Group #34000166 — subgroups

Subgroups — #34000166

Order 18688.

234 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 80 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	#22626292	C₂	1	2	yes	maximal
2	#22609676	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11924031	—	2	2	no	—
4	C₂²⋊#11924031	—	2	2	no	—
5	#17989475	C₄	1	4	yes	complemented
6	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
7	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
8	#17988341	—	2	2	no	—
9	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924031	—	2	4	no	—
11	C₂²×#10596831	D₄	1	2	yes	—
12	C₂×C₄×Dic₇₃	—	2	1	no	—
13	C₂×C₄×Dic₇₃	—	2	1	no	—
14	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
15	C₄×#10596831	C₂×C₄	1	2	yes	—
16	C₄×#10596831	D₄	1	1	yes	characteristic
17	C₄×#10596831	Q₈	1	1	yes	characteristic
18	C₂²×C₈×C₇₃	—	2	1	no	—
19	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₇₃	—	2	1	no	—
21	C₄×C₈×C₇₃	C₈	1	2	yes	—
22	#11924031	—	4	4	no	—
23	C₂²×Dic₇₃	—	2	2	no	—
24	C₂×#10596831	—	4	2	no	—
25	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
26	C₂×#10596831	C₄⋊C₄	1	2	yes	—
27	C₄×Dic₇₃	—	2	4	no	—
28	C₄×Dic₇₃	—	4	2	no	—
29	C₄×Dic₇₃	—	4	2	no	—
30	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	4	2	no	—
35	C₄²×C₇₃	C₄²	1	1	yes	characteristic
36	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
39	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	4	no	—
50	C₂×Dic₇₃	—	4	2	no	—
51	C₂×Dic₇₃	—	4	2	no	—
52	C₈×C₇₃	—	4	2	no	—
53	#10596831	—	2	2	no	—
54	#10596831	—	2	2	no	—
55	C₂²×C₇₃	—	2	1	no	—
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	#336	1	1	yes	characteristic
58	C₂²×C₇₃	#349	1	2	yes	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	H(3,ℤ/4)	1	2	yes	—
64	C₄×C₇₃	#364	1	2	yes	—
65	Dic₇₃	—	4	4	no	—
66	#7143	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	—	2	2	no	—
68	C₂×C₇₃	#1274	1	1	yes	characteristic
69	C₂×C₇₃	#1376	1	1	yes	characteristic
70	C₂×C₇₃	#1377	1	1	yes	characteristic
71	#2163	—	73	2	no	—
72	#1744	—	73	1	no	—
73	C₇₃	#7143	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×C₈	—	73	1	no	—
75	C₂×C₄×C₈	#64408	1	1	yes	characteristic
76	C₄³	—	73	1	no	—
77	C₄⋊C₁₆	—	146	2	no	—
78	C₂²⋊C₁₆	—	146	2	no	—
79	#346	—	73	4	no	—
80	#336	—	146	2	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	73	2	no	—
83	C₂×C₄²	—	73	2	no	—
84	C₂×C₄²	—	146	1	no	—
85	C₂×C₄²	—	146	1	no	—
86	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
87	C₂×C₁₆	—	146	4	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	73	2	no	—
90	C₄×C₈	—	73	1	no	—
91	C₄×C₈	—	73	1	no	—
92	C₄×C₈	#10596833	1	2	yes	—
93	C₂²×C₄	—	146	2	no	—
94	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
95	C₂²×C₄	#11924036	1	1	yes	characteristic
96	C₂²×C₄	#11924038	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	292	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	2	no	—
102	C₄²	—	292	2	no	—
103	C₄²	—	292	2	no	—
104	C₄²	—	146	4	no	—
105	C₄²	C₄×#64408	1	1	yes	characteristic
106	C₄²	C₄⋊#64408	1	1	yes	characteristic
107	C₄²	C₂×#10596833	1	1	yes	characteristic
108	C₄²	#11924037	1	1	yes	characteristic
109	C₁₆	—	292	4	no	—
110	C₂³	#14556648	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	292	2	no	—
115	C₂×C₄	—	292	2	no	—
116	C₂×C₄	—	146	4	no	—
117	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
119	C₂×C₄	#14556646	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
121	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
122	C₂×C₄	#14556658	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	146	2	no	—
125	C₈	—	146	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#17989453	1	1	yes	characteristic
129	C₂²	#17989484	1	2	yes	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	292	4	no	—
135	C₄	#17989454	1	2	yes	—
136	C₄	#17989509	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	#22624755	1	1	yes	characteristic
139	C₂	#22624976	1	1	yes	characteristic
140	C₂	#22624979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626292

C₂

yes

maximal

#22609676

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₂²⋊#11924031

—

#17989475

C₄

yes

complemented

C₂×C₄×#10596831

C₂²

yes

characteristic

C₄²×Dic₇₃

C₄

yes

characteristic

#17988341

—

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂²×#10596831

D₄

yes

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

D₄

yes

characteristic

C₄×#10596831

Q₈

yes

characteristic

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₈

yes

—

#11924031

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₂²×C₇₃

#349

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

H(3,ℤ/4)

yes

—

C₄×C₇₃

#364

yes

—

Dic₇₃

—

#7143

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₇₃

#7143

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

146

—

#346

—

#336

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#10596833

yes

—

C₂²×C₄

—

146

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄²

—

292

—

103

C₄²

—

292

—

104

C₄²

—

146

—

105

C₄²

C₄×#64408

yes

characteristic

106

C₄²

C₄⋊#64408

yes

characteristic

107

C₄²

C₂×#10596833

yes

characteristic

108

C₄²

#11924037

yes

characteristic

109

C₁₆

—

292

—

110

C₂³

#14556648

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

292

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

146

—

117

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

118

C₂×C₄

C₄×#10596833

yes

characteristic

119

C₂×C₄

#14556646

yes

characteristic

120

C₂×C₄

C₂²⋊#10596833

yes

characteristic

121

C₂×C₄

C₄⋊#10596833

yes

characteristic

122

C₂×C₄

#14556658

yes

characteristic

123

C₈

—

124

C₈

—

146

—

125

C₈

—

146

—

126

C₂²

—

127

C₂²

—

128

C₂²

#17989453

yes

characteristic

129

C₂²

#17989484

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

292

—

135

C₄

#17989454

yes

—

136

C₄

#17989509

yes

—

137

C₂

—

138

C₂

#22624755

yes

characteristic

139

C₂

#22624976

yes

characteristic

140

C₂

#22624979

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#64408)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄×#10596831	C₂²
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#7143
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

(C₂×#64408)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle

C₂²×C₇₃

#336

prime radical

C₂³×C₇₃

C₂²⋊C₈

C₂²

C₂×C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. #17989475	C₄	semidirect
73. C₇₃	#7143	semidirect

H ≅ G/N

type

5. #17989475

C₄

semidirect

73. C₇₃

#7143

semidirect

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

subgroup	index
#22626292	2
#22609676	2
#7143	73

subgroup

index

#22626292

#22609676

#7143

Subgroups — #34000166

Conjugacy classes of subgroups

Complements

Maximal subgroups