Finite Group Databasebeta

← #28349027

Subgroups — #28349027

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#20293324C211yesmaximal, characteristic
2#20293319C211yesmaximal, characteristic, complemented
3#20293028C211yesmaximal, characteristic, complemented
4#20292965C211yesmaximal, characteristic, complemented
5#20292956C211yesmaximal, characteristic, complemented
6#20292949C211yesmaximal, characteristic, complemented
7#20292943C211yesmaximal, characteristic, complemented
8#20292936C211yesmaximal, characteristic, complemented
9#20292929C211yesmaximal, characteristic, complemented
10#20292870C211yesmaximal, characteristic, complemented
11SD16×Dic107C211yesmaximal, characteristic, complemented
12#20292703C211yesmaximal, characteristic, complemented
13#20292669C211yesmaximal, characteristic, complemented
14#20292668C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133801C2211yescharacteristic
17#16812104—21no—
18#16812102—21no—
19#16812100—21no—
20#16812100C2211yescharacteristic
21#16812100C2211yescharacteristic
22#16812093—21no—
23#16812090—21no—
24#16812085C2211yescharacteristic
25#16812080—21no—
26#16812080C2211yescharacteristic
27Q8×Dic107C2211yescharacteristic
28D4×Dic107C2211yescharacteristic
29C4×C22⋊D107C2211yescharacteristic
30(C22⋊D107)⋊C4—21no—
31#16812068C2211yescharacteristic
32#16812067C2211yescharacteristic
33#16812065—21no—
34C2×C4⋊Dic107C2211yescharacteristic
35C4⋊C4×D107C2211yescharacteristic
36#16812043C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812041C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812040C2211yescharacteristic
43#16812040C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×C22⋊D107—21no—
60C2×C22⋊D107D411yescharacteristic
61C22×Dic107—21no—
62C22×Dic107—21no—
63C2×C4×D107—21no—
64C2×C4×D107C2311yescharacteristic
65#13133801—22no—
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C4⋊Dic107—21no—
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4D411yescharacteristic
81C22⋊Dic107—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107D411yescharacteristic
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4D411yescharacteristic
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107C2311yescharacteristic
92C2×Q8×C107C2311yescharacteristic
93C2×D4×C107—21no—
94C2×D4×C107C2311yescharacteristic
95C22×C4×C107C2311yescharacteristic
96SD16×C107—22no—
97M16×C107—22no—
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C4⋊C4×C107—21no—
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C23×C107—21no—
106C23×C107C2×D411yescharacteristic
107C22×D107C2×D411yescharacteristic
108Q8×C107—21no—
109Q8×C107—21no—
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—81no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2411yescharacteristic
120C22⋊D107—41no—
121C22⋊D107—42no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—42no—
127C2×Dic107—81no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C4×D107—21no—
132C4×D107—21no—
133C4×D107—81no—
134C8×C107—21no—
135C8×C107#4112yes—
136C107⋊C8—21no—
137C107⋊C8—21no—
138C22×C107—21no—
139C22×C107—21no—
140C22×C107—21no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107C22×D411yescharacteristic
145C2×D107—21no—
146C2×D107—21no—
147C2×D107—21no—
148C4×C107—21no—
149C4×C107—41no—
150C4×C107—41no—
151C4×C107—41no—
152C4×C107C2×#4111yescharacteristic
153C4×C1072−1+411yescharacteristic
154Dic107—21no—
155Dic107—21no—
156Dic107—21no—
157Dic107—41no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161C2×C107—41no—
162C2×C107—81no—
163C2×C107#54711yescharacteristic
164C2×C107#57511yescharacteristic
165C2×C107#57711yescharacteristic
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#478—1071no—
178#468—1071no—
179#468D10711yescharacteristic, complemented
180C4×SD16—1071no—
181#436—1071no—
182#428—1071no—
183#428—1071no—
184C2×SD16C2×D10711yescharacteristic
185C2×M16—1071no—
186C2×M16C2×D10711yescharacteristic
187#126—2141no—
188#126—2141no—
189#125—1071no—
190#125—1071no—
191#125—2141no—
192#123—2141no—
193#123—2141no—
194C22⋊Q8—1071no—
195C22⋊Q8C2×D10711yescharacteristic
196#121—1071no—
197#121—2141no—
198#121C2×D10711yescharacteristic
199C4×Q8—1071no—
200C4×D4—1071no—
201C4×D4—1071no—
202C4×D4—2141no—
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#106—1071no—
217C4⋊C8—1071no—
218#103—1071no—
219#103—1071no—
220#103C2×D10711yescharacteristic
221#102—1071no—
222#102—1071no—
223#102C2×D10711yescharacteristic
224C22⋊C8—1071no—
225C22⋊C8—1071no—
226C4×C8—1071no—
227C2×Q8C22×D10711yescharacteristic
228C2×D4—21no—
229C2×D4—1071no—
230C2×D4—2141no—
231C2×D4C22×D10711yescharacteristic
232C22×C4—1071no—
233C22×C4—2141no—
234C22×C4—2141no—
235C22×C4—2141no—
236C22×C4C22×D10711yescharacteristic
237SD16—22no—
238M16—22no—
239M16—2142no—
240C2×C8—1071no—
241C2×C8—1071no—
242C2×C8C22×D10711yescharacteristic
243C2×C8C22×D10711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—1071no—
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⋊C4—2141no—
258C4⋊C4C22×D10711yescharacteristic
259C4⋊C4C22×D10711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—1071no—
263C22⋊C4—1071no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C42—1071no—
270C42—2141no—
271C42—2141no—
272C42—2141no—
273C23—21no—
274C23—1071no—
275C23D4×D10711yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—21no—
279D4—21no—
280D4—81no—
281D4—4282no—
282D4—4281no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—8561no—
289C2×C4—8561no—
290C2×C4—1071no—
291C2×C4—1071no—
292C2×C4—1071no—
293C2×C4—4282no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4—2141no—
298C2×C4—2141no—
299C2×C4—2141no—
300C2×C4D4×D10711yescharacteristic
301C2×C4C23×D10711yescharacteristic
302C8—21no—
303C8—2141no—
304C8—2141no—
305C8#1313381712yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—2141no—
313C22—2141no—
314C22—2141no—
315C22C2×D4×D10711yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—4281no—
321C4—4281no—
322C4—4281no—
323C4—4281no—
324C4—2141no—
325C4—2141no—
326C4—2141no—
327C4C2×#1313381711yescharacteristic
328C4#1681215611yescharacteristic
329C2—41no—
330C2—81no—
331C2—4281no—
332C2#2029327711yescharacteristic
333C2#2029342611yescharacteristic
334C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
prime radical#20293324C2
hypercenter#468D107
hypocenterC107#3351
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #20293319C2semidirect
3. #20293028C2semidirect
4. #20292965C2semidirect
5. #20292956C2semidirect
6. #20292949C2semidirect
7. #20292943C2semidirect
8. #20292936C2semidirect
9. #20292929C2semidirect
10. #20292870C2semidirect
11. SD16×Dic107C2semidirect
12. #20292703C2semidirect
13. #20292669C2semidirect
14. #20292668C2semidirect
15. #468×C107C2semidirect
168. C107#3351semidirect
179. #468D107semidirect

Maximal subgroups

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

subgroupindex
#202933242
#202933192
#202930282
#202929652
#202929562
#202929492
#202929432
#202929362
#202929292
#202928702
SD16×Dic1072
#202927032
#202926692
#202926682
#468×C1072
#3351107