Finite Group Databasebeta

← #28348296

Subgroups — #28348296

Order 13696.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293317C211yesmaximal, characteristic, complemented
2#20292981C211yesmaximal, characteristic, complemented
3#20292955C211yesmaximal, characteristic, complemented
4#20292948C211yesmaximal, characteristic, complemented
5#20292927C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292854C211yesmaximal, characteristic, complemented
8#20292846C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292778C211yesmaximal, characteristic, complemented
11C4×#13133806C211yesmaximal, characteristic, complemented
12D4×C107⋊C8C211yesmaximal, characteristic, complemented
13#20292623C211yesmaximal, characteristic, complemented
14C2×#16812030C211yesmaximal, characteristic, complemented
15#546×C107C211yesmaximal, characteristic
16C2×#13133808C2211yescharacteristic, complemented
17C2×#13133806C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic, complemented
21#16812102—21no—
22#16812101C2211yescharacteristic, complemented
23#16812099—21no—
24#16812090—21no—
25#16812088C2211yescharacteristic
26#16812088C2211yescharacteristic, complemented
27C4×Dic214C2211yescharacteristic, complemented
28#16812068C2211yescharacteristic, complemented
29#16812065—21no—
30C2×C4⋊Dic107C2211yescharacteristic
31#16812042C2211yescharacteristic, complemented
32#16812040C2211yescharacteristic, complemented
33#16812040C2211yescharacteristic, complemented
34C4⋊(C107⋊C8)C2211yescharacteristic, complemented
35#16812033C2211yescharacteristic, complemented
36#16812033C2211yescharacteristic, complemented
37#16812033C2211yescharacteristic, complemented
38#16812032C2211yescharacteristic, complemented
39#16812030C2211yescharacteristic
40#16812030C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812030C2211yescharacteristic
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×Dic214C2311yescharacteristic
60C22×Dic107—21no—
61#13133808—41no—
62#13133806—22no—
63#13133806—41no—
64C2×C107⋊C8—41no—
65C2×C107⋊C8—41no—
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2311yescharacteristic
70C4⋊Dic107—21no—
71C4⋊Dic107C2311yescharacteristic
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74Dic107⋊C4—21no—
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C22⋊Dic107—21no—
80C22⋊Dic107—21no—
81C22⋊Dic107—21no—
82C4×Dic107—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
118Q8×C107—21no—
119Q8×C107—21no—
120D4×C107—22no—
121D4×C107—21no—
122D4×C107—21no—
123D4×C107—21no—
124D4×C107—42no—
125D4×C107—42no—
126D4×C107—81no—
127D4×C107#4112yes—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—42no—
135C2×C4×C107—81no—
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2411yescharacteristic
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—42no—
146Dic214—21no—
147Dic214—21no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—22no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107—41no—
162C22×C107—41no—
163C22×C107—84no—
164C22×C107#13512yes—
165C22×C107C22×D411yescharacteristic
166C4×C107—21no—
167C4×C107—21no—
168C4×C107—21no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107C2×#4111yescharacteristic
173C4×C107H(5,𝔽2)11yescharacteristic
174Dic107—41no—
175Dic107—41no—
176Dic107—41no—
177Dic107—41no—
178C2×C107—22no—
179C2×C107—41no—
180C2×C107—42no—
181C2×C107—81no—
182C2×C107#54611yescharacteristic
183C2×C107C2×#13511yescharacteristic
184C2×C107#57711yescharacteristic
185#3296—1071nomaximal, Sylow
186C107#329611yescharacteristic, complemented, Sylow
187#552—1071no—
188#546D10711yescharacteristic
189#488—1071no—
190#483—1071no—
191#482—1071no—
192#478—1071no—
193#467—1071no—
194#465—1071no—
195#464—1071no—
196#452—1071no—
197#450—1071no—
198C4×SD16—1071no—
199C8×D4—1071no—
200#416—1071no—
201C2×#102—1071no—
202C2×#41C2×D10711yescharacteristic
203C22×D4C2×D10711yescharacteristic
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8—1071no—
208C22×C8—1071no—
209#126—2141no—
210#125—1071no—
211#125—2141no—
212#124C2×D10711yescharacteristic
213#123—21no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8—1071no—
217C22⋊Q8C2×D10711yescharacteristic
218#121—21no—
219#121C2×D10711yescharacteristic
220#120—21no—
221C4×Q8—1071no—
222C4×D4C2×D10711yescharacteristic
223C4×D4C2×D10711yescharacteristic
224#117—1071no—
225#117—2141no—
226C2×C4⋊C4—1071no—
227C2×C22⋊C4—21no—
228#107—1071no—
229#106—1071no—
230#106—1071no—
231C4⋊C8—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241#102—1071no—
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8—1071no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—1071no—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D10711yescharacteristic
256C2×D4C22×D10711yescharacteristic
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2141no—
261C22×C4C22×D10711yescharacteristic
262C22×C4C22×D10711yescharacteristic
263Q16—4281no—
264SD16—4281no—
265SD16—2142no—
266C2×C8—1071no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—4281no—
271C2×C8—4281no—
272C4⋊C4—21no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4—2141no—
282C4⋊C4C22⋊D10711yescharacteristic
283C4⋊C4C22×D10711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4C22⋊D10711yescharacteristic
293C22⋊C4C22⋊D10711yescharacteristic
294C42—2141no—
295C42—2141no—
296C42C22×D10711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D10711yescharacteristic
303C23C2×C22⋊D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2141no—
307Q8—2141no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1313381712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—4282no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4C2×C22⋊D10711yescharacteristic
330C2×C4C2×C22⋊D10711yescharacteristic
331C2×C4C2×C22⋊D10711yescharacteristic
332C2×C4C2×C22⋊D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—2141no—
335C8—2141no—
336C8—2141no—
337C8—2141no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#1681212912yes—
351C22C22×C22⋊D10711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—4281no—
359C4—4281no—
360C4—4281no—
361C4—4281no—
362C4C2×#1313381711yescharacteristic
363C4#1681215411yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2029327111yescharacteristic
369C2C2×#1681212911yescharacteristic
370C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#546×C107C2
socleC22×C107C22×D4
hypercenter#546D107
hypocenterC107#3296
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. #20293317C2semidirect
2. #20292981C2semidirect
3. #20292955C2semidirect
4. #20292948C2semidirect
5. #20292927C2semidirect
6. #20292866C2semidirect
7. #20292854C2semidirect
8. #20292846C2semidirect
9. #20292789C2semidirect
10. #20292778C2semidirect
11. C4×#13133806C2semidirect
12. D4×C107⋊C8C2semidirect
13. #20292623C2semidirect
14. C2×#16812030C2semidirect
16. C2×#13133808C22semidirect
20. C22×C107⋊C8C22semidirect
22. #16812101C22semidirect
26. #16812088C22semidirect
27. C4×Dic214C22semidirect
28. #16812068C22semidirect
31. #16812042C22semidirect
32. #16812040C22semidirect
33. #16812040C22semidirect
34. C4⋊(C107⋊C8)C22semidirect
35. #16812033C22semidirect
36. #16812033C22semidirect
37. #16812033C22semidirect
38. #16812032C22semidirect
46. C22⋊(C107⋊C8)C22semidirect
47. C4×C107⋊C8C22semidirect
186. C107#3296semidirect

Maximal subgroups

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

subgroupindex
#202933172
#202929812
#202929552
#202929482
#202929272
#202928662
#202928542
#202928462
#202927892
#202927782
C4×#131338062
D4×C107⋊C82
#202926232
C2×#168120302
#546×C1072
#3296107