Group #34034007 — subgroups

Subgroups — #34034007

Order 18688.

200 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22627157	C₂	1	2	yes	maximal
2	#22626989	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11924040	C₂	1	2	yes	maximal
4	#22626948	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#10596833	C₂	1	1	yes	maximal, characteristic
6	#17989690	—	2	2	no	—
7	C₄⋊#11924040	C₂²	1	2	yes	—
8	#17989681	—	2	2	no	—
9	C₄×#11924040	C₂²	1	1	yes	characteristic
10	C₄×#11924040	C₂²	1	1	yes	characteristic
11	Q₈×#10596833	C₂²	1	2	yes	—
12	#17989450	C₄	1	1	yes	characteristic
13	Q₈⋊#10596833	C₄	1	2	yes	—
14	C₈×#10596833	C₂²	1	1	yes	characteristic
15	Q₁₆×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	1	no	—
18	C₂×#11924040	D₄	1	2	yes	—
19	C₄⋊#10596833	—	2	2	no	—
20	C₄⋊#10596833	C₂×C₄	1	2	yes	—
21	C₄×#10596833	—	2	2	no	—
22	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596833	C₂³	1	1	yes	characteristic
24	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
25	#14556525	C₈	1	1	yes	characteristic
26	#14556515	C₈	1	2	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic
29	#11924040	—	2	2	no	—
30	#11924040	—	4	1	no	—
31	#11924040	—	4	2	no	—
32	C₂×#10596833	—	2	2	no	—
33	C₂×#10596833	—	2	1	no	—
34	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
35	C₂×#10596833	#41	1	1	yes	characteristic
36	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
37	C₄⋊Dic₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
39	C₄×Dic₇₃	—	2	2	no	—
40	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
42	Q₁₆×C₇₃	—	2	2	no	—
43	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
44	Q₈×C₇₃	—	2	2	no	—
45	Q₈×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	2	no	—
49	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₇₃	#131	1	1	yes	characteristic
51	#10596833	—	2	1	no	—
52	#10596833	—	4	1	no	—
53	#10596833	—	4	2	no	—
54	#10596833	#137	1	2	yes	—
55	C₈×C₇₃	M₃₂	1	2	yes	—
56	#10596831	—	2	1	no	—
57	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₃	—	4	2	no	—
59	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
60	C₄×C₇₃	#504	1	1	yes	characteristic
61	Dic₇₃	—	2	1	no	—
62	Dic₇₃	—	4	2	no	—
63	Dic₇₃	#441	1	1	yes	characteristic
64	Dic₇₃	#444	1	1	yes	characteristic
65	#12061	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	#1579	1	1	yes	characteristic
67	C₂×C₇₃	#2169	1	1	yes	characteristic
68	C₂×C₇₃	#2171	1	1	yes	characteristic
69	#2183	—	73	2	no	—
70	C₈×Q₁₆	—	73	1	no	—
71	#1367	—	73	1	no	—
72	Q₈⋊C₁₆	—	73	2	no	—
73	#1312	—	73	1	no	—
74	C₇₃	#12061	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	73	2	no	—
76	C₄×Q₁₆	—	73	1	no	—
77	C₄⋊C₁₆	—	73	2	no	—
78	C₄⋊C₁₆	—	146	2	no	—
79	#346	—	146	2	no	—
80	C₄×C₁₆	—	73	1	no	—
81	C₄×C₁₆	—	73	1	no	—
82	#335	—	73	1	no	—
83	Q₈⋊C₈	—	73	2	no	—
84	C₈²	—	73	1	no	—
85	C₂×Q₁₆	#10596842	1	1	yes	characteristic
86	C₄×Q₈	—	73	2	no	—
87	C₂×C₁₆	—	73	2	no	—
88	C₂×C₁₆	—	146	1	no	—
89	C₂×C₁₆	—	146	2	no	—
90	#107	—	73	1	no	—
91	C₄⋊C₈	—	73	2	no	—
92	C₄⋊C₈	—	146	2	no	—
93	#103	—	73	2	no	—
94	C₄×C₈	—	73	1	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	146	2	no	—
98	C₂×Q₈	C₂×#10596842	1	2	yes	—
99	Q₁₆	—	2	2	no	—
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	146	1	no	—
104	C₂×C₈	—	146	2	no	—
105	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
106	C₄⋊C₄	—	73	2	no	—
107	C₄⋊C₄	—	146	2	no	—
108	C₄²	—	73	1	no	—
109	C₄²	—	146	2	no	—
110	C₁₆	—	292	2	no	—
111	C₁₆	—	292	1	no	—
112	C₁₆	—	146	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	73	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	146	2	no	—
119	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
120	C₈	—	292	2	no	—
121	C₈	—	292	1	no	—
122	C₈	—	73	2	no	—
123	C₈	—	146	1	no	—
124	C₈	—	146	1	no	—
125	C₈	#14556687	1	2	yes	—
126	C₂²	D₄×#10596842	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	292	2	no	—
129	C₄	—	73	1	no	—
130	C₄	—	73	1	no	—
131	C₄	—	146	1	no	—
132	C₄	C₂×#14556687	1	1	yes	characteristic
133	C₄	#17989708	1	1	yes	characteristic
134	C₂	#22627061	1	1	yes	characteristic
135	C₂	#22627147	1	1	yes	characteristic
136	C₂	#22627152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627157

C₂

yes

maximal

#22626989

C₂

yes

maximal, characteristic

Q₈⋊#11924040

C₂

yes

maximal

#22626948

C₂

yes

maximal, characteristic

Q₁₆×#10596833

C₂

yes

maximal, characteristic

#17989690

—

C₄⋊#11924040

C₂²

yes

—

#17989681

—

C₄×#11924040

C₂²

yes

characteristic

C₄×#11924040

C₂²

yes

characteristic

Q₈×#10596833

C₂²

yes

—

#17989450

C₄

yes

characteristic

Q₈⋊#10596833

C₄

yes

—

C₈×#10596833

C₂²

yes

characteristic

Q₁₆×Dic₇₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#10596833

—

C₄⋊#10596833

C₂×C₄

yes

—

C₄×#10596833

—

C₄×#10596833

C₂×C₄

yes

characteristic

C₄×#10596833

C₂³

yes

characteristic

Q₈×Dic₇₃

C₂×C₄

yes

—

#14556525

C₈

yes

characteristic

#14556515

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂×Q₁₆×C₇₃

C₈

yes

—

—

—

—

—

C₂×D₄

yes

yes

C₂×C₈

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄×D₄

yes

C₂×Dic₇₃

yes

—

—

—

yes

—

C₈×C₇₃

M₃₂

yes

—

#10596831

—

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₇₃

#12061

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

#346

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

#10596842

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂×Q₈

C₂×#10596842

yes

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₂×C₈

C₂×#10596842

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

146

—

108

C₄²

—

109

C₄²

—

146

—

110

C₁₆

—

292

—

111

C₁₆

—

292

—

112

C₁₆

—

146

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

146

—

119

C₂×C₄

C₂²×#10596842

yes

characteristic

120

C₈

—

292

—

121

C₈

—

292

—

122

C₈

—

123

C₈

—

146

—

124

C₈

—

146

—

125

C₈

#14556687

yes

—

126

C₂²

D₄×#10596842

yes

characteristic

127

C₄

—

128

C₄

—

292

—

129

C₄

—

130

C₄

—

131

C₄

—

146

—

132

C₄

C₂×#14556687

yes

characteristic

133

C₄

#17989708

yes

characteristic

134

C₂

#22627061

yes

characteristic

135

C₂

#22627147

yes

characteristic

136

C₂

#22627152

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#10596842
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₂×Q₁₆×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	C₈×D₄
cosocle	C₄×#10596833	C₂³
hypercenter	C₂×Q₁₆	#10596842
hypocenter	C₇₃	#12061
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#10596842

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10596842

Fitting

C₂×Q₁₆×C₇₃

C₈

socle, prime radical

C₂²×C₇₃

C₈×D₄

C₂³

C₂×Q₁₆

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
74. C₇₃	#12061	semidirect

H ≅ G/N

type

74. C₇₃

#12061

semidirect

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

subgroup	index
#22627157	2
#22626989	2
Q₈⋊#11924040	2
#22626948	2
Q₁₆×#10596833	2
#12061	73

subgroup

index

Q₁₆×#10596833

Subgroups — #34034007

Conjugacy classes of subgroups

Complements

Maximal subgroups