Finite Group Databasebeta

← #28345534

Subgroups — #28345534

Order 13696.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293173C211yesmaximal, characteristic
2#20292703C211yesmaximal, characteristic, complemented
3#20292702C211yesmaximal, characteristic, complemented
4#20292701C211yesmaximal, characteristic, complemented
5#20292698C211yesmaximal, characteristic, complemented
6#20292696C211yesmaximal, characteristic, complemented
7#20292695C211yesmaximal, characteristic, complemented
8C8×C22⋊D107C211yesmaximal, characteristic, complemented
9(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
10#20292681C211yesmaximal, characteristic, complemented
11#20292677C211yesmaximal, characteristic, complemented
12#20292594C211yesmaximal, characteristic, complemented
13#20292592C211yesmaximal, characteristic, complemented
14#20292592C211yesmaximal, characteristic, complemented
15#408×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×#13133802C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812107C411yescharacteristic
24#16812105C411yescharacteristic
25#16812098C411yescharacteristic
26#16812096C411yescharacteristic
27#16812081C411yescharacteristic
28#16812081C411yescharacteristic
29#16812081C411yescharacteristic
30#16812081C411yescharacteristic
31C4×C22⋊D107C2211yescharacteristic
32C4×C22⋊D107C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34C4×C22⋊D107C2211yescharacteristic
35C2×C4×Dic107C2211yescharacteristic
36Dic107⋊C8C2211yescharacteristic
37Dic107⋊C8C2211yescharacteristic
38Dic107⋊C8C2211yescharacteristic
39Dic107⋊C8C2211yescharacteristic
40C22⋊(C107⋊C8)C2211yescharacteristic
41C22⋊(C107⋊C8)C2211yescharacteristic
42C22⋊(C107⋊C8)C2211yescharacteristic
43C22⋊(C107⋊C8)C2211yescharacteristic
44(C2×D107)⋊C8C2211yescharacteristic
45(C2×D107)⋊C8C2211yescharacteristic
46(C2×D107)⋊C8C2211yescharacteristic
47(C2×D107)⋊C8C2211yescharacteristic
48#16812019C2211yescharacteristic
49#16812019C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22×C8×C107C2211yescharacteristic
55C22⋊C8×C107C2211yescharacteristic
56C22⋊C8×C107C2211yescharacteristic
57C22⋊C8×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133818—41no—
60#13133818—41no—
61C2×Dic214C2×C411yescharacteristic
62C2×D428C2×C411yescharacteristic
63C2×C22⋊D107C2×C411yescharacteristic
64C2×C22⋊D107C2×C411yescharacteristic
65C2×C22⋊D107C2×C411yescharacteristic
66C2×C22⋊D107C2×C411yescharacteristic
67C2×C22⋊D107C2×C411yescharacteristic
68C2×C22⋊D107C2×C411yescharacteristic
69C22×Dic107C2×C411yescharacteristic
70C22×Dic107C2×C411yescharacteristic
71C2×C4×D107C2311yescharacteristic
72C2×C4×D107C2311yescharacteristic
73#13133802—41no—
74#13133801—41no—
75#13133801—41no—
76C2×C107⋊C8—41no—
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C2×C107⋊C8C2311yescharacteristic
81C8×D107—41no—
82C8×D107—41no—
83Dic107⋊C4C2×C411yescharacteristic
84Dic107⋊C4C2×C411yescharacteristic
85Dic107⋊C4C2×C411yescharacteristic
86Dic107⋊C4C2×C411yescharacteristic
87C22⋊Dic107C2×C411yescharacteristic
88C22⋊Dic107C2×C411yescharacteristic
89C22⋊Dic107C2×C411yescharacteristic
90C22⋊Dic107C2×C411yescharacteristic
91(C2×D107)⋊C4C2×C411yescharacteristic
92(C2×D107)⋊C4C2×C411yescharacteristic
93(C2×D107)⋊C4C2×C411yescharacteristic
94(C2×D107)⋊C4C2×C411yescharacteristic
95C4×Dic107C2×C411yescharacteristic
96C4×Dic107C2×C411yescharacteristic
97C4×Dic107C2311yescharacteristic
98C4×Dic107C2311yescharacteristic
99#41×C107—41no—
100#41×C107—41no—
101C2×Q8×C107C2×C411yescharacteristic
102C2×D4×C107C2×C411yescharacteristic
103C2×D4×C107C2×C411yescharacteristic
104C2×D4×C107C2×C411yescharacteristic
105C22×C4×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107C22×C4×C107C2311yescharacteristic
108M16×C107—41no—
109C2×C8×C107—41no—
110C2×C8×C107C2311yescharacteristic
111C2×C8×C107C2311yescharacteristic
112C2×C8×C107C2311yescharacteristic
113C2×C8×C107C2311yescharacteristic
114C23×C107C22×C411yescharacteristic
115C23×C107C22×C411yescharacteristic
116C23×C107C22×C411yescharacteristic
117C22×D107C22×C411yescharacteristic
118C22×D107C22×C411yescharacteristic
119Q8×C107—41no—
120D4×C107—41no—
121D4×C107—41no—
122D4×C107—41no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107C22×C411yescharacteristic
130C2×C4×C107C22×C411yescharacteristic
131C2×C4×C107C22×C411yescharacteristic
132C2×C4×C107C2411yescharacteristic
133C22⋊D107—22no—
134C22⋊D107—22no—
135C22⋊D107—41no—
136C22⋊D107—41no—
137C22⋊D107—41no—
138C22⋊D107—41no—
139C2×Dic107—22no—
140C2×Dic107—41no—
141C2×Dic107C22×C411yescharacteristic
142C2×Dic107C22×C411yescharacteristic
143C2×Dic107C22×C411yescharacteristic
144C2×Dic107C22×C411yescharacteristic
145C2×Dic107C22×C411yescharacteristic
146C2×Dic107C22×C411yescharacteristic
147D428—41no—
148C4×D107—21no—
149C4×D107—21no—
150C4×D107—21no—
151C4×D107—21no—
152Dic214—41no—
153C8×C107—21no—
154C8×C107—21no—
155C8×C107—21no—
156C8×C107—21no—
157C107⋊C8—21no—
158C107⋊C8—21no—
159C107⋊C8—21no—
160C107⋊C8—21no—
161C22×C107—21no—
162C22×C107—21no—
163C22×C107—21no—
164C22×C107—21no—
165C22×C107—21no—
166C22×C107—21no—
167C22×C107—21no—
168C22×C107—21no—
169C22×C107—21no—
170C22×C107C23×C411yescharacteristic
171C2×D107—21no—
172C2×D107—21no—
173C2×D107—21no—
174C2×D107—21no—
175C2×D107—21no—
176C2×D107—21no—
177C4×C107—21no—
178C4×C107—21no—
179C4×C107—21no—
180C4×C107H(5,𝔽2)11yescharacteristic
181C4×C107H(5,𝔽2)11yescharacteristic
182Dic107—21no—
183Dic107—21no—
184Dic107—21no—
185Dic107—21no—
186Dic107—21no—
187Dic107#13112yes—
188C2×C107—41no—
189C2×C107—41no—
190C2×C107—41no—
191C2×C107#51811yescharacteristic
192C2×C107C2×#13111yescharacteristic
193C2×C107#56811yescharacteristic
194D107—41no—
195D107—41no—
196#2981—1071nomaximal, Sylow
197C107#298111yescharacteristic, complemented, Sylow
198#532—1071no—
199#436—1071no—
200#436—1071no—
201#435—1071no—
202#435—1071no—
203#435—1071no—
204#435—1071no—
205C8×D4—1071no—
206C8×D4—1071no—
207#432—1071no—
208#431—1071no—
209#408—1071no—
210#408—1071no—
211#408—1071no—
212#408D10711yescharacteristic, complemented
213C2×#41—1071no—
214C2×#41C2×D10711yescharacteristic
215C2×M16—1071no—
216C2×M16—1071no—
217C2×M16—1071no—
218C2×M16C2×D10711yescharacteristic
219C22×C8—1071no—
220C22×C8—1071no—
221C22×C8—1071no—
222C22×C8C2×D10711yescharacteristic
223C4⋊Q8—1071no—
224#127—1071no—
225#124—1071no—
226#124—1071no—
227#121—1071no—
228#121—1071no—
229#121—1071no—
230#121—1071no—
231C4×D4—1071no—
232C4×D4—1071no—
233C4×D4—1071no—
234C4×D4—1071no—
235C2×C42—1071no—
236C4⋊C8—1071no—
237C4⋊C8—1071no—
238C4⋊C8—1071no—
239C4⋊C8—1071no—
240C22⋊C8—1071no—
241C22⋊C8—1071no—
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C22⋊C8—1071no—
245C22⋊C8—1071no—
246C22⋊C8—1071no—
247C22⋊C8—1071no—
248C22⋊C8C2×D10711yescharacteristic
249C22⋊C8C2×D10711yescharacteristic
250C22⋊C8C2×D10711yescharacteristic
251C22⋊C8C2×D10711yescharacteristic
252#97—1071no—
253#97—1071no—
254C4×C8—1071no—
255C4×C8—1071no—
256#41—41no—
257#41—41no—
258#41—4281no—
259#41—4281no—
260C2×Q8—1071no—
261C2×Q8C4×D10711yescharacteristic
262C2×D4—1071no—
263C2×D4—1071no—
264C2×D4—1071no—
265C2×D4—1071no—
266C2×D4—1071no—
267C2×D4—1071no—
268C2×D4—1071no—
269C2×D4C4×D10711yescharacteristic
270C2×D4C4×D10711yescharacteristic
271C2×D4C4×D10711yescharacteristic
272C22×C4—1071no—
273C22×C4—1071no—
274C22×C4—1071no—
275C22×C4—1071no—
276C22×C4C22×D10711yescharacteristic
277C22×C4C22×D10711yescharacteristic
278C22×C4C22×D10711yescharacteristic
279M16—41no—
280M16—4281no—
281M16—4281no—
282M16—4281no—
283C2×C8—41no—
284C2×C8—1071no—
285C2×C8—1071no—
286C2×C8—1071no—
287C2×C8—1071no—
288C2×C8—4281no—
289C2×C8—4281no—
290C2×C8—4281no—
291C2×C8C22×D10711yescharacteristic
292C2×C8C22×D10711yescharacteristic
293C2×C8C22×D10711yescharacteristic
294C2×C8C22×D10711yescharacteristic
295C4⋊C4—1071no—
296C4⋊C4—1071no—
297C4⋊C4—1071no—
298C4⋊C4—1071no—
299C22⋊C4—1071no—
300C22⋊C4—1071no—
301C22⋊C4—1071no—
302C22⋊C4—1071no—
303C22⋊C4—1071no—
304C22⋊C4—1071no—
305C22⋊C4—1071no—
306C22⋊C4—1071no—
307C42—1071no—
308C42—1071no—
309C42—1071no—
310C42—1071no—
311C23—1071no—
312C23—1071no—
313C23C2×C4×D10711yescharacteristic
314C23C2×C4×D10711yescharacteristic
315C23C2×C4×D10711yescharacteristic
316Q8—41no—
317Q8—4281no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—4281no—
322D4—4281no—
323D4—4281no—
324D4—4281no—
325D4—4281no—
326D4—2142no—
327D4—2142no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—1071no—
335C2×C4—1071no—
336C2×C4—1071no—
337C2×C4—1071no—
338C2×C4—1071no—
339C2×C4—1071no—
340C2×C4—4281no—
341C2×C4—2142no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4C2×C4×D10711yescharacteristic
347C2×C4C2×C4×D10711yescharacteristic
348C2×C4C2×C4×D10711yescharacteristic
349C2×C4C23×D10711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—2141no—
355C8—2141no—
356C8—2141no—
357C8—2141no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—2141no—
368C22—2141no—
369C22—2141no—
370C22—2141no—
371C22—2141no—
372C22—2141no—
373C22C22×C4×D10711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—1072no—
378C4—2141no—
379C4—2141no—
380C4—2141no—
381C4—2141no—
382C4—2141no—
383C4#1681215411yescharacteristic
384C4#1681215411yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—4281no—
389C2—4281no—
390C2#2029310011yescharacteristic
391C2#131×D10711yescharacteristic
392C2#2029339211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D107
commutator, socleC22×C107C23×C4
Fitting#408×C107C2
prime radical#20293173C2
cosocleC2×C4×C107C24
hypercenter#408D107
hypocenterC107#2981
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20292703C2semidirect
3. #20292702C2semidirect
4. #20292701C2semidirect
5. #20292698C2semidirect
6. #20292696C2semidirect
7. #20292695C2semidirect
8. C8×C22⋊D107C2semidirect
9. (C22⋊D107)⋊C8C2semidirect
10. #20292681C2semidirect
11. #20292677C2semidirect
12. #20292594C2semidirect
13. #20292592C2semidirect
14. #20292592C2semidirect
15. #408×C107C2semidirect
197. C107#2981semidirect
212. #408D107semidirect

Maximal subgroups

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

subgroupindex
#202931732
#202927032
#202927022
#202927012
#202926982
#202926962
#202926952
C8×C22⋊D1072
(C22⋊D107)⋊C82
#202926812
#202926772
#202925942
#202925922
#408×C1072
#2981107