Finite Group Databasebeta

← #28346723

Subgroups — #28346723

Order 13696.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293248C211yesmaximal, characteristic, complemented
2#20292962C211yesmaximal, characteristic, complemented
3#20292940C211yesmaximal, characteristic, complemented
4#20292922C211yesmaximal, characteristic, complemented
5#20292907C211yesmaximal, characteristic, complemented
6#20292849C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292789C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic
10#20292727C211yesmaximal, characteristic, complemented
11#20292725C211yesmaximal, characteristic, complemented
12C4×#13133808C211yesmaximal, characteristic, complemented
13C4×#13133803C211yesmaximal, characteristic, complemented
14#20292688C211yesmaximal, characteristic, complemented
15#520×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20#16812109C2211yescharacteristic
21#16812104—21no—
22#16812097C2211yescharacteristic
23#16812095—21no—
24#16812091—21no—
25#16812088C2211yescharacteristic
26#16812087—21no—
27#16812083C2211yescharacteristic
28C4×Dic214C2211yescharacteristic
29C4×D428C2211yescharacteristic
30#16812042C2211yescharacteristic
31#16812042C2211yescharacteristic
32#16812042C2211yescharacteristic
33#16812042C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35C4⋊(C107⋊C8)C2211yescharacteristic
36#16812033C2211yescharacteristic
37#16812033C2211yescharacteristic
38#16812032C2211yescharacteristic
39#16812032C2211yescharacteristic
40#16812030C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44C22⋊(C107⋊C8)C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46#16812020C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48C2×#41×C107C2211yescharacteristic
49C4×Q8×C107C2211yescharacteristic
50C4×Q8×C107C2211yescharacteristic
51C4×D4×C107—21no—
52C4×D4×C107—21no—
53C4×D4×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55#117×C107—21no—
56#117×C107C2211yescharacteristic
57C2×C4⋊C4×C107—21no—
58C2×C4⋊C4×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C4×D107—21no—
64#13133808—22no—
65#13133806—22no—
66#13133805—22no—
67#13133803—22no—
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2311yescharacteristic
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C4⋊Dic107—21no—
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77C22⋊Dic107—21no—
78(C2×D107)⋊C4—21no—
79(C2×D107)⋊C4—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82C4×Dic107—21no—
83#41×C107—22no—
84#41×C107—22no—
85C2×Q8×C107C2311yescharacteristic
86C2×D4×C107—21no—
87C2×D4×C107C2311yescharacteristic
88C22×C4×C107—21no—
89C22×C4×C107—21no—
90C22×C4×C107—21no—
91C22×C4×C107—21no—
92C22×C4×C107C2311yescharacteristic
93C4⋊C4×C107—21no—
94C4⋊C4×C107—21no—
95C4⋊C4×C107—21no—
96C4⋊C4×C107D411yescharacteristic
97C4⋊C4×C107D411yescharacteristic
98C4⋊C4×C107C2311yescharacteristic
99C4⋊C4×C107C2311yescharacteristic
100C22⋊C4×C107—21no—
101C22⋊C4×C107—21no—
102C22⋊C4×C107D411yescharacteristic
103C22⋊C4×C107D411yescharacteristic
104C42×C107—21no—
105C42×C107—21no—
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111Q8×C107—21no—
112Q8×C107#4112yes—
113D4×C107—24no—
114D4×C107—21no—
115D4×C107#4112yes—
116C2×C4×C107—24no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—81no—
123C2×C4×C107—81no—
124C2×C4×C107—81no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—81no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135D428—21no—
136D428—21no—
137C4×D107—81no—
138Dic214—21no—
139Dic214—21no—
140Dic214—81no—
141C107⋊C8—21no—
142C107⋊C8—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—22no—
147C22×C107—21no—
148C22×C107—21no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107C22×D411yescharacteristic
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C4×C107—21no—
156C4×C107—22no—
157C4×C107—21no—
158C4×C107—21no—
159C4×C107—21no—
160C4×C107—21no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107C2×#4111yescharacteristic
164C4×C107C2×#4111yescharacteristic
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168C2×C107—41no—
169C2×C107—42no—
170C2×C107#52511yescharacteristic
171C2×C107#57611yescharacteristic
172C2×C107#57811yescharacteristic
173D107—81no—
174#3124—1071nomaximal, Sylow
175C107#312411yescharacteristic, complemented, Sylow
176#542—1071no—
177#520D10711yescharacteristic, complemented
178#484—1071no—
179#480—1071no—
180#477—1071no—
181#474—1071no—
182#464—1071no—
183#463—1071no—
184#452—1071no—
185#449—1071no—
186#440—1071no—
187#440—1071no—
188C4×Q16—1071no—
189C4×D8—1071no—
190#433—1071no—
191C2×#41C2×D10711yescharacteristic
192C2×Q16—1071no—
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×D8—1071no—
196C4⋊Q8—1071no—
197#126—2141no—
198#124—1071no—
199#124—2141no—
200#123—2141no—
201C22⋊Q8—1071no—
202C22⋊Q8—2141no—
203#121—1071no—
204C4×Q8—1071no—
205C4×Q8C2×D10711yescharacteristic
206C4×Q8C2×D10711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1071no—
210C4×D4C2×D10711yescharacteristic
211C4×D4C2×D10711yescharacteristic
212#117—21no—
213#117C2×D10711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D10711yescharacteristic
216#107—1071no—
217#107—1071no—
218#107—1071no—
219#107—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222#103—1071no—
223#103—1071no—
224#103—1071no—
225#103—1071no—
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102—1071no—
230C22⋊C8—1071no—
231C22⋊C8—1071no—
232#97—1071no—
233C4×C8—1071no—
234#41—22no—
235#41—22no—
236C2×Q8—1071no—
237C2×Q8—2141no—
238C2×Q8C22×D10711yescharacteristic
239C2×D4—21no—
240C2×D4—1071no—
241C2×D4—2141no—
242C2×D4C22×D10711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2141no—
248C22×C4C22×D10711yescharacteristic
249Q16—2142no—
250SD16—2142no—
251SD16—2142no—
252D8—2142no—
253C2×C8—1071no—
254C2×C8—1071no—
255C2×C8—1071no—
256C2×C8—1071no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4C22⋊D10711yescharacteristic
266C4⋊C4C22⋊D10711yescharacteristic
267C4⋊C4C22×D10711yescharacteristic
268C4⋊C4C22×D10711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4C22⋊D10711yescharacteristic
277C22⋊C4C22⋊D10711yescharacteristic
278C42—21no—
279C42—21no—
280C42—2141no—
281C42C22×D10711yescharacteristic
282C42C22×D10711yescharacteristic
283C23—21no—
284C23—2141no—
285C23C2×C22⋊D10711yescharacteristic
286Q8—21no—
287Q8—8561no—
288Q8—2141no—
289Q8—2141no—
290Q8#1313381812yes—
291D4—24no—
292D4—21no—
293D4—8561no—
294D4—2141no—
295D4—2141no—
296D4#1313381812yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—8561no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2141no—
310C2×C4C2×C22⋊D10711yescharacteristic
311C2×C4C2×C22⋊D10711yescharacteristic
312C2×C4C2×C22⋊D10711yescharacteristic
313C2×C4C2×C22⋊D10711yescharacteristic
314C2×C4C2×C22⋊D10711yescharacteristic
315C2×C4C23×D10711yescharacteristic
316C8—2141no—
317C8—2141no—
318C8—2141no—
319C8—2141no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22C22×C22⋊D10711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—4281no—
339C4—4281no—
340C4—4281no—
341C4C2×#1313381811yescharacteristic
342C4C2×#1313381811yescharacteristic
343C2—41no—
344C2—42no—
345C2—8561no—
346C2#2029313811yescharacteristic
347C2#2029343011yescharacteristic
348C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#520×C107C2
socleC22×C107C22×D4
prime radical#20292772C2
hypercenter#520D107
hypocenterC107#3124
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20293248C2semidirect
2. #20292962C2semidirect
3. #20292940C2semidirect
4. #20292922C2semidirect
5. #20292907C2semidirect
6. #20292849C2semidirect
7. #20292842C2semidirect
8. #20292789C2semidirect
10. #20292727C2semidirect
11. #20292725C2semidirect
12. C4×#13133808C2semidirect
13. C4×#13133803C2semidirect
14. #20292688C2semidirect
15. #520×C107C2semidirect
175. C107#3124semidirect
177. #520D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202929622
#202929402
#202929222
#202929072
#202928492
#202928422
#202927892
#202927722
#202927272
#202927252
C4×#131338082
C4×#131338032
#202926882
#520×C1072
#3124107