Finite Group Databasebeta

← #28348384

Subgroups — #28348384

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#20293279C211yesmaximal, characteristic, complemented
3#20292976C211yesmaximal, characteristic, complemented
4#20292971C211yesmaximal, characteristic, complemented
5#20292965C211yesmaximal, characteristic, complemented
6#20292916C211yesmaximal, characteristic, complemented
7#20292855C211yesmaximal, characteristic, complemented
8#20292830C211yesmaximal, characteristic, complemented
9#20292771C211yesmaximal, characteristic, complemented
10#20292770C211yesmaximal, characteristic, complemented
11#20292707C211yesmaximal, characteristic, complemented
12C8×C22⋊D107C211yesmaximal, characteristic, complemented
13#20292624C211yesmaximal, characteristic, complemented
14#20292622C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107C2211yescharacteristic
23#16812096—21no—
24#16812095—21no—
25#16812094—21no—
26#16812091—21no—
27#16812089C2211yescharacteristic
28#16812082—21no—
29#16812081—21no—
30#16812080C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32#16812071C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34#16812068C2211yescharacteristic
35#16812067C2211yescharacteristic
36#16812065—21no—
37C4⋊C4×D107—21no—
38#16812042C2211yescharacteristic
39#16812040C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133819—22no—
60#13133819—41no—
61#13133818—22no—
62#13133818—41no—
63C2×Dic214—21no—
64C2×Dic214C2311yescharacteristic
65C2×D428—21no—
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107D411yescharacteristic
70C22×Dic107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75#13133806—41no—
76#13133803—22no—
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—41no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107D411yescharacteristic
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107—21no—
100C2×D4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102SD16×C107—41no—
103C2×C8×C107—41no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107—21no—
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C22⋊C4×C107—21no—
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—21no—
118D4×C107—21no—
119D4×C107—81no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C22⋊D107—22no—
128C22⋊D107—42no—
129C22⋊D107—81no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—42no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—21no—
138D428—42no—
139D428#4112yes—
140C4×D107—21no—
141C4×D107—21no—
142C4×D107—42no—
143C4×D107—42no—
144C4×D107—42no—
145Dic214—21no—
146Dic214—21no—
147Dic214—81no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—41no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107C22×D411yescharacteristic
159C2×D107—21no—
160C2×D107—22no—
161C2×D107—21no—
162C2×D107—21no—
163C2×D107—41no—
164C2×D107—41no—
165C4×C107—21no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107C2×#4111yescharacteristic
170C4×C107H(5,𝔽2)11yescharacteristic
171Dic107—21no—
172Dic107—21no—
173Dic107—41no—
174Dic107—41no—
175Dic107—41no—
176Dic107#13512yes—
177C2×C107—41no—
178C2×C107—81no—
179C2×C107#54611yescharacteristic
180C2×C107C2×#13511yescharacteristic
181C2×C107#57711yescharacteristic
182D107—41no—
183D107—42no—
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#465D10711yescharacteristic, complemented
194#461—1071no—
195#449—1071no—
196#449—1071no—
197C4×D8—1071no—
198C8×D4—1071no—
199#416—1071no—
200#416—1071no—
201C2×#41—1071no—
202C2×#41—1071no—
203C2×SD16—1071no—
204C2×SD16C2×D10711yescharacteristic
205C2×D8—1071no—
206C22×C8—1071no—
207C22×C8C2×D10711yescharacteristic
208C4⋊Q8—1071no—
209C4⋊Q8—1071no—
210#124—2141no—
211#124—2141no—
212#123—2141no—
213#123—2141no—
214C22⋊Q8—1071no—
215C22⋊Q8C2×D10711yescharacteristic
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#106C2×D10711yescharacteristic
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⋊C8—1071no—
243C4×C8—1071no—
244#41—4281no—
245#41—4281no—
246#41—2142no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8—2141no—
250C2×Q8C22×D10711yescharacteristic
251C2×D4—21no—
252C2×D4—1071no—
253C2×D4—1071no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2141no—
257C2×D4C22×D10711yescharacteristic
258C22×C4—1071no—
259C22×C4—2141no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4—2141no—
263C22×C4C22×D10711yescharacteristic
264SD16—41no—
265SD16—4281no—
266D8—2142no—
267C2×C8—41no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—4281no—
271C2×C8C22×D10711yescharacteristic
272C2×C8C22×D10711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—1071no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4C22×D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1071no—
286C22⋊C4—1071no—
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—1071no—
298C23—2141no—
299C23D4×D10711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—8561no—
303Q8—2141no—
304Q8—2141no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—8561no—
309D4—1072no—
310D4—4282no—
311D4—4282no—
312D4—2142no—
313D4—2141no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—1071no—
320C2×C4—1071no—
321C2×C4—1071no—
322C2×C4—4282no—
323C2×C4—4282no—
324C2×C4—4282no—
325C2×C4—4282no—
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—41no—
341C22—41no—
342C22—41no—
343C22—4281no—
344C22—4281no—
345C22—2141no—
346C22—2141no—
347C22—2142no—
348C22—2141no—
349C22C2×D4×D10711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—1072no—
355C4—4281no—
356C4—4281no—
357C4—4281no—
358C4—2141no—
359C4—2141no—
360C4#41×D10711yescharacteristic
361C4#1681215411yescharacteristic
362C2—41no—
363C2—81no—
364C2—4282no—
365C2—4281no—
366C2#2029326611yescharacteristic
367C2#135×D10711yescharacteristic
368C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
hypercenter#465D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20293341C2semidirect
2. #20293279C2semidirect
3. #20292976C2semidirect
4. #20292971C2semidirect
5. #20292965C2semidirect
6. #20292916C2semidirect
7. #20292855C2semidirect
8. #20292830C2semidirect
9. #20292771C2semidirect
10. #20292770C2semidirect
11. #20292707C2semidirect
12. C8×C22⋊D107C2semidirect
13. #20292624C2semidirect
14. #20292622C2semidirect
15. #465×C107C2semidirect
185. C107#3303semidirect
193. #465D107semidirect

Maximal subgroups

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

subgroupindex
#202933412
#202932792
#202929762
#202929712
#202929652
#202929162
#202928552
#202928302
#202927712
#202927702
#202927072
C8×C22⋊D1072
#202926242
#202926222
#465×C1072
#3303107