Finite Group Databasebeta

← #28348312

Subgroups — #28348312

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#20293321C211yesmaximal, characteristic, complemented
2#20293265C211yesmaximal, characteristic, complemented
3#20292983C211yesmaximal, characteristic, complemented
4#20292961C211yesmaximal, characteristic, complemented
5#20292951C211yesmaximal, characteristic, complemented
6#20292925C211yesmaximal, characteristic, complemented
7#20292863C211yesmaximal, characteristic, complemented
8#20292859C211yesmaximal, characteristic, complemented
9#20292845C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292707C211yesmaximal, characteristic, complemented
13C8×C22⋊D107C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#416×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C2×#13133805C2211yescharacteristic, complemented
18C2×D856C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812103—21no—
22#16812100C2211yescharacteristic, complemented
23#16812098C2211yescharacteristic
24#16812092—21no—
25#16812091—21no—
26#16812085C2211yescharacteristic
27#16812084C2211yescharacteristic
28#16812083C2211yescharacteristic
29#16812083C2211yescharacteristic, complemented
30#16812082—21no—
31#16812081—21no—
32#16812077—21no—
33C4×D428—21no—
34D4×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic, complemented
36C4×C22⋊D107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38C2×(C2×D107)⋊C4—21no—
39#16812043C2211yescharacteristic, complemented
40#16812042C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic, complemented
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic, complemented
52C2×#41×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic, complemented
54#117×C107C2211yescharacteristic, complemented
55#103×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C23×D107—21no—
60C2×D428—21no—
61C2×D428C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107D411yescharacteristic
67D4×D107—22no—
68D4×D107—41no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107C2311yescharacteristic
73#13133805—41no—
74D856—41no—
75#13133803—22no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C8×D107—41no—
79C4⋊Dic107—21no—
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4D411yescharacteristic
86C22⋊Dic107—21no—
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×Dic107C2311yescharacteristic
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107—21no—
103C22×C4×C107C2311yescharacteristic
104C2×C8×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C42×C107—21no—
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107—42no—
116C22×D107—42no—
117C22×D107—81no—
118C22×D107C2×D411yescharacteristic
119Q8×C107—21no—
120Q8×C107—21no—
121D4×C107—21no—
122D4×C107—42no—
123D4×C107#4112yes—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—42no—
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2411yescharacteristic
132C22⋊D107—22no—
133C22⋊D107—42no—
134C22⋊D107—81no—
135C22⋊D107—81no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—81no—
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141C2×Dic107C2×D411yescharacteristic
142D428—21no—
143D428—21no—
144D428—21no—
145D428—21no—
146D428—42no—
147C4×D107—21no—
148C4×D107—21no—
149C4×D107—81no—
150C4×D107—81no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—22no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—21no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—41no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—84no—
171C2×D107#13512yes—
172C4×C107—21no—
173C4×C107—41no—
174C4×C107—41no—
175C4×C107—41no—
176C4×C107C2×#4111yescharacteristic
177C4×C107H(5,𝔽2)11yescharacteristic
178Dic107—21no—
179Dic107—21no—
180Dic107—21no—
181Dic107—41no—
182Dic107—41no—
183C2×C107—42no—
184C2×C107—41no—
185C2×C107#54611yescharacteristic
186C2×C107C2×#13511yescharacteristic
187C2×C107#57611yescharacteristic
188D107—22no—
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×D4—1071no—
206#416D10711yescharacteristic, complemented
207C2×#102—1071no—
208C2×#41C2×D10711yescharacteristic
209C22×D4—1071no—
210C2×SD16—1071no—
211C2×D8—1071no—
212C2×D8—1071no—
213C22×C8—1071no—
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—1071no—
230C4×D4—2141no—
231#117C2×D10711yescharacteristic, complemented
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—2141no—
234#107—1071no—
235#107—1071no—
236#106—1071no—
237C4⋊C8—1071no—
238#103—1071no—
239#103C2×D10711yescharacteristic, complemented
240#103C2×D10711yescharacteristic, complemented
241#102—1071no—
242#102—1071no—
243#102—1071no—
244#102—1071no—
245#102—1071no—
246#102C2×D10711yescharacteristic
247#102C2×D10711yescharacteristic
248C22⋊C8—1071no—
249C22⋊C8—1071no—
250C4×C8—1071no—
251C24—2141no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D10711yescharacteristic
255C2×D4—21no—
256C2×D4—1071no—
257C2×D4—1071no—
258C2×D4—1071no—
259C2×D4—4281no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2141no—
264C2×D4—2142no—
265C2×D4C22×D10711yescharacteristic
266C22×C4—21no—
267C22×C4—1071no—
268C22×C4—2141no—
269C22×C4—2141no—
270C22×C4—2141no—
271C22×C4C22×D10711yescharacteristic
272SD16—4281no—
273D8—4281no—
274D8—2142no—
275C2×C8—41no—
276C2×C8—1071no—
277C2×C8—1071no—
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—1071no—
292C22⋊C4—1071no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4—2141no—
300C22⋊C4—2141no—
301C42—21no—
302C42—1071no—
303C23—21no—
304C23—8561no—
305C23—1071no—
306C23—4282no—
307C23—4282no—
308C23—2141no—
309C23—2141no—
310C23D4×D10711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—42no—
315D4—8561no—
316D4—8561no—
317D4—4282no—
318D4—4282no—
319D4—2142no—
320D4—2141no—
321D4—2141no—
322D4—2141no—
323D4—2141no—
324D4#1313381812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—8561no—
332C2×C4—8561no—
333C2×C4—8561no—
334C2×C4—1071no—
335C2×C4—1071no—
336C2×C4—1071no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4—2141no—
341C2×C4D4×D10711yescharacteristic
342C2×C4C23×D10711yescharacteristic
343C8—21no—
344C8—21no—
345C8—2141no—
346C8—2141no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—8564no—
354C22—1072no—
355C22—4281no—
356C22—4281no—
357C22—4281no—
358C22—4281no—
359C22—4281no—
360C22—4281no—
361C22—2141no—
362C22—2141no—
363C22C2×D4×D10711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—4281no—
369C4—4281no—
370C4—2141no—
371C4—2141no—
372C4—2141no—
373C4C2×#1313381811yescharacteristic
374C4#1681215411yescharacteristic
375C2—41no—
376C2—42no—
377C2—8561no—
378C2—8561no—
379C2—2142no—
380C2#2029326711yescharacteristic
381C2#135×D10711yescharacteristic
382C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
hypercenter#416D107
hypocenterC107#3297
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20293321C2semidirect
2. #20293265C2semidirect
3. #20292983C2semidirect
4. #20292961C2semidirect
5. #20292951C2semidirect
6. #20292925C2semidirect
7. #20292863C2semidirect
8. #20292859C2semidirect
9. #20292845C2semidirect
10. #20292772C2semidirect
11. #20292762C2semidirect
12. #20292707C2semidirect
13. C8×C22⋊D107C2semidirect
14. #102×D107C2semidirect
15. #416×C107C2semidirect
17. C2×#13133805C22semidirect
22. #16812100C22semidirect
29. #16812083C22semidirect
35. #16812071C22semidirect
39. #16812043C22semidirect
40. #16812042C22semidirect
41. #16812040C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812033C22semidirect
45. #16812031C22semidirect
49. C22⋊(C107⋊C8)C22semidirect
51. C8×Dic107C22semidirect
53. C22×C8×C107C22semidirect
54. #117×C107C22semidirect
55. #103×C107C22semidirect
56. #103×C107C22semidirect
192. C107#3297semidirect
206. #416D107semidirect
214. C22×C8C2×D107semidirect
231. #117C2×D107semidirect
239. #103C2×D107semidirect
240. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933212
#202932652
#202929832
#202929612
#202929512
#202929252
#202928632
#202928592
#202928452
#202927722
#202927622
#202927072
C8×C22⋊D1072
#102×D1072
#416×C1072
#3297107