Finite Group Databasebeta

← #28349030

Subgroups — #28349030

Order 13696.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293317C211yesmaximal, characteristic
2#20293026C211yesmaximal, characteristic, complemented
3#20292968C211yesmaximal, characteristic, complemented
4#20292965C211yesmaximal, characteristic, complemented
5#20292962C211yesmaximal, characteristic, complemented
6#20292960C211yesmaximal, characteristic, complemented
7#20292934C211yesmaximal, characteristic, complemented
8#20292931C211yesmaximal, characteristic, complemented
9#20292883C211yesmaximal, characteristic, complemented
10#20292883C211yesmaximal, characteristic, complemented
11C4×#13133808C211yesmaximal, characteristic, complemented
12#20292705C211yesmaximal, characteristic
13#20292670C211yesmaximal, characteristic, complemented
14#20292667C211yesmaximal, characteristic, complemented
15#552×C107C211yesmaximal, characteristic
16C2×#13133808C2211yescharacteristic
17C2×#13133802C2211yescharacteristic
18C2×#13133802C2211yescharacteristic
19#16812102—21no—
20#16812101C2211yescharacteristic
21#16812099—21no—
22#16812090—21no—
23#16812088C2211yescharacteristic
24#16812088C2211yescharacteristic
25C4×Dic214C2211yescharacteristic
26#16812068C2211yescharacteristic
27#16812065—21no—
28C2×C4⋊Dic107C2211yescharacteristic
29#16812042C2211yescharacteristic
30#16812042C2211yescharacteristic
31#16812042C2211yescharacteristic
32#16812042C2211yescharacteristic
33#16812042C2211yescharacteristic
34#16812040C2211yescharacteristic
35#16812040C2211yescharacteristic
36#16812040C2211yescharacteristic
37#16812040C2211yescharacteristic
38C4⋊(C107⋊C8)C2211yescharacteristic
39#16812033C2211yescharacteristic
40#16812033C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812032C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812032C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48#126×C107—21no—
49#125×C107—21no—
50#125×C107C2211yescharacteristic
51#123×C107—21no—
52C22⋊Q8×C107C2211yescharacteristic
53C22⋊Q8×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56#117×C107—21no—
57#117×C107C2211yescharacteristic
58C2×C4⋊C4×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C22×Dic107—21no—
61#13133808—22no—
62#13133802—22no—
63#13133802—22no—
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C4⋊Dic107—21no—
69C4⋊Dic107C2311yescharacteristic
70C4⋊Dic107C2311yescharacteristic
71C4⋊Dic107C2311yescharacteristic
72Dic107⋊C4—21no—
73Dic107⋊C4—21no—
74Dic107⋊C4—21no—
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77C22⋊Dic107—21no—
78C22⋊Dic107—21no—
79C22⋊Dic107—21no—
80C4×Dic107—21no—
81C4×Dic107—21no—
82C2×Q8×C107C2311yescharacteristic
83C2×D4×C107D411yescharacteristic
84C22×C4×C107—21no—
85C22×C4×C107C2311yescharacteristic
86C22×C4×C107C2311yescharacteristic
87C4⋊C4×C107—21no—
88C4⋊C4×C107—21no—
89C4⋊C4×C107—21no—
90C4⋊C4×C107—21no—
91C4⋊C4×C107—21no—
92C4⋊C4×C107—21no—
93C4⋊C4×C107D411yescharacteristic
94C4⋊C4×C107C2311yescharacteristic
95C4⋊C4×C107C2311yescharacteristic
96C4⋊C4×C107C2311yescharacteristic
97C22⋊C4×C107—21no—
98C22⋊C4×C107—21no—
99C22⋊C4×C107—21no—
100C22⋊C4×C107D411yescharacteristic
101C22⋊C4×C107D411yescharacteristic
102C42×C107—21no—
103C42×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107C2×D411yescharacteristic
106C23×C107C2×D411yescharacteristic
107Q8×C107—21no—
108Q8×C107—21no—
109D4×C107—41no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—42no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—42no—
129Dic214—21no—
130Dic214—21no—
131C107⋊C8—21no—
132C107⋊C8—21no—
133C107⋊C8—21no—
134C107⋊C8#4112yes—
135C22×C107—21no—
136C22×C107—21no—
137C22×C107—21no—
138C22×C107—21no—
139C22×C107—21no—
140C22×C107—21no—
141C22×C107C22×D411yescharacteristic
142C4×C107—21no—
143C4×C107—21no—
144C4×C107—21no—
145C4×C107—21no—
146C4×C107—41no—
147C4×C107—41no—
148C4×C107—41no—
149C4×C107—41no—
150C4×C107C2×#4111yescharacteristic
151C4×C1072−1+411yescharacteristic
152Dic107—41no—
153Dic107—41no—
154Dic107—41no—
155Dic107—41no—
156C2×C107—41no—
157C2×C107—41no—
158C2×C107#54711yescharacteristic
159C2×C107#57511yescharacteristic
160C2×C107#57811yescharacteristic
161#3352—1071nomaximal, Sylow
162C107#335211yescharacteristic, complemented, Sylow
163#552—1071no—
164#552D10711yescharacteristic
165#499—1071no—
166#485—1071no—
167#485—1071no—
168#484—1071no—
169#484—1071no—
170#479—1071no—
171#479—1071no—
172#470—1071no—
173#470—1071no—
174C4×Q16—1071no—
175#436—1071no—
176#428—1071no—
177#428—1071no—
178C2×Q16—1071no—
179C2×M16—1071no—
180C2×M16—1071no—
181#126—21no—
182#126—2141no—
183#125—21no—
184#125—1071no—
185#125—2141no—
186#125C2×D10711yescharacteristic
187#123—21no—
188#123—2141no—
189C22⋊Q8—1071no—
190C22⋊Q8—1071no—
191C22⋊Q8C2×D10711yescharacteristic
192C22⋊Q8C2×D10711yescharacteristic
193C4×Q8—1071no—
194C4×Q8C2×D10711yescharacteristic
195C4×D4C2×D10711yescharacteristic
196#117—21no—
197#117—1071no—
198#117—2141no—
199#117C2×D10711yescharacteristic
200C2×C4⋊C4—1071no—
201C2×C4⋊C4C2×D10711yescharacteristic
202#107—1071no—
203#107—1071no—
204#107—1071no—
205#107—1071no—
206#107—1071no—
207#106—1071no—
208#106—1071no—
209#106—1071no—
210#106—1071no—
211C4⋊C8—1071no—
212#103—1071no—
213#103—1071no—
214#103—1071no—
215#103—1071no—
216#103—1071no—
217#103—1071no—
218C22⋊C8—1071no—
219C22⋊C8—1071no—
220C4×C8—1071no—
221C2×Q8—1071no—
222C2×Q8C22×D10711yescharacteristic
223C2×D4C22⋊D10711yescharacteristic
224C22×C4—21no—
225C22×C4—2141no—
226C22×C4C22×D10711yescharacteristic
227C22×C4C22×D10711yescharacteristic
228Q16—2142no—
229M16—2142no—
230M16—2142no—
231C2×C8—1071no—
232C2×C8—1071no—
233C2×C8—1071no—
234C2×C8—1071no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—1071no—
242C4⋊C4—1071no—
243C4⋊C4—1071no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4C22⋊D10711yescharacteristic
251C4⋊C4C22×D10711yescharacteristic
252C4⋊C4C22×D10711yescharacteristic
253C4⋊C4C22×D10711yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—2141no—
258C22⋊C4—2141no—
259C22⋊C4—2141no—
260C22⋊C4C22⋊D10711yescharacteristic
261C22⋊C4C22⋊D10711yescharacteristic
262C42—21no—
263C42—21no—
264C42—2141no—
265C42—2141no—
266C42C22×D10711yescharacteristic
267C23C2×C22⋊D10711yescharacteristic
268C23C2×C22⋊D10711yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—2141no—
272Q8—2141no—
273D4—41no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—4282no—
284C2×C4—2141no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4C2×C22⋊D10711yescharacteristic
289C2×C4C2×C22⋊D10711yescharacteristic
290C2×C4C2×C22⋊D10711yescharacteristic
291C2×C4C2×C22⋊D10711yescharacteristic
292C2×C4C23×D10711yescharacteristic
293C8—1072no—
294C8—2141no—
295C8—2141no—
296C8—2141no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22C22×C22⋊D10711yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—4281no—
313C4—4281no—
314C4—4281no—
315C4—4281no—
316C4#41×D10711yescharacteristic
317C4#1681215611yescharacteristic
318C2—41no—
319C2—41no—
320C2#2029327611yescharacteristic
321C2#2029342511yescharacteristic
322C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#552×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×C107D4
hypercenter#552D107
hypocenterC107#3352
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20293026C2semidirect
3. #20292968C2semidirect
4. #20292965C2semidirect
5. #20292962C2semidirect
6. #20292960C2semidirect
7. #20292934C2semidirect
8. #20292931C2semidirect
9. #20292883C2semidirect
10. #20292883C2semidirect
11. C4×#13133808C2semidirect
13. #20292670C2semidirect
14. #20292667C2semidirect
162. C107#3352semidirect

Maximal subgroups

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

subgroupindex
#202933172
#202930262
#202929682
#202929652
#202929622
#202929602
#202929342
#202929312
#202928832
C4×#131338082
#202927052
#202926702
#202926672
#552×C1072
#3352107