Group #34033948 — subgroups

Subgroups — #34033948

Order 18688.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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	Q₁₆×#10596842	C₂	1	2	yes	maximal
2	#22626998	C₂	1	1	yes	maximal, characteristic
3	#22626978	C₂	1	2	yes	maximal
4	#22626946	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×#64408	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596842	—	2	2	no	—
7	#64408⋊C₁₆	C₂²	1	1	yes	characteristic
8	#17989677	C₂²	1	2	yes	—
9	Q₈⋊#10596842	—	2	2	no	—
10	C₈×#10596842	C₂²	1	1	yes	characteristic
11	Q₁₆×#64408	C₂²	1	2	yes	—
12	#17989513	C₄	1	1	yes	characteristic
13	#17989500	C₄	1	2	yes	—
14	C₁₆×#64408	C₂²	1	1	yes	characteristic
15	Q₃₂×D₇₃	C₄	1	1	yes	characteristic
16	#14556686	—	2	1	no	—
17	C₄⋊#10596842	—	2	2	no	—
18	C₄⋊#10596842	—	4	2	no	—
19	C₄×#10596842	—	4	2	no	—
20	C₄×#10596842	D₄	1	1	yes	characteristic
21	C₄×#10596842	D₄	1	1	yes	characteristic
22	Q₈×#64408	—	2	2	no	—
23	#14556664	C₂×C₄	1	1	yes	characteristic
24	#14556660	C₂×C₄	1	2	yes	—
25	#14556654	—	2	2	no	—
26	C₈×#64408	C₂³	1	1	yes	characteristic
27	Q₁₆×D₇₃	C₂×C₄	1	2	yes	—
28	Dic₅₈₄	C₈	1	1	yes	characteristic, complemented
29	#14556537	C₈	1	2	yes	complemented
30	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
31	Q₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
32	C₂×#10596842	—	2	1	no	—
33	C₂×#10596842	—	4	2	no	—
34	C₂×#10596842	D₈	1	2	yes	—
35	#11924040	—	2	2	no	—
36	#11924036	#41	1	1	yes	characteristic
37	C₄⋊#64408	—	2	2	no	—
38	C₄⋊#64408	—	4	2	no	—
39	C₄×#64408	—	4	2	no	—
40	C₄×#64408	C₂×D₄	1	1	yes	characteristic
41	#11924031	—	2	1	no	—
42	Q₈×D₇₃	—	2	2	no	—
43	Dic₂₉₂	C₂×C₈	1	2	yes	—
44	#11924018	—	2	2	no	—
45	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
46	#11924003	C₂×C₈	1	1	yes	characteristic
47	Q₁₆×C₇₃	C₂×C₈	1	2	yes	—
48	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×#64408	—	4	2	no	—
50	C₂×#64408	C₂×D₈	1	1	yes	characteristic
51	C₂×#64408	#135	1	1	yes	characteristic
52	#10596842	—	2	2	no	—
53	#10596842	—	4	1	no	—
54	#10596842	—	8	2	no	—
55	Q₈×C₇₃	—	2	2	no	—
56	C₄×D₇₃	—	4	2	no	—
57	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
58	Dic₁₄₆	—	2	2	no	—
59	Dic₁₄₆	—	4	2	no	—
60	#10596833	—	2	1	no	—
61	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
62	#10596831	#131	1	1	yes	characteristic
63	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
64	#64408	—	2	1	no	—
65	#64408	—	8	2	no	—
66	#64408	#510	1	2	yes	—
67	C₄×C₇₃	—	4	2	no	—
68	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
69	Dic₇₃	—	4	2	no	—
70	Dic₇₃	#443	1	1	yes	characteristic
71	#12044	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
73	D₇₃	#2176	1	1	yes	characteristic
74	D₇₃	#2179	1	1	yes	characteristic
75	C₄×Q₃₂	—	73	1	no	—
76	C₈×Q₁₆	—	73	2	no	—
77	#1372	—	73	1	no	—
78	#1332	—	73	2	no	—
79	#1311	—	73	1	no	—
80	C₇₃	#12044	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₃₂	—	73	1	no	—
82	C₈×Q₈	—	146	2	no	—
83	C₄×Q₁₆	—	73	2	no	—
84	#366	—	73	1	no	—
85	#358	—	73	2	no	—
86	C₄×C₁₆	—	73	1	no	—
87	C₄×C₁₆	—	73	1	no	—
88	#335	—	73	2	no	—
89	Q₈⋊C₈	—	146	2	no	—
90	C₈²	—	73	1	no	—
91	C₂×Q₁₆	—	73	2	no	—
92	C₄×Q₈	—	146	2	no	—
93	Q₃₂	—	73	2	no	—
94	Q₃₂	—	73	1	no	—
95	Q₃₂	#10596842	1	1	yes	characteristic, complemented
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	—	146	1	no	—
99	#107	—	73	2	no	—
100	C₄⋊C₈	—	292	2	no	—
101	C₄⋊C₈	—	146	2	no	—
102	#103	—	146	2	no	—
103	C₄×C₈	—	292	2	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₂×Q₈	—	146	2	no	—
108	Q₁₆	—	73	2	no	—
109	Q₁₆	—	146	2	no	—
110	Q₁₆	C₂×#10596842	1	2	yes	—
111	C₂×C₈	—	292	2	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	146	1	no	—
116	C₄⋊C₄	—	292	2	no	—
117	C₄⋊C₄	—	146	2	no	—
118	C₄²	—	292	2	no	—
119	C₄²	—	73	1	no	—
120	C₁₆	—	73	1	no	—
121	C₁₆	—	146	1	no	—
122	C₁₆	—	146	2	no	—
123	C₁₆	C₂×#10596842	1	1	yes	characteristic
124	Q₈	—	2	2	no	—
125	Q₈	—	292	2	no	—
126	Q₈	—	146	2	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	—	292	2	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₈	—	292	1	no	—
133	C₈	—	584	2	no	—
134	C₈	—	73	1	no	—
135	C₈	—	146	1	no	—
136	C₈	—	146	2	no	—
137	C₈	C₂²×#10596842	1	1	yes	characteristic
138	C₂²	—	73	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	292	2	no	—
141	C₄	—	584	2	no	—
142	C₄	—	73	2	no	—
143	C₄	—	73	1	no	—
144	C₄	—	146	1	no	—
145	C₄	D₄×#10596842	1	1	yes	characteristic
146	C₂	—	73	1	no	—
147	C₂	—	73	1	no	—
148	C₂	D₈×#10596842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#10596842

C₂

yes

maximal

#22626998

C₂

yes

maximal, characteristic

#22626978

C₂

yes

maximal

#22626946

C₂

yes

maximal, characteristic

Q₃₂×#64408

C₂

yes

maximal, characteristic

Q₈×#10596842

—

#64408⋊C₁₆

C₂²

yes

characteristic

#17989677

C₂²

yes

—

Q₈⋊#10596842

—

C₈×#10596842

C₂²

yes

characteristic

Q₁₆×#64408

C₂²

yes

—

#17989513

C₄

yes

characteristic

#17989500

C₄

yes

—

C₁₆×#64408

C₂²

yes

characteristic

Q₃₂×D₇₃

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#10596842

D₄

yes

characteristic

Q₈×#64408

—

#14556664

C₂×C₄

yes

characteristic

#14556660

C₂×C₄

yes

—

#14556654

—

C₈×#64408

C₂³

yes

characteristic

Q₁₆×D₇₃

C₂×C₄

yes

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556537

C₈

yes

complemented

C₁₆×D₇₃

C₂×C₄

yes

characteristic

Q₃₂×C₇₃

C₈

yes

characteristic, complemented

C₂×#10596842

—

C₂×#10596842

—

C₂×#10596842

D₈

yes

—

—

yes

—

—

—

C₂×D₄

yes

characteristic

#11924031

—

Q₈×D₇₃

—

Dic₂₉₂

C₂×C₈

yes

—

#11924018

—

C₈×D₇₃

C₂²×C₄

yes

characteristic

#11924003

C₂×C₈

yes

characteristic

Q₁₆×C₇₃

C₂×C₈

yes

—

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×#64408

—

C₂×#64408

C₂×D₈

yes

yes

—

—

—

Q₈×C₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₄×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

C₂²×C₈

yes

yes

C₂×D₇₃

C₄×D₈

yes

—

—

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

—

C₂×C₇₃

C₈×D₈

yes

D₇₃

yes

D₇₃

yes

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₇₃

#12044

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

146

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

—

C₂×Q₁₆

—

C₄×Q₈

—

146

—

Q₃₂

—

Q₃₂

—

Q₃₂

#10596842

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#107

—

100

C₄⋊C₈

—

292

—

101

C₄⋊C₈

—

146

—

102

#103

—

146

—

103

C₄×C₈

—

292

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₂×Q₈

—

146

—

108

Q₁₆

—

109

Q₁₆

—

146

—

110

Q₁₆

C₂×#10596842

yes

—

111

C₂×C₈

—

292

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

292

—

117

C₄⋊C₄

—

146

—

118

C₄²

—

292

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

146

—

122

C₁₆

—

146

—

123

C₁₆

C₂×#10596842

yes

characteristic

124

Q₈

—

125

Q₈

—

292

—

126

Q₈

—

146

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₈

—

292

—

133

C₈

—

584

—

134

C₈

—

135

C₈

—

146

—

136

C₈

—

146

—

137

C₈

C₂²×#10596842

yes

characteristic

138

C₂²

—

139

C₄

—

140

C₄

—

292

—

141

C₄

—

584

—

142

C₄

—

143

C₄

—

144

C₄

—

146

—

145

C₄

D₄×#10596842

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

D₈×#10596842

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#10596842
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₈	C₂²×#10596842
Fitting	Q₃₂×C₇₃	C₈
socle	C₂×C₇₃	C₈×D₈
prime radical	C₂×D₇₃	C₄×D₈
cosocle	C₈×#64408	C₂³
hypercenter	Q₃₂	#10596842
hypocenter	C₇₃	#12044
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

D₈×#10596842

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₈

C₂²×#10596842

Fitting

Q₃₂×C₇₃

C₈

socle

C₂×C₇₃

C₈×D₈

prime radical

C₂×D₇₃

C₄×D₈

C₂³

Q₃₂

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
28. Dic₅₈₄	C₈	semidirect
29. #14556537	C₈	semidirect
31. Q₃₂×C₇₃	C₈	semidirect
80. C₇₃	#12044	semidirect
95. Q₃₂	#10596842	semidirect

H ≅ G/N

type

28. Dic₅₈₄

C₈

semidirect

29. #14556537

C₈

semidirect

31. Q₃₂×C₇₃

C₈

semidirect

80. C₇₃

#12044

semidirect

95. Q₃₂

#10596842

semidirect

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

subgroup	index
Q₁₆×#10596842	2
#22626998	2
#22626978	2
#22626946	2
Q₃₂×#64408	2
#12044	73

subgroup

index

Q₁₆×#10596842

Q₃₂×#64408

Subgroups — #34033948

Conjugacy classes of subgroups

Complements

Maximal subgroups