Finite Group Databasebeta

← #28348382

Subgroups — #28348382

Order 13696.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293341C211yesmaximal, characteristic, complemented
2#20293276C211yesmaximal, characteristic, complemented
3#20292976C211yesmaximal, characteristic, complemented
4#20292969C211yesmaximal, characteristic, complemented
5#20292967C211yesmaximal, characteristic, complemented
6#20292915C211yesmaximal, characteristic, complemented
7#20292854C211yesmaximal, characteristic, complemented
8#20292853C211yesmaximal, characteristic, complemented
9#20292830C211yesmaximal, characteristic, complemented
10#20292769C211yesmaximal, characteristic, complemented
11D8×Dic107C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13#20292623C211yesmaximal, characteristic, complemented
14#20292620C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C22×C107⋊C8C2211yescharacteristic
19C2×C8×D107C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812096—21no—
23#16812095—21no—
24#16812092—21no—
25#16812090—21no—
26#16812088C2211yescharacteristic
27#16812085C2211yescharacteristic
28#16812081—21no—
29#16812081—21no—
30#16812080C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32D4×Dic107C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34#16812068C2211yescharacteristic
35#16812067C2211yescharacteristic
36#16812065—21no—
37C2×Dic107⋊C4—21no—
38#16812043C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×D8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133817—22no—
60#13133817—41no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107D411yescharacteristic
66C22×Dic107—21no—
67C22×Dic107—21no—
68C22×Dic107—21no—
69C2×C4×D107C2311yescharacteristic
70#13133806—41no—
71C2×C107⋊C8—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C8×D107—41no—
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107D411yescharacteristic
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4D411yescharacteristic
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96#41×C107—22no—
97#41×C107—41no—
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107—21no—
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107C2311yescharacteristic
105SD16×C107—41no—
106D8×C107—22no—
107C2×C8×C107C2311yescharacteristic
108C2×C8×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C23×C107—21no—
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—21no—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—42no—
121D4×C107—81no—
122D4×C107#4112yes—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—42no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—22no—
131C22⋊D107—42no—
132C22⋊D107—81no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—42no—
138C2×Dic107—42no—
139C2×Dic107—42no—
140C2×Dic107C2×D411yescharacteristic
141C2×Dic107C2×D411yescharacteristic
142C2×Dic107C2×D411yescharacteristic
143C4×D107—21no—
144C4×D107—21no—
145Dic214—21no—
146Dic214—21no—
147Dic214—81no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—22no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—21no—
165C4×C107—21no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107C2×#4111yescharacteristic
169C4×C107H(5,𝔽2)11yescharacteristic
170Dic107—21no—
171Dic107—21no—
172Dic107—41no—
173Dic107—41no—
174Dic107—41no—
175Dic107—41no—
176Dic107#13512yes—
177C2×C107—41no—
178C2×C107—42no—
179C2×C107—81no—
180C2×C107#54611yescharacteristic
181C2×C107C2×#13511yescharacteristic
182C2×C107#57711yescharacteristic
183D107—41no—
184#3303—1071nomaximal, Sylow
185C107#330311yescharacteristic, complemented, Sylow
186#555—1071no—
187#547—1071no—
188#487—1071no—
189#485—1071no—
190#485—1071no—
191#476—1071no—
192#465—1071no—
193#465—1071no—
194#461—1071no—
195#449—1071no—
196#449D10711yescharacteristic, complemented
197C4×D8—1071no—
198C8×D4—1071no—
199#416—1071no—
200#416—1071no—
201C2×#41—1071no—
202C2×#41C2×D10711yescharacteristic
203C2×SD16—1071no—
204C2×SD16C2×D10711yescharacteristic
205C2×D8C2×D10711yescharacteristic
206C22×C8—1071no—
207C22×C8—1071no—
208C4⋊Q8—1071no—
209C4⋊Q8—1071no—
210#124—2141no—
211#124—2141no—
212#123—2141no—
213#123—2141no—
214C22⋊Q8—1071no—
215C22⋊Q8—1071no—
216#121—1071no—
217#121—2141no—
218#121—2141no—
219#121C2×D10711yescharacteristic
220C4×D4—1071no—
221C4×D4—1071no—
222C4×D4—1071no—
223#117—1071no—
224#117—1071no—
225#117—2141no—
226C2×C4⋊C4—2141no—
227#107—1071no—
228#106—1071no—
229#106—1071no—
230C4⋊C8—1071no—
231#103—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#103C2×D10711yescharacteristic
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102C2×D10711yescharacteristic
241C22⋊C8—1071no—
242C22⋊C8C2×D10711yescharacteristic
243C4×C8—1071no—
244#41—22no—
245#41—41no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8—2141no—
250C2×Q8C22×D10711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—1071no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4C22×D10711yescharacteristic
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—1071no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4—2141no—
263C22×C4C22×D10711yescharacteristic
264SD16—41no—
265SD16—4281no—
266D8—22no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—4281no—
270C2×C8—4281no—
271C2×C8C22×D10711yescharacteristic
272C2×C8C22×D10711yescharacteristic
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—1071no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4—2141no—
282C4⋊C4C22×D10711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1071no—
285C22⋊C4—1071no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C42—1071no—
294C42—2141no—
295C42—2141no—
296C23—21no—
297C23—21no—
298C23—1071no—
299C23D4×D10711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—8561no—
303Q8—2141no—
304Q8—2141no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—8561no—
311D4—4282no—
312D4—2142no—
313D4#1313381712yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—1071no—
320C2×C4—1071no—
321C2×C4—1071no—
322C2×C4—4282no—
323C2×C4—4282no—
324C2×C4—4282no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4D4×D10711yescharacteristic
332C2×C4C23×D10711yescharacteristic
333C8—21no—
334C8—21no—
335C8—2141no—
336C8—2141no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2141no—
347C22—2141no—
348C22—2141no—
349C22C2×D4×D10711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—1072no—
354C4—4281no—
355C4—4281no—
356C4—4281no—
357C4—4281no—
358C4—2141no—
359C4—2141no—
360C4C2×#1313381711yescharacteristic
361C4#1681215411yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—4281no—
366C2#2029326411yescharacteristic
367C2#135×D10711yescharacteristic
368C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20293341C2semidirect
2. #20293276C2semidirect
3. #20292976C2semidirect
4. #20292969C2semidirect
5. #20292967C2semidirect
6. #20292915C2semidirect
7. #20292854C2semidirect
8. #20292853C2semidirect
9. #20292830C2semidirect
10. #20292769C2semidirect
11. D8×Dic107C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. #20292623C2semidirect
14. #20292620C2semidirect
15. #449×C107C2semidirect
185. C107#3303semidirect
196. #449D107semidirect

Maximal subgroups

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

subgroupindex
#202933412
#202932762
#202929762
#202929692
#202929672
#202929152
#202928542
#202928532
#202928302
#202927692
D8×Dic1072
(C22⋊D107)⋊C82
#202926232
#202926202
#449×C1072
#3303107