Finite Group Databasebeta

← #28349529

Subgroups — #28349529

Order 13696.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293362C211yesmaximal, characteristic, complemented
2#20293350C211yesmaximal, characteristic, complemented
3#20293021C211yesmaximal, characteristic, complemented
4#20293016C211yesmaximal, characteristic
5#20293011C211yesmaximal, characteristic, complemented
6#20292850C211yesmaximal, characteristic, complemented
7#20292850C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292843C211yesmaximal, characteristic, complemented
10#20292833C211yesmaximal, characteristic, complemented
11#20292828C211yesmaximal, characteristic, complemented
12#20292740C211yesmaximal, characteristic, complemented
13#20292735C211yesmaximal, characteristic, complemented
14C4×#13133807C211yesmaximal, characteristic, complemented
15#446×C107C211yesmaximal, characteristic, complemented
16C2×Dic428C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133807C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21C2×#13133807C2211yescharacteristic
22C2×#13133807C2211yescharacteristic
23C2×D856C2211yescharacteristic
24C2×D856C2211yescharacteristic
25#16812109C2211yescharacteristic
26#16812106C2211yescharacteristic
27#16812104—21no—
28#16812104—21no—
29#16812098—21no—
30#16812097C2211yescharacteristic
31#16812097C2211yescharacteristic
32#16812097C2211yescharacteristic
33#16812097C2211yescharacteristic
34#16812094—21no—
35#16812094—21no—
36#16812085—21no—
37C4×Dic214C2211yescharacteristic
38C4×Dic214C2211yescharacteristic
39C4×Dic214C2211yescharacteristic
40C4×D428C2211yescharacteristic
41C4×D428C2211yescharacteristic
42C4×D428C2211yescharacteristic
43#16812071—21no—
44#16812067—21no—
45#16812041C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52C4×Q8×C107C2211yescharacteristic
53C4⋊C8×C107C2211yescharacteristic
54C4⋊C8×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic
56#97×C107C2211yescharacteristic
57#97×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428—21no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70Dic428—22no—
71Dic428—41no—
72#13133807—22no—
73#13133807—41no—
74#13133807—41no—
75#13133807—41no—
76#13133807—41no—
77D856—22no—
78D856—41no—
79C4⋊Dic107—21no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107—21no—
100C2×Q8×C107D411yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107D411yescharacteristic
106C4⋊C4×C107D411yescharacteristic
107C4⋊C4×C107D411yescharacteristic
108C42×C107C2311yescharacteristic
109C42×C107C2311yescharacteristic
110C42×C107C2311yescharacteristic
111C22×D107—21no—
112C22×D107—21no—
113C22×D107—21no—
114Q8×C107—41no—
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2411yescharacteristic
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127D428—21no—
128D428—21no—
129D428—21no—
130D428—21no—
131D428—21no—
132D428—21no—
133D428—42no—
134D428—81no—
135C4×D107—42no—
136C4×D107—81no—
137C4×D107—81no—
138Dic214—21no—
139Dic214—21no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143Dic214—21no—
144C8×C107—21no—
145C8×C107—21no—
146C8×C107—21no—
147C8×C107#4112yes—
148C22×C107C22×D411yescharacteristic
149C2×D107—41no—
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C4×C107—21no—
159C4×C107—21no—
160C4×C107—21no—
161C4×C107—21no—
162C4×C107—21no—
163C4×C107—21no—
164C4×C107C2×#4111yescharacteristic
165C4×C107H(5,𝔽2)11yescharacteristic
166Dic107—41no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171C2×C107#55011yescharacteristic
172C2×C107#57511yescharacteristic
173C2×C107#57711yescharacteristic
174D107—81no—
175D107—81no—
176D107—81no—
177#3401—1071nomaximal, Sylow
178C107#340111yescharacteristic, complemented, Sylow
179#559—1071no—
180#556—1071no—
181#497—1071no—
182#496—1071no—
183#495—1071no—
184#464—1071no—
185#464—1071no—
186#463—1071no—
187#463—1071no—
188#461—1071no—
189#460—1071no—
190#446D10711yescharacteristic, complemented
191#442—1071no—
192#441—1071no—
193C4×SD16—1071no—
194C2×Q16—1071no—
195C2×Q16—1071no—
196C2×SD16—1071no—
197C2×SD16—1071no—
198C2×SD16—1071no—
199C2×SD16—1071no—
200C2×SD16—1071no—
201C2×D8—1071no—
202C2×D8—1071no—
203C4⋊Q8—1071no—
204#127—1071no—
205#126—2141no—
206#126—2141no—
207#124—1071no—
208#124—1071no—
209#124—1071no—
210#124—1071no—
211#124—2141no—
212#123—2141no—
213#123—2141no—
214C22⋊Q8—2141no—
215C4×Q8—1071no—
216C4×Q8—1071no—
217C4×Q8—1071no—
218C4×Q8C2×D10711yescharacteristic
219C4×D4—1071no—
220C4×D4—1071no—
221C4×D4—1071no—
222C4×D4—2141no—
223#117—2141no—
224#106—1071no—
225C4⋊C8C2×D10711yescharacteristic
226C4⋊C8C2×D10711yescharacteristic
227C4⋊C8C2×D10711yescharacteristic
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#97C2×D10711yescharacteristic
235#97C2×D10711yescharacteristic
236C4×C8C2×D10711yescharacteristic
237C2×Q8—1071no—
238C2×Q8—1071no—
239C2×Q8—1071no—
240C2×Q8D42811yescharacteristic
241C2×D4—1071no—
242C2×D4—1071no—
243C2×D4—1071no—
244C2×D4—2141no—
245C2×D4—2141no—
246C22×C4—2141no—
247C22×C4—2141no—
248C22×C4—2141no—
249Q16—4281no—
250Q16—2142no—
251SD16—4281no—
252SD16—4281no—
253SD16—4281no—
254SD16—4281no—
255SD16—2142no—
256D8—4281no—
257D8—2142no—
258C2×C8C22×D10711yescharacteristic
259C2×C8C22×D10711yescharacteristic
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C4⋊C4—1071no—
263C4⋊C4—1071no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4D42811yescharacteristic
270C4⋊C4D42811yescharacteristic
271C4⋊C4D42811yescharacteristic
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C42—2141no—
284C42—2141no—
285C42—2141no—
286C42C22×D10711yescharacteristic
287C42C22×D10711yescharacteristic
288C42C22×D10711yescharacteristic
289C23—2141no—
290C23—2141no—
291C23—2141no—
292Q8—41no—
293Q8—2141no—
294Q8—2141no—
295Q8—2141no—
296Q8—2141no—
297Q8—2141no—
298Q8—2141no—
299D4—8561no—
300D4—4282no—
301D4—2141no—
302D4—2141no—
303D4—2141no—
304D4—2141no—
305D4—2141no—
306D4—2141no—
307C2×C4—8561no—
308C2×C4—8561no—
309C2×C4—4282no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4C2×D42811yescharacteristic
316C2×C4C2×D42811yescharacteristic
317C2×C4C2×D42811yescharacteristic
318C2×C4C2×D42811yescharacteristic
319C2×C4C2×D42811yescharacteristic
320C2×C4C2×D42811yescharacteristic
321C2×C4C23×D10711yescharacteristic
322C8—21no—
323C8—21no—
324C8—21no—
325C8#1313381912yes—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22—4281no—
330C22—4281no—
331C22—4281no—
332C22—4281no—
333C22—4281no—
334C22—4281no—
335C22C22×D42811yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—4281no—
343C4—4281no—
344C4—4281no—
345C4—4281no—
346C4—4281no—
347C4C2×#1313381911yescharacteristic
348C4#1681215511yescharacteristic
349C2—8561no—
350C2—8561no—
351C2—8561no—
352C2#2029329911yescharacteristic
353C2#2029342711yescharacteristic
354C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#446×C107C2
socleC22×C107C22×D4
prime radical#20293016C2
hypercenter#446D107
hypocenterC107#3401
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #20293362C2semidirect
2. #20293350C2semidirect
3. #20293021C2semidirect
5. #20293011C2semidirect
6. #20292850C2semidirect
7. #20292850C2semidirect
8. #20292843C2semidirect
9. #20292843C2semidirect
10. #20292833C2semidirect
11. #20292828C2semidirect
12. #20292740C2semidirect
13. #20292735C2semidirect
14. C4×#13133807C2semidirect
15. #446×C107C2semidirect
178. C107#3401semidirect
190. #446D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202933502
#202930212
#202930162
#202930112
#202928502
#202928432
#202928332
#202928282
#202927402
#202927352
C4×#131338072
#446×C1072
#3401107