Finite Group Databasebeta

← #28346382

Subgroups — #28346382

Order 13696.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293134C211yesmaximal, characteristic
2C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
3#20292961C212yesmaximal, complemented
4#20292956C212yesmaximal, complemented
5#20292944C212yesmaximal, complemented
6#20292939C212yesmaximal, complemented
7#20292669C211yesmaximal, characteristic, complemented
8#20292667C211yesmaximal, characteristic, complemented
9#20292628C211yesmaximal, characteristic, complemented
10#20292590C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107—21no—
13C22×C4×D107C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15#16812090—21no—
16#16812085C2211yescharacteristic
17#16812085C2211yescharacteristic
18#16812080C2211yescharacteristic
19#16812080C2211yescharacteristic
20#16812077—21no—
21C4×C22⋊D107—21no—
22C4×C22⋊D107—21no—
23#16812068C2211yescharacteristic
24C2×C4⋊Dic107C2211yescharacteristic
25C2×Dic107⋊C4—21no—
26C4⋊C4×D107—21no—
27C4⋊C4×D107—21no—
28C4⋊C4×D107C2212yes—
29C4⋊C4×D107C2212yes—
30#16812043C2212yescomplemented
31#16812042C2212yescomplemented
32#16812041C2212yescomplemented
33#16812040C2212yescomplemented
34#16812033C2212yescomplemented
35#16812030C2212yescomplemented
36(C2×D107)⋊C8C2212yes—
37(C2×D107)⋊C8C2212yes—
38C2×#41×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yescomplemented
42#102×C107C2212yescomplemented
43C23×D107D411yescharacteristic
44C2×C22⋊D107—21no—
45C2×C22⋊D107—21no—
46C22×Dic107—21no—
47C22×Dic107D411yescharacteristic
48C2×C4×D107—22no—
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52C2×C4×D107—44no—
53C2×C4×D107D411yescharacteristic
54C2×C4×D107D411yescharacteristic
55C2×C4×D107C2311yescharacteristic
56C2×C4×D107C2311yescharacteristic
57#13133802—22no—
58C2×C107⋊C8C2312yes—
59C4⋊Dic107—21no—
60C4⋊Dic107C2312yes—
61C4⋊Dic107C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63Dic107⋊C4—22no—
64Dic107⋊C4—22no—
65Dic107⋊C4—21no—
66Dic107⋊C4—21no—
67C22⋊Dic107—21no—
68C22⋊Dic107—21no—
69C22⋊Dic107—21no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—21no—
73#41×C107—22no—
74#41×C107—22no—
75C2×Q8×C107C2311yescharacteristic
76C2×D4×C107—21no—
77C2×D4×C107C2311yescharacteristic
78C22×C4×C107—21no—
79C22×C4×C107—21no—
80C22×C4×C107C2311yescharacteristic
81M16×C107—22no—
82C2×C8×C107C2312yes—
83C4⋊C4×C107—21no—
84C4⋊C4×C107C2312yes—
85C23×C107—21no—
86C23×C107C2×D411yescharacteristic
87C22×D107—22no—
88C22×D107—22no—
89C22×D107—22no—
90C22×D107C2×D411yescharacteristic
91C22×D107C2×D411yescharacteristic
92Q8×C107—21no—
93Q8×C107—21no—
94D4×C107—21no—
95D4×C107—21no—
96D4×C107—42no—
97C2×C4×C107—22no—
98C2×C4×C107—21no—
99C2×C4×C107—42no—
100C2×C4×C107—42no—
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107#4112yes—
103C2×C4×C107#4112yes—
104C2×C4×C107C2411yescharacteristic
105C22⋊D107—42no—
106C22⋊D107—42no—
107C2×Dic107—22no—
108C2×Dic107—21no—
109C2×Dic107—21no—
110C2×Dic107—22no—
111C2×Dic107—42no—
112C2×Dic107C2×D411yescharacteristic
113C2×Dic107C2×D411yescharacteristic
114C4×D107—22no—
115C4×D107—22no—
116C4×D107—21no—
117C4×D107—21no—
118C4×D107—21no—
119C4×D107—21no—
120C4×D107—44no—
121C4×D107—44no—
122C8×C107—22no—
123C107⋊C8—22no—
124C22×C107—21no—
125C22×C107—21no—
126C22×C107—41no—
127C22×C107—41no—
128C22×C107—41no—
129C22×C107#12312yes—
130C22×C107C22×D411yescharacteristic
131C2×D107—28no—
132C2×D107—21no—
133C2×D107—21no—
134C2×D107—21no—
135C2×D107—21no—
136C2×D107AGL(1,ℤ/8)12yes—
137C2×D107#13712yes—
138C4×C107—41no—
139C4×C107—42no—
140C4×C107#12312yes—
141C4×C107C2×#4111yescharacteristic
142C4×C107C2×#4111yescharacteristic
143Dic107—21no—
144Dic107—21no—
145Dic107—42no—
146Dic107—41no—
147Dic107—41no—
148C2×C107—22no—
149C2×C107—81no—
150C2×C107C2×#12311yescharacteristic
151C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C107C2×#13711yescharacteristic
153D107—22no—
154D107—22no—
155#3092—1071nomaximal, Sylow
156C107#309211yescharacteristic, complemented, Sylow
157#525—1071no—
158C22×C4⋊C4—1071no—
159#484—1072no—
160#483—1072no—
161#481—1072no—
162#480—1072no—
163#428—1071no—
164#428—1071no—
165#417—1071no—
166#417D10711yescharacteristic, complemented
167#407—1071no—
168C2×#41C2×D10711yescharacteristic
169C23×C4—1071no—
170C23×C4—2141no—
171C2×M16—1071no—
172C2×M16C2×D10711yescharacteristic
173#123—2141no—
174C22⋊Q8—1071no—
175C22⋊Q8—1071no—
176#121—1071no—
177#121—1071no—
178#120—2141no—
179C4×D4—2141no—
180C4×D4—2141no—
181#117—1071no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4—1072no—
184C2×C4⋊C4—1072no—
185C2×C4⋊C4—2141no—
186C2×C4⋊C4—2141no—
187C2×C4⋊C4—2141no—
188C2×C4⋊C4C2×D10711yescharacteristic
189#107—1072no—
190#107—1072no—
191#106—1072no—
192#106—1072no—
193#103—1072no—
194#103C2×D10712yescomplemented
195#102—1072no—
196#102C2×D10712yescomplemented
197C22⋊C8—1072no—
198C22⋊C8—1072no—
199C24—1071no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D10711yescharacteristic
203C2×D4—21no—
204C2×D4—2141no—
205C2×D4—2141no—
206C2×D4C22×D10711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—1071no—
210C22×C4—1071no—
211C22×C4—1071no—
212C22×C4—1071no—
213C22×C4—1071no—
214C22×C4—4284no—
215C22×C4—2142no—
216C22×C4—2142no—
217C22×C4—2141no—
218C22×C4—2142no—
219C22×C4—2142no—
220C22×C4C22×D10711yescharacteristic
221M16—22no—
222M16—2142no—
223C2×C8—1072no—
224C2×C8C22×D10712yes—
225C4⋊C4—21no—
226C4⋊C4—1072no—
227C4⋊C4—1071no—
228C4⋊C4—1071no—
229C4⋊C4—2142no—
230C4⋊C4—2142no—
231C4⋊C4—2141no—
232C4⋊C4—2141no—
233C4⋊C4—2141no—
234C4⋊C4C22×D10712yes—
235C22⋊C4—2141no—
236C22⋊C4—2141no—
237C22⋊C4—2141no—
238C22⋊C4—2141no—
239C22⋊C4—2141no—
240C42—2141no—
241C23—21no—
242C23—1071no—
243C23—1071no—
244C23—2142no—
245C23—2142no—
246C23—2142no—
247C23D4×D10711yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—4282no—
254D4—4282no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1071no—
260C2×C4—1071no—
261C2×C4—4284no—
262C2×C4—4284no—
263C2×C4—4282no—
264C2×C4—2141no—
265C2×C4—2141no—
266C2×C4—2142no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4—2141no—
274C2×C4D4×D10711yescharacteristic
275C2×C4#1313381712yes—
276C2×C4#1313381812yes—
277C2×C4C23×D10711yescharacteristic
278C8—22no—
279C8—2142no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—1072no—
286C22—1072no—
287C22—2148no—
288C22—2141no—
289C22—2141no—
290C22—2141no—
291C22—2141no—
292C22#1681209112yes—
293C22C2×D4×D10711yescharacteristic
294C4—41no—
295C4—42no—
296C4—4282no—
297C4—4281no—
298C4—4281no—
299C4—2141no—
300C4—2141no—
301C4#1681209112yes—
302C4C2×#1313381711yescharacteristic
303C4C2×#1313381811yescharacteristic
304C2—22no—
305C2—81no—
306C2—2142no—
307C2—2142no—
308C2C2×#1681209111yescharacteristic
309C2AGL(1,ℤ/8)×D10711yescharacteristic
310C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radical#20293134C2
hypercenter#417D107
hypocenterC107#3092
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D107C2semidirect
3. #20292961C2semidirect
4. #20292956C2semidirect
5. #20292944C2semidirect
6. #20292939C2semidirect
7. #20292669C2semidirect
8. #20292667C2semidirect
9. #20292628C2semidirect
10. #20292590C2semidirect
11. #417×C107C2semidirect
30. #16812043C22semidirect
31. #16812042C22semidirect
32. #16812041C22semidirect
33. #16812040C22semidirect
34. #16812033C22semidirect
35. #16812030C22semidirect
41. #103×C107C22semidirect
42. #102×C107C22semidirect
156. C107#3092semidirect
166. #417D107semidirect
194. #103C2×D107semidirect
196. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931342
C2×C4⋊C4×D1072
#202929612
#202929562
#202929442
#202929392
#202926692
#202926672
#202926282
#202925902
#417×C1072
#3092107