Finite Group Databasebeta

← #37544964

Subgroups — #37544964

Order 23168.

328 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041834C211yesmaximal, characteristic, complemented
2#26041814C211yesmaximal, characteristic
3#26041689C211yesmaximal, characteristic, complemented
4#26041641C211yesmaximal, characteristic, complemented
5#26041639C211yesmaximal, characteristic, complemented
6#26041623C212yesmaximal, complemented
7#26041606C212yesmaximal
8#26041585C212yesmaximal, complemented
9#26041578C212yesmaximal, complemented
10#26041334C211yesmaximal, characteristic, complemented
11#501×C181C211yesmaximal, characteristic, complemented
12#19153816C2211yescharacteristic
13#19153811—21no—
14#19153807C2212yes—
15#19153804C2211yescharacteristic
16#19153801—21no—
17#19153796—21no—
18#19153796C2211yescharacteristic
19#19153792—21no—
20#19153791C2211yescharacteristic
21Q8×Dic181—21no—
22#19153781—21no—
23#19153781C2211yescharacteristic
24#19153778—21no—
25#19153778C2211yescharacteristic
26#19153774—21no—
27#19153774C2212yes—
28C42×D181C2211yescharacteristic
29#19153749C2212yes—
30#19153747C2212yes—
31#19153745C2212yes—
32#19153741C2212yes—
33#19153739C2212yes—
34#19153738C2212yes—
35#19153736C2212yes—
36#19153728C2212yes—
37#19153727C2211yescharacteristic
38C4⋊Q8×C181C2211yescharacteristic
39#125×C181C2211yescharacteristic
40#107×C181C2212yes—
41#106×C181C2212yes—
42#97×C181C2211yescharacteristic
43C2×Dic362—21no—
44C2×Dic362C2311yescharacteristic
45C2×D724—21no—
46C2×D724C2311yescharacteristic
47C2×C4×D181—21no—
48C2×C4×D181D411yescharacteristic
49C2×C4×D181D411yescharacteristic
50C2×C4×D181C2311yescharacteristic
51C2×#13089678C2312yes—
52C4⋊Dic181—21no—
53C4⋊Dic181—22no—
54C4⋊Dic181—21no—
55C4⋊Dic181C2312yes—
56#15668428—22no—
57#15668428—21no—
58#15668426—22no—
59#15668426—21no—
60#15668426—21no—
61#15668426—21no—
62#15668426—21no—
63#15668426—21no—
64C4×Dic181—21no—
65C4×Dic181—22no—
66C4×Dic181D411yescharacteristic
67C4×Dic181D411yescharacteristic
68C4×Dic181C2311yescharacteristic
69C2×Q8×C181—21no—
70C2×C8×C181C2312yes—
71C4⋊C4×C181—21no—
72C4⋊C4×C181—21no—
73C4⋊C4×C181—21no—
74C4⋊C4×C181C2311yescharacteristic
75C4⋊C4×C181C2312yes—
76C4⋊C4×C181C2311yescharacteristic
77C42×C181C2311yescharacteristic
78C22×D181—21no—
79C22×D181C2×D411yescharacteristic
80Dic362—22no—
81Dic362—42no—
82D724—22no—
83D724—42no—
84C2×Dic181—21no—
85C2×Dic181—22no—
86C2×Dic181C2×D411yescharacteristic
87C2×Dic181C2×D411yescharacteristic
88C2×Dic181C2×D411yescharacteristic
89C4×D181—22no—
90C4×D181—41no—
91C4×D181—41no—
92C4×D181—42no—
93#13089678—22no—
94Q8×C181—42no—
95C2×C4×C181—21no—
96C2×C4×C181—21no—
97C2×C4×C181—22no—
98C2×C4×C181C2×D411yescharacteristic
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2411yescharacteristic
101C8×C181—22no—
102C22×C181C22×D411yescharacteristic
103C2×D181—21no—
104C2×D181—21no—
105C2×D181—21no—
106C2×D181—41no—
107C2×D181—41no—
108C2×D181—41no—
109C4×C181—21no—
110C4×C181—21no—
111C4×C181—42no—
112C4×C181—41no—
113C4×C181—41no—
114C4×C1812−1+412yes—
115Dic181—21no—
116Dic181—21no—
117Dic181—21no—
118Dic181—41no—
119Dic181—42no—
120C2×C181#53611yescharacteristic
121C2×C181#57511yescharacteristic
122C2×C181#57511yescharacteristic
123D181—41no—
124D181—81no—
125C181#324011yescharacteristic, complemented, Sylow
126#3240—1811nomaximal, Sylow
127#533—1811no—
128#529—1811no—
129#501—1811no—
130#501D18111yescharacteristic, complemented
131#489—1811no—
132#489—1811no—
133#485—1812no—
134#482—1812no—
135#479—1812no—
136#478—1812no—
137#432—1811no—
138C4⋊Q8—1811no—
139C4⋊Q8C2×D18111yescharacteristic
140#126—3621no—
141#125—1812no—
142#125C2×D18111yescharacteristic
143#124—1811no—
144#123—3621no—
145C22⋊Q8—3621no—
146C22⋊Q8—3621no—
147C22⋊Q8—1811no—
148#121—1811no—
149C4×Q8—3621no—
150C4×Q8—3621no—
151C4×Q8—1811no—
152C4×D4—3621no—
153C4×D4—1811no—
154#117—3621no—
155#117—1812no—
156C2×C42—1811no—
157#107—1812no—
158#107C2×D18112yes—
159#106—1812no—
160#106C2×D18112yes—
161C4⋊C8—1812no—
162#103—1812no—
163#103—1812no—
164#102—1812no—
165#102—1812no—
166C22⋊C8—1812no—
167#97—1811no—
168#97C2×D18111yescharacteristic
169C2×Q8—21no—
170C2×Q8—3621no—
171C2×Q8—1811no—
172C2×D4—3621no—
173C2×D4—1811no—
174C22×C4—3621no—
175C22×C4—1811no—
176C22×C4—1811no—
177C22×C4—1811no—
178C2×C8—1812no—
179C2×C8C22×D18112yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—3621no—
184C4⋊C4—3622no—
185C4⋊C4—3622no—
186C4⋊C4—3621no—
187C4⋊C4—3621no—
188C4⋊C4—1812no—
189C4⋊C4C22×D18112yes—
190C4⋊C4C22×D18111yescharacteristic
191C4⋊C4C22×D18111yescharacteristic
192C22⋊C4—3622no—
193C22⋊C4—3621no—
194C22⋊C4—3621no—
195C22⋊C4—3621no—
196C22⋊C4—3621no—
197C22⋊C4—3621no—
198C42—3621no—
199C42—3622no—
200C42—1811no—
201C42—1811no—
202C42—1811no—
203C42C22×D18111yescharacteristic
204C23—3621no—
205C23—1811no—
206Q8—42no—
207Q8—3622no—
208Q8—7242no—
209D4—3622no—
210D4—7242no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—3622no—
215C2×C4—3621no—
216C2×C4—3622no—
217C2×C4—1811no—
218C2×C4—1811no—
219C2×C4—1811no—
220C2×C4—7241no—
221C2×C4—7241no—
222C2×C4—7242no—
223C2×C4D4×D18111yescharacteristic
224C2×C4D4×D18111yescharacteristic
225C2×C4C23×D18111yescharacteristic
226C8—22no—
227C8—3622no—
228C22—3621no—
229C22—3621no—
230C22—3621no—
231C22—7241no—
232C22—7241no—
233C22—7241no—
234C22C2×D4×D18111yescharacteristic
235C4—21no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—3621no—
241C4—3621no—
242C4—3621no—
243C4—7242no—
244C4—7241no—
245C4#1915386412yes—
246C2—14481no—
247C2—7241no—
248C2#2604185411yescharacteristic
249C2#2604207511yescharacteristic
250C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#501×C181C2
socleC22×C181C22×D4
prime radical#19153791C22
hypercenter#501D181
hypocenterC181#3240
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041689C2semidirect
4. #26041641C2semidirect
5. #26041639C2semidirect
6. #26041623C2semidirect
8. #26041585C2semidirect
9. #26041578C2semidirect
10. #26041334C2semidirect
11. #501×C181C2semidirect
125. C181#3240semidirect
130. #501D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418142
#260416892
#260416412
#260416392
#260416232
#260416062
#260415852
#260415782
#260413342
#501×C1812
#3240181