Finite Group Databasebeta

← #28346680

Subgroups — #28346680

Order 13696.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293240C211yesmaximal, characteristic, complemented
2#20293106C211yesmaximal, characteristic
3#20292964C211yesmaximal, characteristic, complemented
4#20292959C211yesmaximal, characteristic, complemented
5#20292931C211yesmaximal, characteristic, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292832C211yesmaximal, characteristic, complemented
8#20292790C211yesmaximal, characteristic
9#20292790C211yesmaximal, characteristic
10#20292733C211yesmaximal, characteristic, complemented
11#20292724C211yesmaximal, characteristic, complemented
12#20292721C211yesmaximal, characteristic, complemented
13#20292718C211yesmaximal, characteristic, complemented
14#20292687C211yesmaximal, characteristic, complemented
15#478×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×Dic428C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×#13133806C2211yescharacteristic
21#16812109C2211yescharacteristic
22#16812104—21no—
23#16812104—21no—
24#16812100—21no—
25#16812100—21no—
26#16812089C2211yescharacteristic
27#16812089C2211yescharacteristic
28C4×Dic214C2211yescharacteristic
29#16812074C2211yescharacteristic
30#16812074C2211yescharacteristic
31#16812074C2211yescharacteristic
32D4×Dic107C2211yescharacteristic
33C4×C22⋊D107—21no—
34(C22⋊D107)⋊C4—21no—
35#16812066C2211yescharacteristic
36#16812065—21no—
37C2×Dic107⋊C4—21no—
38C4⋊C4×D107C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812040C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812032C2211yescharacteristic
48(C2×D107)⋊C8C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52#125×C107C2211yescharacteristic
53C4×D4×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—22no—
60#13133817—22no—
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C2×Dic214C2311yescharacteristic
64C2×C22⋊D107—21no—
65C22×Dic107—21no—
66C22×Dic107—21no—
67C2×C4×D107—21no—
68C2×C4×D107C2311yescharacteristic
69Dic428—22no—
70Dic428—22no—
71#13133806—22no—
72#13133806—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4D411yescharacteristic
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4D411yescharacteristic
91(C2×D107)⋊C4D411yescharacteristic
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107C2×D411yescharacteristic
110D4×C107—21no—
111D4×C107#4112yes—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—81no—
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C22⋊D107—24no—
120C2×Dic107—24no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—81no—
126C2×Dic107C2×D411yescharacteristic
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C4×D107—21no—
130C4×D107—21no—
131C4×D107—81no—
132Dic214—21no—
133Dic214—21no—
134Dic214—21no—
135Dic214—21no—
136Dic214—21no—
137Dic214#4112yes—
138C8×C107—21no—
139C8×C107—21no—
140C107⋊C8—21no—
141C107⋊C8—21no—
142C22×C107—22no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107C22×D411yescharacteristic
146C2×D107—21no—
147C2×D107—21no—
148C2×D107—21no—
149C4×C107—21no—
150C4×C107—21no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107—41no—
154C4×C107C2×#4111yescharacteristic
155C4×C107C2×#4111yescharacteristic
156Dic107—22no—
157Dic107—21no—
158Dic107—21no—
159Dic107—21no—
160Dic107—41no—
161Dic107—41no—
162Dic107—41no—
163C2×C107—42no—
164C2×C107#52511yescharacteristic
165C2×C107#57711yescharacteristic
166C2×C107#57811yescharacteristic
167D107—41no—
168#3120—1071nomaximal, Sylow
169C107#312011yescharacteristic, complemented, Sylow
170#541—1071no—
171#520—1071no—
172#484—1071no—
173#483—1071no—
174#479—1071no—
175#478D10711yescharacteristic, complemented
176#462—1071no—
177#461—1071no—
178#452—1071no—
179#452—1071no—
180#441—1071no—
181#440—1071no—
182C4×Q16—1071no—
183C4×SD16—1071no—
184#433—1071no—
185C2×#41—1071no—
186C2×Q16—1071no—
187C2×Q16—1071no—
188C2×SD16—1071no—
189C2×SD16—1071no—
190C4⋊Q8—1071no—
191#126—2141no—
192#126—2141no—
193#125—2141no—
194#125—2141no—
195#125C2×D10711yescharacteristic
196C22⋊Q8—1071no—
197C22⋊Q8—1071no—
198C4×Q8—1071no—
199C4×Q8—1071no—
200C4×Q8—1071no—
201C4×Q8—1071no—
202C4×D4—1071no—
203C4×D4—2141no—
204C4×D4—2141no—
205C4×D4C2×D10711yescharacteristic
206#117—1071no—
207#117—2141no—
208C2×C4⋊C4—1071no—
209C2×C4⋊C4—2141no—
210#107—1071no—
211#107C2×D10711yescharacteristic
212#106—1071no—
213#106C2×D10711yescharacteristic
214C4⋊C8—1071no—
215C4⋊C8C2×D10711yescharacteristic
216#103—1071no—
217#103—1071no—
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#103—1071no—
222#102C2×D10711yescharacteristic
223#102C2×D10711yescharacteristic
224C22⋊C8—1071no—
225C22⋊C8—1071no—
226#97—1071no—
227C4×C8—1071no—
228#41—2142no—
229#41—2142no—
230C2×Q8—1071no—
231C2×Q8—1071no—
232C2×Q8—1071no—
233C2×D4—2141no—
234C2×D4C22×D10711yescharacteristic
235C22×C4—21no—
236C22×C4—1071no—
237C22×C4—2141no—
238C22×C4—2141no—
239C22×C4—2141no—
240Q16—2142no—
241Q16—2142no—
242SD16—2142no—
243SD16—2142no—
244C2×C8—1071no—
245C2×C8—1071no—
246C2×C8C22×D10711yescharacteristic
247C2×C8C22×D10711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—1071no—
251C4⋊C4—1071no—
252C4⋊C4—1071no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4C22×D10711yescharacteristic
263C4⋊C4C22×D10711yescharacteristic
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—1071no—
267C22⋊C4—1071no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C42—1071no—
272C42—2141no—
273C42—2141no—
274C42—2141no—
275C42—2141no—
276C42C22×D10711yescharacteristic
277C23—21no—
278C23—1071no—
279Q8—1072no—
280Q8—2141no—
281Q8—2141no—
282Q8—2141no—
283Q8—2141no—
284Q8—2141no—
285D4—21no—
286D4—2144no—
287D4#1313381912yes—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—8561no—
293C2×C4—8561no—
294C2×C4—1071no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—2141no—
298C2×C4—2141no—
299C2×C4—2141no—
300C2×C4—2144no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4D4×D10711yescharacteristic
305C2×C4D4×D10711yescharacteristic
306C2×C4C23×D10711yescharacteristic
307C8—21no—
308C8—21no—
309C8—2141no—
310C8—2141no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—2141no—
315C22—2141no—
316C22—2141no—
317C22C2×D4×D10711yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—4281no—
324C4—4281no—
325C4—4281no—
326C4—2141no—
327C4—2142no—
328C4—2141no—
329C4—2141no—
330C4#41×D10711yescharacteristic
331C4C2×#1313381911yescharacteristic
332C2—42no—
333C2—4281no—
334C2#2029314011yescharacteristic
335C2#2029343711yescharacteristic
336C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133817C22
hypercenter#478D107
hypocenterC107#3120
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20293240C2semidirect
3. #20292964C2semidirect
4. #20292959C2semidirect
5. #20292931C2semidirect
6. #20292838C2semidirect
7. #20292832C2semidirect
10. #20292733C2semidirect
11. #20292724C2semidirect
12. #20292721C2semidirect
13. #20292718C2semidirect
14. #20292687C2semidirect
15. #478×C107C2semidirect
169. C107#3120semidirect
175. #478D107semidirect

Maximal subgroups

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

subgroupindex
#202932402
#202931062
#202929642
#202929592
#202929312
#202928382
#202928322
#202927902
#202927332
#202927242
#202927212
#202927182
#202926872
#478×C1072
#3120107