Finite Group Databasebeta

← #28346697

Subgroups — #28346697

Order 13696.

370 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#20293251C211yesmaximal, characteristic
2#20293103C211yesmaximal, characteristic, complemented
3#20292971C211yesmaximal, characteristic, complemented
4#20292953C211yesmaximal, characteristic
5#20292919C211yesmaximal, characteristic, complemented
6#20292850C211yesmaximal, characteristic, complemented
7#20292849C211yesmaximal, characteristic, complemented
8#20292784C211yesmaximal, characteristic, complemented
9#20292767C211yesmaximal, characteristic
10#20292733C211yesmaximal, characteristic, complemented
11#20292730C211yesmaximal, characteristic, complemented
12#20292719C211yesmaximal, characteristic, complemented
13#20292714C211yesmaximal, characteristic, complemented
14#20292687C211yesmaximal, characteristic, complemented
15#475×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133807C2211yescharacteristic
20C2×#13133805C2211yescharacteristic
21#16812104—21no—
22#16812098—21no—
23#16812097C2211yescharacteristic
24#16812094—21no—
25#16812089C2211yescharacteristic
26#16812085—21no—
27#16812084C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic
29C4×Dic214—21no—
30C4×Dic214C2211yescharacteristic
31#16812074—21no—
32#16812074C2211yescharacteristic
33#16812074C2211yescharacteristic
34#16812071C2211yescharacteristic
35#16812067—21no—
36#16812067C2211yescharacteristic
37#16812066—21no—
38#16812066C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48(C2×D107)⋊C8C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C4⋊Q8×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54#106×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62Q8×D107—22no—
63Q8×D107—22no—
64C2×D428—21no—
65C2×D428C2311yescharacteristic
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107C2311yescharacteristic
69Dic428—22no—
70#13133808—22no—
71#13133807—22no—
72#13133805—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4D411yescharacteristic
90(C2×D107)⋊C4D411yescharacteristic
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C2×Q8×C107—21no—
98C2×Q8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C22×D107—21no—
109C22×D107C2×D411yescharacteristic
110Q8×C107—21no—
111Q8×C107—81no—
112Q8×C107#4112yes—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2411yescharacteristic
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126D428—21no—
127D428—21no—
128D428—81no—
129C4×D107—24no—
130C4×D107—21no—
131C4×D107—21no—
132C4×D107—81no—
133Dic214—24no—
134Dic214—21no—
135Dic214—21no—
136Dic214—21no—
137Dic214#4112yes—
138C8×C107—21no—
139C8×C107—21no—
140C107⋊C8—21no—
141C107⋊C8—21no—
142C22×C107C22×D411yescharacteristic
143C2×D107—21no—
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—41no—
147C2×D107—41no—
148C2×D107—41no—
149C4×C107—22no—
150C4×C107—21no—
151C4×C107—21no—
152C4×C107—41no—
153C4×C107—41no—
154C4×C107—41no—
155C4×C107C2×#4111yescharacteristic
156C4×C107C2×#4111yescharacteristic
157Dic107—21no—
158Dic107—22no—
159Dic107—21no—
160Dic107—21no—
161Dic107—41no—
162Dic107—41no—
163C2×C107#52511yescharacteristic
164C2×C107#57711yescharacteristic
165C2×C107#57811yescharacteristic
166D107—41no—
167D107—81no—
168#3121—1071nomaximal, Sylow
169C107#312111yescharacteristic, complemented, Sylow
170#542—1071no—
171#519—1071no—
172#485—1071no—
173#482—1071no—
174#477—1071no—
175#475D10711yescharacteristic, complemented
176#464—1071no—
177#464—1071no—
178#451—1071no—
179#448—1071no—
180#441—1071no—
181#440—1071no—
182C4×Q16—1071no—
183C4×SD16—1071no—
184#433—1071no—
185C22×Q8—1071no—
186C2×Q16—1071no—
187C2×Q16—1071no—
188C2×SD16—1071no—
189C2×SD16—1071no—
190C4⋊Q8C2×D10711yescharacteristic
191#126—2141no—
192#124—1071no—
193#124—2141no—
194#123—2141no—
195C22⋊Q8—1071no—
196C22⋊Q8—2141no—
197#121—1071no—
198C4×Q8—1071no—
199C4×Q8—1071no—
200C4×Q8—1071no—
201C4×Q8—1071no—
202C4×Q8—2141no—
203C4×Q8—2141no—
204C4×Q8C2×D10711yescharacteristic
205C4×D4—1071no—
206#117—1071no—
207#117—1071no—
208#117—2141no—
209#117—2141no—
210#107—1071no—
211#107—1071no—
212#106C2×D10711yescharacteristic
213#106C2×D10711yescharacteristic
214C4⋊C8—1071no—
215C4⋊C8C2×D10711yescharacteristic
216#103—1071no—
217#103—1071no—
218#103—1071no—
219#103—1071no—
220#103C2×D10711yescharacteristic
221#103C2×D10711yescharacteristic
222#102—1071no—
223#102—1071no—
224C22⋊C8—1071no—
225C22⋊C8—1071no—
226#97—1071no—
227C4×C8—1071no—
228C2×Q8—21no—
229C2×Q8—1071no—
230C2×Q8—1071no—
231C2×Q8—2141no—
232C2×Q8—2142no—
233C2×Q8—2142no—
234C2×Q8C22×D10711yescharacteristic
235C2×D4—1071no—
236C2×D4—2141no—
237C22×C4—1071no—
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⋊C4C22×D10711yescharacteristic
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—1071no—
262C22⋊C4—1071no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C42—21no—
270C42—1071no—
271C42—2141no—
272C42—2141no—
273C42—2141no—
274C42—2141no—
275C42—2141no—
276C42C22×D10711yescharacteristic
277C23—1071no—
278C23—2141no—
279Q8—21no—
280Q8—81no—
281Q8—1072no—
282Q8—2144no—
283Q8—2141no—
284Q8—2141no—
285Q8—2141no—
286Q8#1313381912yes—
287D4—8561no—
288D4—2141no—
289D4—2141no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—8561no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—2144no—
299C2×C4—2141no—
300C2×C4—2141no—
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—4281no—
312C22—4281no—
313C22—4281no—
314C22—2141no—
315C22—2141no—
316C22—2141no—
317C22C2×D4×D10711yescharacteristic
318C4—22no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4281no—
325C4—4281no—
326C4—2141no—
327C4—2142no—
328C4—2141no—
329C4—2141no—
330C4#41×D10711yescharacteristic
331C4C2×#1313381911yescharacteristic
332C2—8561no—
333C2—4281no—
334C2#2029314011yescharacteristic
335C2#2029343311yescharacteristic
336C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#475×C107C2
socleC22×C107C22×D4
prime radical#16812084C22
hypercenter#475D107
hypocenterC107#3121
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20293103C2semidirect
3. #20292971C2semidirect
5. #20292919C2semidirect
6. #20292850C2semidirect
7. #20292849C2semidirect
8. #20292784C2semidirect
10. #20292733C2semidirect
11. #20292730C2semidirect
12. #20292719C2semidirect
13. #20292714C2semidirect
14. #20292687C2semidirect
15. #475×C107C2semidirect
169. C107#3121semidirect
175. #475D107semidirect

Maximal subgroups

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

subgroupindex
#202932512
#202931032
#202929712
#202929532
#202929192
#202928502
#202928492
#202927842
#202927672
#202927332
#202927302
#202927192
#202927142
#202926872
#475×C1072
#3121107