Finite Group Databasebeta

← #28349031

Subgroups — #28349031

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#20293319C211yesmaximal, characteristic
2#20293311C211yesmaximal, characteristic
3#20293027C211yesmaximal, characteristic, complemented
4#20292966C211yesmaximal, characteristic, complemented
5#20292965C211yesmaximal, characteristic, complemented
6#20292961C211yesmaximal, characteristic, complemented
7#20292960C211yesmaximal, characteristic, complemented
8#20292936C211yesmaximal, characteristic, complemented
9#20292932C211yesmaximal, characteristic, complemented
10#20292882C211yesmaximal, characteristic, complemented
11Q16×Dic107C211yesmaximal, characteristic, complemented
12#20292703C211yesmaximal, characteristic
13#20292669C211yesmaximal, characteristic, complemented
14#20292666C211yesmaximal, characteristic, complemented
15#470×C107C211yesmaximal, characteristic, complemented
16C2×#13133801C2211yescharacteristic
17#16812104—21no—
18#16812102—21no—
19#16812100—21no—
20#16812100C2211yescharacteristic
21#16812100C2211yescharacteristic
22#16812099—21no—
23#16812093—21no—
24#16812091—21no—
25#16812085C2211yescharacteristic
26#16812085C2211yescharacteristic
27Q8×Dic107C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic
29C4×C22⋊D107C2211yescharacteristic
30#16812068C2211yescharacteristic
31#16812067—21no—
32#16812067C2211yescharacteristic
33#16812065—21no—
34C2×C4⋊Dic107C2211yescharacteristic
35C4⋊C4×D107C2211yescharacteristic
36#16812043C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812041C2211yescharacteristic
43#16812040C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812033C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×C22⋊D107D411yescharacteristic
60C22×Dic107—21no—
61C2×C4×D107—21no—
62C2×C4×D107C2311yescharacteristic
63#13133801—22no—
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C4⋊Dic107—21no—
67C4⋊Dic107—21no—
68C4⋊Dic107—21no—
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107C2311yescharacteristic
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4D411yescharacteristic
81C22⋊Dic107—21no—
82C22⋊Dic107D411yescharacteristic
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4D411yescharacteristic
87C4×Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107C2311yescharacteristic
92C2×Q8×C107C2311yescharacteristic
93C2×Q8×C107C2311yescharacteristic
94C22×C4×C107C2311yescharacteristic
95Q16×C107—22no—
96M16×C107—22no—
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107—21no—
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C23×C107C2×D411yescharacteristic
106C22×D107C2×D411yescharacteristic
107Q8×C107—21no—
108Q8×C107—21no—
109Q8×C107—21no—
110Q8×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×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C22⋊D107—41no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—42no—
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128C4×D107—21no—
129C4×D107—21no—
130C4×D107—42no—
131C8×C107—21no—
132C8×C107#4112yes—
133C107⋊C8—21no—
134C107⋊C8—21no—
135C22×C107—21no—
136C22×C107—21no—
137C22×C107—21no—
138C22×C107C22×D411yescharacteristic
139C2×D107—21no—
140C2×D107—21no—
141C2×D107—21no—
142C4×C107—21no—
143C4×C107—41no—
144C4×C107—41no—
145C4×C107—41no—
146C4×C107—41no—
147C4×C107C2×#4111yescharacteristic
148C4×C1072−1+411yescharacteristic
149Dic107—21no—
150Dic107—21no—
151Dic107—21no—
152Dic107—41no—
153Dic107—41no—
154Dic107—41no—
155Dic107—41no—
156C2×C107—41no—
157C2×C107#54711yescharacteristic
158C2×C107#57511yescharacteristic
159C2×C107#57811yescharacteristic
160D107—41no—
161#3352—1071nomaximal, Sylow
162C107#335211yescharacteristic, complemented, Sylow
163#552—1071no—
164#552—1071no—
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#470D10711yescharacteristic, complemented
174C4×Q16—1071no—
175#436—1071no—
176#428—1071no—
177#428—1071no—
178C2×Q16C2×D10711yescharacteristic
179C2×M16—1071no—
180C2×M16C2×D10711yescharacteristic
181#126—2141no—
182#126—2141no—
183#125—1071no—
184#125—1071no—
185#125—2141no—
186#125—2141no—
187#123—2141no—
188#123—2141no—
189C22⋊Q8—1071no—
190C22⋊Q8—1071no—
191C22⋊Q8C2×D10711yescharacteristic
192C22⋊Q8C2×D10711yescharacteristic
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×D4—1071no—
196#117—1071no—
197#117—1071no—
198#117—2141no—
199#117—2141no—
200C2×C4⋊C4—1071no—
201C2×C4⋊C4—1071no—
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#106C2×D10711yescharacteristic
211C4⋊C8—1071no—
212#103—1071no—
213#103—1071no—
214#103—1071no—
215#103—1071no—
216#103C2×D10711yescharacteristic
217#103C2×D10711yescharacteristic
218C22⋊C8—1071no—
219C22⋊C8—1071no—
220C4×C8—1071no—
221C2×Q8C22×D10711yescharacteristic
222C2×Q8C22×D10711yescharacteristic
223C2×D4—1071no—
224C22×C4—1071no—
225C22×C4—2141no—
226C22×C4—2141no—
227C22×C4C22×D10711yescharacteristic
228Q16—22no—
229M16—22no—
230M16—2142no—
231C2×C8—1071no—
232C2×C8—1071no—
233C2×C8C22×D10711yescharacteristic
234C2×C8C22×D10711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—1071no—
238C4⋊C4—1071no—
239C4⋊C4—1071no—
240C4⋊C4—1071no—
241C4⋊C4—1071no—
242C4⋊C4—2141no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4C22×D10711yescharacteristic
253C4⋊C4C22×D10711yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—1071no—
257C22⋊C4—1071no—
258C22⋊C4—2141no—
259C22⋊C4—2141no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C42—1071no—
263C42—2141no—
264C42—2141no—
265C42—2141no—
266C42—2141no—
267C23—1071no—
268C23D4×D10711yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—21no—
272Q8—21no—
273D4—4281no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—1071no—
281C2×C4—1071no—
282C2×C4—1071no—
283C2×C4—4282no—
284C2×C4—4282no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4D4×D10711yescharacteristic
292C2×C4C23×D10711yescharacteristic
293C8—21no—
294C8—2141no—
295C8—2141no—
296C8#1313381712yes—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—2141no—
301C22—2141no—
302C22—2141no—
303C22C2×D4×D10711yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—4281no—
310C4—4281no—
311C4—4281no—
312C4—4281no—
313C4—2141no—
314C4—2141no—
315C4—2141no—
316C4C2×#1313381711yescharacteristic
317C4#1681215611yescharacteristic
318C2—41no—
319C2—4281no—
320C2#2029327711yescharacteristic
321C2#2029342611yescharacteristic
322C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
prime radicalC2×C22⋊D107D4
hypercenter#470D107
hypocenterC107#3352
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. #20293027C2semidirect
4. #20292966C2semidirect
5. #20292965C2semidirect
6. #20292961C2semidirect
7. #20292960C2semidirect
8. #20292936C2semidirect
9. #20292932C2semidirect
10. #20292882C2semidirect
11. Q16×Dic107C2semidirect
13. #20292669C2semidirect
14. #20292666C2semidirect
15. #470×C107C2semidirect
162. C107#3352semidirect
173. #470D107semidirect

Maximal subgroups

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

subgroupindex
#202933192
#202933112
#202930272
#202929662
#202929652
#202929612
#202929602
#202929362
#202929322
#202928822
Q16×Dic1072
#202927032
#202926692
#202926662
#470×C1072
#3352107