Finite Group Databasebeta

← #28348448

Subgroups — #28348448

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293288C211yesmaximal, characteristic, complemented
2#20293278C211yesmaximal, characteristic, complemented
3#20292988C211yesmaximal, characteristic, complemented
4#20292961C211yesmaximal, characteristic, complemented
5#20292940C211yesmaximal, characteristic, complemented
6#20292920C211yesmaximal, characteristic, complemented
7#20292876C211yesmaximal, characteristic, complemented
8#20292863C211yesmaximal, characteristic, complemented
9#20292845C211yesmaximal, characteristic, complemented
10#20292775C211yesmaximal, characteristic, complemented
11#20292766C211yesmaximal, characteristic, complemented
12#20292730C211yesmaximal, characteristic, complemented
13#20292698C211yesmaximal, characteristic, complemented
14#20292627C211yesmaximal, characteristic, complemented
15C2×#103×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818—21no—
18C2×#13133805C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×D856C2211yescharacteristic
21C2×#13133801C2211yescharacteristic
22#16812108C2211yescharacteristic
23#16812098—21no—
24#16812098C2211yescharacteristic
25#16812094—21no—
26#16812092—21no—
27#16812087—21no—
28#16812085—21no—
29#16812085C2211yescharacteristic
30#16812084C2211yescharacteristic
31#16812083—21no—
32#16812083C2211yescharacteristic
33#16812083C2211yescharacteristic
34Q8×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic
36C4×C22⋊D107—21no—
37C4×C22⋊D107C2211yescharacteristic
38C4⋊C4×D107C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812042C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812019C2211yescharacteristic
52C22×Q8×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C2×C4⋊C4×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59#13133819—22no—
60#13133819—41no—
61#13133818—44no—
62C2×Dic214—21no—
63C2×D428—21no—
64C2×D428—21no—
65C2×D428C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107D411yescharacteristic
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133805—22no—
75#13133805—41no—
76D856—41no—
77#13133801—41no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107D411yescharacteristic
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4D411yescharacteristic
96C4×Dic107—21no—
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107—21no—
99C2×Q8×C107—22no—
100C2×Q8×C107—41no—
101C2×Q8×C107C2311yescharacteristic
102C2×Q8×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107—21no—
105C22×C4×C107C2311yescharacteristic
106C2×C8×C107—41no—
107C2×C8×C107C2311yescharacteristic
108C2×C8×C107C2311yescharacteristic
109C4⋊C4×C107—21no—
110C4⋊C4×C107C2311yescharacteristic
111C4⋊C4×C107C2311yescharacteristic
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107C2×D411yescharacteristic
116Q8×C107—21no—
117Q8×C107—21no—
118Q8×C107—21no—
119Q8×C107—42no—
120Q8×C107#4112yescomplemented
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—42no—
128C2×C4×C107—42no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—22no—
132C22⋊D107—42no—
133C22⋊D107—42no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138C2×Dic107C2×D411yescharacteristic
139D428—21no—
140D428—21no—
141D428—21no—
142D428—21no—
143D428—42no—
144D428—42no—
145C4×D107—21no—
146C4×D107—21no—
147C4×D107—42no—
148C4×D107—42no—
149C4×D107—42no—
150Dic214—42no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107AGL(1,ℤ/8)12yes—
158C22×C107C22×D411yescharacteristic
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—21no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—41no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C4×C107—22no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107—41no—
173C4×C107C2×#4111yescharacteristic
174C4×C107H(5,𝔽2)11yescharacteristic
175Dic107—21no—
176Dic107—21no—
177Dic107—21no—
178Dic107—41no—
179Dic107—41no—
180C2×C107—22no—
181C2×C107#54611yescharacteristic
182C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C107#57811yescharacteristic
184D107—41no—
185D107—81no—
186D107—81no—
187#3308—1071nomaximal, Sylow
188C107#330811yescharacteristic, complemented, Sylow
189#548—1071no—
190#547—1071no—
191#489—1071no—
192#484—1071no—
193#480—1071no—
194#477—1071no—
195#468—1071no—
196#466—1071no—
197#463—1071no—
198#449—1071no—
199#448—1071no—
200#440—1071no—
201#435—1071no—
202#417—1071no—
203C2×#103D10711yescharacteristic, complemented
204C2×#41—1071no—
205C2×#41—2141no—
206C22×Q8C2×D10711yescharacteristic
207C2×SD16—1071no—
208C2×SD16—1071no—
209C2×D8—1071no—
210C2×M16—1071no—
211C22×C8C2×D10711yescharacteristic
212C4⋊Q8—1071no—
213#124—1071no—
214#124—2141no—
215#123—2141no—
216#123—2141no—
217C22⋊Q8—1071no—
218C22⋊Q8—2141no—
219C22⋊Q8—2141no—
220#121—1071no—
221#121—1071no—
222#121—1071no—
223#121—2141no—
224C4×Q8—1071no—
225C4×D4—1071no—
226C4×D4—1071no—
227C4×D4—2141no—
228C2×C4⋊C4—1071no—
229C2×C4⋊C4C2×D10711yescharacteristic
230#107—1071no—
231#107—1071no—
232#107—1071no—
233C4⋊C8—1071no—
234#103—1071no—
235#103—1071no—
236#103C2×D10711yescharacteristic
237#103C2×D10711yescharacteristic
238#103C2×D10711yescharacteristic
239#103C2×D10711yescharacteristic
240#102—1071no—
241#102—1071no—
242#102—1071no—
243#102—1071no—
244C22⋊C8—1071no—
245C22⋊C8—1071no—
246#97—1071no—
247#41—4284no—
248#41—4281no—
249#41—2142no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—2141no—
254C2×Q8C22×D10711yescharacteristic
255C2×Q8C22×D10711yescharacteristic
256C2×D4—1071no—
257C2×D4—1071no—
258C2×D4—1071no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1071no—
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4—2141no—
269C22×C4C22×D10711yescharacteristic
270SD16—4281no—
271SD16—2142no—
272D8—4281no—
273M16—4281no—
274C2×C8—41no—
275C2×C8—1071no—
276C2×C8—1071no—
277C2×C8C22×D10711yescharacteristic
278C2×C8C22×D10711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—1071no—
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—2141no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4C22×D10711yescharacteristic
288C4⋊C4C22×D10711yescharacteristic
289C22⋊C4—1071no—
290C22⋊C4—1071no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C42—1071no—
299C42—2141no—
300C23—1071no—
301C23—2141no—
302C23—2141no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—4282no—
309Q8#1313381812yescomplemented
310D4—4282no—
311D4—4282no—
312D4—4282no—
313D4—4282no—
314D4—2142no—
315D4—2141no—
316D4—2141no—
317D4—2141no—
318D4—2141no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—1071no—
328C2×C4—1071no—
329C2×C4—1071no—
330C2×C4—4282no—
331C2×C4—4282no—
332C2×C4—4282no—
333C2×C4—2141no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4D4×D10711yescharacteristic
338C2×C4C23×D10711yescharacteristic
339C8—21no—
340C8—21no—
341C8—2141no—
342C8—2141no—
343C22—21no—
344C22—21no—
345C22—4281no—
346C22—4281no—
347C22—4281no—
348C22—4281no—
349C22—4281no—
350C22—4281no—
351C22—2141no—
352C22—2141no—
353C22—2141no—
354C22#1681213612yes—
355C22C2×D4×D10711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—4281no—
362C4—4281no—
363C4—2141no—
364C4—2141no—
365C4—2141no—
366C4C2×#1313381811yescharacteristic
367C4#1681215411yescharacteristic
368C2—22no—
369C2—8561no—
370C2—8561no—
371C2—4281no—
372C2#2029326711yescharacteristic
373C2C2×#1681213611yescharacteristic
374C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC22×C107C22×D4
hypercenterC2×#103D107
hypocenterC107#3308
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #20293288C2semidirect
2. #20293278C2semidirect
3. #20292988C2semidirect
4. #20292961C2semidirect
5. #20292940C2semidirect
6. #20292920C2semidirect
7. #20292876C2semidirect
8. #20292863C2semidirect
9. #20292845C2semidirect
10. #20292775C2semidirect
11. #20292766C2semidirect
12. #20292730C2semidirect
13. #20292698C2semidirect
14. #20292627C2semidirect
15. C2×#103×C107C2semidirect
120. Q8×C107#41semidirect
188. C107#3308semidirect
203. C2×#103D107semidirect
309. Q8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202932882
#202932782
#202929882
#202929612
#202929402
#202929202
#202928762
#202928632
#202928452
#202927752
#202927662
#202927302
#202926982
#202926272
C2×#103×C1072
#3308107