Finite Group Databasebeta

← #28347015

Subgroups — #28347015

Order 13696.

412 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#20293179C211yesmaximal, characteristic, complemented
2#20293172C211yesmaximal, characteristic
3#20293024C211yesmaximal, characteristic, complemented
4#20292998C211yesmaximal, characteristic, complemented
5#20292976C211yesmaximal, characteristic, complemented
6#20292972C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292856C211yesmaximal, characteristic, complemented
9#20292854C211yesmaximal, characteristic, complemented
10#20292718C211yesmaximal, characteristic, complemented
11SD16×Dic107C211yesmaximal, characteristic, complemented
12#20292714C211yesmaximal, characteristic, complemented
13#20292712C211yesmaximal, characteristic, complemented
14C2×C8×Dic107C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×#13133807C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812108C2211yescharacteristic
23#16812107C2211yescharacteristic
24#16812105C2211yescharacteristic
25#16812099—21no—
26#16812095—21no—
27#16812088C2211yescharacteristic
28#16812086—21no—
29#16812083C2211yescharacteristic
30#16812081—21no—
31Q8×Dic107C2211yescharacteristic
32#16812074—21no—
33#16812074C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35#16812071C2211yescharacteristic
36(C22⋊D107)⋊C4—21no—
37#16812065—21no—
38C2×C4×Dic107C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48C8×Dic107C2211yescharacteristic
49C8×Dic107C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63C2×D428C2311yescharacteristic
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C22×Dic107—21no—
68C22×Dic107D411yescharacteristic
69C22×Dic107D411yescharacteristic
70C2×C4×D107—21no—
71#13133807—22no—
72#13133806—22no—
73#13133805—22no—
74C2×C107⋊C8—22no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107D411yescharacteristic
93C4×Dic107D411yescharacteristic
94C4×Dic107C2311yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100SD16×C107—22no—
101C2×C8×C107—22no—
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107C2311yescharacteristic
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C23×C107—21no—
110C23×C107C2×D411yescharacteristic
111C22×D107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114D4×C107—21no—
115D4×C107—21no—
116D4×C107—42no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C22⋊D107—42no—
125C22⋊D107—42no—
126C22⋊D107—42no—
127C2×Dic107—22no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—42no—
132C2×Dic107SD1614yescomplemented
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—21no—
138D428—21no—
139C4×D107—42no—
140Dic214—21no—
141Dic214—21no—
142Dic214—42no—
143Dic214—42no—
144C8×C107—21no—
145C8×C107#4112yescomplemented
146C107⋊C8—21no—
147C107⋊C8#4112yescomplemented
148C22×C107—21no—
149C22×C107—21no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107C2×SD1612yes—
154C22×C107C22×D411yescharacteristic
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C4×C107—21no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107C2×#4111yescharacteristic
163C4×C107C2×#4111yescharacteristic
164Dic107—21no—
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168Dic107C2×SD1612yes—
169Dic107C2×SD1612yes—
170Dic107#13512yes—
171C2×C107—22no—
172C2×C107—81no—
173C2×C107#53211yescharacteristic
174C2×C107C22×SD1611yescharacteristic
175C2×C107C2×#13511yescharacteristic
176D107—81no—
177#3167—1071nomaximal, Sylow
178C107#316711yescharacteristic, complemented, Sylow
179#533—1071no—
180#532—1071no—
181#498—1071no—
182#492—1071no—
183#487—1071no—
184#486—1071no—
185#465—1071no—
186#465—1071no—
187#465—1071no—
188#465D10711yescharacteristic, complemented
189C4×SD16—1071no—
190C4×SD16—1071no—
191C4×SD16—1071no—
192C4×SD16—1071no—
193C2×C4×C8—1071no—
194C2×#41—2141no—
195C2×SD16—1071no—
196C2×SD16—1071no—
197C2×SD16—1071no—
198C2×SD16C2×D10711yescharacteristic
199C22×C8—1071no—
200C22×C8C2×D10711yescharacteristic
201C4⋊Q8—1071no—
202C4⋊Q8—1071no—
203C4⋊Q8—1071no—
204#127—1071no—
205#125—2141no—
206#124—2141no—
207C22⋊Q8—1071no—
208C22⋊Q8—2141no—
209C22⋊Q8C2×D10711yescharacteristic
210#121—1071no—
211#121—2141no—
212#121C2×D10711yescharacteristic
213C4×Q8—1071no—
214C4×Q8—1071no—
215C4×Q8—2141no—
216C4×D4—1071no—
217C4×D4—1071no—
218C4×D4—2141no—
219#117—2141no—
220C2×C42—1071no—
221#106—1071no—
222#106—1071no—
223#106—1071no—
224#106C2×D10711yescharacteristic
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103C2×D10711yescharacteristic
229#102—1071no—
230#102—1071no—
231#102—1071no—
232#102C2×D10711yescharacteristic
233C4×C8—1071no—
234C4×C8—1071no—
235C4×C8—1071no—
236C4×C8—1071no—
237#41—4284no—
238C2×Q8—1071no—
239C2×Q8—2141no—
240C2×Q8—2141no—
241C2×Q8C22×D10711yescharacteristic
242C2×D4—21no—
243C2×D4—1071no—
244C2×D4—2141no—
245C2×D4—2141no—
246C2×D4—2141no—
247C2×D4C22×D10711yescharacteristic
248C22×C4—1071no—
249C22×C4—1071no—
250C22×C4—2141no—
251C22×C4—2141no—
252C22×C4C22×D10711yescharacteristic
253SD16—22no—
254SD16—2142no—
255SD16—2142no—
256SD16—2142no—
257C2×C8—22no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8—2142no—
261C2×C8C22×D10711yescharacteristic
262C2×C8C22×D10711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4C22×D10711yescharacteristic
274C4⋊C4C22×D10711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C42—1071no—
282C42—1071no—
283C42—1071no—
284C42—1071no—
285C42—2141no—
286C42—2141no—
287C23—21no—
288C23—2141no—
289C23D4×D10711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—4282no—
293Q8—4282no—
294Q8—2141no—
295Q8—2141no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—4282no—
300D4—4282no—
301D4—4282no—
302D4—2141no—
303D4—2141no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1074no—
310C2×C4—1071no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—1071no—
314C2×C4—4282no—
315C2×C4—4282no—
316C2×C4—2142no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4D4×D10711yescharacteristic
321C2×C4C23×D10711yescharacteristic
322C8—21no—
323C8—1072no—
324C8—2141no—
325C8#1313381712yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—4281no—
332C22—4281no—
333C22—4281no—
334C22SD16×D10712yes—
335C22C2×D4×D10711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—1072no—
341C4—1072no—
342C4—1072no—
343C4—4281no—
344C4—4281no—
345C4—4281no—
346C4—2141no—
347C4#41×D10711yescharacteristic
348C4C2×#1313381711yescharacteristic
349C2—22no—
350C2—81no—
351C2—8561no—
352C2#2029317211yescharacteristic
353C2C2×SD16×D10711yescharacteristic
354C2#135×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
prime radical#20293172C2
hypercenter#465D107
hypocenterC107#3167
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. #20293179C2semidirect
3. #20293024C2semidirect
4. #20292998C2semidirect
5. #20292976C2semidirect
6. #20292972C2semidirect
7. #20292858C2semidirect
8. #20292856C2semidirect
9. #20292854C2semidirect
10. #20292718C2semidirect
11. SD16×Dic107C2semidirect
12. #20292714C2semidirect
13. #20292712C2semidirect
14. C2×C8×Dic107C2semidirect
15. #465×C107C2semidirect
132. C2×Dic107SD16semidirect
145. C8×C107#41semidirect
147. C107⋊C8#41semidirect
178. C107#3167semidirect
188. #465D107semidirect
325. C8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202931792
#202931722
#202930242
#202929982
#202929762
#202929722
#202928582
#202928562
#202928542
#202927182
SD16×Dic1072
#202927142
#202927122
C2×C8×Dic1072
#465×C1072
#3167107