Finite Group Databasebeta

← #28348316

Subgroups — #28348316

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293327C211yesmaximal, characteristic, complemented
2#20293272C211yesmaximal, characteristic, complemented
3#20292982C211yesmaximal, characteristic, complemented
4#20292964C211yesmaximal, characteristic, complemented
5#20292954C211yesmaximal, characteristic, complemented
6#20292927C211yesmaximal, characteristic, complemented
7#20292862C211yesmaximal, characteristic, complemented
8#20292855C211yesmaximal, characteristic, complemented
9#20292842C211yesmaximal, characteristic, complemented
10#20292769C211yesmaximal, characteristic, complemented
11#20292759C211yesmaximal, characteristic, complemented
12C8×D428C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15C4×D8×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×#13133806C2211yescharacteristic, complemented
19C2×#13133803C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic, complemented
22#16812102—21no—
23#16812101C2211yescharacteristic, complemented
24#16812097C2211yescharacteristic
25#16812093—21no—
26#16812092—21no—
27#16812089C2211yescharacteristic
28#16812084C2211yescharacteristic
29#16812083—21no—
30#16812082—21no—
31#16812080C2211yescharacteristic
32#16812080C2211yescharacteristic, complemented
33#16812079—21no—
34C4×D428C2211yescharacteristic
35C4×C22⋊D107—21no—
36#16812067C2211yescharacteristic, complemented
37C4⋊C4×D107C2211yescharacteristic
38C2×(C2×D107)⋊C4—21no—
39#16812042C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41C4⋊(C107⋊C8)C2211yescharacteristic, complemented
42#16812034C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812032C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52C2×D8×C107C2211yescharacteristic
53C4×D4×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic, complemented
55#107×C107C2211yescharacteristic, complemented
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic, complemented
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133817—22no—
61#13133817—41no—
62C2×Dic214C2311yescharacteristic
63C2×D428—21no—
64C2×D428D411yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70D4×D107—22no—
71D4×D107—41no—
72C22×Dic107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75C2×C4×D107C2311yescharacteristic
76#13133806—41no—
77#13133803—41no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—41no—
81C8×D107—41no—
82C4⋊Dic107—21no—
83C4⋊Dic107D411yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97(C2×D107)⋊C4D411yescharacteristic
98C4×Dic107—21no—
99C2×D4×C107C2311yescharacteristic
100C2×D4×C107C2311yescharacteristic
101C22×C4×C107—21no—
102C22×C4×C107—21no—
103D8×C107—22no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C22⋊C4×C107—21no—
109C22⋊C4×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C23×C107—21no—
113C22×D107—21no—
114C22×D107—42no—
115C22×D107—42no—
116C22×D107—81no—
117C22×D107C2×D411yescharacteristic
118C22×D107C2×D411yescharacteristic
119D4×C107—21no—
120D4×C107—21no—
121D4×C107—21no—
122D4×C107#4112yes—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—81no—
126C2×C4×C107—81no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—42no—
131C22⋊D107—42no—
132C22⋊D107—42no—
133C22⋊D107—81no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—42no—
138C2×Dic107C2×D411yescharacteristic
139C2×Dic107C2×D411yescharacteristic
140D428—22no—
141D428—21no—
142D428—21no—
143D428—81no—
144C4×D107—21no—
145C4×D107—21no—
146C4×D107—21no—
147C4×D107—21no—
148C4×D107—81no—
149Dic214—21no—
150Dic214—21no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—22no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—21no—
166C2×D107—21no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—84no—
171C2×D107#13512yes—
172C4×C107—21no—
173C4×C107—21no—
174C4×C107—41no—
175C4×C107—41no—
176C4×C107C2×#4111yescharacteristic
177C4×C107H(5,𝔽2)11yescharacteristic
178Dic107—21no—
179Dic107—21no—
180Dic107—41no—
181Dic107—41no—
182Dic107—41no—
183C2×C107—42no—
184C2×C107—81no—
185C2×C107#54611yescharacteristic
186C2×C107C2×#13511yescharacteristic
187C2×C107#57611yescharacteristic
188D107—22no—
189D107—41no—
190D107—81no—
191#3297—1071nomaximal, Sylow
192C107#329711yescharacteristic, complemented, Sylow
193#553—1071no—
194#546—1071no—
195#488—1071no—
196#484—1071no—
197#482—1071no—
198#478—1071no—
199#466—1071no—
200#465—1071no—
201#463—1071no—
202#449—1071no—
203#447—1071no—
204C4×D8D10711yescharacteristic, complemented
205C8×D4—1071no—
206#416—1071no—
207C2×#102—1071no—
208C2×#41—1071no—
209C22×D4—1071no—
210C2×SD16—1071no—
211C2×D8—1071no—
212C2×D8C2×D10711yescharacteristic
213C22×C8—1071no—
214C22×C8—1071no—
215#126—2141no—
216#125—1071no—
217#124—1071no—
218#123—2141no—
219#123—2141no—
220C22⋊Q8—1071no—
221#121—1071no—
222#121—1071no—
223#121—1071no—
224#121—2141no—
225#121—2141no—
226#120—2141no—
227C4×D4—1071no—
228C4×D4—2141no—
229C4×D4C2×D10711yescharacteristic
230C4×D4C2×D10711yescharacteristic, complemented
231#117—1071no—
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—2141no—
234#107—1071no—
235#107C2×D10711yescharacteristic, complemented
236#106—1071no—
237C4⋊C8—1071no—
238#103—1071no—
239#103—1071no—
240#103—1071no—
241#102—1071no—
242#102—1071no—
243#102—1071no—
244#102—1071no—
245#102—1071no—
246#102C2×D10711yescharacteristic
247#102C2×D10711yescharacteristic, complemented
248C22⋊C8—1071no—
249C22⋊C8—1071no—
250C4×C8C2×D10711yescharacteristic, complemented
251C24—2141no—
252#41—4281no—
253#41—2142no—
254C2×Q8—1071no—
255C2×D4—1071no—
256C2×D4—1071no—
257C2×D4—4281no—
258C2×D4—2141no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2142no—
264C2×D4C22×D10711yescharacteristic
265C2×D4C22×D10711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—2141no—
271C22×C4—2141no—
272SD16—4281no—
273D8—22no—
274D8—4281no—
275C2×C8—1071no—
276C2×C8—1071no—
277C2×C8—4281no—
278C2×C8—4281no—
279C2×C8C22×D10711yescharacteristic
280C2×C8C22×D10711yescharacteristic
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—1071no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4—2141no—
288C4⋊C4C22×D10711yescharacteristic
289C4⋊C4C22×D10711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—1071no—
293C22⋊C4—1071no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4—2141no—
300C22⋊C4—2141no—
301C42—2141no—
302C42C22×D10711yescharacteristic
303C23—21no—
304C23—21no—
305C23—8561no—
306C23—1071no—
307C23—1071no—
308C23—4282no—
309C23—4282no—
310C23—2141no—
311Q8—2141no—
312Q8—2141no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—8561no—
317D4—8561no—
318D4—4282no—
319D4—4282no—
320D4—4282no—
321D4—2142no—
322D4—2141no—
323D4—2141no—
324D4#1313381812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—8561no—
330C2×C4—1071no—
331C2×C4—1071no—
332C2×C4—4282no—
333C2×C4—2141no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4D4×D10711yescharacteristic
341C2×C4D4×D10711yescharacteristic
342C2×C4C23×D10711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2141no—
346C8—2141no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—8564no—
354C22—1072no—
355C22—4281no—
356C22—4281no—
357C22—4281no—
358C22—2141no—
359C22—2141no—
360C22—2141no—
361C22—2141no—
362C22—2141no—
363C22C2×D4×D10711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—4281no—
369C4—4281no—
370C4—4281no—
371C4—2141no—
372C4—2141no—
373C4C2×#1313381811yescharacteristic
374C4#1681215511yescharacteristic
375C2—42no—
376C2—81no—
377C2—8561no—
378C2—4281no—
379C2—2142no—
380C2#2029327311yescharacteristic
381C2#135×D10711yescharacteristic
382C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×D8×C107C2
socleC22×C107C22×D4
hypercenterC4×D8D107
hypocenterC107#3297
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20293327C2semidirect
2. #20293272C2semidirect
3. #20292982C2semidirect
4. #20292964C2semidirect
5. #20292954C2semidirect
6. #20292927C2semidirect
7. #20292862C2semidirect
8. #20292855C2semidirect
9. #20292842C2semidirect
10. #20292769C2semidirect
11. #20292759C2semidirect
12. C8×D428C2semidirect
13. #20292620C2semidirect
14. #102×D107C2semidirect
15. C4×D8×C107C2semidirect
18. C2×#13133806C22semidirect
21. C2×C8×D107C22semidirect
23. #16812101C22semidirect
32. #16812080C22semidirect
36. #16812067C22semidirect
39. #16812042C22semidirect
40. #16812040C22semidirect
41. C4⋊(C107⋊C8)C22semidirect
42. #16812034C22semidirect
43. #16812034C22semidirect
44. #16812032C22semidirect
51. (C2×D107)⋊C8C22semidirect
54. C4×D4×C107C22semidirect
55. #107×C107C22semidirect
57. #102×C107C22semidirect
58. C4×C8×C107C22semidirect
192. C107#3297semidirect
204. C4×D8D107semidirect
230. C4×D4C2×D107semidirect
235. #107C2×D107semidirect
247. #102C2×D107semidirect
250. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933272
#202932722
#202929822
#202929642
#202929542
#202929272
#202928622
#202928552
#202928422
#202927692
#202927592
C8×D4282
#202926202
#102×D1072
C4×D8×C1072
#3297107