Finite Group Databasebeta

← #28349180

Subgroups — #28349180

Order 13696.

372 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#20293349C211yesmaximal, characteristic, complemented
2D4×Dic214C211yesmaximal, characteristic, complemented
3#20292988C211yesmaximal, characteristic, complemented
4#20292987C211yesmaximal, characteristic, complemented
5#20292976C211yesmaximal, characteristic, complemented
6#20292922C211yesmaximal, characteristic, complemented
7#20292921C211yesmaximal, characteristic, complemented
8#20292906C211yesmaximal, characteristic, complemented
9#20292844C211yesmaximal, characteristic
10#20292842C211yesmaximal, characteristic
11#20292830C211yesmaximal, characteristic, complemented
12#20292755C211yesmaximal, characteristic, complemented
13#20292724C211yesmaximal, characteristic, complemented
14#20292707C211yesmaximal, characteristic
15#440×C107C211yesmaximal, characteristic, complemented
16C22×Dic214—21no—
17C2×#13133806C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19#16812109C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107C2211yescharacteristic
23#16812104—21no—
24#16812103—21no—
25#16812098C2211yescharacteristic
26#16812097C2211yescharacteristic
27#16812088—21no—
28#16812087—21no—
29#16812086—21no—
30#16812086—21no—
31C4×Dic214C2211yescharacteristic
32#16812074C2211yescharacteristic
33D4×Dic107C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic
36#16812066—21no—
37#16812042C2211yescharacteristic
38#16812042C2211yescharacteristic
39C4⋊(C107⋊C8)C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812029C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214—44no—
63C2×Dic214D411yescharacteristic
64C2×Dic214C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C22×Dic107—21no—
67C22×Dic107—21no—
68C2×C4×D107—21no—
69#13133806—22no—
70#13133803—22no—
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4D411yescharacteristic
83Dic107⋊C4D411yescharacteristic
84C22⋊Dic107—21no—
85C22⋊Dic107—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—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C4×Dic107C2311yescharacteristic
94C2×Q8×C107C2311yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C22×C4×C107—21no—
97SD16×C107—22no—
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C42×C107—21no—
105C42×C107C2311yescharacteristic
106C23×C107—21no—
107C22×D107—21no—
108Q8×C107—21no—
109Q8×C107—21no—
110D4×C107—21no—
111D4×C107#4112yes—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—42no—
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—42no—
123C2×Dic107—42no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128D428—21no—
129D428—21no—
130C4×D107—42no—
131Dic214—22no—
132Dic214—21no—
133Dic214—21no—
134Dic214—42no—
135Dic214—42no—
136Dic214—42no—
137C8×C107—21no—
138C8×C107—21no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C22×C107—22no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107C22×D411yescharacteristic
145C2×D107—41no—
146C2×D107—41no—
147C2×D107—41no—
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107C2×#4111yescharacteristic
154C4×C1072−1+411yescharacteristic
155Dic107—21no—
156Dic107—21no—
157Dic107—21no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161Dic107AGL(1,ℤ/8)12yes—
162C2×C107—42no—
163C2×C107#54811yescharacteristic
164C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C107#57811yescharacteristic
166D107—81no—
167#3367—1071nomaximal, Sylow
168C107#336711yescharacteristic, complemented, Sylow
169#556—1071no—
170D4×Q8—1071no—
171#489—1071no—
172#489—1071no—
173#487—1071no—
174#477—1071no—
175#477—1071no—
176#474—1071no—
177#463—1071no—
178#463—1071no—
179#461—1071no—
180#446—1071no—
181#440—1071no—
182#440D10711yescharacteristic, complemented
183C4×D8—1071no—
184C22×Q8—2141no—
185C2×SD16—1071no—
186C2×SD16C2×D10711yescharacteristic
187C2×D8—1071no—
188C4⋊Q8—1071no—
189C4⋊Q8—1071no—
190C4⋊Q8—1071no—
191C4⋊Q8—1071no—
192#126—2141no—
193#126—2141no—
194#124—1071no—
195#124—1071no—
196C22⋊Q8—2141no—
197C22⋊Q8—2141no—
198C22⋊Q8—2141no—
199C22⋊Q8—2141no—
200C4×Q8—1071no—
201C4×Q8—1071no—
202C4×Q8C2×D10711yescharacteristic
203C4×D4—1071no—
204C4×D4—1071no—
205C4×D4—1071no—
206C4×D4C2×D10711yescharacteristic
207#117—2141no—
208#107—1071no—
209#107—1071no—
210#107C2×D10711yescharacteristic
211C4⋊C8—1071no—
212C4⋊C8—1071no—
213C4⋊C8—1071no—
214#103—1071no—
215#103—1071no—
216#103—1071no—
217#103—1071no—
218#103—1071no—
219#103C2×D10711yescharacteristic
220#102—1071no—
221#102—1071no—
222#102—1071no—
223#102C2×D10711yescharacteristic
224#97—1071no—
225#97C2×D10711yescharacteristic
226C4×C8—1071no—
227C2×Q8—1071no—
228C2×Q8—1071no—
229C2×Q8—4284no—
230C2×Q8—2141no—
231C2×Q8—2141no—
232C2×Q8—2141no—
233C2×Q8C22×D10711yescharacteristic
234C2×D4—1071no—
235C2×D4C22×D10711yescharacteristic
236C22×C4—21no—
237C22×C4—2141no—
238C22×C4—2141no—
239C22×C4—2141no—
240SD16—22no—
241SD16—2142no—
242D8—2142no—
243C2×C8—1071no—
244C2×C8—1071no—
245C2×C8C22×D10711yescharacteristic
246C2×C8C22×D10711yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—1071no—
249C4⋊C4—1071no—
250C4⋊C4—1071no—
251C4⋊C4—1071no—
252C4⋊C4—1071no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2141no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C42—21no—
270C42—1071no—
271C42—1071no—
272C42—2141no—
273C42C22×D10711yescharacteristic
274C23—21no—
275C23—2141no—
276Q8—21no—
277Q8—21no—
278Q8—4282no—
279Q8—4282no—
280Q8—4282no—
281Q8—2142no—
282Q8—2141no—
283Q8—2141no—
284D4—21no—
285D4—2141no—
286D4—2141no—
287D4#1313381812yes—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—1071no—
293C2×C4—1071no—
294C2×C4—1071no—
295C2×C4—1071no—
296C2×C4—4282no—
297C2×C4—4282no—
298C2×C4—4282no—
299C2×C4—2141no—
300C2×C4—2141no—
301C2×C4—2141no—
302C2×C4D4×D10711yescharacteristic
303C2×C4D4×D10711yescharacteristic
304C2×C4C23×D10711yescharacteristic
305C8—21no—
306C8—21no—
307C8—2141no—
308C8—2141no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—4281no—
313C22—4281no—
314C22—4281no—
315C22C2×D4×D10711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—1072no—
322C4—4281no—
323C4—4281no—
324C4—4281no—
325C4—2141no—
326C4—2141no—
327C4—2141no—
328C4C2×#1313381811yescharacteristic
329C4#1681215611yescharacteristic
330C2—42no—
331C2—8561no—
332C2#2029328611yescharacteristic
333C2AGL(1,ℤ/8)×D10711yescharacteristic
334C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#440D107
hypocenterC107#3367
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #20293349C2semidirect
2. D4×Dic214C2semidirect
3. #20292988C2semidirect
4. #20292987C2semidirect
5. #20292976C2semidirect
6. #20292922C2semidirect
7. #20292921C2semidirect
8. #20292906C2semidirect
11. #20292830C2semidirect
12. #20292755C2semidirect
13. #20292724C2semidirect
15. #440×C107C2semidirect
168. C107#3367semidirect
182. #440D107semidirect

Maximal subgroups

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

subgroupindex
#202933492
D4×Dic2142
#202929882
#202929872
#202929762
#202929222
#202929212
#202929062
#202928442
#202928422
#202928302
#202927552
#202927242
#202927072
#440×C1072
#3367107