Finite Group Databasebeta

← #28348313

Subgroups — #28348313

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#20293323C211yesmaximal, characteristic, complemented
2#20293273C211yesmaximal, characteristic, complemented
3#20292982C211yesmaximal, characteristic, complemented
4#20292963C211yesmaximal, characteristic, complemented
5#20292952C211yesmaximal, characteristic, complemented
6#20292928C211yesmaximal, characteristic, complemented
7#20292863C211yesmaximal, characteristic, complemented
8#20292856C211yesmaximal, characteristic, complemented
9#20292843C211yesmaximal, characteristic, complemented
10#20292774C211yesmaximal, characteristic, complemented
11#20292763C211yesmaximal, characteristic, complemented
12C4×D856C211yesmaximal, characteristic, complemented
13#20292624C211yesmaximal, characteristic, complemented
14C2×#16812031C211yesmaximal, characteristic, complemented
15C8×D4×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C2×#13133807C2211yescharacteristic, complemented
19C2×D856C2211yescharacteristic
20C2×D856C2211yescharacteristic
21#16812104—21no—
22#16812101C2211yescharacteristic, complemented
23#16812097C2211yescharacteristic
24#16812094—21no—
25#16812091—21no—
26#16812088C2211yescharacteristic
27#16812083C2211yescharacteristic
28#16812083C2211yescharacteristic
29#16812083C2211yescharacteristic, complemented
30#16812082—21no—
31#16812081—21no—
32#16812077—21no—
33C4×D428C2211yescharacteristic
34C4×D428C2211yescharacteristic, complemented
35(C22⋊D107)⋊C4—21no—
36#16812068C2211yescharacteristic, complemented
37C2×C4⋊Dic107C2211yescharacteristic
38C22⋊C4×D107—21no—
39#16812043C2211yescharacteristic, complemented
40#16812043C2211yescharacteristic, complemented
41#16812041C2211yescharacteristic, complemented
42#16812034C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic, complemented
52C22×C8×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic, complemented
54C4×D4×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic, complemented
56C22⋊C8×C107C2211yescharacteristic
57C22⋊C8×C107C2211yescharacteristic, complemented
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133818—22no—
61#13133818—41no—
62C2×Dic214C2311yescharacteristic
63C2×D428—22no—
64C2×D428—21no—
65C2×D428—41no—
66C2×D428C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C22×Dic107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—21no—
77#13133807—41no—
78D856—22no—
79D856—41no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—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)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97C4×Dic107—21no—
98C2×D4×C107D411yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C2×C8×C107—41no—
102C2×C8×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107D411yescharacteristic
108C22⋊C4×C107D411yescharacteristic
109C22⋊C4×C107D411yescharacteristic
110C42×C107C2311yescharacteristic
111C23×C107C2×D411yescharacteristic
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—21no—
116C22×D107—42no—
117C22×D107—42no—
118C22×D107—81no—
119D4×C107—22no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—42no—
130C22⋊D107—42no—
131C22⋊D107—81no—
132C22⋊D107—81no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—81no—
139D428—21no—
140D428—21no—
141D428—21no—
142D428—21no—
143D428—21no—
144D428—42no—
145D428#4112yes—
146C4×D107—42no—
147C4×D107—81no—
148C4×D107—81no—
149Dic214—21no—
150Dic214—21no—
151C8×C107—21no—
152C8×C107—21no—
153C8×C107—21no—
154C8×C107—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—21no—
159C22×C107—21no—
160C22×C107#13512yes—
161C22×C107C22×D411yescharacteristic
162C2×D107—22no—
163C2×D107—41no—
164C2×D107—41no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—84no—
172C4×C107—21no—
173C4×C107—21no—
174C4×C107—21no—
175C4×C107—21no—
176C4×C107C2×#4111yescharacteristic
177C4×C107H(5,𝔽2)11yescharacteristic
178Dic107—41no—
179Dic107—41no—
180Dic107—41no—
181Dic107—41no—
182Dic107—41no—
183C2×C107—22no—
184C2×C107—41no—
185C2×C107#54611yescharacteristic
186C2×C107C2×#13511yescharacteristic
187C2×C107#57611yescharacteristic
188D107—42no—
189D107—81no—
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×D8—1071no—
205C8×D4D10711yescharacteristic, complemented
206#416—1071no—
207C2×#102—1071no—
208C2×#41—1071no—
209C22×D4—1071no—
210C2×SD16—1071no—
211C2×D8—1071no—
212C2×D8—1071no—
213C22×C8C2×D10711yescharacteristic
214C22×C8C2×D10711yescharacteristic, complemented
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—1071no—
229C4×D4—2141no—
230C4×D4C2×D10711yescharacteristic
231#117—1071no—
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—2141no—
234#107—1071no—
235#107—1071no—
236#106—1071no—
237C4⋊C8C2×D10711yescharacteristic, complemented
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#102—1071no—
247#102—1071no—
248C22⋊C8C2×D10711yescharacteristic
249C22⋊C8C2×D10711yescharacteristic, complemented
250C4×C8C2×D10711yescharacteristic, complemented
251C24—2141no—
252#41—4281no—
253#41—2142no—
254C2×Q8—1071no—
255C2×D4—1071no—
256C2×D4—1071no—
257C2×D4—1071no—
258C2×D4—4281no—
259C2×D4—2142no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2141no—
264C2×D4—2141no—
265C2×D4D42811yescharacteristic
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4—2141no—
269C22×C4—2141no—
270C22×C4C22×D10711yescharacteristic
271C22×C4C22×D10711yescharacteristic
272SD16—4281no—
273D8—4281no—
274D8—2142no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D10711yescharacteristic
278C2×C8C22×D10711yescharacteristic
279C2×C8C22×D10711yescharacteristic
280C2×C8C22×D10711yescharacteristic
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—1071no—
284C4⋊C4—1071no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4—2141no—
288C4⋊C4—2141no—
289C4⋊C4D42811yescharacteristic
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4D42811yescharacteristic
300C22⋊C4D42811yescharacteristic
301C42—2141no—
302C42C22×D10711yescharacteristic
303C23—8561no—
304C23—4282no—
305C23—4282no—
306C23—2141no—
307C23—2141no—
308C23—2141no—
309C23C2×D42811yescharacteristic
310C23C2×D42811yescharacteristic
311Q8—2141no—
312Q8—2141no—
313D4—22no—
314D4—8561no—
315D4—8561no—
316D4—1072no—
317D4—4282no—
318D4—4282no—
319D4—4282no—
320D4—2141no—
321D4—2141no—
322D4—2141no—
323D4—2141no—
324D4—2141no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—8561no—
330C2×C4—8561no—
331C2×C4—8561no—
332C2×C4—4282no—
333C2×C4—2141no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4C2×D42811yescharacteristic
339C2×C4C2×D42811yescharacteristic
340C2×C4C2×D42811yescharacteristic
341C2×C4C2×D42811yescharacteristic
342C2×C4C23×D10711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—8564no—
353C22—4281no—
354C22—4281no—
355C22—4281no—
356C22—4281no—
357C22—4281no—
358C22—4281no—
359C22—4281no—
360C22—4281no—
361C22—2142no—
362C22#1681213112yes—
363C22C22×D42811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—4281no—
369C4—4281no—
370C4—4281no—
371C4—4281no—
372C4—4281no—
373C4#41×D10711yescharacteristic
374C4#1681215411yescharacteristic
375C2—22no—
376C2—41no—
377C2—8561no—
378C2—8561no—
379C2—4282no—
380C2#2029327211yescharacteristic
381C2C2×#1681213111yescharacteristic
382C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
hypercenterC8×D4D107
hypocenterC107#3297
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293323C2semidirect
2. #20293273C2semidirect
3. #20292982C2semidirect
4. #20292963C2semidirect
5. #20292952C2semidirect
6. #20292928C2semidirect
7. #20292863C2semidirect
8. #20292856C2semidirect
9. #20292843C2semidirect
10. #20292774C2semidirect
11. #20292763C2semidirect
12. C4×D856C2semidirect
13. #20292624C2semidirect
14. C2×#16812031C2semidirect
15. C8×D4×C107C2semidirect
18. C2×#13133807C22semidirect
22. #16812101C22semidirect
29. #16812083C22semidirect
34. C4×D428C22semidirect
36. #16812068C22semidirect
39. #16812043C22semidirect
40. #16812043C22semidirect
41. #16812041C22semidirect
42. #16812034C22semidirect
43. #16812034C22semidirect
44. #16812034C22semidirect
51. #16812031C22semidirect
53. C22×C8×C107C22semidirect
55. C4⋊C8×C107C22semidirect
57. C22⋊C8×C107C22semidirect
58. C4×C8×C107C22semidirect
192. C107#3297semidirect
205. C8×D4D107semidirect
214. C22×C8C2×D107semidirect
237. C4⋊C8C2×D107semidirect
249. C22⋊C8C2×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
#202933232
#202932732
#202929822
#202929632
#202929522
#202929282
#202928632
#202928562
#202928432
#202927742
#202927632
C4×D8562
#202926242
C2×#168120312
C8×D4×C1072
#3297107