Finite Group Databasebeta

← #28403130

Subgroups — #28403130

Order 14464.

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#20299292C211yesmaximal, characteristic
2#20299283C211yesmaximal, characteristic, complemented
3#20298999C211yesmaximal, characteristic, complemented
4#20298939C211yesmaximal, characteristic, complemented
5#20298930C211yesmaximal, characteristic, complemented
6#20298925C211yesmaximal, characteristic, complemented
7#20298916C211yesmaximal, characteristic, complemented
8#20298904C211yesmaximal, characteristic, complemented
9#20298897C211yesmaximal, characteristic, complemented
10#20298845C211yesmaximal, characteristic, complemented
11#20298844C211yesmaximal, characteristic, complemented
12#20298685C211yesmaximal, characteristic, complemented
13#20298674C211yesmaximal, characteristic, complemented
14#20298637C211yesmaximal, characteristic, complemented
15#428×C113C211yesmaximal, characteristic, complemented
16C2×#14268636C2211yescharacteristic
17C2×#14268630C2211yescharacteristic
18#16813617—21no—
19#16813616—21no—
20#16813613C2211yescharacteristic
21#16813613C2211yescharacteristic
22#16813612—21no—
23#16813605—21no—
24#16813604—21no—
25#16813602C2211yescharacteristic
26#16813601C2211yescharacteristic
27#16813597C2211yescharacteristic
28#16813596C2211yescharacteristic
29#16813595—21no—
30#16813587C2211yescharacteristic
31C4×D452—21no—
32#16813584C2211yescharacteristic
33C4×C22⋊D113C2211yescharacteristic
34#16813581—21no—
35#16813580C2211yescharacteristic
36C4⋊C4×D113C2211yescharacteristic
37#16813556C2211yescharacteristic
38#16813555C2211yescharacteristic
39#16813553C2211yescharacteristic
40#16813553C2211yescharacteristic
41#16813553C2211yescharacteristic
42#16813551C2211yescharacteristic
43#16813547C2211yescharacteristic
44#16813545C2211yescharacteristic
45#16813545C2211yescharacteristic
46#16813544C2211yescharacteristic
47#16813542C2211yescharacteristic
48#16813542C2211yescharacteristic
49C22⋊#11744751C2211yescharacteristic
50#16813534C2211yescharacteristic
51C8×Dic113C2211yescharacteristic
52C2×M16×C113C2211yescharacteristic
53#117×C113C2211yescharacteristic
54C2×C4⋊C4×C113C2211yescharacteristic
55#107×C113C2211yescharacteristic
56#107×C113C2211yescharacteristic
57#106×C113C2211yescharacteristic
58#106×C113C2211yescharacteristic
59C2×Dic226C2311yescharacteristic
60C2×D452—21no—
61C2×D452C2311yescharacteristic
62C2×C22⋊D113—21no—
63C2×C22⋊D113D411yescharacteristic
64C2×C4×D113—21no—
65C2×C4×D113—21no—
66C2×C4×D113C2311yescharacteristic
67#14268636—22no—
68#14268630—22no—
69C2×#11744751C2311yescharacteristic
70C2×#11744751C2311yescharacteristic
71C4⋊Dic113—21no—
72C4⋊Dic113—21no—
73C4⋊Dic113—21no—
74C4⋊Dic113C2311yescharacteristic
75C4⋊Dic113C2311yescharacteristic
76#14268626—21no—
77#14268626—21no—
78#14268626—21no—
79#14268626—21no—
80#14268626—21no—
81#14268626D411yescharacteristic
82C22⋊Dic113—21no—
83C22⋊Dic113D411yescharacteristic
84#14268624—21no—
85#14268624—21no—
86#14268624—21no—
87#14268624—21no—
88#14268624—21no—
89#14268624D411yescharacteristic
90C4×Dic113—21no—
91C4×Dic113—21no—
92C4×Dic113C2311yescharacteristic
93C22×C4×C113—21no—
94C22×C4×C113C2311yescharacteristic
95M16×C113—22no—
96C2×C8×C113C2311yescharacteristic
97C2×C8×C113C2311yescharacteristic
98C4⋊C4×C113—21no—
99C4⋊C4×C113C2311yescharacteristic
100C4⋊C4×C113C2311yescharacteristic
101C4⋊C4×C113C2311yescharacteristic
102C4⋊C4×C113C2311yescharacteristic
103C22⋊C4×C113—21no—
104C42×C113—21no—
105C23×C113C2×D411yescharacteristic
106C22×D113—21no—
107C22×D113C2×D411yescharacteristic
108C2×C4×C113—21no—
109C2×C4×C113—21no—
110C2×C4×C113—21no—
111C2×C4×C113—21no—
112C2×C4×C113—21no—
113C2×C4×C113—21no—
114C2×C4×C113—42no—
115C2×C4×C113C2×D411yescharacteristic
116C2×C4×C113C2411yescharacteristic
117C22⋊D113—41no—
118C22⋊D113—81no—
119C2×Dic113—21no—
120C2×Dic113—21no—
121C2×Dic113—21no—
122C2×Dic113C2×D411yescharacteristic
123C2×Dic113C2×D411yescharacteristic
124C2×Dic113C2×D411yescharacteristic
125D452—21no—
126D452—21no—
127D452—42no—
128C4×D113—21no—
129C4×D113—21no—
130C4×D113—81no—
131C4×D113—81no—
132Dic226—21no—
133Dic226—21no—
134C8×C113—21no—
135C8×C113#4112yes—
136#11744751—21no—
137#11744751—21no—
138C22×C113—21no—
139C22×C113—21no—
140C22×C113—21no—
141C22×C113C22×D411yescharacteristic
142C2×D113—21no—
143C2×D113—21no—
144C2×D113—21no—
145C2×D113—41no—
146C2×D113—41no—
147C2×D113—41no—
148C4×C113—21no—
149C4×C113—41no—
150C4×C113—41no—
151C4×C113—41no—
152C4×C113—41no—
153C4×C113C2×#4111yescharacteristic
154C4×C1132−1+411yescharacteristic
155Dic113—21no—
156Dic113—21no—
157Dic113—21no—
158Dic113—41no—
159Dic113—41no—
160Dic113—41no—
161C2×C113—41no—
162C2×C113#54711yescharacteristic
163C2×C113#57511yescharacteristic
164C2×C113#57711yescharacteristic
165D113—41no—
166D113—81no—
167#3351—1131nomaximal, Sylow
168C113#335111yescharacteristic, complemented, Sylow
169#553—1131no—
170#552—1131no—
171#499—1131no—
172#485—1131no—
173#483—1131no—
174#482—1131no—
175#481—1131no—
176#479—1131no—
177#478—1131no—
178#468—1131no—
179#468—1131no—
180C4×SD16—1131no—
181#436—1131no—
182#428—1131no—
183#428D11311yescharacteristic, complemented
184C2×SD16—1131no—
185C2×M16—1131no—
186C2×M16C2×D11311yescharacteristic
187#126—2261no—
188#126—2261no—
189#125—1131no—
190#125—1131no—
191#125—2261no—
192#123—2261no—
193#123—2261no—
194C22⋊Q8—1131no—
195C22⋊Q8—1131no—
196#121—1131no—
197#121—1131no—
198#121—2261no—
199C4×Q8—1131no—
200C4×D4—1131no—
201C4×D4—1131no—
202C4×D4—2261no—
203#117—1131no—
204#117—2261no—
205#117C2×D11311yescharacteristic
206C2×C4⋊C4—1131no—
207C2×C4⋊C4C2×D11311yescharacteristic
208#107—1131no—
209#107—1131no—
210#107C2×D11311yescharacteristic
211#107C2×D11311yescharacteristic
212#106—1131no—
213#106—1131no—
214#106—1131no—
215#106C2×D11311yescharacteristic
216#106C2×D11311yescharacteristic
217C4⋊C8—1131no—
218#103—1131no—
219#103—1131no—
220#103—1131no—
221#102—1131no—
222#102—1131no—
223#102—1131no—
224C22⋊C8—1131no—
225C22⋊C8—1131no—
226C4×C8—1131no—
227C2×Q8—1131no—
228C2×D4—1131no—
229C2×D4—1131no—
230C2×D4—2261no—
231C2×D4—2261no—
232C22×C4—21no—
233C22×C4—1131no—
234C22×C4—2261no—
235C22×C4—2261no—
236C22×C4C22×D11311yescharacteristic
237SD16—2262no—
238M16—22no—
239M16—2262no—
240C2×C8—1131no—
241C2×C8—1131no—
242C2×C8C22×D11311yescharacteristic
243C2×C8C22×D11311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—1131no—
246C4⋊C4—1131no—
247C4⋊C4—1131no—
248C4⋊C4—2261no—
249C4⋊C4—2261no—
250C4⋊C4—2261no—
251C4⋊C4—2261no—
252C4⋊C4—2261no—
253C4⋊C4—2261no—
254C4⋊C4—2261no—
255C4⋊C4—2261no—
256C4⋊C4C22×D11311yescharacteristic
257C4⋊C4C22×D11311yescharacteristic
258C4⋊C4C22×D11311yescharacteristic
259C4⋊C4C22×D11311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—1131no—
262C22⋊C4—1131no—
263C22⋊C4—2261no—
264C22⋊C4—2261no—
265C22⋊C4—2261no—
266C22⋊C4—2261no—
267C22⋊C4—2261no—
268C22⋊C4—2261no—
269C42—21no—
270C42—1131no—
271C42—2261no—
272C42—2261no—
273C23—1131no—
274C23—2261no—
275C23D4×D11311yescharacteristic
276Q8—2261no—
277Q8—2261no—
278D4—9041no—
279D4—4522no—
280D4—4521no—
281D4—2261no—
282D4—2261no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—1131no—
291C2×C4—1131no—
292C2×C4—1131no—
293C2×C4—9041no—
294C2×C4—9041no—
295C2×C4—2261no—
296C2×C4—2261no—
297C2×C4—2261no—
298C2×C4—2261no—
299C2×C4—2261no—
300C2×C4D4×D11311yescharacteristic
301C2×C4C23×D11311yescharacteristic
302C8—21no—
303C8—2261no—
304C8—2261no—
305C8#1426864712yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—4521no—
310C22—4521no—
311C22—4521no—
312C22—2261no—
313C22—2261no—
314C22—2261no—
315C22C2×D4×D11311yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—4521no—
322C4—4521no—
323C4—4521no—
324C4—2261no—
325C4—2261no—
326C4—2261no—
327C4C2×#1426864711yescharacteristic
328C4#1681367011yescharacteristic
329C2—41no—
330C2—9041no—
331C2—4521no—
332C2#2029924911yescharacteristic
333C2#2029939411yescharacteristic
334C2#2029940711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#428×C113C2
socleC22×C113C22×D4
prime radical#20299292C2
hypercenter#428D113
hypocenterC113#3351
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #20299283C2semidirect
3. #20298999C2semidirect
4. #20298939C2semidirect
5. #20298930C2semidirect
6. #20298925C2semidirect
7. #20298916C2semidirect
8. #20298904C2semidirect
9. #20298897C2semidirect
10. #20298845C2semidirect
11. #20298844C2semidirect
12. #20298685C2semidirect
13. #20298674C2semidirect
14. #20298637C2semidirect
15. #428×C113C2semidirect
168. C113#3351semidirect
183. #428D113semidirect

Maximal subgroups

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

subgroupindex
#202992922
#202992832
#202989992
#202989392
#202989302
#202989252
#202989162
#202989042
#202988972
#202988452
#202988442
#202986852
#202986742
#202986372
#428×C1132
#3351113