Finite Group Databasebeta

← #28346324

Subgroups — #28346324

Order 13696.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293257C211yesmaximal, characteristic, complemented
2#20293106C211yesmaximal, characteristic
3#20292933C211yesmaximal, characteristic, complemented
4#20292931C211yesmaximal, characteristic, complemented
5#20292929C211yesmaximal, characteristic, complemented
6#20292917C211yesmaximal, characteristic, complemented
7#20292915C211yesmaximal, characteristic, complemented
8#20292913C211yesmaximal, characteristic, complemented
9#20292911C211yesmaximal, characteristic, complemented
10#20292668C211yesmaximal, characteristic, complemented
11#106×D107C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292625C211yesmaximal, characteristic
14#20292620C211yesmaximal, characteristic
15#478×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×#13133801C2211yescharacteristic
18C2×C8×D107C2211yescharacteristic
19#16812109C2211yescharacteristic
20#16812108—21no—
21#16812108C2211yescharacteristic
22#16812100C2211yescharacteristic
23#16812099—21no—
24#16812089—21no—
25#16812087—21no—
26C4×Dic214C2211yescharacteristic
27#16812074C2211yescharacteristic
28D4×Dic107C2211yescharacteristic
29C4×C22⋊D107—21no—
30(C22⋊D107)⋊C4—21no—
31#16812067C2211yescharacteristic
32#16812066C2211yescharacteristic
33#16812065—21no—
34C2×Dic107⋊C4—21no—
35C4⋊C4×D107C2211yescharacteristic
36C4⋊C4×D107C2211yescharacteristic
37#16812042C2211yescharacteristic
38#16812041C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812040C2211yescharacteristic
43C4⋊(C107⋊C8)C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812032C2211yescharacteristic
50#16812030C2211yescharacteristic
51#16812030C2211yescharacteristic
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#13133817Q814yescomplemented
61C2×Dic214—21no—
62C2×Dic214—21no—
63C2×Dic214C2311yescharacteristic
64C2×C22⋊D107—21no—
65C22×Dic107—21no—
66C22×Dic107—21no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107C2311yescharacteristic
70#13133801—22no—
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C8×D107—22no—
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
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)⋊C4D411yescharacteristic
90(C2×D107)⋊C4D411yescharacteristic
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C22×C4×C107—21no—
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107—21no—
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C42×C107C2311yescharacteristic
106C23×C107—21no—
107C22×D107C2×D411yescharacteristic
108D4×C107—21no—
109D4×C107C2×Q812yes—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—81no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C22⋊D107—24no—
118C2×Dic107—21no—
119C2×Dic107—24no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—81no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C4×D107—81no—
128C4×D107—81no—
129C4×D107C2×Q812yes—
130C4×D107#4112yes—
131Dic214—21no—
132Dic214—81no—
133Dic214—81no—
134Dic214C2×Q812yes—
135C8×C107—21no—
136C8×C107—21no—
137C107⋊C8—21no—
138C107⋊C8—21no—
139C22×C107—22no—
140C22×C107—41no—
141C22×C107—41no—
142C22×C107C22×D411yescharacteristic
143C2×D107—21no—
144C2×D107—21no—
145C2×D107C22⋊Q812yes—
146C4×C107—21no—
147C4×C107—21no—
148C4×C107—41no—
149C4×C107—41no—
150C4×C107—41no—
151C4×C107C22×Q811yescharacteristic
152C4×C107C2×#4111yescharacteristic
153Dic107—22no—
154Dic107—21no—
155Dic107—21no—
156Dic107—41no—
157Dic107—41no—
158Dic107—41no—
159Dic107C22⋊Q812yes—
160C2×C107—42no—
161C2×C107C2×C22⋊Q811yescharacteristic
162C2×C107#57711yescharacteristic
163C2×C107#57711yescharacteristic
164D107—22no—
165#3084—1071nomaximal, Sylow
166C107#308411yescharacteristic, complemented, Sylow
167#544—1071no—
168#520—1071no—
169#479—1071no—
170#479—1071no—
171#478—1071no—
172#478D10711yescharacteristic, complemented
173#476—1071no—
174#476—1071no—
175#475—1071no—
176#475—1071no—
177#428—1071no—
178C2×#106—1071no—
179#424—1071no—
180#417—1071no—
181#416—1071no—
182C2×#41—1071no—
183C2×M16—1071no—
184C22×C8—1071no—
185C4⋊Q8—1071no—
186C4⋊Q8—1071no—
187C4⋊Q8—2141no—
188#125—1071no—
189#125—2141no—
190#125C2×D10711yescharacteristic
191C22⋊Q8—2141no—
192C22⋊Q8—2141no—
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×D4—1071no—
196C4×D4—2141no—
197C4×D4—2141no—
198C4×D4C2×D10711yescharacteristic
199#117—1071no—
200#117—1071no—
201#117—2141no—
202C2×C4⋊C4—1071no—
203C2×C4⋊C4—1071no—
204C2×C4⋊C4—2141no—
205#107—1071no—
206#107C2×D10711yescharacteristic
207#106—1071no—
208#106—1071no—
209#106—1071no—
210#106—1071no—
211#106—1071no—
212#106C2×D10711yescharacteristic
213C4⋊C8—1071no—
214C4⋊C8—1071no—
215C4⋊C8—1071no—
216C4⋊C8C2×D10711yescharacteristic
217#103—1071no—
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#102—1071no—
222#102—1071no—
223#102C2×D10711yescharacteristic
224#102C2×D10711yescharacteristic
225#41—1074no—
226#41—2142no—
227C2×Q8—1071no—
228C2×Q8—2141no—
229C2×Q8—2141no—
230C2×D4—2141no—
231C2×D4C22×D10711yescharacteristic
232C22×C4—21no—
233C22×C4—1071no—
234C22×C4—2141no—
235C22×C4—2141no—
236C22×C4—2141no—
237C22×C4—2141no—
238M16—2142no—
239C2×C8—1071no—
240C2×C8—1071no—
241C2×C8—2142no—
242C2×C8C22×D10711yescharacteristic
243C2×C8C22×D10711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—1071no—
247C4⋊C4—1071no—
248C4⋊C4—1071no—
249C4⋊C4—1071no—
250C4⋊C4—1071no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C4⋊C4C22×D10711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—1071no—
264C22⋊C4—1071no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C42—1071no—
268C42—2141no—
269C42—2141no—
270C42—2141no—
271C42C22×D10711yescharacteristic
272C23—21no—
273C23—1071no—
274Q8—8561no—
275Q8—8561no—
276Q8—1072no—
277Q8—2141no—
278D4—21no—
279D4—2144no—
280D4Q8×D10712yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—8561no—
286C2×C4—8561no—
287C2×C4—8561no—
288C2×C4—1071no—
289C2×C4—1072no—
290C2×C4—1072no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—2144no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4—2141no—
298C2×C4D4×D10711yescharacteristic
299C2×C4D4×D10711yescharacteristic
300C2×C4C23×D10711yescharacteristic
301C8—21no—
302C8—21no—
303C8—2141no—
304C8—2141no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—1072no—
309C22—2141no—
310C22—2141no—
311C22C2×D4×D10711yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—1072no—
318C4—4281no—
319C4—4281no—
320C4—4281no—
321C4—2142no—
322C4—2141no—
323C4—2141no—
324C4C2×Q8×D10711yescharacteristic
325C4#41×D10711yescharacteristic
326C2—42no—
327C2—2142no—
328C2C22⋊Q8×D10711yescharacteristic
329C2#2029343411yescharacteristic
330C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133817C22
hypercenter#478D107
hypocenterC107#3084
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. #20293257C2semidirect
3. #20292933C2semidirect
4. #20292931C2semidirect
5. #20292929C2semidirect
6. #20292917C2semidirect
7. #20292915C2semidirect
8. #20292913C2semidirect
9. #20292911C2semidirect
10. #20292668C2semidirect
11. #106×D107C2semidirect
12. #20292652C2semidirect
15. #478×C107C2semidirect
60. #13133817Q8semidirect
166. C107#3084semidirect
172. #478D107semidirect

Maximal subgroups

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

subgroupindex
#202932572
#202931062
#202929332
#202929312
#202929292
#202929172
#202929152
#202929132
#202929112
#202926682
#106×D1072
#202926522
#202926252
#202926202
#478×C1072
#3084107