Finite Group Databasebeta

← #28347024

Subgroups — #28347024

Order 13696.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293163C211yesmaximal, characteristic, complemented
2#20293158C211yesmaximal, characteristic, complemented
3#20293027C211yesmaximal, characteristic, complemented
4#20293008C211yesmaximal, characteristic, complemented
5#20292983C211yesmaximal, characteristic, complemented
6#20292980C211yesmaximal, characteristic, complemented
7#20292866C211yesmaximal, characteristic, complemented
8#20292861C211yesmaximal, characteristic
9#20292856C211yesmaximal, characteristic, complemented
10#20292719C211yesmaximal, characteristic, complemented
11SD16×Dic107C211yesmaximal, characteristic, complemented
12#20292714C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14C2×C8×Dic107C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20#16812104—21no—
21#16812102—21no—
22#16812100C2211yescharacteristic
23#16812100C2211yescharacteristic
24#16812098C2211yescharacteristic
25#16812096C2211yescharacteristic
26#16812093—21no—
27#16812090—21no—
28#16812088C2211yescharacteristic
29#16812083C2211yescharacteristic
30Q8×Dic107C2211yescharacteristic
31#16812074C2211yescharacteristic
32D4×Dic107—21no—
33D4×Dic107C2211yescharacteristic
34#16812071C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36#16812067—21no—
37#16812065—21no—
38C2×C4×Dic107C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48C8×Dic107C2211yescharacteristic
49C8×Dic107C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C22×Dic107—21no—
63C22×Dic107D411yescharacteristic
64C22×Dic107D411yescharacteristic
65C2×C4×D107—21no—
66#13133808—22no—
67#13133807—22no—
68#13133803—22no—
69C2×C107⋊C8—22no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107—21no—
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81C22⋊Dic107—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107D411yescharacteristic
92C4×Dic107D411yescharacteristic
93C4×Dic107C2311yescharacteristic
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×D4×C107—21no—
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99SD16×C107—22no—
100C2×C8×C107—22no—
101C2×C8×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C4⋊C4×C107—21no—
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—42no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2411yescharacteristic
123C22⋊D107—42no—
124C2×Dic107—22no—
125C2×Dic107—22no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—42no—
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134D428—21no—
135D428—21no—
136C4×D107—42no—
137Dic214—21no—
138Dic214—21no—
139C8×C107—21no—
140C8×C107#4112yes—
141C107⋊C8—21no—
142C107⋊C8#4112yes—
143C22×C107—21no—
144C22×C107—21no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107#13512yes—
149C22×C107C22×D411yescharacteristic
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C4×C107—21no—
154C4×C107—41no—
155C4×C107—41no—
156C4×C107—41no—
157C4×C107C2×#4111yescharacteristic
158C4×C107C2×#4111yescharacteristic
159Dic107—21no—
160Dic107—21no—
161Dic107—21no—
162Dic107—41no—
163Dic107—41no—
164Dic107—41no—
165Dic107#13512yes—
166C2×C107—22no—
167C2×C107—81no—
168C2×C107#53211yescharacteristic
169C2×C107C2×#13511yescharacteristic
170C2×C107C2×#13511yescharacteristic
171D107—81no—
172#3168—1071nomaximal, Sylow
173C107#316811yescharacteristic, complemented, Sylow
174#529—1071no—
175#529—1071no—
176#499—1071no—
177#495—1071no—
178#488—1071no—
179#488—1071no—
180#467—1071no—
181#466—1071no—
182#465—1071no—
183#465D10711yescharacteristic, complemented
184C4×Q16—1071no—
185C4×SD16—1071no—
186C4×SD16—1071no—
187C4×D8—1071no—
188C2×C4×C8—1071no—
189C2×Q16—1071no—
190C2×SD16—1071no—
191C2×SD16C2×D10711yescharacteristic
192C2×D8—1071no—
193C22×C8—1071no—
194C22×C8C2×D10711yescharacteristic
195#126—2141no—
196#126—2141no—
197#125—1071no—
198#125—1071no—
199#124—1071no—
200#124—1071no—
201#123—2141no—
202#123—2141no—
203C22⋊Q8—1071no—
204C22⋊Q8C2×D10711yescharacteristic
205#121—1071no—
206#121C2×D10711yescharacteristic
207C4×Q8—1071no—
208C4×Q8—1071no—
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—2141no—
212C4×D4—2141no—
213#117—2141no—
214#117—2141no—
215C2×C42—1071no—
216#107—1071no—
217#107—1071no—
218#106—1071no—
219#106C2×D10711yescharacteristic
220#103—1071no—
221#103—1071no—
222#103—1071no—
223#103C2×D10711yescharacteristic
224#102—1071no—
225#102—1071no—
226#102—1071no—
227#102C2×D10711yescharacteristic
228C4×C8—1071no—
229C4×C8—1071no—
230C4×C8—1071no—
231C4×C8—1071no—
232C2×Q8—1071no—
233C2×Q8C22×D10711yescharacteristic
234C2×D4—21no—
235C2×D4—1071no—
236C2×D4—2141no—
237C2×D4C22×D10711yescharacteristic
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—2141no—
241C22×C4—2141no—
242C22×C4C22×D10711yescharacteristic
243Q16—2142no—
244SD16—22no—
245SD16—2142no—
246D8—2142no—
247C2×C8—22no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8—2142no—
251C2×C8C22×D10711yescharacteristic
252C2×C8C22×D10711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1071no—
255C4⋊C4—1071no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4C22×D10711yescharacteristic
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C42—1071no—
276C42—1071no—
277C42—1071no—
278C42—1071no—
279C42—2141no—
280C42—2141no—
281C23—21no—
282C23—2141no—
283C23D4×D10711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—2141no—
287Q8—2141no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—4282no—
292D4—2141no—
293D4—2141no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—1071no—
300C2×C4—1071no—
301C2×C4—1071no—
302C2×C4—1071no—
303C2×C4—4282no—
304C2×C4—4282no—
305C2×C4—2142no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2142no—
310C2×C4D4×D10711yescharacteristic
311C2×C4C23×D10711yescharacteristic
312C8—21no—
313C8—1072no—
314C8—2141no—
315C8#1313381712yes—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—4281no—
322C22—4281no—
323C22—4281no—
324C22#1681213012yes—
325C22C2×D4×D10711yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—1072no—
331C4—4281no—
332C4—4281no—
333C4—4281no—
334C4—2141no—
335C4—2141no—
336C4—2141no—
337C4#41×D10711yescharacteristic
338C4C2×#1313381711yescharacteristic
339C2—22no—
340C2—81no—
341C2—8561no—
342C2#2029317211yescharacteristic
343C2#135×D10711yescharacteristic
344C2C2×#1681213011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
prime radical#20292861C2
hypercenter#465D107
hypocenterC107#3168
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20293163C2semidirect
2. #20293158C2semidirect
3. #20293027C2semidirect
4. #20293008C2semidirect
5. #20292983C2semidirect
6. #20292980C2semidirect
7. #20292866C2semidirect
9. #20292856C2semidirect
10. #20292719C2semidirect
11. SD16×Dic107C2semidirect
12. #20292714C2semidirect
13. #20292707C2semidirect
14. C2×C8×Dic107C2semidirect
15. #465×C107C2semidirect
173. C107#3168semidirect
183. #465D107semidirect

Maximal subgroups

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

subgroupindex
#202931632
#202931582
#202930272
#202930082
#202929832
#202929802
#202928662
#202928612
#202928562
#202927192
SD16×Dic1072
#202927142
#202927072
C2×C8×Dic1072
#465×C1072
#3168107