Group #34021324 — subgroups

Subgroups — #34021324

Order 18688.

1350 conjugacy classes of subgroups; only the 504 normal subgroups are shown.

normal subgroups only

This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 504 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×#17989591	C₂	1	6	yes	maximal
2	#22625863	C₂	1	8	yes	maximal, complemented
3	C₄³×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924038	C₂²	1	4	yes	—
5	#17989591	—	2	24	no	—
6	C₂×C₄⋊#10596833	C₂²	1	24	yes	—
7	C₂×C₄²×D₇₃	C₄	1	4	yes	—
8	C₂×C₄²×D₇₃	C₂²	1	3	yes	—
9	C₄²×Dic₇₃	C₄	1	3	yes	—
10	C₄²×Dic₇₃	C₂²	1	4	yes	—
11	C₄³×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924038	—	2	24	no	—
13	C₂²×#10596833	C₂³	1	8	yes	—
14	C₄⋊#10596833	—	2	48	no	—
15	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
16	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
18	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
19	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
20	C₄²×D₇₃	—	2	16	no	—
21	C₄²×D₇₃	D₄	1	12	yes	—
22	C₄²×D₇₃	Q₈	1	12	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
25	#11924038	—	4	16	no	—
26	C₂×#10596833	—	2	24	no	—
27	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
29	C₂²×Dic₇₃	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×D₇₃	—	2	24	no	—
31	C₂×C₄×D₇₃	C₄⋊C₄	1	24	yes	—
32	C₂×C₄×D₇₃	C₂×D₄	1	6	yes	—
33	C₂×C₄×D₇₃	C₂×Q₈	1	6	yes	—
34	C₄×Dic₇₃	—	2	24	no	—
35	C₄×Dic₇₃	C₄⋊C₄	1	12	yes	—
36	C₄×Dic₇₃	C₂×D₄	1	12	yes	—
37	C₄×Dic₇₃	C₂×Q₈	1	12	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
40	C₄²×C₇₃	C₄⋊C₄	1	12	yes	—
41	C₄²×C₇₃	M₁₆	1	16	yes	complemented
42	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
43	C₂²×D₇₃	—	2	4	no	—
44	C₂²×D₇₃	C₂×C₄⋊C₄	1	6	yes	—
45	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	6	yes	—
46	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	12	yes	—
47	C₂×C₄×C₇₃	C₂×M₁₆	1	24	yes	—
48	C₂×Dic₇₃	—	2	12	no	—
49	C₂×Dic₇₃	C₂×C₄⋊C₄	1	12	yes	—
50	C₂×Dic₇₃	C₂²×D₄	1	3	yes	—
51	C₂×Dic₇₃	C₂²×Q₈	1	3	yes	—
52	C₄×D₇₃	—	2	48	no	—
53	C₄×D₇₃	#127	1	4	yes	—
54	C₄×D₇₃	C₄⋊Q₈	1	12	yes	—
55	#10596833	—	4	8	no	—
56	C₂²×C₇₃	C₂²×C₄⋊C₄	1	3	yes	—
57	C₂²×C₇₃	C₂²×M₁₆	1	4	yes	—
58	C₂×D₇₃	—	2	12	no	—
59	C₂×D₇₃	#384	1	4	yes	—
60	C₄×C₇₃	#384	1	4	yes	—
61	C₄×C₇₃	#423	1	24	yes	complemented
62	Dic₇₃	—	2	12	no	—
63	Dic₇₃	C₂×#127	1	1	yes	characteristic
64	Dic₇₃	C₂×C₄⋊Q₈	1	3	yes	—
65	#20791	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
67	C₂×C₇₃	C₂×#423	1	6	yes	—
68	D₇₃	—	2	4	no	—
69	C₂×#423	—	73	6	no	—
70	C₂×C₄³	—	73	1	no	—
71	#1842	—	73	8	no	—
72	C₇₃	#20791	1	1	yes	characteristic, complemented, Sylow
73	C₂²×M₁₆	—	73	4	no	—
74	C₂²×C₄²	—	73	3	no	—
75	C₂²×C₄²	—	73	4	no	—
76	#423	—	146	24	no	—
77	C₂×C₄⋊C₈	—	73	24	no	—
78	C₄³	—	73	4	no	—
79	C₄³	—	73	3	no	—
80	C₄³	#64408	1	1	yes	characteristic
81	C₂³×C₄	—	73	1	no	—
82	C₂³×C₄	—	73	6	no	—
83	C₂×M₁₆	—	146	24	no	—
84	C₂²×C₈	—	73	8	no	—
85	C₂×C₄²	—	73	3	no	—
86	C₂×C₄²	—	73	6	no	—
87	C₂×C₄²	—	73	12	no	—
88	C₂×C₄²	—	73	12	no	—
89	C₂×C₄²	—	73	12	no	—
90	C₂×C₄²	—	146	16	no	—
91	C₂×C₄²	C₂×#64408	1	3	yes	—
92	C₂×C₄²	C₂×#64408	1	4	yes	—
93	C₄⋊C₈	—	146	48	no	—
94	C₂⁴	—	73	1	no	—
95	C₂²×C₄	—	73	1	no	—
96	C₂²×C₄	—	73	6	no	—
97	C₂²×C₄	—	73	6	no	—
98	C₂²×C₄	—	73	6	no	—
99	C₂²×C₄	—	73	24	no	—
100	C₂²×C₄	—	146	24	no	—
101	C₂²×C₄	C₂²×#64408	1	6	yes	—
102	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
103	M₁₆	—	292	16	no	—
104	C₂×C₈	—	146	24	no	—
105	C₄²	—	73	12	no	—
106	C₄²	—	73	12	no	—
107	C₄²	—	73	12	no	—
108	C₄²	—	146	24	no	—
109	C₄²	C₄⋊#64408	1	12	yes	—
110	C₄²	#11924038	1	16	yes	complemented
111	C₂³	—	73	6	no	—
112	C₂³	—	146	4	no	—
113	C₂³	C₂³×#64408	1	1	yes	characteristic
114	C₂×C₄	—	73	3	no	—
115	C₂×C₄	—	73	3	no	—
116	C₂×C₄	—	73	4	no	—
117	C₂×C₄	—	73	12	no	—
118	C₂×C₄	—	73	12	no	—
119	C₂×C₄	—	146	12	no	—
120	C₂×C₄	—	146	48	no	—
121	C₂×C₄	C₂×C₄⋊#64408	1	6	yes	—
122	C₂×C₄	C₂×C₄⋊#64408	1	12	yes	—
123	C₂×C₄	C₂×#11924038	1	24	yes	—
124	C₈	—	292	8	no	—
125	C₂²	—	73	4	no	—
126	C₂²	—	146	12	no	—
127	C₂²	C₂²×C₄⋊#64408	1	3	yes	—
128	C₂²	C₂²×#11924038	1	4	yes	—
129	C₄	—	73	1	no	—
130	C₄	—	73	3	no	—
131	C₄	—	146	12	no	—
132	C₄	#17989533	1	4	yes	—
133	C₄	#17989591	1	24	yes	complemented
134	C₂	—	146	4	no	—
135	C₂	C₂×#17989533	1	1	yes	characteristic
136	C₂	C₂×#17989591	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989591

C₂

yes

maximal

#22625863

C₂

yes

maximal, complemented

C₄³×D₇₃

C₂

yes

maximal, characteristic

C₂²×#11924038

C₂²

yes

—

#17989591

—

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄²×D₇₃

C₄

yes

—

C₂×C₄²×D₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₄

yes

—

C₄²×Dic₇₃

C₂²

yes

—

C₄³×C₇₃

C₄

yes

characteristic

C₂×#11924038

—

C₂²×#10596833

C₂³

yes

—

C₄⋊#10596833

—

C₂²×C₄×D₇₃

C₂×C₄

yes

—

C₂²×C₄×D₇₃

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

—

C₄²×D₇₃

—

C₄²×D₇₃

D₄

yes

—

C₄²×D₇₃

Q₈

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

#11924038

—

C₂×#10596833

—

C₂³×D₇₃

C₂²×C₄

yes

characteristic

C₂²×Dic₇₃

C₂²×C₄

yes

—

C₂²×Dic₇₃

C₂⁴

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₂×D₄

yes

—

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂²×C₄

yes

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

complemented

C₂³×C₇₃

C₂³×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

C₂²×D₄

yes

—

C₂×Dic₇₃

C₂²×Q₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

#127

yes

—

C₄×D₇₃

C₄⋊Q₈

yes

—

#10596833

—

C₂²×C₇₃

C₂²×C₄⋊C₄

yes

—

C₂²×C₇₃

C₂²×M₁₆

yes

—

C₂×D₇₃

—

C₂×D₇₃

#384

yes

—

C₄×C₇₃

#384

yes

—

C₄×C₇₃

#423

yes

complemented

Dic₇₃

—

Dic₇₃

C₂×#127

yes

characteristic

Dic₇₃

C₂×C₄⋊Q₈

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₂×#423

—

C₂×C₄³

—

#1842

—

C₇₃

#20791

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

—

C₂²×C₄²

—

#423

—

146

—

C₂×C₄⋊C₈

—

C₄³

—

C₄³

—

C₄³

#64408

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×#64408

yes

—

C₂×C₄²

C₂×#64408

yes

—

C₄⋊C₈

—

146

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

146

—

101

C₂²×C₄

C₂²×#64408

yes

—

102

C₂²×C₄

C₂²×#64408

yes

characteristic

103

M₁₆

—

292

—

104

C₂×C₈

—

146

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

146

—

109

C₄²

C₄⋊#64408

yes

—

110

C₄²

#11924038

yes

complemented

111

C₂³

—

112

C₂³

—

146

—

113

C₂³

C₂³×#64408

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

C₂×C₄⋊#64408

yes

—

122

C₂×C₄

C₂×C₄⋊#64408

yes

—

123

C₂×C₄

C₂×#11924038

yes

—

124

C₈

—

292

—

125

C₂²

—

126

C₂²

—

146

—

127

C₂²

C₂²×C₄⋊#64408

yes

—

128

C₂²

C₂²×#11924038

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

146

—

132

C₄

#17989533

yes

—

133

C₄

#17989591

yes

complemented

134

C₂

—

146

—

135

C₂

C₂×#17989533

yes

characteristic

136

C₂

C₂×#17989591

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×#64408
commutator, socle	C₂³×C₇₃	C₂³×C₄
Fitting	C₄³×C₇₃	C₄
prime radical	C₂³×D₇₃	C₂²×C₄
cosocle	C₂²×Dic₇₃	C₂⁴
hypercenter	C₄³	#64408
hypocenter	C₇₃	#20791
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×#64408

commutator, socle

C₂³×C₇₃

C₂³×C₄

Fitting

C₄³×C₇₃

C₄

prime radical

C₂³×D₇₃

C₂²×C₄

cosocle

C₂²×Dic₇₃

C₂⁴

C₄³

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
2. #22625863	C₂	semidirect
41. C₄²×C₇₃	M₁₆	semidirect
61. C₄×C₇₃	#423	semidirect
72. C₇₃	#20791	semidirect
110. C₄²	#11924038	semidirect
133. C₄	#17989591	semidirect

H ≅ G/N

type

2. #22625863

C₂

semidirect

41. C₄²×C₇₃

M₁₆

semidirect

61. C₄×C₇₃

#423

semidirect

72. C₇₃

#20791

semidirect

110. C₄²

#11924038

semidirect

133. C₄

#17989591

semidirect

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

subgroup	index
C₂×#17989591	2
#22625863	2
C₄³×D₇₃	2
#20791	73

subgroup

index

C₂×#17989591

#22625863

C₄³×D₇₃

#20791

Subgroups — #34021324

Conjugacy classes of subgroups

Complements

Maximal subgroups