Group #34034959 — subgroups

Subgroups — #34034959

Order 18688.

526 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 297 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	Q₈×#11924040	C₂	1	8	yes	maximal, complemented
2	#22627128	C₂	1	6	yes	maximal
3	C₂×Q₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924040	C₂²	1	12	yes	—
5	C₂²⋊#11924040	C₂²	1	4	yes	—
6	C₄×#11924040	C₂²	1	12	yes	—
7	Q₈×#10596833	C₄	1	4	yes	—
8	Q₈×#10596833	C₂²	1	4	yes	—
9	C₂×C₄⋊#10596833	C₄	1	3	yes	—
10	C₂×C₄×#10596833	C₂²	1	3	yes	—
11	C₂×Q₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924040	C₂³	1	8	yes	—
13	C₂²×#10596833	C₂×C₄	1	3	yes	—
14	C₂²×#10596833	C₂³	1	1	yes	characteristic
15	C₄⋊#10596833	C₂×C₄	1	12	yes	—
16	C₄×#10596833	C₂×C₄	1	6	yes	—
17	C₄×#10596833	C₂³	1	6	yes	—
18	Q₈×Dic₇₃	C₈	1	4	yes	—
19	Q₈×Dic₇₃	C₂×C₄	1	4	yes	—
20	C₂×C₄⋊Dic₇₃	C₈	1	3	yes	—
21	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
22	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
23	#11924040	—	2	8	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂×#10596833	—	4	3	no	—
26	C₂×#10596833	C₂²×C₄	1	6	yes	—
27	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
28	C₂×#10596833	C₂⁴	1	1	yes	characteristic
29	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
30	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	C₂×C₈	1	12	yes	—
32	C₄×Dic₇₃	C₂×C₈	1	6	yes	—
33	C₄×Dic₇₃	C₂²×C₄	1	6	yes	—
34	C₂×Q₈×C₇₃	C₁₆	1	8	yes	complemented
35	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
37	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
38	Q₈×C₇₃	C₂×C₁₆	1	16	yes	—
39	C₂×C₄×C₇₃	C₂×C₁₆	1	12	yes	—
40	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
41	C₂×Dic₇₃	—	2	1	no	—
42	C₂×Dic₇₃	—	2	1	no	—
43	C₂×Dic₇₃	—	4	3	no	—
44	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
45	C₂×Dic₇₃	C₂²×C₈	1	6	yes	—
46	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
47	#10596833	—	2	1	no	—
48	#10596833	—	2	6	no	—
49	#10596833	2₋¹⁺⁴	1	2	yes	—
50	C₂²×C₇₃	—	2	1	no	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
53	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
54	C₄×C₇₃	C₂²×C₁₆	1	12	yes	—
55	Dic₇₃	—	2	6	no	—
56	Dic₇₃	—	2	1	no	—
57	Dic₇₃	#519	1	1	yes	characteristic
58	Dic₇₃	#568	1	1	yes	characteristic
59	#33496	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	—	2	2	no	—
61	C₂×C₇₃	#2982	1	1	yes	characteristic
62	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
63	C₂×C₇₃	#3407	1	1	yes	characteristic
64	C₂×C₈×Q₈	—	73	1	no	—
65	Q₈×C₁₆	—	73	8	no	—
66	#2162	—	73	6	no	—
67	C₇₃	#33496	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	—	73	1	no	—
69	C₈×Q₈	—	73	4	no	—
70	C₈×Q₈	—	73	4	no	—
71	C₂×C₄⋊C₈	—	73	3	no	—
72	C₂×C₄×C₈	—	73	3	no	—
73	C₄⋊C₁₆	—	73	12	no	—
74	C₂²⋊C₁₆	—	73	4	no	—
75	C₄×C₁₆	—	73	12	no	—
76	C₂²×Q₈	#10596842	1	1	yes	characteristic
77	C₂²×C₈	—	73	3	no	—
78	C₂²×C₈	—	73	1	no	—
79	C₄×Q₈	—	73	4	no	—
80	C₄×Q₈	—	73	4	no	—
81	C₂×C₄⋊C₄	—	73	3	no	—
82	C₂×C₄²	—	73	3	no	—
83	C₂×C₁₆	—	73	8	no	—
84	C₄⋊C₈	—	73	12	no	—
85	C₄×C₈	—	73	6	no	—
86	C₄×C₈	—	73	6	no	—
87	C₂×Q₈	#11924040	1	8	yes	complemented
88	C₂×Q₈	C₂×#10596842	1	4	yes	—
89	C₂²×C₄	—	73	3	no	—
90	C₂²×C₄	—	73	1	no	—
91	C₂²×C₄	C₂×#10596842	1	3	yes	—
92	C₂×C₈	—	292	3	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	73	1	no	—
95	C₂×C₈	—	73	6	no	—
96	C₂×C₈	—	146	2	no	—
97	C₄⋊C₄	—	73	12	no	—
98	C₄²	—	73	6	no	—
99	C₄²	—	73	6	no	—
100	C₁₆	—	146	8	no	—
101	C₂³	C₂²×#10596842	1	1	yes	characteristic
102	Q₈	C₂×#11924040	1	16	yes	—
103	C₂×C₄	—	292	3	no	—
104	C₂×C₄	—	73	1	no	—
105	C₂×C₄	—	73	1	no	—
106	C₂×C₄	—	73	6	no	—
107	C₂×C₄	—	146	1	no	—
108	C₂×C₄	—	146	1	no	—
109	C₂×C₄	C₂×#11924040	1	12	yes	—
110	C₂×C₄	C₂²×#10596842	1	6	yes	—
111	C₈	—	73	2	no	—
112	C₈	—	146	6	no	—
113	C₈	—	146	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₂²×#11924040	1	2	yes	—
117	C₂²	C₂³×#10596842	1	1	yes	characteristic
118	C₄	—	73	1	no	—
119	C₄	—	73	1	no	—
120	C₄	—	146	1	no	—
121	C₄	—	146	6	no	—
122	C₄	C₂²×#11924040	1	12	yes	—
123	C₂	—	2	2	no	—
124	C₂	#22627174	1	1	yes	characteristic
125	C₂	C₂³×#11924040	1	1	yes	characteristic
126	C₂	#22627186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11924040

C₂

yes

maximal, complemented

#22627128

C₂

yes

maximal

C₂×Q₈×#10596833

C₂

yes

maximal, characteristic

C₄⋊#11924040

C₂²

yes

—

C₂²⋊#11924040

C₂²

yes

—

C₄×#11924040

C₂²

yes

—

Q₈×#10596833

C₄

yes

—

Q₈×#10596833

C₂²

yes

—

C₂×C₄⋊#10596833

C₄

yes

—

C₂×C₄×#10596833

C₂²

yes

—

C₂×Q₈×Dic₇₃

C₄

yes

characteristic

C₂×#11924040

C₂³

yes

—

C₂²×#10596833

C₂×C₄

yes

—

C₂²×#10596833

C₂³

yes

characteristic

C₄⋊#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂³

yes

—

Q₈×Dic₇₃

C₈

yes

—

Q₈×Dic₇₃

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₃

C₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂²×Q₈×C₇₃

C₈

yes

—

—

—

C₂²×C₄

yes

—

C₂×#10596833

C₂²×C₄

yes

characteristic

C₂×#10596833

C₂⁴

yes

characteristic

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

—

C₂×Q₈×C₇₃

C₁₆

yes

complemented

C₂×Q₈×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

Q₈×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₂³×C₄

yes

—

—

2₋¹⁺⁴

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₁₆

yes

—

C₂²×C₇₃

C₂³×C₈

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#2982

yes

characteristic

C₂×C₇₃

C₂³×C₁₆

yes

characteristic

C₂×C₇₃

#3407

yes

characteristic

C₂×C₈×Q₈

—

Q₈×C₁₆

—

#2162

—

C₇₃

#33496

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×Q₈

#10596842

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

#11924040

yes

complemented

C₂×Q₈

C₂×#10596842

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#10596842

yes

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄²

—

C₄²

—

100

C₁₆

—

146

—

101

C₂³

C₂²×#10596842

yes

characteristic

102

Q₈

C₂×#11924040

yes

—

103

C₂×C₄

—

292

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

146

—

108

C₂×C₄

—

146

—

109

C₂×C₄

C₂×#11924040

yes

—

110

C₂×C₄

C₂²×#10596842

yes

—

111

C₈

—

112

C₈

—

146

—

113

C₈

—

146

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂²×#11924040

yes

—

117

C₂²

C₂³×#10596842

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

146

—

121

C₄

—

146

—

122

C₄

C₂²×#11924040

yes

—

123

C₂

—

124

C₂

#22627174

yes

characteristic

125

C₂

C₂³×#11924040

yes

characteristic

126

C₂

#22627186

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator	C₂×C₇₃	C₂³×C₁₆
Fitting	C₂²×Q₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₂²×Q₈	#10596842
hypocenter	C₇₃	#33496
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#10596842

commutator

C₂×C₇₃

C₂³×C₁₆

Fitting

C₂²×Q₈×C₇₃

C₈

socle

C₂²×C₇₃

C₂³×C₈

prime radical

C₂³×C₇₃

C₂²×C₈

C₂⁴

C₂²×Q₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. Q₈×#11924040	C₂	semidirect
34. C₂×Q₈×C₇₃	C₁₆	semidirect
67. C₇₃	#33496	semidirect
87. C₂×Q₈	#11924040	semidirect

H ≅ G/N

type

1. Q₈×#11924040

C₂

semidirect

34. C₂×Q₈×C₇₃

C₁₆

semidirect

67. C₇₃

#33496

semidirect

87. C₂×Q₈

#11924040

semidirect

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

subgroup	index
Q₈×#11924040	2
#22627128	2
C₂×Q₈×#10596833	2
#33496	73

subgroup

index

Subgroups — #34034959

Conjugacy classes of subgroups

Complements

Maximal subgroups