Finite Group Databasebeta

← #28349028

Subgroups — #28349028

Order 13696.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293327C211yesmaximal, characteristic
2#20293315C211yesmaximal, characteristic, complemented
3#20293026C211yesmaximal, characteristic, complemented
4#20292970C211yesmaximal, characteristic, complemented
5#20292959C211yesmaximal, characteristic, complemented
6#20292949C211yesmaximal, characteristic, complemented
7#20292944C211yesmaximal, characteristic, complemented
8#20292935C211yesmaximal, characteristic, complemented
9#20292872C211yesmaximal, characteristic, complemented
10#20292872C211yesmaximal, characteristic, complemented
11C4×#13133806C211yesmaximal, characteristic, complemented
12#20292704C211yesmaximal, characteristic, complemented
13#20292668C211yesmaximal, characteristic, complemented
14#20292666C211yesmaximal, characteristic, complemented
15#478×C107C211yesmaximal, characteristic, complemented
16C2×#13133806C2211yescharacteristic
17C2×#13133801C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19#16812104—21no—
20#16812102—21no—
21#16812101C2211yescharacteristic
22#16812100—21no—
23#16812094—21no—
24#16812092—21no—
25#16812089C2211yescharacteristic
26#16812089C2211yescharacteristic
27#16812082—21no—
28#16812080C2211yescharacteristic
29#16812080C2211yescharacteristic
30C4×Dic214C2211yescharacteristic
31C4×D428C2211yescharacteristic
32C4×C22⋊D107—21no—
33#16812067—21no—
34#16812067C2211yescharacteristic
35#16812067C2211yescharacteristic
36C4⋊C4×D107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812040C2211yescharacteristic
43#16812040C2211yescharacteristic
44#16812040C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812030C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C4×C107⋊C8C2211yescharacteristic
52#125×C107C2211yescharacteristic
53C4×D4×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C2×D428D411yescharacteristic
61C2×C22⋊D107—21no—
62C2×C22⋊D107—21no—
63C2×C4×D107—21no—
64C2×C4×D107—21no—
65C2×C4×D107C2311yescharacteristic
66C2×C4×D107C2311yescharacteristic
67#13133806—22no—
68#13133801—22no—
69#13133801—22no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—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—
94C2×D4×C107C2311yescharacteristic
95C22×C4×C107—21no—
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C4⋊C4×C107—21no—
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C22×D107C2×D411yescharacteristic
107C22×D107C2×D411yescharacteristic
108D4×C107—21no—
109D4×C107—21no—
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—42no—
118C22⋊D107—81no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125D428—41no—
126C4×D107—21no—
127C4×D107—21no—
128C4×D107—21no—
129C4×D107—21no—
130C4×D107—42no—
131C4×D107—81no—
132Dic214—21no—
133Dic214—21no—
134C8×C107—21no—
135C8×C107—21no—
136C107⋊C8—21no—
137C107⋊C8#4112yes—
138C22×C107—41no—
139C22×C107—41no—
140C22×C107—41no—
141C22×C107C22×D411yescharacteristic
142C2×D107—21no—
143C2×D107—21no—
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—21no—
147C2×D107—21no—
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107C2×#4111yescharacteristic
154C4×C1072−1+411yescharacteristic
155Dic107—21no—
156Dic107—21no—
157Dic107—41no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161C2×C107—81no—
162C2×C107#54711yescharacteristic
163C2×C107#57511yescharacteristic
164C2×C107#57711yescharacteristic
165D107—41no—
166D107—41no—
167#3351—1071nomaximal, Sylow
168C107#335111yescharacteristic, complemented, Sylow
169#553—1071no—
170#552—1071no—
171#499—1071no—
172#485—1071no—
173#483—1071no—
174#482—1071no—
175#481—1071no—
176#479—1071no—
177#478D10711yescharacteristic, complemented
178#468—1071no—
179#468—1071no—
180C4×SD16—1071no—
181#436—1071no—
182#428—1071no—
183#428—1071no—
184C2×SD16—1071no—
185C2×M16—1071no—
186C2×M16—1071no—
187#126—2141no—
188#126—2141no—
189#125—1071no—
190#125—2141no—
191#125C2×D10711yescharacteristic
192#123—2141no—
193#123—2141no—
194C22⋊Q8—1071no—
195C22⋊Q8—1071no—
196#121—1071no—
197#121—1071no—
198#121—2141no—
199C4×Q8—1071no—
200C4×D4—1071no—
201C4×D4—2141no—
202C4×D4C2×D10711yescharacteristic
203#117—1071no—
204#117—1071no—
205#117—2141no—
206C2×C4⋊C4—1071no—
207C2×C4⋊C4—1071no—
208#107—1071no—
209#107—1071no—
210#107—1071no—
211#107C2×D10711yescharacteristic
212#106—1071no—
213#106—1071no—
214#106—1071no—
215#106—1071no—
216#106C2×D10711yescharacteristic
217C4⋊C8C2×D10711yescharacteristic
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#102—1071no—
222#102C2×D10711yescharacteristic
223#102C2×D10711yescharacteristic
224C22⋊C8—1071no—
225C22⋊C8—1071no—
226C4×C8—1071no—
227C2×Q8—1071no—
228C2×D4—1071no—
229C2×D4—2141no—
230C2×D4—2141no—
231C2×D4C22×D10711yescharacteristic
232C22×C4—21no—
233C22×C4—1071no—
234C22×C4—1071no—
235C22×C4—2141no—
236C22×C4—2141no—
237SD16—2142no—
238M16—2142no—
239M16—2142no—
240C2×C8—1071no—
241C2×C8—1071no—
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—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4C22×D10711yescharacteristic
258C4⋊C4C22×D10711yescharacteristic
259C4⋊C4C22×D10711yescharacteristic
260C22⋊C4—21no—
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—2141no—
270C42—2141no—
271C42—2141no—
272C42C22×D10711yescharacteristic
273C23—21no—
274C23—1071no—
275C23—1071no—
276Q8—2141no—
277Q8—2141no—
278D4—21no—
279D4—21no—
280D4—8561no—
281D4—4281no—
282D4—4282no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—8561no—
288C2×C4—1071no—
289C2×C4—1071no—
290C2×C4—4282no—
291C2×C4—2141no—
292C2×C4—2141no—
293C2×C4—2141no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4—2141no—
298C2×C4—2141no—
299C2×C4D4×D10711yescharacteristic
300C2×C4D4×D10711yescharacteristic
301C2×C4C23×D10711yescharacteristic
302C8—21no—
303C8—21no—
304C8—1072no—
305C8—2141no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—2141no—
310C22—2141no—
311C22—2141no—
312C22—2141no—
313C22—2141no—
314C22—2141no—
315C22C2×D4×D10711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—4281no—
322C4—4281no—
323C4—4281no—
324C4—4281no—
325C4—2141no—
326C4—2141no—
327C4#41×D10711yescharacteristic
328C4#1681215711yescharacteristic
329C2—81no—
330C2—4281no—
331C2—4281no—
332C2#2029327911yescharacteristic
333C2#2029342411yescharacteristic
334C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radical#20293327C2
hypercenter#478D107
hypocenterC107#3351
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
2. #20293315C2semidirect
3. #20293026C2semidirect
4. #20292970C2semidirect
5. #20292959C2semidirect
6. #20292949C2semidirect
7. #20292944C2semidirect
8. #20292935C2semidirect
9. #20292872C2semidirect
10. #20292872C2semidirect
11. C4×#13133806C2semidirect
12. #20292704C2semidirect
13. #20292668C2semidirect
14. #20292666C2semidirect
15. #478×C107C2semidirect
168. C107#3351semidirect
177. #478D107semidirect

Maximal subgroups

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

subgroupindex
#202933272
#202933152
#202930262
#202929702
#202929592
#202929492
#202929442
#202929352
#202928722
C4×#131338062
#202927042
#202926682
#202926662
#478×C1072
#3351107