Finite Group Databasebeta

← #28348357

Subgroups — #28348357

Order 13696.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293286C211yesmaximal, characteristic, complemented
2#20292984C211yesmaximal, characteristic, complemented
3#20292968C211yesmaximal, characteristic, complemented
4#20292945C211yesmaximal, characteristic, complemented
5#20292931C211yesmaximal, characteristic, complemented
6#20292861C211yesmaximal, characteristic, complemented
7#20292854C211yesmaximal, characteristic, complemented
8#20292842C211yesmaximal, characteristic, complemented
9#20292770C211yesmaximal, characteristic, complemented
10#20292765C211yesmaximal, characteristic, complemented
11C4×#13133806C211yesmaximal, characteristic, complemented
12D4×C107⋊C8C211yesmaximal, characteristic, complemented
13#20292621C211yesmaximal, characteristic, complemented
14C2×#16812032C211yesmaximal, characteristic, complemented
15#553×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23#16812097C2211yescharacteristic
24#16812095—21no—
25#16812088C2211yescharacteristic
26#16812087—21no—
27#16812083C2211yescharacteristic
28#16812081—21no—
29C4×Dic214C2211yescharacteristic
30(C22⋊D107)⋊C4—21no—
31#16812042C2211yescharacteristic
32#16812040C2211yescharacteristic
33#16812040C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35#16812032C2211yescharacteristic
36#16812032C2211yescharacteristic
37#16812032C2211yescharacteristic
38#16812032C2211yescharacteristic
39#16812032C2211yescharacteristic
40#16812032C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44#16812029C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48#126×C107—21no—
49#125×C107C2211yescharacteristic
50#123×C107—21no—
51#121×C107—21no—
52#121×C107C2211yescharacteristic
53#121×C107C2211yescharacteristic
54C4×D4×C107—21no—
55C4×D4×C107C2211yescharacteristic
56C4×D4×C107C2211yescharacteristic
57#117×C107C2211yescharacteristic
58C2×C4⋊C4×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61C2×Dic214—22no—
62C2×Dic214—21no—
63C2×Dic214—41no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71#13133806—22no—
72#13133806—41no—
73#13133803—41no—
74C2×C107⋊C8—41no—
75C2×C107⋊C8—41no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C4×Dic107—21no—
88C2×D4×C107—21no—
89C2×D4×C107—21no—
90C2×D4×C107D411yescharacteristic
91C2×D4×C107C2311yescharacteristic
92C22×C4×C107—21no—
93C22×C4×C107—21no—
94C22×C4×C107C2311yescharacteristic
95C22×C4×C107C2311yescharacteristic
96C4⋊C4×C107—21no—
97C4⋊C4×C107—21no—
98C4⋊C4×C107—21no—
99C4⋊C4×C107D411yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C22⋊C4×C107D411yescharacteristic
108C22⋊C4×C107D411yescharacteristic
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115D4×C107—22no—
116D4×C107—21no—
117D4×C107—21no—
118D4×C107—42no—
119D4×C107—81no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—81no—
128C2×C4×C107—81no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2411yescharacteristic
134C22⋊D107—42no—
135C22⋊D107—42no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—42no—
140D428—21no—
141D428—21no—
142C4×D107—42no—
143Dic214—21no—
144Dic214—21no—
145Dic214—21no—
146Dic214—42no—
147Dic214#4112yes—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107#13512yes—
161C22×C107C22×D411yescharacteristic
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—41no—
165C4×C107—21no—
166C4×C107—21no—
167C4×C107—21no—
168C4×C107—21no—
169C4×C107—41no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107C2×#4111yescharacteristic
173C4×C107H(5,𝔽2)11yescharacteristic
174Dic107—22no—
175Dic107—41no—
176Dic107—41no—
177C2×C107—22no—
178C2×C107—41no—
179C2×C107—81no—
180C2×C107#54611yescharacteristic
181C2×C107C2×#13511yescharacteristic
182C2×C107#57711yescharacteristic
183D107—81no—
184#3301—1071nomaximal, Sylow
185C107#330111yescharacteristic, complemented, Sylow
186#553D10711yescharacteristic, complemented
187#548—1071no—
188#488—1071no—
189#485—1071no—
190#481—1071no—
191#479—1071no—
192#466—1071no—
193#465—1071no—
194#463—1071no—
195#449—1071no—
196#448—1071no—
197C4×SD16—1071no—
198C8×D4—1071no—
199#416—1071no—
200C2×#103—1071no—
201C2×#41—1071no—
202C22×Q8—1071no—
203C2×SD16—1071no—
204C2×SD16—1071no—
205C2×D8—1071no—
206C22×C8—1071no—
207C22×C8—1071no—
208#126—21no—
209#125C2×D10711yescharacteristic
210#124—1071no—
211#124—2141no—
212#123—21no—
213C22⋊Q8—1071no—
214C22⋊Q8—2141no—
215#121—21no—
216#121—1071no—
217#121—2141no—
218#121C2×D10711yescharacteristic
219#121C2×D10711yescharacteristic
220C4×Q8—1071no—
221C4×D4—21no—
222C4×D4—2141no—
223C4×D4C2×D10711yescharacteristic
224C4×D4C2×D10711yescharacteristic
225#117C2×D10711yescharacteristic
226C2×C4⋊C4C2×D10711yescharacteristic
227#107—1071no—
228#106—1071no—
229#106—1071no—
230C4⋊C8—1071no—
231#103—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#103—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241C22⋊C8—1071no—
242C22⋊C8—1071no—
243C4×C8—1071no—
244#41—4281no—
245#41—2142no—
246C2×Q8—1071no—
247C2×Q8—1071no—
248C2×Q8—4281no—
249C2×Q8—2142no—
250C2×Q8—2141no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—1071no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4C22⋊D10711yescharacteristic
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
263C22×C4C22×D10711yescharacteristic
264SD16—4281no—
265SD16—2142no—
266D8—4281no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—1071no—
271C2×C8—4281no—
272C2×C8—4281no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1071no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4C22⋊D10711yescharacteristic
280C4⋊C4C22×D10711yescharacteristic
281C4⋊C4C22×D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4C22⋊D10711yescharacteristic
292C22⋊C4C22⋊D10711yescharacteristic
293C42—21no—
294C42—2141no—
295C42C22×D10711yescharacteristic
296C23—21no—
297C23—2141no—
298C23C2×C22⋊D10711yescharacteristic
299C23C2×C22⋊D10711yescharacteristic
300Q8—1072no—
301Q8—4282no—
302Q8—2141no—
303Q8—2141no—
304Q8—2141no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—4282no—
311D4—4282no—
312D4—2141no—
313D4—2141no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—4282no—
324C2×C4—4282no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4C2×C22⋊D10711yescharacteristic
329C2×C4C2×C22⋊D10711yescharacteristic
330C2×C4C2×C22⋊D10711yescharacteristic
331C2×C4C2×C22⋊D10711yescharacteristic
332C2×C4C23×D10711yescharacteristic
333C8—2141no—
334C8—2141no—
335C8—2141no—
336C8—2141no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—4281no—
346C22—4281no—
347C22—4281no—
348C22#1681212912yes—
349C22C22×C22⋊D10711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4281no—
358C4—4281no—
359C4—2142no—
360C4#41×D10711yescharacteristic
361C4#1681215511yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—8561no—
366C2#2029326511yescharacteristic
367C2C2×#1681212911yescharacteristic
368C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#553×C107C2
socleC22×C107C22×D4
hypercenter#553D107
hypocenterC107#3301
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20293286C2semidirect
2. #20292984C2semidirect
3. #20292968C2semidirect
4. #20292945C2semidirect
5. #20292931C2semidirect
6. #20292861C2semidirect
7. #20292854C2semidirect
8. #20292842C2semidirect
9. #20292770C2semidirect
10. #20292765C2semidirect
11. C4×#13133806C2semidirect
12. D4×C107⋊C8C2semidirect
13. #20292621C2semidirect
14. C2×#16812032C2semidirect
15. #553×C107C2semidirect
185. C107#3301semidirect
186. #553D107semidirect

Maximal subgroups

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

subgroupindex
#202932862
#202929842
#202929682
#202929452
#202929312
#202928612
#202928542
#202928422
#202927702
#202927652
C4×#131338062
D4×C107⋊C82
#202926212
C2×#168120322
#553×C1072
#3301107