Finite Group Databasebeta

← #28349147

Subgroups — #28349147

Order 13696.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293353C211yesmaximal, characteristic, complemented
2#20293349C211yesmaximal, characteristic
3#20292983C211yesmaximal, characteristic
4#20292982C211yesmaximal, characteristic
5#20292976C211yesmaximal, characteristic, complemented
6#20292934C211yesmaximal, characteristic, complemented
7#20292933C211yesmaximal, characteristic, complemented
8#20292906C211yesmaximal, characteristic
9#20292851C211yesmaximal, characteristic
10#20292849C211yesmaximal, characteristic
11#20292835C211yesmaximal, characteristic, complemented
12C8×Dic214C211yesmaximal, characteristic, complemented
13#20292719C211yesmaximal, characteristic, complemented
14#20292712C211yesmaximal, characteristic
15C4×Q16×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18#16812108—21no—
19#16812108C2211yescharacteristic
20#16812107C2211yescharacteristic
21#16812104—21no—
22#16812103—21no—
23#16812101—21no—
24#16812101C2211yescharacteristic
25#16812100—21no—
26#16812100C2211yescharacteristic
27#16812098C2211yescharacteristic
28#16812097C2211yescharacteristic
29#16812087—21no—
30Q8×Dic107C2211yescharacteristic
31Q8×Dic107C2211yescharacteristic
32C4×Dic214—21no—
33C4×Dic214C2211yescharacteristic
34#16812074C2211yescharacteristic
35#16812071C2211yescharacteristic
36#16812066—21no—
37#16812042C2211yescharacteristic
38#16812040C2211yescharacteristic
39C4⋊(C107⋊C8)C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812029C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C4×D107—21no—
64#13133808—22no—
65#13133805—22no—
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C4⋊Dic107—21no—
69C4⋊Dic107—21no—
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⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107C2311yescharacteristic
92C4×Dic107C2311yescharacteristic
93C2×Q8×C107C2311yescharacteristic
94C2×Q8×C107C2311yescharacteristic
95Q16×C107—22no—
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C4⋊C4×C107—21no—
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C42×C107—21no—
103C42×C107—21no—
104C42×C107C2311yescharacteristic
105C22×D107—21no—
106Q8×C107—21no—
107Q8×C107—21no—
108Q8×C107—21no—
109Q8×C107#4112yes—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107C2×D411yescharacteristic
121C2×Dic107C2×D411yescharacteristic
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124D428—21no—
125D428—21no—
126C4×D107—42no—
127Dic214—22no—
128Dic214—21no—
129Dic214—21no—
130Dic214—42no—
131C8×C107—21no—
132C8×C107—21no—
133C107⋊C8—21no—
134C107⋊C8—21no—
135C22×C107C22×D411yescharacteristic
136C2×D107—41no—
137C2×D107—41no—
138C2×D107—41no—
139C4×C107—22no—
140C4×C107—21no—
141C4×C107—21no—
142C4×C107—41no—
143C4×C107—41no—
144C4×C107—41no—
145C4×C107C2×#4111yescharacteristic
146C4×C1072−1+411yescharacteristic
147Dic107—21no—
148Dic107—21no—
149Dic107—21no—
150Dic107—41no—
151Dic107—41no—
152Dic107—41no—
153Dic107#13512yes—
154C2×C107#54811yescharacteristic
155C2×C107C2×#13511yescharacteristic
156C2×C107#57811yescharacteristic
157D107—81no—
158#3364—1071nomaximal, Sylow
159C107#336411yescharacteristic, complemented, Sylow
160#557—1071no—
161#556—1071no—
162#488—1071no—
163#488—1071no—
164#487—1071no—
165#479—1071no—
166#479—1071no—
167#474—1071no—
168#464—1071no—
169#464—1071no—
170#462—1071no—
171C8×Q8—1071no—
172C4×Q16—1071no—
173C4×Q16D10711yescharacteristic, complemented
174C4×SD16—1071no—
175C2×Q16—1071no—
176C2×Q16C2×D10711yescharacteristic
177C2×SD16—1071no—
178C4⋊Q8—1071no—
179C4⋊Q8—1071no—
180C4⋊Q8—2141no—
181#126—2141no—
182#126—2141no—
183#125—1071no—
184#125—1071no—
185#125—2141no—
186#125—2141no—
187#124—1071no—
188#124—1071no—
189C22⋊Q8—2141no—
190C4×Q8—1071no—
191C4×Q8—1071no—
192C4×Q8—1071no—
193C4×Q8—1071no—
194C4×Q8—2141no—
195C4×Q8C2×D10711yescharacteristic
196C4×Q8C2×D10711yescharacteristic
197C4×D4—1071no—
198#117—2141no—
199#107—1071no—
200#107C2×D10711yescharacteristic
201#106—1071no—
202C4⋊C8—1071no—
203C4⋊C8—1071no—
204C4⋊C8—1071no—
205#103—1071no—
206#103—1071no—
207#103—1071no—
208#103—1071no—
209#103—1071no—
210#103C2×D10711yescharacteristic
211#103C2×D10711yescharacteristic
212#102—1071no—
213#102—1071no—
214#102—1071no—
215C4×C8—1071no—
216C4×C8—1071no—
217C4×C8C2×D10711yescharacteristic
218C2×Q8—1071no—
219C2×Q8—1071no—
220C2×Q8—2141no—
221C2×Q8C22×D10711yescharacteristic
222C2×Q8C22×D10711yescharacteristic
223C2×D4—1071no—
224C22×C4—2141no—
225Q16—22no—
226Q16—2142no—
227SD16—2142no—
228C2×C8—1071no—
229C2×C8—1071no—
230C2×C8C22×D10711yescharacteristic
231C2×C8C22×D10711yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—1071no—
235C4⋊C4—1071no—
236C4⋊C4—1071no—
237C4⋊C4—1071no—
238C4⋊C4—1071no—
239C4⋊C4—2141no—
240C4⋊C4—2141no—
241C4⋊C4—2141no—
242C4⋊C4—2141no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4C22×D10711yescharacteristic
250C4⋊C4C22×D10711yescharacteristic
251C22⋊C4—2141no—
252C22⋊C4—2141no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C22⋊C4—2141no—
256C42—21no—
257C42—21no—
258C42—1071no—
259C42—1071no—
260C42—2141no—
261C42—2141no—
262C42—2141no—
263C42C22×D10711yescharacteristic
264C23—2141no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—4282no—
269Q8—2142no—
270Q8—2141no—
271Q8—2141no—
272Q8#1313381812yes—
273D4—2141no—
274D4—2141no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—1071no—
280C2×C4—1071no—
281C2×C4—1071no—
282C2×C4—1071no—
283C2×C4—4282no—
284C2×C4—2141no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4D4×D10711yescharacteristic
288C2×C4D4×D10711yescharacteristic
289C2×C4C23×D10711yescharacteristic
290C8—21no—
291C8—21no—
292C8—2141no—
293C8—2141no—
294C22—4281no—
295C22—4281no—
296C22—4281no—
297C22C2×D4×D10711yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—1072no—
305C4—4281no—
306C4—4281no—
307C4—4281no—
308C4—2141no—
309C4—2141no—
310C4—2141no—
311C4C2×#1313381811yescharacteristic
312C4#1681215611yescharacteristic
313C2—8561no—
314C2#2029328611yescharacteristic
315C2#135×D10711yescharacteristic
316C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×Q16×C107C2
socleC22×C107C22×D4
prime radicalC2×D428C23
hypercenterC4×Q16D107
hypocenterC107#3364
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #20293353C2semidirect
5. #20292976C2semidirect
6. #20292934C2semidirect
7. #20292933C2semidirect
11. #20292835C2semidirect
12. C8×Dic214C2semidirect
13. #20292719C2semidirect
15. C4×Q16×C107C2semidirect
159. C107#3364semidirect
173. C4×Q16D107semidirect

Maximal subgroups

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

subgroupindex
#202933532
#202933492
#202929832
#202929822
#202929762
#202929342
#202929332
#202929062
#202928512
#202928492
#202928352
C8×Dic2142
#202927192
#202927122
C4×Q16×C1072
#3364107