Finite Group Databasebeta

← #28347065

Subgroups — #28347065

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#20293181C211yesmaximal, characteristic, complemented
2#20293164C211yesmaximal, characteristic
3#20293035C211yesmaximal, characteristic, complemented
4#20293020C211yesmaximal, characteristic, complemented
5#20292994C211yesmaximal, characteristic, complemented
6#20292988C211yesmaximal, characteristic, complemented
7#20292884C211yesmaximal, characteristic, complemented
8#20292882C211yesmaximal, characteristic, complemented
9#20292876C211yesmaximal, characteristic
10#20292874C211yesmaximal, characteristic
11C4×#13133808C211yesmaximal, characteristic, complemented
12C4×#13133807C211yesmaximal, characteristic, complemented
13C4×#13133805C211yesmaximal, characteristic, complemented
14C4×#13133801C211yesmaximal, characteristic, complemented
15C4×Q16×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C2×#13133801C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107C2211yescharacteristic
23#16812104—21no—
24#16812100C2211yescharacteristic
25#16812099—21no—
26#16812098C2211yescharacteristic
27#16812094—21no—
28#16812089C2211yescharacteristic
29#16812087—21no—
30#16812085C2211yescharacteristic
31#16812085C2211yescharacteristic
32#16812084C2211yescharacteristic
33C4×Dic214—21no—
34C4×Dic214C2211yescharacteristic
35C4×D428—21no—
36C4×D428C2211yescharacteristic
37#16812066—21no—
38#16812066—21no—
39C42×D107C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812040C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812019C2211yescharacteristic
50#16812019C2211yescharacteristic
51C4×C107⋊C8C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428—21no—
62C2×D428C2311yescharacteristic
63C2×C4×D107—21no—
64C2×C4×D107D411yescharacteristic
65C2×C4×D107D411yescharacteristic
66C2×C4×D107C2311yescharacteristic
67#13133808—22no—
68#13133807—22no—
69#13133805—22no—
70#13133801—22no—
71#13133801—22no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92C4×Dic107—21no—
93C4×Dic107D411yescharacteristic
94C4×Dic107D411yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98Q16×C107—22no—
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C42×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C22×D107—21no—
109C22×D107C2×D411yescharacteristic
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×C107C2×D411yescharacteristic
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2411yescharacteristic
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127D428—21no—
128D428—21no—
129D428—42no—
130C4×D107—22no—
131C4×D107—21no—
132C4×D107—21no—
133C4×D107—41no—
134C4×D107—42no—
135Dic214—21no—
136Dic214—21no—
137Dic214—42no—
138C8×C107—21no—
139C8×C107#4112yes—
140C107⋊C8—21no—
141C107⋊C8#4112yes—
142C22×C107C22×D411yescharacteristic
143C2×D107—21no—
144C2×D107—21no—
145C2×D107—21no—
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×C107#13712yes—
155C4×C107C2×#4111yescharacteristic
156C4×C107C2×#4111yescharacteristic
157Dic107—21no—
158Dic107—21no—
159Dic107—21no—
160Dic107—41no—
161Dic107—41no—
162Dic107—41no—
163C2×C107#53211yescharacteristic
164C2×C107C2×#13711yescharacteristic
165C2×C107#57511yescharacteristic
166D107—41no—
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#470—1071no—
178#468—1071no—
179#468—1071no—
180C4×Q16—1071no—
181C4×Q16D10711yescharacteristic, complemented
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×M16—1071no—
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—1071no—
200C22⋊Q8—1071no—
201C22⋊Q8—2141no—
202#121—1071no—
203C4×Q8—1071no—
204C4×Q8—2141no—
205C4×Q8C2×D10711yescharacteristic
206C4×Q8C2×D10711yescharacteristic
207C4×D4—1071no—
208C4×D4—2141no—
209#117—2141no—
210#117—2141no—
211C2×C42—1071no—
212#107—1071no—
213#107C2×D10711yescharacteristic
214#106—1071no—
215#106—1071no—
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×C8C2×D10711yescharacteristic
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—1071no—
237C22×C4—2141no—
238Q16—22no—
239Q16—2142no—
240SD16—2142no—
241SD16—2142no—
242M16—2142no—
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—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C42—21no—
271C42—21no—
272C42—1071no—
273C42—1071no—
274C42—1071no—
275C42—2141no—
276C42C22×D10711yescharacteristic
277C23—1071no—
278C23—2141no—
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—1071no—
294C2×C4—1071no—
295C2×C4—1071no—
296C2×C4—4281no—
297C2×C4—4282no—
298C2×C4—2141no—
299C2×C4—2141no—
300C2×C4—2141no—
301C2×C4—2142no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4D4×D10711yescharacteristic
305C2×C4D4×D10711yescharacteristic
306C2×C4C23×D10711yescharacteristic
307C8—21no—
308C8—1072no—
309C8—2141no—
310C8#1313381812yes—
311C22—4281no—
312C22—4281no—
313C22—4281no—
314C22—2141no—
315C22—2141no—
316C22—2141no—
317C22C2×D4×D10711yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—4281no—
324C4—4281no—
325C4—4281no—
326C4—2141no—
327C4—2141no—
328C4—2141no—
329C4#1681214112yes—
330C4#41×D10711yescharacteristic
331C4C2×#1313381811yescharacteristic
332C2—8561no—
333C2—4281no—
334C2#2029317411yescharacteristic
335C2C2×#1681214111yescharacteristic
336C2#2029342411yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×Q16×C107C2
socleC22×C107C22×D4
prime radical#16812084C22
hypercenterC4×Q16D107
hypocenterC107#3172
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #20293181C2semidirect
3. #20293035C2semidirect
4. #20293020C2semidirect
5. #20292994C2semidirect
6. #20292988C2semidirect
7. #20292884C2semidirect
8. #20292882C2semidirect
11. C4×#13133808C2semidirect
12. C4×#13133807C2semidirect
13. C4×#13133805C2semidirect
14. C4×#13133801C2semidirect
15. C4×Q16×C107C2semidirect
169. C107#3172semidirect
181. C4×Q16D107semidirect

Maximal subgroups

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

subgroupindex
#202931812
#202931642
#202930352
#202930202
#202929942
#202929882
#202928842
#202928822
#202928762
#202928742
C4×#131338082
C4×#131338072
C4×#131338052
C4×#131338012
C4×Q16×C1072
#3172107