×C₈×Dic₇₃

#12590

Subgroups — #34012313

Conjugacy classes of subgroups

Complements

Maximal subgroups