Finite Group Databasebeta

← #37544956

Subgroups — #37544956

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#26041808C211yesmaximal, characteristic
3#26041688C211yesmaximal, characteristic, complemented
4#26041685C211yesmaximal, characteristic, complemented
5#26041638C211yesmaximal, characteristic, complemented
6#26041618C212yesmaximal, complemented
7#26041601C212yesmaximal
8#26041588C212yesmaximal, complemented
9#26041581C212yesmaximal, complemented
10#26041334C211yesmaximal, characteristic, complemented
11#489×C181C211yesmaximal, characteristic, complemented
12#19153816C2211yescharacteristic
13#19153809—21no—
14#19153808C2211yescharacteristic
15#19153807C2212yes—
16#19153798—21no—
17#19153796—21no—
18#19153792—21no—
19#19153792C2211yescharacteristic
20#19153787C2211yescharacteristic
21Q8×Dic181—21no—
22Q8×Dic181C2211yescharacteristic
23#19153781—21no—
24D4×Dic181C2211yescharacteristic
25#19153776—21no—
26#19153774C2212yes—
27#19153772—21no—
28C42×D181C2211yescharacteristic
29#19153750C2212yes—
30#19153749C2212yes—
31#19153748C2212yes—
32#19153747C2212yes—
33#19153745C2212yes—
34#19153740C2212yes—
35#19153737C2212yes—
36#19153728C2212yes—
37#19153727C2211yescharacteristic
38C4⋊Q8×C181C2211yescharacteristic
39#124×C181C2211yescharacteristic
40#103×C181C2212yes—
41#102×C181C2212yes—
42#97×C181C2211yescharacteristic
43C2×Dic362—21no—
44C2×C22⋊D181—21no—
45C22×Dic181—21no—
46C2×C4×D181D411yescharacteristic
47C2×C4×D181D411yescharacteristic
48C2×C4×D181C2311yescharacteristic
49C2×#13089678C2312yes—
50C4⋊Dic181—22no—
51C4⋊Dic181—21no—
52C4⋊Dic181—21no—
53C4⋊Dic181C2312yes—
54C4⋊Dic181C2311yescharacteristic
55C4⋊Dic181C2311yescharacteristic
56#15668428—21no—
57#15668428—22no—
58#15668428—21no—
59#15668428—21no—
60C22⋊Dic181—21no—
61C22⋊Dic181—21no—
62#15668426—22no—
63#15668426—21no—
64C4×Dic181—21no—
65C4×Dic181—22no—
66C4×Dic181D411yescharacteristic
67C4×Dic181D411yescharacteristic
68C4×Dic181C2311yescharacteristic
69C2×Q8×C181—21no—
70C2×Q8×C181C2311yescharacteristic
71C2×D4×C181C2311yescharacteristic
72C2×C8×C181C2312yes—
73C4⋊C4×C181—21no—
74C4⋊C4×C181C2312yes—
75C22⋊C4×C181—21no—
76C22⋊C4×C181—21no—
77C42×C181C2311yescharacteristic
78C22×D181C2×D411yescharacteristic
79Dic362—42no—
80C22⋊D181—42no—
81C2×Dic181—22no—
82C2×Dic181—21no—
83C2×Dic181—21no—
84C2×Dic181—42no—
85C2×Dic181C2×D411yescharacteristic
86C2×Dic181C2×D411yescharacteristic
87C2×Dic181C2×D411yescharacteristic
88C4×D181—22no—
89C4×D181—41no—
90C4×D181—41no—
91#13089678—22no—
92C23×C181—21no—
93Q8×C181—22no—
94Q8×C181—42no—
95D4×C181—22no—
96C2×C4×C181—21no—
97C2×C4×C181—22no—
98C2×C4×C181C2×D411yescharacteristic
99C2×C4×C181C2×D411yescharacteristic
100C2×C4×C181C2411yescharacteristic
101C8×C181—22no—
102C22×C181—41no—
103C22×C181—41no—
104C22×C181—41no—
105C22×C181C22×D411yescharacteristic
106C2×D181—21no—
107C2×D181—21no—
108C2×D181—21no—
109C4×C181—21no—
110C4×C181—21no—
111C4×C181—41no—
112C4×C181—42no—
113C4×C1812−1+412yes—
114Dic181—21no—
115Dic181—21no—
116Dic181—21no—
117Dic181—41no—
118Dic181—41no—
119Dic181—42no—
120C2×C181—81no—
121C2×C181#53611yescharacteristic
122C2×C181#57511yescharacteristic
123C2×C181#57511yescharacteristic
124D181—41no—
125C181#324011yescharacteristic, complemented, Sylow
126#3240—1811nomaximal, Sylow
127#533—1811no—
128#529—1811no—
129#501—1811no—
130#501—1811no—
131#489—1811no—
132#489D18111yescharacteristic, complemented
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#125—1811no—
143#124C2×D18111yescharacteristic
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#107—1812no—
159#106—1812no—
160#106—1812no—
161C4⋊C8—1812no—
162#103—1812no—
163#103C2×D18112yes—
164#102—1812no—
165#102C2×D18112yes—
166C22⋊C8—1812no—
167#97—1811no—
168#97C2×D18111yescharacteristic
169C2×Q8—21no—
170C2×Q8—3621no—
171C2×Q8C22×D18111yescharacteristic
172C2×D4—3621no—
173C2×D4C22×D18111yescharacteristic
174C22×C4—3621no—
175C22×C4—1811no—
176C22×C4—1811no—
177C22×C4—1811no—
178C2×C8—1812no—
179C2×C8C22×D18112yes—
180C4⋊C4—21no—
181C4⋊C4—3621no—
182C4⋊C4—3622no—
183C4⋊C4—3622no—
184C4⋊C4—3621no—
185C4⋊C4—3621no—
186C4⋊C4—3621no—
187C4⋊C4—3621no—
188C4⋊C4—1812no—
189C4⋊C4—1811no—
190C4⋊C4—1811no—
191C4⋊C4C22×D18112yes—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—3622no—
195C22⋊C4—3621no—
196C22⋊C4—3621no—
197C22⋊C4—3621no—
198C42—3621no—
199C42—3622no—
200C42—1811no—
201C42—1811no—
202C42—1811no—
203C42C22×D18111yescharacteristic
204C23—21no—
205C23—1811no—
206Q8—22no—
207Q8—42no—
208Q8—7242no—
209D4—22no—
210D4—7242no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—3622no—
214C2×C4—3621no—
215C2×C4—3622no—
216C2×C4—3621no—
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—41no—
229C22—41no—
230C22—41no—
231C22—3621no—
232C22—3621no—
233C22—3621no—
234C22C2×D4×D18111yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—41no—
239C4—3621no—
240C4—3621no—
241C4—3621no—
242C4—7241no—
243C4—7241no—
244C4—7242no—
245C4#1915386312yes—
246C2—81no—
247C2—7241no—
248C2#2604184911yescharacteristic
249C2#2604207511yescharacteristic
250C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#489×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#489D181
hypocenterC181#3240
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #26041834C2semidirect
3. #26041688C2semidirect
4. #26041685C2semidirect
5. #26041638C2semidirect
6. #26041618C2semidirect
8. #26041588C2semidirect
9. #26041581C2semidirect
10. #26041334C2semidirect
11. #489×C181C2semidirect
125. C181#3240semidirect
132. #489D181semidirect

Maximal subgroups

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

subgroupindex
#260418342
#260418082
#260416882
#260416852
#260416382
#260416182
#260416012
#260415882
#260415812
#260413342
#489×C1812
#3240181