Finite Group Databasebeta

← #28348311

Subgroups — #28348311

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#20292981C211yesmaximal, characteristic, complemented
3#20292960C211yesmaximal, characteristic, complemented
4#20292948C211yesmaximal, characteristic, complemented
5#20292927C211yesmaximal, characteristic, complemented
6#20292861C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic, complemented
10#20292761C211yesmaximal, characteristic, complemented
11C4×#13133803C211yesmaximal, characteristic, complemented
12D4×C107⋊C8C211yesmaximal, characteristic, complemented
13#20292623C211yesmaximal, characteristic, complemented
14C2×#16812030C211yesmaximal, characteristic, complemented
15#546×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic, complemented
17C2×#13133803C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic, complemented
21#16812104—21no—
22#16812101C2211yescharacteristic, complemented
23#16812091—21no—
24#16812083C2211yescharacteristic
25#16812083C2211yescharacteristic, complemented
26#16812081—21no—
27C4×D428C2211yescharacteristic, complemented
28(C22⋊D107)⋊C4—21no—
29#16812068C2211yescharacteristic, complemented
30C2×C4⋊Dic107C2211yescharacteristic
31#16812042C2211yescharacteristic, complemented
32#16812042C2211yescharacteristic, complemented
33#16812040C2211yescharacteristic, complemented
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35#16812033C2211yescharacteristic, complemented
36#16812033C2211yescharacteristic, complemented
37#16812033C2211yescharacteristic, complemented
38#16812030C2211yescharacteristic
39#16812030C2211yescharacteristic
40#16812030C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812029C2211yescharacteristic, complemented
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic, complemented
47C4×C107⋊C8C2211yescharacteristic, complemented
48C2×#41×C107C2211yescharacteristic
49C22×D4×C107C2211yescharacteristic
50#124×C107C2211yescharacteristic
51#123×C107—21no—
52C22⋊Q8×C107C2211yescharacteristic
53#121×C107—21no—
54#121×C107C2211yescharacteristic
55#120×C107—21no—
56C4×D4×C107C2211yescharacteristic
57C4×D4×C107C2211yescharacteristic
58C2×C22⋊C4×C107—21no—
59C2×D428C2311yescharacteristic
60C2×C22⋊D107—21no—
61C2×C22⋊D107—21no—
62C22×Dic107—21no—
63C2×C4×D107—21no—
64#13133805—41no—
65#13133803—22no—
66#13133803—41no—
67C2×C107⋊C8—41no—
68C2×C107⋊C8—41no—
69C2×C107⋊C8C2311yescharacteristic
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C22⋊Dic107—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83C4×Dic107—21no—
84C24×C107—21no—
85#41×C107—22no—
86#41×C107—41no—
87C2×Q8×C107C2311yescharacteristic
88C2×D4×C107—22no—
89C2×D4×C107—21no—
90C2×D4×C107—21no—
91C2×D4×C107—21no—
92C2×D4×C107—41no—
93C2×D4×C107D411yescharacteristic
94C2×D4×C107C2311yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C22×C4×C107—21no—
97C22×C4×C107—21no—
98C22×C4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107D411yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C22⋊C4×C107D411yescharacteristic
109C22⋊C4×C107D411yescharacteristic
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C23×C107—21no—
113C23×C107—42no—
114C23×C107—42no—
115C23×C107—81no—
116C23×C107C2×D411yescharacteristic
117C23×C107C2×D411yescharacteristic
118C22×D107—21no—
119Q8×C107—21no—
120Q8×C107—21no—
121D4×C107—22no—
122D4×C107—21no—
123D4×C107—21no—
124D4×C107—21no—
125D4×C107—42no—
126D4×C107—42no—
127D4×C107—81no—
128D4×C107#4112yes—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—42no—
136C2×C4×C107—81no—
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2×D411yescharacteristic
141C2×C4×C107C2411yescharacteristic
142C22⋊D107—42no—
143C22⋊D107—81no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—81no—
148D428—21no—
149D428—21no—
150C4×D107—81no—
151C107⋊C8—21no—
152C107⋊C8—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—22no—
159C22×C107—21no—
160C22×C107—21no—
161C22×C107—41no—
162C22×C107—41no—
163C22×C107—41no—
164C22×C107—41no—
165C22×C107—41no—
166C22×C107—84no—
167C22×C107#13512yes—
168C22×C107C22×D411yescharacteristic
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C4×C107—21no—
173C4×C107—21no—
174C4×C107—21no—
175C4×C107—21no—
176C4×C107—41no—
177C4×C107—41no—
178C4×C107C2×#4111yescharacteristic
179C4×C107H(5,𝔽2)11yescharacteristic
180Dic107—41no—
181Dic107—41no—
182Dic107—41no—
183C2×C107—22no—
184C2×C107—41no—
185C2×C107—42no—
186C2×C107—81no—
187C2×C107#54611yescharacteristic
188C2×C107C2×#13511yescharacteristic
189C2×C107#57611yescharacteristic
190D107—81no—
191#3297—1071nomaximal, Sylow
192C107#329711yescharacteristic, complemented, Sylow
193#553—1071no—
194#546D10711yescharacteristic, complemented
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#416—1071no—
207C2×#102—1071no—
208C2×#41C2×D10711yescharacteristic
209C22×D4C2×D10711yescharacteristic
210C2×SD16—1071no—
211C2×D8—1071no—
212C2×D8—1071no—
213C22×C8—1071no—
214C22×C8—1071no—
215#126—2141no—
216#125—1071no—
217#124C2×D10711yescharacteristic
218#123—21no—
219#123—2141no—
220C22⋊Q8C2×D10711yescharacteristic
221#121—21no—
222#121—1071no—
223#121—1071no—
224#121—2141no—
225#121C2×D10711yescharacteristic
226#120—21no—
227C4×D4—1071no—
228C4×D4—2141no—
229C4×D4C2×D10711yescharacteristic
230C4×D4C2×D10711yescharacteristic
231#117—1071no—
232C2×C4⋊C4—1071no—
233C2×C22⋊C4—21no—
234#107—1071no—
235#107—1071no—
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#102—1071no—
247#102—1071no—
248C22⋊C8—1071no—
249C22⋊C8—1071no—
250C4×C8—1071no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D10711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—1071no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4C22⋊D10711yescharacteristic
264C2×D4C22×D10711yescharacteristic
265C2×D4C22×D10711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2141no—
269C22×C4—2141no—
270C22×C4C22×D10711yescharacteristic
271C22×C4C22×D10711yescharacteristic
272SD16—4281no—
273D8—4281no—
274D8—2142no—
275C2×C8—1071no—
276C2×C8—1071no—
277C2×C8—1071no—
278C2×C8—1071no—
279C2×C8—4281no—
280C2×C8—4281no—
281C4⋊C4—21no—
282C4⋊C4—1071no—
283C4⋊C4—1071no—
284C4⋊C4—1071no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4—2141no—
288C4⋊C4C22⋊D10711yescharacteristic
289C4⋊C4C22×D10711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C22⋊C4—2141no—
299C22⋊C4C22⋊D10711yescharacteristic
300C22⋊C4C22⋊D10711yescharacteristic
301C42—2141no—
302C42C22×D10711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—2141no—
309C23C2×C22⋊D10711yescharacteristic
310C23C2×C22⋊D10711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—8561no—
321D4—4282no—
322D4—2141no—
323D4—2141no—
324D4#1313381712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—8561no—
334C2×C4—8561no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4C2×C22⋊D10711yescharacteristic
339C2×C4C2×C22⋊D10711yescharacteristic
340C2×C4C2×C22⋊D10711yescharacteristic
341C2×C4C2×C22⋊D10711yescharacteristic
342C2×C4C23×D10711yescharacteristic
343C8—2141no—
344C8—2141no—
345C8—2141no—
346C8—2141no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—4281no—
360C22—4281no—
361C22—4281no—
362C22#1681212912yes—
363C22C22×C22⋊D10711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—4281no—
371C4—4281no—
372C4—4281no—
373C4C2×#1313381711yescharacteristic
374C4#1681215411yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—8561no—
380C2#2029327111yescharacteristic
381C2C2×#1681212911yescharacteristic
382C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#546×C107C2
socleC22×C107C22×D4
hypercenter#546D107
hypocenterC107#3297
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #20293323C2semidirect
2. #20292981C2semidirect
3. #20292960C2semidirect
4. #20292948C2semidirect
5. #20292927C2semidirect
6. #20292861C2semidirect
7. #20292858C2semidirect
8. #20292839C2semidirect
9. #20292772C2semidirect
10. #20292761C2semidirect
11. C4×#13133803C2semidirect
12. D4×C107⋊C8C2semidirect
13. #20292623C2semidirect
14. C2×#16812030C2semidirect
15. #546×C107C2semidirect
16. C2×#13133805C22semidirect
20. C22×C107⋊C8C22semidirect
22. #16812101C22semidirect
25. #16812083C22semidirect
27. C4×D428C22semidirect
29. #16812068C22semidirect
31. #16812042C22semidirect
32. #16812042C22semidirect
33. #16812040C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. #16812033C22semidirect
36. #16812033C22semidirect
37. #16812033C22semidirect
44. #16812029C22semidirect
46. C22⋊(C107⋊C8)C22semidirect
47. C4×C107⋊C8C22semidirect
192. C107#3297semidirect
194. #546D107semidirect

Maximal subgroups

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

subgroupindex
#202933232
#202929812
#202929602
#202929482
#202929272
#202928612
#202928582
#202928392
#202927722
#202927612
C4×#131338032
D4×C107⋊C82
#202926232
C2×#168120302
#546×C1072
#3297107