Finite Group Databasebeta

← #28349152

Subgroups — #28349152

Order 13696.

348 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293359C211yesmaximal, characteristic
2#20293353C211yesmaximal, characteristic, complemented
3#20292983C211yesmaximal, characteristic, complemented
4#20292981C211yesmaximal, characteristic, complemented
5#20292972C211yesmaximal, characteristic
6#20292928C211yesmaximal, characteristic, complemented
7#20292925C211yesmaximal, characteristic, complemented
8#20292910C211yesmaximal, characteristic, complemented
9#20292851C211yesmaximal, characteristic, complemented
10#20292825C211yesmaximal, characteristic
11#20292751C211yesmaximal, characteristic, complemented
12Q16×Dic107C211yesmaximal, characteristic, complemented
13SD16×Dic107C211yesmaximal, characteristic, complemented
14C4×#13133805C211yesmaximal, characteristic, complemented
15#464×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic
17#16812108—21no—
18#16812108C2211yescharacteristic
19#16812105C2211yescharacteristic
20#16812101—21no—
21#16812101C2211yescharacteristic
22#16812100—21no—
23#16812100C2211yescharacteristic
24#16812098C2211yescharacteristic
25#16812095—21no—
26#16812093—21no—
27#16812091—21no—
28Q8×Dic107C2211yescharacteristic
29Q8×Dic107C2211yescharacteristic
30Q8×Dic107C2211yescharacteristic
31C4×Dic214—21no—
32C4×Dic214C2211yescharacteristic
33C4×D428C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36#16812043C2211yescharacteristic
37#16812041C2211yescharacteristic
38#16812040C2211yescharacteristic
39Dic107⋊C8C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812033C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
49C4×C107⋊C8C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54#124×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C22×Dic107—21no—
65C2×C4×D107—21no—
66#13133805—22no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107D411yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C4×Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107C2311yescharacteristic
91C4×Dic107C2311yescharacteristic
92C2×Q8×C107C2311yescharacteristic
93C2×Q8×C107C2311yescharacteristic
94C2×D4×C107C2311yescharacteristic
95Q16×C107—22no—
96SD16×C107—22no—
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C42×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C22×D107—21no—
107Q8×C107—21no—
108Q8×C107—21no—
109Q8×C107—21no—
110Q8×C107#4112yes—
111D4×C107—21no—
112D4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C22⋊D107—42no—
120C22⋊D107—81no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—81no—
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129D428—21no—
130D428—21no—
131C4×D107—81no—
132Dic214—22no—
133Dic214—42no—
134C8×C107—21no—
135C8×C107—21no—
136C107⋊C8—21no—
137C107⋊C8—21no—
138C22×C107—41no—
139C22×C107—41no—
140C22×C107—41no—
141C22×C107C22×D411yescharacteristic
142C2×D107—41no—
143C2×D107—41no—
144C2×D107—41no—
145C4×C107—22no—
146C4×C107—21no—
147C4×C107—21no—
148C4×C107—41no—
149C4×C107—41no—
150C4×C107C2×#4111yescharacteristic
151C4×C1072−1+411yescharacteristic
152Dic107—21no—
153Dic107—21no—
154Dic107—21no—
155Dic107—41no—
156Dic107—41no—
157Dic107—41no—
158Dic107#13512yes—
159C2×C107—81no—
160C2×C107#54811yescharacteristic
161C2×C107C2×#13511yescharacteristic
162C2×C107#57711yescharacteristic
163D107—81no—
164#3365—1071nomaximal, Sylow
165C107#336511yescharacteristic, complemented, Sylow
166#559—1071no—
167#557—1071no—
168#488—1071no—
169#488—1071no—
170#486—1071no—
171#478—1071no—
172#478—1071no—
173#475—1071no—
174#464—1071no—
175#464D10711yescharacteristic, complemented
176#460—1071no—
177C8×Q8—1071no—
178C4×Q16—1071no—
179C4×SD16—1071no—
180C4×SD16—1071no—
181C2×Q16C2×D10711yescharacteristic
182C2×SD16—1071no—
183C2×SD16C2×D10711yescharacteristic
184C4⋊Q8—1071no—
185C4⋊Q8—2141no—
186#127—1071no—
187#125—1071no—
188#125—1071no—
189#125—2141no—
190#125—2141no—
191#124—1071no—
192#124—2141no—
193#124C2×D10711yescharacteristic
194#123—2141no—
195#123—2141no—
196C4×Q8—1071no—
197C4×Q8—1071no—
198C4×Q8—1071no—
199C4×Q8—1071no—
200C4×Q8—2141no—
201C4×Q8C2×D10711yescharacteristic
202C4×D4—1071no—
203C4×D4—1071no—
204C4×D4—2141no—
205#107—1071no—
206#106—1071no—
207#106—1071no—
208C4⋊C8—1071no—
209C4⋊C8—1071no—
210C4⋊C8C2×D10711yescharacteristic
211#103—1071no—
212#103—1071no—
213#103—1071no—
214#103C2×D10711yescharacteristic
215#102—1071no—
216#102—1071no—
217#102—1071no—
218#102—1071no—
219#102—1071no—
220#102C2×D10711yescharacteristic
221C4×C8—1071no—
222C4×C8—1071no—
223C4×C8—1071no—
224C2×Q8—1071no—
225C2×Q8—2141no—
226C2×Q8C22×D10711yescharacteristic
227C2×Q8C22×D10711yescharacteristic
228C2×D4—1071no—
229C2×D4—2141no—
230C2×D4—2141no—
231C2×D4C22×D10711yescharacteristic
232C22×C4—2141no—
233C22×C4—2141no—
234Q16—22no—
235SD16—22no—
236SD16—2142no—
237C2×C8—1071no—
238C2×C8—1071no—
239C2×C8C22×D10711yescharacteristic
240C2×C8C22×D10711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—1071no—
243C4⋊C4—1071no—
244C4⋊C4—1071no—
245C4⋊C4—1071no—
246C4⋊C4—1071no—
247C4⋊C4—1071no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4C22×D10711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—2141no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C42—21no—
264C42—1071no—
265C42—1071no—
266C42—2141no—
267C42—2141no—
268C42—2141no—
269C42C22×D10711yescharacteristic
270C23—21no—
271C23—2141no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—4282no—
276Q8—2142no—
277Q8#1313381712yes—
278D4—21no—
279D4—21no—
280D4—8561no—
281D4—4282no—
282D4—2141no—
283D4—2141no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—8561no—
288C2×C4—8561no—
289C2×C4—1071no—
290C2×C4—1071no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—2141no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4D4×D10711yescharacteristic
297C2×C4D4×D10711yescharacteristic
298C2×C4C23×D10711yescharacteristic
299C8—21no—
300C8—21no—
301C8—2141no—
302C8—2141no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—4281no—
307C22—4281no—
308C22—4281no—
309C22C2×D4×D10711yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—41no—
314C4—41no—
315C4—1072no—
316C4—4281no—
317C4—4281no—
318C4—4281no—
319C4—2141no—
320C4—2141no—
321C4—2141no—
322C4C2×#1313381711yescharacteristic
323C4#1681215611yescharacteristic
324C2—81no—
325C2—8561no—
326C2#2029328311yescharacteristic
327C2#135×D10711yescharacteristic
328C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#464×C107C2
socleC22×C107C22×D4
prime radical#16812105C22
hypercenter#464D107
hypocenterC107#3365
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #20293353C2semidirect
3. #20292983C2semidirect
4. #20292981C2semidirect
6. #20292928C2semidirect
7. #20292925C2semidirect
8. #20292910C2semidirect
9. #20292851C2semidirect
11. #20292751C2semidirect
12. Q16×Dic107C2semidirect
13. SD16×Dic107C2semidirect
14. C4×#13133805C2semidirect
15. #464×C107C2semidirect
165. C107#3365semidirect
175. #464D107semidirect

Maximal subgroups

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

subgroupindex
#202933592
#202933532
#202929832
#202929812
#202929722
#202929282
#202929252
#202929102
#202928512
#202928252
#202927512
Q16×Dic1072
SD16×Dic1072
C4×#131338052
#464×C1072
#3365107