Finite Group Databasebeta

← #37546301

Subgroups — #37546301

Order 23168.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041997C211yesmaximal, characteristic, complemented
2#26041930C211yesmaximal, characteristic
3#26041683C211yesmaximal, characteristic, complemented
4#26041616C211yesmaximal, characteristic, complemented
5#26041614C211yesmaximal, characteristic, complemented
6#26041605C211yesmaximal, characteristic, complemented
7#26041603C211yesmaximal, characteristic, complemented
8#26041563C211yesmaximal, characteristic, complemented
9#26041558C211yesmaximal, characteristic, complemented
10#26041526C211yesmaximal, characteristic, complemented
11#26041525C211yesmaximal, characteristic, complemented
12#26041366C211yesmaximal, characteristic, complemented
13#26041355C211yesmaximal, characteristic, complemented
14#26041318C211yesmaximal, characteristic, complemented
15#428×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×#15668438C2211yescharacteristic
18C2×#15668432C2211yescharacteristic
19#19153815C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153810—21no—
22#19153809—21no—
23#19153802—21no—
24#19153801—21no—
25#19153799—21no—
26#19153796C2211yescharacteristic
27#19153795C2211yescharacteristic
28#19153794—21no—
29#19153791C2211yescharacteristic
30#19153790C2211yescharacteristic
31C4×Dic362—21no—
32#19153781C2211yescharacteristic
33#19153778C2211yescharacteristic
34C4×C22⋊D181C2211yescharacteristic
35#19153774C2211yescharacteristic
36C4⋊C4×D181C2211yescharacteristic
37#19153750C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×M16×C181C2211yescharacteristic
53#117×C181C2211yescharacteristic
54C2×C4⋊C4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic
58#106×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×D724—21no—
64C2×D724C2311yescharacteristic
65C2×C22⋊D181—21no—
66C2×C22⋊D181D411yescharacteristic
67C2×C4×D181—21no—
68C2×C4×D181—21no—
69C2×C4×D181C2311yescharacteristic
70#15668438—22no—
71#15668432—22no—
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C4⋊Dic181—21no—
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428D411yescharacteristic
83C22⋊Dic181—21no—
84C22⋊Dic181D411yescharacteristic
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426—21no—
90#15668426D411yescharacteristic
91C4×Dic181—21no—
92C4×Dic181—21no—
93C4×Dic181C2311yescharacteristic
94C22×C4×C181—21no—
95C22×C4×C181C2311yescharacteristic
96M16×C181—22no—
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C42×C181—21no—
106C22×D181—21no—
107C22×D181C2×D411yescharacteristic
108Dic362—21no—
109Dic362—21no—
110Dic362—42no—
111Dic362—42no—
112D724—21no—
113D724—21no—
114D724—42no—
115C22⋊D181—22no—
116C22⋊D181—42no—
117C2×Dic181—21no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122C2×Dic181C2×D411yescharacteristic
123C4×D181—21no—
124C4×D181—21no—
125C4×D181—42no—
126C4×D181—42no—
127#13089678—21no—
128#13089678—21no—
129C23×C181C2×D411yescharacteristic
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181—42no—
137C2×C4×C181C2×D411yescharacteristic
138C2×C4×C181C2411yescharacteristic
139C8×C181—21no—
140C8×C181#4112yescomplemented
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—21no—
144C22×C181C22×D411yescharacteristic
145C2×D181—21no—
146C2×D181—21no—
147C2×D181—21no—
148C2×D181—41no—
149C2×D181—41no—
150C2×D181—41no—
151C4×C181—21no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181—41no—
155C4×C181—41no—
156C4×C181C2×#4111yescharacteristic
157C4×C1812−1+411yescharacteristic
158Dic181—21no—
159Dic181—21no—
160Dic181—41no—
161Dic181—41no—
162Dic181—41no—
163Dic181AGL(1,ℤ/8)12yes—
164C2×C181—41no—
165C2×C181#54711yescharacteristic
166C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C181#57811yescharacteristic
168D181—41no—
169D181—81no—
170C181#335311yescharacteristic, complemented, Sylow
171#3353—1811nomaximal, Sylow
172#556—1811no—
173#547—1811no—
174#500—1811no—
175#484—1811no—
176#484—1811no—
177#482—1811no—
178#482—1811no—
179#475—1811no—
180#474—1811no—
181#468—1811no—
182#468—1811no—
183C4×SD16—1811no—
184#436—1811no—
185#428—1811no—
186#428D18111yescharacteristic, complemented
187C2×#41—3621no—
188C2×SD16—1811no—
189C2×M16—1811no—
190C2×M16C2×D18111yescharacteristic
191C4⋊Q8—1811no—
192C4⋊Q8—1811no—
193#126—3621no—
194#126—3621no—
195#124—3621no—
196#123—3621no—
197#123—3621no—
198C22⋊Q8—3621no—
199C22⋊Q8—1811no—
200C22⋊Q8—1811no—
201#121—1811no—
202#121—1811no—
203C4×Q8—3621no—
204C4×Q8—1811no—
205C4×D4—1811no—
206C4×D4—1811no—
207#117—1811no—
208#117C2×D18111yescharacteristic
209C2×C4⋊C4—1811no—
210C2×C4⋊C4C2×D18111yescharacteristic
211#107—1811no—
212#107—1811no—
213#107—1811no—
214#107—1811no—
215#107C2×D18111yescharacteristic
216#107C2×D18111yescharacteristic
217#106—1811no—
218#106C2×D18111yescharacteristic
219#106C2×D18111yescharacteristic
220C4⋊C8—1811no—
221#103—1811no—
222#103—1811no—
223#103—1811no—
224#102—1811no—
225#102—1811no—
226#102—1811no—
227C22⋊C8—1811no—
228C22⋊C8—1811no—
229C4×C8—1811no—
230#41—7244no—
231C2×Q8—3621no—
232C2×Q8—3621no—
233C2×Q8—1811no—
234C2×D4—3621no—
235C2×D4—3621no—
236C2×D4—1811no—
237C2×D4—1811no—
238C22×C4—21no—
239C22×C4—3621no—
240C22×C4—3621no—
241C22×C4—1811no—
242C22×C4C22×D18111yescharacteristic
243SD16—3622no—
244M16—22no—
245M16—3622no—
246C2×C8—1811no—
247C2×C8—1811no—
248C2×C8C22×D18111yescharacteristic
249C2×C8C22×D18111yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—3621no—
256C4⋊C4—3621no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4—1811no—
260C4⋊C4C22×D18111yescharacteristic
261C4⋊C4C22×D18111yescharacteristic
262C4⋊C4C22×D18111yescharacteristic
263C4⋊C4C22×D18111yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—3621no—
269C22⋊C4—3621no—
270C22⋊C4—3621no—
271C22⋊C4—1811no—
272C22⋊C4—1811no—
273C42—21no—
274C42—3621no—
275C42—3621no—
276C42—1811no—
277C23—3621no—
278C23—1811no—
279C23D4×D18111yescharacteristic
280Q8—3621no—
281Q8—3621no—
282Q8—7242no—
283Q8—7242no—
284D4—3622no—
285D4—3621no—
286D4—3621no—
287D4—7242no—
288D4—7242no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
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×C4—7242no—
305C2×C4—7242no—
306C2×C4D4×D18111yescharacteristic
307C2×C4C23×D18111yescharacteristic
308C8—21no—
309C8—3621no—
310C8—3621no—
311C8#1566844912yescomplemented
312C22—21no—
313C22—21no—
314C22—21no—
315C22—3621no—
316C22—3621no—
317C22—3621no—
318C22—7241no—
319C22—7241no—
320C22—7241no—
321C22C2×D4×D18111yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3621no—
328C4—3621no—
329C4—1812no—
330C4—7241no—
331C4—7241no—
332C4—7241no—
333C4C2×#1566844911yescharacteristic
334C4#1915386411yescharacteristic
335C2—41no—
336C2—14481no—
337C2—7241no—
338C2#2604193011yescharacteristic
339C2AGL(1,ℤ/8)×D18111yescharacteristic
340C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
prime radical#26041930C2
hypercenter#428D181
hypocenterC181#3353
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #26041997C2semidirect
3. #26041683C2semidirect
4. #26041616C2semidirect
5. #26041614C2semidirect
6. #26041605C2semidirect
7. #26041603C2semidirect
8. #26041563C2semidirect
9. #26041558C2semidirect
10. #26041526C2semidirect
11. #26041525C2semidirect
12. #26041366C2semidirect
13. #26041355C2semidirect
14. #26041318C2semidirect
15. #428×C181C2semidirect
140. C8×C181#41semidirect
170. C181#3353semidirect
186. #428D181semidirect
311. C8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260419972
#260419302
#260416832
#260416162
#260416142
#260416052
#260416032
#260415632
#260415582
#260415262
#260415252
#260413662
#260413552
#260413182
#428×C1812
#3353181