Finite Group Databasebeta

← #28347062

Subgroups — #28347062

Order 13696.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293179C211yesmaximal, characteristic, complemented
2#20293163C211yesmaximal, characteristic
3#20293034C211yesmaximal, characteristic, complemented
4#20293020C211yesmaximal, characteristic, complemented
5#20292994C211yesmaximal, characteristic, complemented
6#20292988C211yesmaximal, characteristic, complemented
7#20292883C211yesmaximal, characteristic, complemented
8#20292875C211yesmaximal, characteristic
9#20292873C211yesmaximal, characteristic
10Q16×Dic107C211yesmaximal, characteristic, complemented
11#20292719C211yesmaximal, characteristic, complemented
12#20292714C211yesmaximal, characteristic, complemented
13#20292712C211yesmaximal, characteristic, complemented
14M16×Dic107C211yesmaximal, characteristic, complemented
15#470×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133802C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812104—21no—
23#16812100C2211yescharacteristic
24#16812099—21no—
25#16812098C2211yescharacteristic
26#16812093—21no—
27#16812088C2211yescharacteristic
28#16812086—21no—
29#16812083C2211yescharacteristic
30Q8×Dic107C2211yescharacteristic
31Q8×Dic107C2211yescharacteristic
32#16812074—21no—
33#16812074C2211yescharacteristic
34#16812071C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36#16812067—21no—
37#16812065—21no—
38C2×C4×Dic107C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812019C2211yescharacteristic
49#16812019C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C22×Dic107D411yescharacteristic
64C22×Dic107D411yescharacteristic
65C2×C4×D107—21no—
66#13133808—22no—
67#13133807—22no—
68#13133805—22no—
69#13133802—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—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107D411yescharacteristic
92C4×Dic107D411yescharacteristic
93C4×Dic107C2311yescharacteristic
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C22×C4×C107C2311yescharacteristic
98Q16×C107—22no—
99M16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107—21no—
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C23×C107C2×D411yescharacteristic
109C22×D107—21no—
110Q8×C107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2411yescharacteristic
122C22⋊D107—42no—
123C2×Dic107—22no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—41no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132D428—21no—
133D428—21no—
134C4×D107—42no—
135Dic214—21no—
136Dic214—21no—
137Dic214—42no—
138C8×C107—21no—
139C8×C107#4112yes—
140C107⋊C8—21no—
141C107⋊C8#4112yes—
142C22×C107—21no—
143C22×C107—21no—
144C22×C107—21no—
145C22×C107C22×D411yescharacteristic
146C2×D107—41no—
147C2×D107—41no—
148C2×D107—41no—
149C4×C107—21no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107—41no—
154C4×C107C2×#4111yescharacteristic
155C4×C107C2×#4111yescharacteristic
156Dic107—21no—
157Dic107—21no—
158Dic107—21no—
159Dic107—41no—
160Dic107—41no—
161Dic107—41no—
162Dic107#13712yes—
163C2×C107—41no—
164C2×C107#53211yescharacteristic
165C2×C107C2×#13711yescharacteristic
166C2×C107#57511yescharacteristic
167D107—81no—
168#3172—1071nomaximal, Sylow
169C107#317211yescharacteristic, complemented, Sylow
170#533—1071no—
171#529—1071no—
172#501—1071no—
173#497—1071no—
174#491—1071no—
175#489—1071no—
176#470—1071no—
177#470D10711yescharacteristic, complemented
178#468—1071no—
179#468—1071no—
180C4×Q16—1071no—
181C4×Q16—1071no—
182C4×SD16—1071no—
183C4×SD16—1071no—
184C4×M16—1071no—
185C2×Q16—1071no—
186C2×Q16C2×D10711yescharacteristic
187C2×SD16—1071no—
188C2×SD16—1071no—
189C2×M16—1071no—
190C2×M16C2×D10711yescharacteristic
191C4⋊Q8—1071no—
192C4⋊Q8—1071no—
193#126—2141no—
194#125—1071no—
195#125—2141no—
196#124—1071no—
197#123—2141no—
198C22⋊Q8—1071no—
199C22⋊Q8—2141no—
200C22⋊Q8C2×D10711yescharacteristic
201C22⋊Q8C2×D10711yescharacteristic
202#121—1071no—
203C4×Q8—1071no—
204C4×Q8—1071no—
205C4×Q8—1071no—
206C4×Q8—2141no—
207C4×D4—1071no—
208C4×D4—2141no—
209#117—2141no—
210#117—2141no—
211C2×C42—1071no—
212#107—1071no—
213#107—1071no—
214#106—1071no—
215#106C2×D10711yescharacteristic
216#103—1071no—
217#103—1071no—
218#103—1071no—
219#103—1071no—
220#103C2×D10711yescharacteristic
221#103C2×D10711yescharacteristic
222#102—1071no—
223#102—1071no—
224#97—1071no—
225#97—1071no—
226C4×C8—1071no—
227C4×C8—1071no—
228C2×Q8—1071no—
229C2×Q8—2141no—
230C2×Q8C22×D10711yescharacteristic
231C2×Q8C22×D10711yescharacteristic
232C2×D4—1071no—
233C2×D4—2141no—
234C22×C4—1071no—
235C22×C4—1071no—
236C22×C4—2141no—
237C22×C4C22×D10711yescharacteristic
238Q16—22no—
239Q16—2142no—
240SD16—2142no—
241SD16—2142no—
242M16—22no—
243M16—2142no—
244C2×C8—1071no—
245C2×C8—1071no—
246C2×C8C22×D10711yescharacteristic
247C2×C8C22×D10711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—1071no—
251C4⋊C4—1071no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4C22×D10711yescharacteristic
262C4⋊C4C22×D10711yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C42—1071no—
271C42—1071no—
272C42—1071no—
273C42—1071no—
274C42—2141no—
275C42—2141no—
276C42—2141no—
277C23—2141no—
278C23D4×D10711yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—4282no—
284Q8—2141no—
285Q8—2141no—
286D4—4282no—
287D4—2141no—
288D4—2141no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—1071no—
299C2×C4—4282no—
300C2×C4—4281no—
301C2×C4—2142no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4D4×D10711yescharacteristic
306C2×C4C23×D10711yescharacteristic
307C8—21no—
308C8—1072no—
309C8—2141no—
310C8#1313381712yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—4281no—
315C22—4281no—
316C22—4281no—
317C22C2×D4×D10711yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—1072no—
324C4—4281no—
325C4—4281no—
326C4—4281no—
327C4—2141no—
328C4—2141no—
329C4—2141no—
330C4#41×D10711yescharacteristic
331C4C2×#1313381711yescharacteristic
332C2—41no—
333C2—8561no—
334C2#2029317211yescharacteristic
335C2#137×D10711yescharacteristic
336C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
prime radical#16812083C22
hypercenter#470D107
hypocenterC107#3172
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #20293179C2semidirect
3. #20293034C2semidirect
4. #20293020C2semidirect
5. #20292994C2semidirect
6. #20292988C2semidirect
7. #20292883C2semidirect
10. Q16×Dic107C2semidirect
11. #20292719C2semidirect
12. #20292714C2semidirect
13. #20292712C2semidirect
14. M16×Dic107C2semidirect
15. #470×C107C2semidirect
169. C107#3172semidirect
177. #470D107semidirect

Maximal subgroups

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

subgroupindex
#202931792
#202931632
#202930342
#202930202
#202929942
#202929882
#202928832
#202928752
#202928732
Q16×Dic1072
#202927192
#202927142
#202927122
M16×Dic1072
#470×C1072
#3172107