Finite Group Databasebeta

← #28349224

Subgroups — #28349224

Order 13696.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293363C211yesmaximal, characteristic
2#20293354C211yesmaximal, characteristic, complemented
3#20292985C211yesmaximal, characteristic, complemented
4#20292980C211yesmaximal, characteristic, complemented
5#20292931C211yesmaximal, characteristic, complemented
6#20292929C211yesmaximal, characteristic, complemented
7#20292915C211yesmaximal, characteristic, complemented
8#20292848C211yesmaximal, characteristic, complemented
9#20292848C211yesmaximal, characteristic, complemented
10#20292824C211yesmaximal, characteristic
11#20292755C211yesmaximal, characteristic, complemented
12#20292735C211yesmaximal, characteristic, complemented
13#20292724C211yesmaximal, characteristic, complemented
14#20292718C211yesmaximal, characteristic, complemented
15#490×C107C211yesmaximal, characteristic
16C2×Dic428C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19#16812108—21no—
20#16812108C2211yescharacteristic
21#16812100C2211yescharacteristic
22#16812099—21no—
23#16812099—21no—
24#16812096—21no—
25#16812096C2211yescharacteristic
26#16812096C2211yescharacteristic
27#16812090—21no—
28#16812090—21no—
29C4×Dic214C2211yescharacteristic
30C4×Dic214C2211yescharacteristic
31C4×Dic214C2211yescharacteristic
32#16812074—21no—
33#16812074C2211yescharacteristic
34#16812074C2211yescharacteristic
35D4×Dic107—21no—
36D4×Dic107C2211yescharacteristic
37D4×Dic107C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52#127×C107C2211yescharacteristic
53#125×C107C2211yescharacteristic
54#102×C107C2211yescharacteristic
55#102×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C22×Dic107—21no—
64C22×Dic107—21no—
65Dic428—22no—
66#13133806—22no—
67#13133806—22no—
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2311yescharacteristic
70C4⋊Dic107—21no—
71C4⋊Dic107D411yescharacteristic
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74Dic107⋊C4—21no—
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4D411yescharacteristic
81Dic107⋊C4D411yescharacteristic
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C4×Dic107C2311yescharacteristic
94C2×D4×C107—21no—
95C2×D4×C107—21no—
96C2×D4×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C23×C107—21no—
107D4×C107—21no—
108D4×C107—21no—
109D4×C107—21no—
110D4×C107—21no—
111D4×C107—42no—
112D4×C107—81no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2411yescharacteristic
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—81no—
123C2×Dic107—81no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128Dic214—21no—
129Dic214—21no—
130Dic214—21no—
131Dic214—41no—
132Dic214—42no—
133Dic214#4112yes—
134C8×C107—21no—
135C8×C107—21no—
136C107⋊C8—21no—
137C107⋊C8—21no—
138C22×C107—41no—
139C22×C107—41no—
140C22×C107—41no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107C22×D411yescharacteristic
145C4×C107—21no—
146C4×C107—21no—
147C4×C107—41no—
148C4×C107—41no—
149C4×C107C2×#4111yescharacteristic
150C4×C1072−1+411yescharacteristic
151Dic107—22no—
152Dic107—21no—
153Dic107—21no—
154Dic107—21no—
155Dic107—21no—
156Dic107—41no—
157Dic107—41no—
158Dic107—41no—
159C2×C107—81no—
160C2×C107—81no—
161C2×C107#54811yescharacteristic
162C2×C107#57511yescharacteristic
163C2×C107#57711yescharacteristic
164#3371—1071nomaximal, Sylow
165C107#337111yescharacteristic, complemented, Sylow
166#559—1071no—
167#557—1071no—
168#490D10711yescharacteristic
169#489—1071no—
170#488—1071no—
171#479—1071no—
172#478—1071no—
173#476—1071no—
174#464—1071no—
175#464—1071no—
176#460—1071no—
177#446—1071no—
178#441—1071no—
179#440—1071no—
180C4×SD16—1071no—
181C2×Q16—1071no—
182C2×SD16—1071no—
183C2×SD16—1071no—
184C4⋊Q8—1071no—
185C4⋊Q8—2141no—
186#127C2×D10711yescharacteristic
187#125—1071no—
188#125—2141no—
189#125—2141no—
190#125C2×D10711yescharacteristic
191#124—1071no—
192#124—1071no—
193#124—2141no—
194#123—2141no—
195#123—2141no—
196C4×Q8—1071no—
197C4×Q8—1071no—
198C4×Q8—1071no—
199C4×Q8—1071no—
200C4×Q8—1071no—
201C4×Q8—2141no—
202C4×D4—1071no—
203C4×D4—1071no—
204C4×D4—2141no—
205#107—1071no—
206#106—1071no—
207#106—1071no—
208C4⋊C8—1071no—
209C4⋊C8—1071no—
210C4⋊C8—1071no—
211#103—1071no—
212#103—1071no—
213#103—1071no—
214#103—1071no—
215#102—1071no—
216#102—1071no—
217#102C2×D10711yescharacteristic
218#102C2×D10711yescharacteristic
219#102C2×D10711yescharacteristic
220#102C2×D10711yescharacteristic
221#97—1071no—
222#97C2×D10711yescharacteristic
223C4×C8—1071no—
224C2×Q8—1071no—
225C2×Q8—1071no—
226C2×Q8—1071no—
227C2×Q8—2141no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C22×D10711yescharacteristic
231C2×D4C22×D10711yescharacteristic
232C22×C4—2141no—
233C22×C4—2141no—
234Q16—2142no—
235SD16—2142no—
236SD16—2142no—
237C2×C8—1071no—
238C2×C8—1071no—
239C2×C8C22×D10711yescharacteristic
240C2×C8C22×D10711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—1071no—
244C4⋊C4—1071no—
245C4⋊C4—1071no—
246C4⋊C4—1071no—
247C4⋊C4—1071no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4C22×D10711yescharacteristic
256C4⋊C4C22×D10711yescharacteristic
257C22⋊C4—2141no—
258C22⋊C4—2141no—
259C22⋊C4—2141no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C42—1071no—
264C42—1071no—
265C42—2141no—
266C42—2141no—
267C42—2141no—
268C42—2141no—
269C42C22×D10711yescharacteristic
270C23—21no—
271C23—21no—
272Q8—1072no—
273Q8—4281no—
274Q8—4282no—
275Q8—2141no—
276Q8—2141no—
277Q8—2141no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—8561no—
287C2×C4—8561no—
288C2×C4—1071no—
289C2×C4—1071no—
290C2×C4—1071no—
291C2×C4—1071no—
292C2×C4—2141no—
293C2×C4—2141no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4D4×D10711yescharacteristic
297C2×C4D4×D10711yescharacteristic
298C2×C4C23×D10711yescharacteristic
299C8—21no—
300C8—21no—
301C8—2141no—
302C8—2141no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×D4×D10711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—4281no—
315C4—4281no—
316C4—4281no—
317C4—2141no—
318C4—2141no—
319C4—2141no—
320C4—2141no—
321C4—2142no—
322C4#41×D10711yescharacteristic
323C4#1681215611yescharacteristic
324C2—81no—
325C2—81no—
326C2#2029328511yescharacteristic
327C2#2029342311yescharacteristic
328C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#490×C107C2
socleC22×C107C22×D4
prime radical#127×C107C22
hypercenter#490D107
hypocenterC107#3371
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #20293354C2semidirect
3. #20292985C2semidirect
4. #20292980C2semidirect
5. #20292931C2semidirect
6. #20292929C2semidirect
7. #20292915C2semidirect
8. #20292848C2semidirect
9. #20292848C2semidirect
11. #20292755C2semidirect
12. #20292735C2semidirect
13. #20292724C2semidirect
14. #20292718C2semidirect
165. C107#3371semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933542
#202929852
#202929802
#202929312
#202929292
#202929152
#202928482
#202928242
#202927552
#202927352
#202927242
#202927182
#490×C1072
#3371107