Finite Group Databasebeta

← #28347891

Subgroups — #28347891

Order 13696.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293242C211yesmaximal, characteristic
2#20293240C211yesmaximal, characteristic, complemented
3#20292931C211yesmaximal, characteristic, complemented
4#20292921C211yesmaximal, characteristic, complemented
5#20292911C211yesmaximal, characteristic, complemented
6#20292886C211yesmaximal, characteristic, complemented
7#20292872C211yesmaximal, characteristic
8#20292869C211yesmaximal, characteristic, complemented
9#20292855C211yesmaximal, characteristic
10#20292848C211yesmaximal, characteristic, complemented
11#20292848C211yesmaximal, characteristic, complemented
12#20292838C211yesmaximal, characteristic, complemented
13#20292832C211yesmaximal, characteristic, complemented
14#20292652C211yesmaximal, characteristic, complemented
15#478×C107C211yesmaximal, characteristic, complemented
16C2×Dic428C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×#13133801C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22#16812109C2211yescharacteristic
23#16812108C2211yescharacteristic
24#16812104—21no—
25#16812104—21no—
26#16812102—21no—
27#16812100—21no—
28#16812100—21no—
29#16812096C2211yescharacteristic
30#16812095—21no—
31#16812092—21no—
32#16812089C2211yescharacteristic
33#16812089C2211yescharacteristic
34#16812089C2211yescharacteristic
35#16812088—21no—
36#16812080C2211yescharacteristic
37C4×Dic214C2211yescharacteristic
38#16812074C2211yescharacteristic
39#16812074C2211yescharacteristic
40#16812066C2211yescharacteristic
41#16812042C2211yescharacteristic
42#16812040C2211yescharacteristic
43C4⋊(C107⋊C8)C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812032C2211yescharacteristic
50#16812032C2211yescharacteristic
51#16812032C2211yescharacteristic
52#125×C107C2211yescharacteristic
53C4×D4×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C4×D107C2311yescharacteristic
65Dic428—41no—
66Dic428—41no—
67#13133806—41no—
68#13133806—41no—
69#13133801—41no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C8×D107—41no—
73C4⋊Dic107—21no—
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92(C2×D107)⋊C4D411yescharacteristic
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107C2×D411yescharacteristic
110D4×C107—21no—
111D4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—81no—
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C22⋊D107—81no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C4×D107—21no—
128C4×D107—21no—
129Dic214—21no—
130Dic214—21no—
131Dic214—21no—
132Dic214—21no—
133Dic214—21no—
134Dic214—21no—
135Dic214—81no—
136C8×C107—21no—
137C8×C107—21no—
138C107⋊C8—21no—
139C107⋊C8—21no—
140C22×C107—41no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107C22×D411yescharacteristic
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—21no—
147C4×C107—21no—
148C4×C107—21no—
149C4×C107—41no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107H(5,𝔽2)11yescharacteristic
153C4×C1072−1+411yescharacteristic
154Dic107—21no—
155Dic107—21no—
156Dic107—21no—
157Dic107—41no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161C2×C107—81no—
162C2×C107#53711yescharacteristic
163C2×C107#57711yescharacteristic
164C2×C107#57811yescharacteristic
165D107—41no—
166#3258—1071nomaximal, Sylow
167C107#325811yescharacteristic, complemented, Sylow
168#542—1071no—
169#541—1071no—
170#479—1071no—
171#478D10711yescharacteristic, complemented
172#477—1071no—
173#475—1071no—
174#470—1071no—
175#468—1071no—
176#467—1071no—
177#465—1071no—
178#464—1071no—
179#464—1071no—
180#462—1071no—
181#461—1071no—
182#424—1071no—
183C2×Q16—1071no—
184C2×Q16—1071no—
185C2×SD16—1071no—
186C2×SD16—1071no—
187C2×M16—1071no—
188C22×C8—1071no—
189C4⋊Q8—1071no—
190C4⋊Q8—1071no—
191#126—2141no—
192#126—2141no—
193#126—2141no—
194#125—2141no—
195#125—2141no—
196#125C2×D10711yescharacteristic
197#124—1071no—
198#124—2141no—
199#123—2141no—
200C22⋊Q8—1071no—
201C22⋊Q8—1071no—
202C22⋊Q8—1071no—
203C22⋊Q8—2141no—
204#121—1071no—
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×Q8—1071no—
208C4×D4C2×D10711yescharacteristic
209#117—1071no—
210#107—1071no—
211#107C2×D10711yescharacteristic
212#106—1071no—
213#106C2×D10711yescharacteristic
214C4⋊C8—1071no—
215C4⋊C8—1071no—
216C4⋊C8—1071no—
217C4⋊C8C2×D10711yescharacteristic
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#103—1071no—
222#103—1071no—
223#103—1071no—
224#102C2×D10711yescharacteristic
225#102C2×D10711yescharacteristic
226C2×Q8—1071no—
227C2×Q8—1071no—
228C2×Q8—1071no—
229C2×Q8—2141no—
230C2×D4—2141no—
231C2×D4C22×D10711yescharacteristic
232C22×C4—21no—
233C22×C4—1071no—
234Q16—4281no—
235Q16—4281no—
236SD16—4281no—
237SD16—4281no—
238M16—4281no—
239C2×C8—1071no—
240C2×C8—1071no—
241C2×C8—4281no—
242C2×C8C22×D10711yescharacteristic
243C2×C8C22×D10711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—1071no—
247C4⋊C4—1071no—
248C4⋊C4—1071no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4C22×D10711yescharacteristic
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—1071no—
263C22⋊C4—1071no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C42—1071no—
271C42—2141no—
272C42—2141no—
273C42—2141no—
274C42C22×D10711yescharacteristic
275C23—21no—
276C23—1071no—
277Q8—8561no—
278Q8—2141no—
279Q8—2141no—
280Q8—2141no—
281Q8—2141no—
282Q8—2141no—
283Q8—2141no—
284D4—21no—
285D4—21no—
286D4—8561no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—81no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—1071no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4—2141no—
298C2×C4—2141no—
299C2×C4—2141no—
300C2×C4D4×D10711yescharacteristic
301C2×C4D4×D10711yescharacteristic
302C2×C4C23×D10711yescharacteristic
303C8—21no—
304C8—21no—
305C8—2141no—
306C8—2141no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—2141no—
311C22—2141no—
312C22—2141no—
313C22C2×D4×D10711yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—4281no—
320C4—4281no—
321C4—4281no—
322C4—4281no—
323C4—2141no—
324C4—2141no—
325C4—2141no—
326C4#1681215511yescharacteristic
327C4#1681215711yescharacteristic
328C2—81no—
329C2—4281no—
330C2#2029320711yescharacteristic
331C2#2029343711yescharacteristic
332C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radical#16812080C22
hypercenter#478D107
hypocenterC107#3258
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
2. #20293240C2semidirect
3. #20292931C2semidirect
4. #20292921C2semidirect
5. #20292911C2semidirect
6. #20292886C2semidirect
8. #20292869C2semidirect
10. #20292848C2semidirect
11. #20292848C2semidirect
12. #20292838C2semidirect
13. #20292832C2semidirect
14. #20292652C2semidirect
15. #478×C107C2semidirect
167. C107#3258semidirect
171. #478D107semidirect

Maximal subgroups

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

subgroupindex
#202932422
#202932402
#202929312
#202929212
#202929112
#202928862
#202928722
#202928692
#202928552
#202928482
#202928382
#202928322
#202926522
#478×C1072
#3258107