Finite Group Databasebeta

← #37546282

Subgroups — #37546282

Order 23168.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041977C211yesmaximal, characteristic
2#26041968C211yesmaximal, characteristic, complemented
3#26041680C211yesmaximal, characteristic, complemented
4#26041622C211yesmaximal, characteristic, complemented
5#26041609C211yesmaximal, characteristic, complemented
6#26041603C211yesmaximal, characteristic, complemented
7#26041599C211yesmaximal, characteristic, complemented
8#26041589C211yesmaximal, characteristic, complemented
9#26041582C211yesmaximal, 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×#15668438C2211yescharacteristic
17C2×#15668432C2211yescharacteristic
18#19153811—21no—
19#19153809—21no—
20#19153808—21no—
21#19153807C2211yescharacteristic
22#19153807C2211yescharacteristic
23#19153801—21no—
24#19153798—21no—
25#19153796C2211yescharacteristic
26#19153795C2211yescharacteristic
27#19153791C2211yescharacteristic
28#19153790—21no—
29#19153790C2211yescharacteristic
30#19153781C2211yescharacteristic
31#19153778C2211yescharacteristic
32C4×C22⋊D181—21no—
33C4×C22⋊D181C2211yescharacteristic
34#19153774C2211yescharacteristic
35#19153773—21no—
36C4⋊C4×D181C2211yescharacteristic
37#19153750C2211yescharacteristic
38#19153749C2211yescharacteristic
39#19153747C2211yescharacteristic
40#19153747C2211yescharacteristic
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
59C2×Dic362C2311yescharacteristic
60C2×D724—21no—
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181D411yescharacteristic
64C2×C4×D181—21no—
65C2×C4×D181—21no—
66C2×C4×D181C2311yescharacteristic
67#15668438—22no—
68#15668432—22no—
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181—21no—
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82C22⋊Dic181—21no—
83C22⋊Dic181D411yescharacteristic
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87#15668426—21no—
88#15668426—21no—
89#15668426D411yescharacteristic
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181C2311yescharacteristic
93C22×C4×C181—21no—
94C22×C4×C181C2311yescharacteristic
95M16×C181—22no—
96C2×C8×C181C2311yescharacteristic
97C2×C8×C181C2311yescharacteristic
98C4⋊C4×C181—21no—
99C4⋊C4×C181C2311yescharacteristic
100C4⋊C4×C181C2311yescharacteristic
101C4⋊C4×C181C2311yescharacteristic
102C4⋊C4×C181C2311yescharacteristic
103C22⋊C4×C181—21no—
104C42×C181—21no—
105C22×D181—21no—
106C22×D181C2×D411yescharacteristic
107Dic362—21no—
108Dic362—21no—
109D724—21no—
110D724—21no—
111D724—81no—
112C22⋊D181—41no—
113C22⋊D181—42no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C4×D181—21no—
121C4×D181—21no—
122C4×D181—42no—
123C4×D181—81no—
124#13089678—21no—
125#13089678—21no—
126C23×C181C2×D411yescharacteristic
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—81no—
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2411yescharacteristic
136C8×C181—21no—
137C8×C181#4112yes—
138C22×C181—21no—
139C22×C181—21no—
140C22×C181—21no—
141C22×C181C22×D411yescharacteristic
142C2×D181—21no—
143C2×D181—21no—
144C2×D181—21no—
145C2×D181—41no—
146C2×D181—41no—
147C2×D181—41no—
148C4×C181—21no—
149C4×C181—41no—
150C4×C181—41no—
151C4×C181—41no—
152C4×C181—41no—
153C4×C181C2×#4111yescharacteristic
154C4×C1812−1+411yescharacteristic
155Dic181—21no—
156Dic181—21no—
157Dic181—21no—
158Dic181—41no—
159Dic181—41no—
160Dic181—41no—
161C2×C181—41no—
162C2×C181#54711yescharacteristic
163C2×C181#57511yescharacteristic
164C2×C181#57711yescharacteristic
165D181—41no—
166D181—81no—
167C181#335111yescharacteristic, complemented, Sylow
168#3351—1811nomaximal, Sylow
169#553—1811no—
170#552—1811no—
171#499—1811no—
172#485—1811no—
173#483—1811no—
174#482—1811no—
175#481—1811no—
176#479—1811no—
177#478—1811no—
178#468—1811no—
179#468—1811no—
180C4×SD16—1811no—
181#436—1811no—
182#428—1811no—
183#428D18111yescharacteristic, complemented
184C2×SD16—1811no—
185C2×M16—1811no—
186C2×M16C2×D18111yescharacteristic
187#126—3621no—
188#126—3621no—
189#125—3621no—
190#125—1811no—
191#125—1811no—
192#123—3621no—
193#123—3621no—
194C22⋊Q8—1811no—
195C22⋊Q8—1811no—
196#121—3621no—
197#121—1811no—
198#121—1811no—
199C4×Q8—1811no—
200C4×D4—3621no—
201C4×D4—1811no—
202C4×D4—1811no—
203#117—3621no—
204#117—1811no—
205#117C2×D18111yescharacteristic
206C2×C4⋊C4—1811no—
207C2×C4⋊C4C2×D18111yescharacteristic
208#107—1811no—
209#107—1811no—
210#107C2×D18111yescharacteristic
211#107C2×D18111yescharacteristic
212#106—1811no—
213#106—1811no—
214#106—1811no—
215#106C2×D18111yescharacteristic
216#106C2×D18111yescharacteristic
217C4⋊C8—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#102—1811no—
222#102—1811no—
223#102—1811no—
224C22⋊C8—1811no—
225C22⋊C8—1811no—
226C4×C8—1811no—
227C2×Q8—1811no—
228C2×D4—3621no—
229C2×D4—3621no—
230C2×D4—1811no—
231C2×D4—1811no—
232C22×C4—21no—
233C22×C4—3621no—
234C22×C4—3621no—
235C22×C4—1811no—
236C22×C4C22×D18111yescharacteristic
237SD16—3622no—
238M16—22no—
239M16—3622no—
240C2×C8—1811no—
241C2×C8—1811no—
242C2×C8C22×D18111yescharacteristic
243C2×C8C22×D18111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—1811no—
254C4⋊C4—1811no—
255C4⋊C4—1811no—
256C4⋊C4C22×D18111yescharacteristic
257C4⋊C4C22×D18111yescharacteristic
258C4⋊C4C22×D18111yescharacteristic
259C4⋊C4C22×D18111yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—3621no—
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—1811no—
273C23—3621no—
274C23—1811no—
275C23D4×D18111yescharacteristic
276Q8—3621no—
277Q8—3621no—
278D4—3621no—
279D4—3621no—
280D4—14481no—
281D4—7241no—
282D4—7242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—81no—
290C2×C4—3621no—
291C2×C4—3621no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—14481no—
296C2×C4—1811no—
297C2×C4—1811no—
298C2×C4—1811no—
299C2×C4—7242no—
300C2×C4D4×D18111yescharacteristic
301C2×C4C23×D18111yescharacteristic
302C8—21no—
303C8—3621no—
304C8—3621no—
305C8#1566844912yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—3621no—
310C22—3621no—
311C22—3621no—
312C22—7241no—
313C22—7241no—
314C22—7241no—
315C22C2×D4×D18111yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3621no—
322C4—3621no—
323C4—3621no—
324C4—7241no—
325C4—7241no—
326C4—7241no—
327C4C2×#1566844911yescharacteristic
328C4#1915386411yescharacteristic
329C2—41no—
330C2—14481no—
331C2—7241no—
332C2#2604193011yescharacteristic
333C2#2604207811yescharacteristic
334C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
prime radical#26041977C2
hypercenter#428D181
hypocenterC181#3351
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
2. #26041968C2semidirect
3. #26041680C2semidirect
4. #26041622C2semidirect
5. #26041609C2semidirect
6. #26041603C2semidirect
7. #26041599C2semidirect
8. #26041589C2semidirect
9. #26041582C2semidirect
10. #26041526C2semidirect
11. #26041525C2semidirect
12. #26041366C2semidirect
13. #26041355C2semidirect
14. #26041318C2semidirect
15. #428×C181C2semidirect
167. C181#3351semidirect
183. #428D181semidirect

Maximal subgroups

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

subgroupindex
#260419772
#260419682
#260416802
#260416222
#260416092
#260416032
#260415992
#260415892
#260415822
#260415262
#260415252
#260413662
#260413552
#260413182
#428×C1812
#3351181