Finite Group Databasebeta

← #28349191

Subgroups — #28349191

Order 13696.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293353C211yesmaximal, characteristic, complemented
2#20293298C211yesmaximal, characteristic
3#20292992C211yesmaximal, characteristic, complemented
4#20292990C211yesmaximal, characteristic, complemented
5#20292972C211yesmaximal, characteristic, complemented
6#20292930C211yesmaximal, characteristic, complemented
7#20292927C211yesmaximal, characteristic, complemented
8#20292906C211yesmaximal, characteristic, complemented
9#20292827C211yesmaximal, characteristic, complemented
10#20292825C211yesmaximal, characteristic, complemented
11#20292825C211yesmaximal, characteristic, complemented
12#20292755C211yesmaximal, characteristic, complemented
13#20292730C211yesmaximal, characteristic, complemented
14#20292712C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×#13133818—21no—
17C2×#13133805C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19#16812108—21no—
20#16812108C2211yescharacteristic
21#16812106C2211yescharacteristic
22#16812105C2211yescharacteristic
23#16812105C2211yescharacteristic
24#16812101—21no—
25#16812101C2211yescharacteristic
26#16812100—21no—
27#16812100C2211yescharacteristic
28#16812083—21no—
29#16812082—21no—
30#16812082—21no—
31Q8×Dic107C2211yescharacteristic
32Q8×Dic107C2211yescharacteristic
33C4×Dic214—21no—
34C4×Dic214C2211yescharacteristic
35#16812071C2211yescharacteristic
36#16812071C2211yescharacteristic
37#16812042C2211yescharacteristic
38#16812040C2211yescharacteristic
39C4⋊(C107⋊C8)C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812031C2211yescharacteristic
43#16812031C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
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
59#13133818—44no—
60C2×Dic214—21no—
61C2×Dic214D411yescharacteristic
62C2×D428—21no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72#13133805—22no—
73#13133805—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107D411yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87Dic107⋊C4D411yescharacteristic
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90C4×Dic107—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—
108C22×D107—21no—
109Q8×C107—21no—
110Q8×C107#4112yescomplemented
111D4×C107—21no—
112D4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—42no—
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2411yescharacteristic
120C22⋊D107—42no—
121C22⋊D107—42no—
122C22⋊D107—81no—
123C22⋊D107—81no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130D428—21no—
131D428—21no—
132D428—21no—
133D428—21no—
134D428—42no—
135D428—81no—
136C4×D107—42no—
137C4×D107—42no—
138Dic214—22no—
139Dic214—42no—
140C8×C107—21no—
141C8×C107—21no—
142C107⋊C8—21no—
143C107⋊C8—21no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107C22×D411yescharacteristic
148C2×D107—41no—
149C2×D107—41no—
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C4×C107—22no—
155C4×C107—21no—
156C4×C107—21no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107C2×#4111yescharacteristic
160C4×C1072−1+411yescharacteristic
161Dic107—21no—
162Dic107—21no—
163Dic107—21no—
164Dic107—41no—
165Dic107—41no—
166Dic107AGL(1,ℤ/8)12yes—
167C2×C107—81no—
168C2×C107#54811yescharacteristic
169C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C107#57711yescharacteristic
171D107—81no—
172D107—81no—
173#3368—1071nomaximal, Sylow
174C107#336811yescharacteristic, complemented, Sylow
175#557—1071no—
176#550—1071no—
177#490—1071no—
178#490—1071no—
179#486—1071no—
180#478—1071no—
181#478—1071no—
182#474—1071no—
183#460—1071no—
184#460—1071no—
185#460—1071no—
186#446—1071no—
187#440—1071no—
188#440D10711yescharacteristic, complemented
189C4×SD16—1071no—
190C2×#41—2141no—
191C2×SD16—1071no—
192C2×SD16—1071no—
193C2×SD16C2×D10711yescharacteristic
194C4⋊Q8—1071no—
195C4⋊Q8—2141no—
196#127—1071no—
197#127—1071no—
198#127—1071no—
199#125—1071no—
200#125—1071no—
201#125—2141no—
202#125—2141no—
203#121—2141no—
204#121—2141no—
205#121—2141no—
206C4×Q8—1071no—
207C4×Q8—1071no—
208C4×Q8—1071no—
209C4×Q8—2141no—
210C4×Q8C2×D10711yescharacteristic
211C4×D4—1071no—
212C4×D4—1071no—
213C4×D4C2×D10711yescharacteristic
214#107—1071no—
215#107C2×D10711yescharacteristic
216#106—1071no—
217C4⋊C8—1071no—
218C4⋊C8—1071no—
219C4⋊C8—1071no—
220#103C2×D10711yescharacteristic
221#102—1071no—
222#102—1071no—
223#102—1071no—
224#102—1071no—
225#102—1071no—
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102C2×D10711yescharacteristic
230#97—1071no—
231#97C2×D10711yescharacteristic
232C4×C8—1071no—
233#41—4284no—
234C2×Q8—1071no—
235C2×Q8—2141no—
236C2×Q8C22×D10711yescharacteristic
237C2×D4—1071no—
238C2×D4—1071no—
239C2×D4—2141no—
240C2×D4—2141no—
241C2×D4—2141no—
242C2×D4—2141no—
243C2×D4—2141no—
244C2×D4—2141no—
245C2×D4C22×D10711yescharacteristic
246C22×C4—21no—
247C22×C4—2141no—
248C22×C4—2141no—
249SD16—22no—
250SD16—2142no—
251SD16—2142no—
252C2×C8—1071no—
253C2×C8—1071no—
254C2×C8C22×D10711yescharacteristic
255C2×C8C22×D10711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4C22×D10711yescharacteristic
270C4⋊C4C22×D10711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C42—21no—
275C42—1071no—
276C42—1071no—
277C42—2141no—
278C42—2141no—
279C42C22×D10711yescharacteristic
280C23—21no—
281C23—2141no—
282C23—2141no—
283Q8—21no—
284Q8—4282no—
285Q8—2142no—
286Q8#1313381812yescomplemented
287D4—21no—
288D4—21no—
289D4—8561no—
290D4—8561no—
291D4—8561no—
292D4—4282no—
293D4—4282no—
294D4—4282no—
295D4—2141no—
296D4—2141no—
297D4—2141no—
298D4—2141no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—1071no—
304C2×C4—1071no—
305C2×C4—1071no—
306C2×C4—1071no—
307C2×C4—4282no—
308C2×C4—4282no—
309C2×C4—2141no—
310C2×C4—2141no—
311C2×C4D4×D10711yescharacteristic
312C2×C4D4×D10711yescharacteristic
313C2×C4C23×D10711yescharacteristic
314C8—21no—
315C8—21no—
316C8—2141no—
317C8—2141no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4281no—
322C22—4281no—
323C22—4281no—
324C22—4281no—
325C22—4281no—
326C22—4281no—
327C22C2×D4×D10711yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—1072no—
334C4—4281no—
335C4—4281no—
336C4—2141no—
337C4—2141no—
338C4—2141no—
339C4C2×#1313381811yescharacteristic
340C4#1681215611yescharacteristic
341C2—81no—
342C2—8561no—
343C2—8561no—
344C2#2029328611yescharacteristic
345C2AGL(1,ℤ/8)×D10711yescharacteristic
346C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radical#20293298C2
hypercenter#440D107
hypocenterC107#3368
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. #20293353C2semidirect
3. #20292992C2semidirect
4. #20292990C2semidirect
5. #20292972C2semidirect
6. #20292930C2semidirect
7. #20292927C2semidirect
8. #20292906C2semidirect
9. #20292827C2semidirect
10. #20292825C2semidirect
11. #20292825C2semidirect
12. #20292755C2semidirect
13. #20292730C2semidirect
14. #20292712C2semidirect
15. #440×C107C2semidirect
110. Q8×C107#41semidirect
174. C107#3368semidirect
188. #440D107semidirect
286. Q8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202933532
#202932982
#202929922
#202929902
#202929722
#202929302
#202929272
#202929062
#202928272
#202928252
#202927552
#202927302
#202927122
#440×C1072
#3368107