Finite Group Databasebeta

← #28349526

Subgroups — #28349526

Order 13696.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293359C211yesmaximal, characteristic, complemented
2#20293351C211yesmaximal, characteristic, complemented
3#20293013C211yesmaximal, characteristic
4#20293012C211yesmaximal, characteristic, complemented
5#20292851C211yesmaximal, characteristic, complemented
6#20292851C211yesmaximal, characteristic, complemented
7#20292844C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292829C211yesmaximal, characteristic, complemented
10#20292825C211yesmaximal, characteristic, complemented
11#20292755C211yesmaximal, characteristic, complemented
12#20292740C211yesmaximal, characteristic, complemented
13#20292734C211yesmaximal, characteristic, complemented
14SD16×Dic107C211yesmaximal, characteristic, complemented
15#497×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×D856C2211yescharacteristic
20C2×#13133803C2211yescharacteristic
21#16812105C2211yescharacteristic
22#16812104—21no—
23#16812104—21no—
24#16812098C2211yescharacteristic
25#16812098C2211yescharacteristic
26#16812098C2211yescharacteristic
27#16812095—21no—
28#16812093—21no—
29#16812091—21no—
30#16812089—21no—
31Q8×Dic107C2211yescharacteristic
32Q8×Dic107C2211yescharacteristic
33Q8×Dic107C2211yescharacteristic
34C4×Dic214C2211yescharacteristic
35C4×D428C2211yescharacteristic
36C4×D428C2211yescharacteristic
37D4×Dic107C2211yescharacteristic
38(C22⋊D107)⋊C4—21no—
39#16812067—21no—
40#16812041C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×Q16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55C2×SD16×C107C2211yescharacteristic
56C4⋊Q8×C107C2211yescharacteristic
57#124×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214D411yescharacteristic
60C2×D428C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C22×Dic107—21no—
65C2×C4×D107—21no—
66C2×C4×D107—21no—
67#13133805—41no—
68#13133805—41no—
69#13133805—41no—
70D856—22no—
71#13133803—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107D411yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4D411yescharacteristic
81Dic107⋊C4D411yescharacteristic
82C22⋊Dic107—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107C2311yescharacteristic
100Q16×C107—22no—
101Q16×C107—41no—
102SD16×C107—22no—
103SD16×C107—41no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107—21no—
107C4⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C22⋊C4×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C22×D107—21no—
113C22×D107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107—21no—
118Q8×C107—21no—
119Q8×C107—21no—
120D4×C107—21no—
121D4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2411yescharacteristic
128C22⋊D107—42no—
129C22⋊D107—81no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—81no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—21no—
138D428—21no—
139D428—21no—
140D428—21no—
141C4×D107—42no—
142C4×D107—81no—
143Dic214—41no—
144C8×C107—21no—
145C8×C107#4112yes—
146C107⋊C8—21no—
147C107⋊C8—21no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107C22×D411yescharacteristic
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C4×C107—21no—
159C4×C107—21no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107C2×#4111yescharacteristic
164C4×C107H(5,𝔽2)11yescharacteristic
165Dic107—21no—
166Dic107—21no—
167Dic107—21no—
168Dic107—21no—
169Dic107—41no—
170Dic107—41no—
171C2×C107—81no—
172C2×C107#55011yescharacteristic
173C2×C107#57511yescharacteristic
174C2×C107#57711yescharacteristic
175D107—81no—
176D107—81no—
177#3401—1071nomaximal, Sylow
178C107#340111yescharacteristic, complemented, Sylow
179#559—1071no—
180#556—1071no—
181#497D10711yescharacteristic, complemented
182#496—1071no—
183#495—1071no—
184#464—1071no—
185#464—1071no—
186#463—1071no—
187#463—1071no—
188#461—1071no—
189#460—1071no—
190#446—1071no—
191#442—1071no—
192#441—1071no—
193C4×SD16—1071no—
194C2×Q16C2×D10711yescharacteristic
195C2×Q16C2×D10711yescharacteristic
196C2×SD16—1071no—
197C2×SD16—1071no—
198C2×SD16—1071no—
199C2×SD16C2×D10711yescharacteristic
200C2×SD16C2×D10711yescharacteristic
201C2×D8—1071no—
202C2×D8—1071no—
203C4⋊Q8C2×D10711yescharacteristic
204#127—1071no—
205#126—2141no—
206#126—2141no—
207#124—1071no—
208#124—1071no—
209#124—1071no—
210#124—2141no—
211#124C2×D10711yescharacteristic
212#123—2141no—
213#123—2141no—
214C22⋊Q8—2141no—
215C4×Q8—1071no—
216C4×Q8—1071no—
217C4×Q8—1071no—
218C4×Q8—1071no—
219C4×D4—1071no—
220C4×D4—1071no—
221C4×D4—1071no—
222C4×D4—2141no—
223#117—2141no—
224#106—1071no—
225C4⋊C8—1071no—
226C4⋊C8—1071no—
227C4⋊C8—1071no—
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#97—1071no—
235#97C2×D10711yescharacteristic
236C4×C8—1071no—
237C2×Q8—1071no—
238C2×Q8C22×D10711yescharacteristic
239C2×Q8C22×D10711yescharacteristic
240C2×Q8C22×D10711yescharacteristic
241C2×D4—1071no—
242C2×D4—1071no—
243C2×D4—2141no—
244C2×D4—2141no—
245C2×D4C22×D10711yescharacteristic
246C22×C4—2141no—
247C22×C4—2141no—
248C22×C4—2141no—
249Q16—22no—
250Q16—41no—
251SD16—22no—
252SD16—41no—
253SD16—4281no—
254SD16—4281no—
255SD16—4281no—
256D8—4281no—
257D8—2142no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—1071no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C42—1071no—
284C42—1071no—
285C42—2141no—
286C42—2141no—
287C42—2141no—
288C42C22×D10711yescharacteristic
289C23—21no—
290C23—2141no—
291C23—2141no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—4281no—
299D4—21no—
300D4—21no—
301D4—8561no—
302D4—4282no—
303D4—2141no—
304D4—2141no—
305D4—2141no—
306D4—2141no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—8561no—
311C2×C4—8561no—
312C2×C4—1071no—
313C2×C4—1071no—
314C2×C4—1071no—
315C2×C4—1071no—
316C2×C4—4282no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4D4×D10711yescharacteristic
320C2×C4D4×D10711yescharacteristic
321C2×C4C23×D10711yescharacteristic
322C8—21no—
323C8—2141no—
324C8—2141no—
325C8#1313381712yes—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—4281no—
330C22—4281no—
331C22—4281no—
332C22—4281no—
333C22—4281no—
334C22—4281no—
335C22C2×D4×D10711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—4281no—
342C4—4281no—
343C4—2141no—
344C4—2141no—
345C4—2141no—
346C4—2141no—
347C4C2×#1313381711yescharacteristic
348C4#1681215411yescharacteristic
349C2—81no—
350C2—8561no—
351C2—8561no—
352C2#2029329711yescharacteristic
353C2#2029342611yescharacteristic
354C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#497×C107C2
socleC22×C107C22×D4
prime radical#20293013C2
hypercenter#497D107
hypocenterC107#3401
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #20293359C2semidirect
2. #20293351C2semidirect
4. #20293012C2semidirect
5. #20292851C2semidirect
6. #20292851C2semidirect
7. #20292844C2semidirect
8. #20292839C2semidirect
9. #20292829C2semidirect
10. #20292825C2semidirect
11. #20292755C2semidirect
12. #20292740C2semidirect
13. #20292734C2semidirect
14. SD16×Dic107C2semidirect
15. #497×C107C2semidirect
178. C107#3401semidirect
181. #497D107semidirect

Maximal subgroups

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

subgroupindex
#202933592
#202933512
#202930132
#202930122
#202928512
#202928442
#202928392
#202928292
#202928252
#202927552
#202927402
#202927342
SD16×Dic1072
#497×C1072
#3401107