Finite Group Databasebeta

← #28345504

Subgroups — #28345504

Order 13696.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293134C211yesmaximal, 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#20292685C211yesmaximal, characteristic, complemented
11#20292685C211yesmaximal, characteristic, complemented
12#20292594C211yesmaximal, characteristic, complemented
13#20292589C211yesmaximal, characteristic, complemented
14C22⋊C8×D107C211yesmaximal, characteristic, complemented
15#408×C107C211yesmaximal, characteristic, complemented
16C22×C4×D107C2211yescharacteristic
17C2×#13133802C2211yescharacteristic, complemented
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic, complemented
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812090C411yescharacteristic
24#16812090C411yescharacteristic
25#16812085C411yescharacteristic
26#16812085C411yescharacteristic
27#16812080C411yescharacteristic
28#16812080C411yescharacteristic
29#16812077C411yescharacteristic
30#16812077C411yescharacteristic
31C4×C22⋊D107C2211yescharacteristic
32C4×C22⋊D107C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34C4×C22⋊D107C2211yescharacteristic
35#16812068C2211yescharacteristic
36Dic107⋊C8C2211yescharacteristic, complemented
37Dic107⋊C8C2211yescharacteristic, complemented
38Dic107⋊C8C2211yescharacteristic, complemented
39Dic107⋊C8C2211yescharacteristic, complemented
40C22⋊(C107⋊C8)C2211yescharacteristic
41C22⋊(C107⋊C8)C2211yescharacteristic
42C22⋊(C107⋊C8)C2211yescharacteristic, complemented
43C22⋊(C107⋊C8)C2211yescharacteristic, complemented
44(C2×D107)⋊C8C2211yescharacteristic
45(C2×D107)⋊C8C2211yescharacteristic
46(C2×D107)⋊C8C2211yescharacteristic
47(C2×D107)⋊C8C2211yescharacteristic
48#16812019C2211yescharacteristic, complemented
49#16812019C2211yescharacteristic, complemented
50C8×Dic107C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×#41×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic, complemented
54C22×C8×C107C2211yescharacteristic, complemented
55C22⋊C8×C107C2211yescharacteristic
56C22⋊C8×C107C2211yescharacteristic
57C22⋊C8×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic, complemented
59C23×D107C2×C411yescharacteristic
60C2×C22⋊D107C2×C411yescharacteristic
61C2×C22⋊D107C2×C411yescharacteristic
62C2×C22⋊D107C2×C411yescharacteristic
63C2×C22⋊D107C2×C411yescharacteristic
64C22×Dic107C2×C411yescharacteristic
65C2×C4×D107—41no—
66C2×C4×D107—41no—
67C2×C4×D107C2×C411yescharacteristic
68C2×C4×D107C2×C411yescharacteristic
69C2×C4×D107C2311yescharacteristic
70C2×C4×D107C2311yescharacteristic
71#13133802—41no—
72#13133801—41no—
73#13133801—41no—
74C2×C107⋊C8—41no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—41no—
80C8×D107—41no—
81C4⋊Dic107C2×C411yescharacteristic
82C4⋊Dic107C2×C411yescharacteristic
83Dic107⋊C4C2×C411yescharacteristic
84Dic107⋊C4C2×C411yescharacteristic
85Dic107⋊C4C2×C411yescharacteristic
86Dic107⋊C4C2×C411yescharacteristic
87C22⋊Dic107C2×C411yescharacteristic
88C22⋊Dic107C2×C411yescharacteristic
89C22⋊Dic107C2×C411yescharacteristic
90C22⋊Dic107C2×C411yescharacteristic
91C22⋊Dic107C2×C411yescharacteristic
92C22⋊Dic107C2×C411yescharacteristic
93(C2×D107)⋊C4C2×C411yescharacteristic
94(C2×D107)⋊C4C2×C411yescharacteristic
95(C2×D107)⋊C4C2×C411yescharacteristic
96(C2×D107)⋊C4C2×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×D107—22no—
118C22×D107—41no—
119C22×D107—41no—
120C22×D107C22×C411yescharacteristic
121C22×D107C22×C411yescharacteristic
122Q8×C107—41no—
123D4×C107—41no—
124D4×C107—41no—
125D4×C107—41no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107C22×C411yescharacteristic
133C2×C4×C107C22×C411yescharacteristic
134C2×C4×C107C22×C411yescharacteristic
135C2×C4×C107C2411yescharacteristic
136C22⋊D107—22no—
137C22⋊D107—22no—
138C22⋊D107—41no—
139C22⋊D107—41no—
140C2×Dic107—41no—
141C2×Dic107C22×C411yescharacteristic
142C2×Dic107C22×C411yescharacteristic
143C2×Dic107C22×C411yescharacteristic
144C2×Dic107C22×C411yescharacteristic
145C2×Dic107C22×C411yescharacteristic
146C2×Dic107C22×C411yescharacteristic
147C4×D107—21no—
148C4×D107—21no—
149C4×D107—21no—
150C4×D107—21no—
151C4×D107—41no—
152C4×D107—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—44no—
177C2×D107#13112yes—
178C4×C107—21no—
179C4×C107—21no—
180C4×C107—21no—
181C4×C107H(5,𝔽2)11yescharacteristic
182C4×C107H(5,𝔽2)11yescharacteristic
183Dic107—21no—
184Dic107—21no—
185Dic107—21no—
186Dic107—21no—
187Dic107—21no—
188Dic107—21no—
189C2×C107—41no—
190C2×C107—41no—
191C2×C107—41no—
192C2×C107#51811yescharacteristic
193C2×C107C2×#13111yescharacteristic
194C2×C107#56811yescharacteristic
195D107—22no—
196D107—41no—
197#2976—1071nomaximal, Sylow
198C107#297611yescharacteristic, complemented, Sylow
199#525—1071no—
200#436—1071no—
201#436—1071no—
202#435—1071no—
203#435—1071no—
204#435—1071no—
205#435—1071no—
206C8×D4—1071no—
207C8×D4—1071no—
208#433—1071no—
209#433—1071no—
210#408—1071no—
211#408D10711yescharacteristic, complemented
212#407—1071no—
213C2×C22⋊C8—1071no—
214C2×#41C2×D10711yescharacteristic
215C23×C4—1071no—
216C2×M16—1071no—
217C2×M16—1071no—
218C2×M16—1071no—
219C2×M16C2×D10711yescharacteristic, complemented
220C22×C8—1071no—
221C22×C8—1071no—
222C22×C8—1071no—
223C22×C8C2×D10711yescharacteristic, complemented
224#123—1071no—
225#123—1071no—
226C22⋊Q8—1071no—
227C22⋊Q8—1071no—
228#121—1071no—
229#121—1071no—
230#120—1071no—
231#120—1071no—
232C4×D4—1071no—
233C4×D4—1071no—
234C4×D4—1071no—
235C4×D4—1071no—
236#117—1071no—
237C4⋊C8—1071no—
238C4⋊C8—1071no—
239C4⋊C8—1071no—
240C4⋊C8—1071no—
241C22⋊C8—1071no—
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C22⋊C8—1071no—
245C22⋊C8—1071no—
246C22⋊C8—1071no—
247C22⋊C8—1071no—
248C22⋊C8—1071no—
249C22⋊C8C2×D10711yescharacteristic
250C22⋊C8C2×D10711yescharacteristic
251C22⋊C8C2×D10711yescharacteristic, complemented
252C22⋊C8C2×D10711yescharacteristic, complemented
253#97—1071no—
254#97—1071no—
255C4×C8—1071no—
256C4×C8—1071no—
257C24—1071no—
258#41—41no—
259#41—41no—
260C2×Q8C4×D10711yescharacteristic
261C2×D4—1071no—
262C2×D4—1071no—
263C2×D4—1071no—
264C2×D4—1071no—
265C2×D4C4×D10711yescharacteristic
266C2×D4C4×D10711yescharacteristic
267C2×D4C4×D10711yescharacteristic
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—1071no—
271C22×C4—1071no—
272C22×C4—1071no—
273C22×C4—4281no—
274C22×C4—4281no—
275C22×C4C22×D10711yescharacteristic
276C22×C4C22×D10711yescharacteristic
277C22×C4C22×D10711yescharacteristic
278M16—41no—
279M16—4281no—
280M16—4281no—
281M16—4281no—
282C2×C8—41no—
283C2×C8—1071no—
284C2×C8—1071no—
285C2×C8—1071no—
286C2×C8—1071no—
287C2×C8—4281no—
288C2×C8—4281no—
289C2×C8—4281no—
290C2×C8C22×D10711yescharacteristic
291C2×C8C22×D10711yescharacteristic
292C2×C8C22×D10711yescharacteristic
293C2×C8C22×D10711yescharacteristic
294C4⋊C4—1071no—
295C4⋊C4—1071no—
296C4⋊C4—1071no—
297C4⋊C4—1071no—
298C4⋊C4—1071no—
299C4⋊C4—1071no—
300C22⋊C4—1071no—
301C22⋊C4—1071no—
302C22⋊C4—1071no—
303C22⋊C4—1071no—
304C22⋊C4—1071no—
305C22⋊C4—1071no—
306C22⋊C4—1071no—
307C22⋊C4—1071no—
308C22⋊C4—1071no—
309C22⋊C4—1071no—
310C42—1071no—
311C42—1071no—
312C23—1071no—
313C23—1071no—
314C23—4281no—
315C23—4281no—
316C23—2142no—
317C23C2×C4×D10711yescharacteristic
318C23C2×C4×D10711yescharacteristic
319C23C2×C4×D10711yescharacteristic
320Q8—41no—
321D4—41no—
322D4—41no—
323D4—41no—
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—4281no—
342C2×C4—4281no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4—2141no—
347C2×C4C2×C4×D10711yescharacteristic
348C2×C4C2×C4×D10711yescharacteristic
349C2×C4C2×C4×D10711yescharacteristic
350C2×C4C23×D10711yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C8—2141no—
356C8—2141no—
357C8—2141no—
358C8—2141no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—1072no—
369C22—4284no—
370C22—2141no—
371C22—2141no—
372C22—2141no—
373C22—2141no—
374C22—2141no—
375C22C22×C4×D10711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—2141no—
380C4—2141no—
381C4—2141no—
382C4—2141no—
383C4—2141no—
384C4—2141no—
385C4#1681215411yescharacteristic
386C4#1681215411yescharacteristic
387C2—41no—
388C2—41no—
389C2—41no—
390C2—4281no—
391C2—2142no—
392C2#2029310011yescharacteristic
393C2#131×D10711yescharacteristic
394C2#2029339211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D107
commutator, socleC22×C107C23×C4
Fitting#408×C107C2
prime radical#20293134C2
cosocleC2×C4×C107C24
hypercenter#408D107
hypocenterC107#2976
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. #20292685C2semidirect
11. #20292685C2semidirect
12. #20292594C2semidirect
13. #20292589C2semidirect
14. C22⋊C8×D107C2semidirect
15. #408×C107C2semidirect
17. C2×#13133802C22semidirect
20. C22×C107⋊C8C22semidirect
36. Dic107⋊C8C22semidirect
37. Dic107⋊C8C22semidirect
38. Dic107⋊C8C22semidirect
39. Dic107⋊C8C22semidirect
42. C22⋊(C107⋊C8)C22semidirect
43. C22⋊(C107⋊C8)C22semidirect
48. #16812019C22semidirect
49. #16812019C22semidirect
50. C8×Dic107C22semidirect
51. C8×Dic107C22semidirect
53. C2×M16×C107C22semidirect
54. C22×C8×C107C22semidirect
57. C22⋊C8×C107C22semidirect
58. C22⋊C8×C107C22semidirect
198. C107#2976semidirect
211. #408D107semidirect
219. C2×M16C2×D107semidirect
223. C22×C8C2×D107semidirect
251. C22⋊C8C2×D107semidirect
252. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931342
#202927032
#202927022
#202927012
#202926982
#202926962
#202926952
C8×C22⋊D1072
(C22⋊D107)⋊C82
#202926852
#202925942
#202925892
C22⋊C8×D1072
#408×C1072
#2976107