Finite Group Databasebeta

← #37543948

Subgroups — #37543948

Order 23168.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041894C211yesmaximal, characteristic
2#26041755C211yesmaximal, characteristic, complemented
3#26041622C211yesmaximal, characteristic, complemented
4#26041602C211yesmaximal, characteristic
5#26041573C211yesmaximal, characteristic, complemented
6#26041562C211yesmaximal, characteristic, complemented
7#26041500C211yesmaximal, characteristic, complemented
8#26041499C211yesmaximal, characteristic, complemented
9#26041436C211yesmaximal, characteristic, complemented
10#26041416C211yesmaximal, characteristic
11#26041387C211yesmaximal, characteristic, complemented
12C4×#15668439C211yesmaximal, characteristic, complemented
13C4×#15668437C211yesmaximal, characteristic, complemented
14#26041341C211yesmaximal, characteristic, complemented
15#440×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×Dic724C2211yescharacteristic
18C2×#15668439C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153809—21no—
22#19153803C2211yescharacteristic
23#19153802—21no—
24#19153799—21no—
25#19153796C2211yescharacteristic
26#19153795—21no—
27#19153787C2211yescharacteristic
28Q8×Dic181C2211yescharacteristic
29C4×Dic362—21no—
30C4×Dic362C2211yescharacteristic
31C4×Dic362C2211yescharacteristic
32#19153781—21no—
33#19153781C2211yescharacteristic
34#19153774—21no—
35#19153774C2211yescharacteristic
36#19153773—21no—
37#19153773C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153748C2211yescharacteristic
40#19153747C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51C4×#13089678C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362C2311yescharacteristic
62C2×Dic362C2311yescharacteristic
63Q8×D181—22no—
64Q8×D181—22no—
65C2×C22⋊D181—21no—
66C2×C4×D181—21no—
67C2×C4×D181C2311yescharacteristic
68Dic724—22no—
69#15668439—22no—
70#15668437—22no—
71C2×#13089678C2311yescharacteristic
72C2×#13089678C2311yescharacteristic
73C4⋊Dic181—21no—
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82#15668428D411yescharacteristic
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426D411yescharacteristic
89#15668426D411yescharacteristic
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181—21no—
93C4×Dic181—21no—
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96C2×Q8×C181C2311yescharacteristic
97C2×D4×C181C2311yescharacteristic
98C22×C4×C181—21no—
99SD16×C181—22no—
100C2×C8×C181C2311yescharacteristic
101C2×C8×C181C2311yescharacteristic
102C4⋊C4×C181—21no—
103C4⋊C4×C181C2311yescharacteristic
104C4⋊C4×C181C2311yescharacteristic
105C22⋊C4×C181—21no—
106C42×C181—21no—
107C42×C181C2311yescharacteristic
108C22×D181C2×D411yescharacteristic
109Dic362—24no—
110Dic362—21no—
111Dic362—21no—
112Dic362—21no—
113Dic362—81no—
114Dic362#4112yes—
115C22⋊D181—81no—
116C2×Dic181—21no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C2×Dic181C2×D411yescharacteristic
123C4×D181—24no—
124C4×D181—21no—
125C4×D181—21no—
126#13089678—21no—
127#13089678—21no—
128C23×C181—21no—
129Q8×C181—21no—
130Q8×C181#4112yes—
131D4×C181—21no—
132D4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—81no—
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2×D411yescharacteristic
139C2×C4×C181C2411yescharacteristic
140C8×C181—21no—
141C8×C181—21no—
142C22×C181—41no—
143C22×C181—41no—
144C22×C181—41no—
145C22×C181C22×D411yescharacteristic
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—21no—
149C4×C181—22no—
150C4×C181—21no—
151C4×C181—21no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181C2×#4111yescharacteristic
155C4×C181C2×#4111yescharacteristic
156Dic181—22no—
157Dic181—21no—
158Dic181—21no—
159Dic181—21no—
160Dic181—41no—
161Dic181—41no—
162Dic181—41no—
163C2×C181—81no—
164C2×C181#52511yescharacteristic
165C2×C181#57711yescharacteristic
166C2×C181#57811yescharacteristic
167D181—41no—
168C181#312111yescharacteristic, complemented, Sylow
169#3121—1811nomaximal, Sylow
170#542—1811no—
171#519—1811no—
172#485—1811no—
173#482—1811no—
174#477—1811no—
175#475—1811no—
176#464—1811no—
177#464—1811no—
178#451—1811no—
179#448—1811no—
180#441—1811no—
181#440D18111yescharacteristic, complemented
182C4×Q16—1811no—
183C4×SD16—1811no—
184#433—1811no—
185C22×Q8—1811no—
186C2×Q16—1811no—
187C2×Q16—1811no—
188C2×SD16—1811no—
189C2×SD16C2×D18111yescharacteristic
190C4⋊Q8—1811no—
191#126—3621no—
192#124—3621no—
193#124—1811no—
194#123—3621no—
195C22⋊Q8—3621no—
196C22⋊Q8—1811no—
197#121—1811no—
198C4×Q8—3621no—
199C4×Q8—3621no—
200C4×Q8—1811no—
201C4×Q8—1811no—
202C4×Q8—1811no—
203C4×Q8—1811no—
204C4×Q8C2×D18111yescharacteristic
205C4×D4C2×D18111yescharacteristic
206#117—3621no—
207#117—3621no—
208#117—1811no—
209#117—1811no—
210#107—1811no—
211#107C2×D18111yescharacteristic
212#106—1811no—
213#106—1811no—
214C4⋊C8—1811no—
215C4⋊C8—1811no—
216#103—1811no—
217#103—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#103C2×D18111yescharacteristic
222#102—1811no—
223#102C2×D18111yescharacteristic
224C22⋊C8—1811no—
225C22⋊C8—1811no—
226#97C2×D18111yescharacteristic
227C4×C8—1811no—
228C2×Q8—3621no—
229C2×Q8—3621no—
230C2×Q8—3622no—
231C2×Q8—3622no—
232C2×Q8—1811no—
233C2×Q8—1811no—
234C2×Q8C22×D18111yescharacteristic
235C2×D4—3621no—
236C2×D4C22×D18111yescharacteristic
237C22×C4—21no—
238C22×C4—3621no—
239C22×C4—1811no—
240Q16—3622no—
241Q16—3622no—
242SD16—22no—
243SD16—3622no—
244C2×C8—1811no—
245C2×C8—1811no—
246C2×C8C22×D18111yescharacteristic
247C2×C8C22×D18111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4C22×D18111yescharacteristic
260C4⋊C4C22×D18111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—1811no—
268C22⋊C4—1811no—
269C42—21no—
270C42—3621no—
271C42—3621no—
272C42—3621no—
273C42—3621no—
274C42—3621no—
275C42—1811no—
276C42C22×D18111yescharacteristic
277C23—21no—
278C23—1811no—
279Q8—21no—
280Q8—3624no—
281Q8—3621no—
282Q8—3621no—
283Q8—3621no—
284Q8—14481no—
285Q8—1812no—
286Q8#1566844912yes—
287D4—21no—
288D4—21no—
289D4—14481no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—81no—
294C2×C4—3624no—
295C2×C4—3621no—
296C2×C4—3621no—
297C2×C4—3621no—
298C2×C4—3621no—
299C2×C4—3621no—
300C2×C4—3621no—
301C2×C4—1811no—
302C2×C4—1811no—
303C2×C4—1811no—
304C2×C4D4×D18111yescharacteristic
305C2×C4D4×D18111yescharacteristic
306C2×C4C23×D18111yescharacteristic
307C8—21no—
308C8—21no—
309C8—3621no—
310C8—3621no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—3621no—
315C22—3621no—
316C22—3621no—
317C22C2×D4×D18111yescharacteristic
318C4—22no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—3622no—
324C4—3621no—
325C4—3621no—
326C4—3621no—
327C4—7241no—
328C4—7241no—
329C4—7241no—
330C4#41×D18111yescharacteristic
331C4C2×#1566844911yescharacteristic
332C2—81no—
333C2—7241no—
334C2#2604178811yescharacteristic
335C2#2604208911yescharacteristic
336C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#440×C181C2
socleC22×C181C22×D4
prime radical#19153787C22
hypercenter#440D181
hypocenterC181#3121
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #26041755C2semidirect
3. #26041622C2semidirect
5. #26041573C2semidirect
6. #26041562C2semidirect
7. #26041500C2semidirect
8. #26041499C2semidirect
9. #26041436C2semidirect
11. #26041387C2semidirect
12. C4×#15668439C2semidirect
13. C4×#15668437C2semidirect
14. #26041341C2semidirect
15. #440×C181C2semidirect
168. C181#3121semidirect
181. #440D181semidirect

Maximal subgroups

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

subgroupindex
#260418942
#260417552
#260416222
#260416022
#260415732
#260415622
#260415002
#260414992
#260414362
#260414162
#260413872
C4×#156684392
C4×#156684372
#260413412
#440×C1812
#3121181