Finite Group Databasebeta

← #28346372

Subgroups — #28346372

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293134C211yesmaximal, characteristic
2#117×D107C211yesmaximal, characteristic, complemented
3#20292967C211yesmaximal, characteristic, complemented
4#20292966C211yesmaximal, characteristic, complemented
5#20292961C211yesmaximal, characteristic, complemented
6#20292956C211yesmaximal, characteristic, complemented
7#20292950C211yesmaximal, characteristic, complemented
8#20292949C211yesmaximal, characteristic, complemented
9#20292944C211yesmaximal, characteristic, complemented
10#20292939C211yesmaximal, characteristic, complemented
11#20292665C211yesmaximal, characteristic, complemented
12#20292663C211yesmaximal, characteristic, complemented
13#20292623C211yesmaximal, characteristic, complemented
14C2×(C2×D107)⋊C8C211yesmaximal, characteristic, complemented
15#416×C107C211yesmaximal, characteristic, complemented
16C22×C4×D107C2211yescharacteristic
17C22×C107⋊C8C2211yescharacteristic
18#16812090—21no—
19#16812085C2211yescharacteristic
20#16812085C2211yescharacteristic
21#16812080C2211yescharacteristic
22#16812080C2211yescharacteristic
23#16812077—21no—
24C4×C22⋊D107—21no—
25C4×C22⋊D107—21no—
26#16812068C2211yescharacteristic
27#16812068C2211yescharacteristic
28#16812067C2211yescharacteristic
29#16812067C2211yescharacteristic
30#16812066—21no—
31#16812065—21no—
32C4⋊C4×D107C2211yescharacteristic
33C4⋊C4×D107C2211yescharacteristic
34C22⋊C4×D107—21no—
35C42×D107—21no—
36#16812043C2211yescharacteristic, complemented
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812041C2211yescharacteristic, complemented
42#16812040C2211yescharacteristic, complemented
43#16812040C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812030C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic, complemented
48(C2×D107)⋊C8C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54#117×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59C23×D107D411yescharacteristic
60C2×C22⋊D107—21no—
61C2×C22⋊D107—21no—
62C22×Dic107D411yescharacteristic
63C2×C4×D107—22no—
64C2×C4×D107—22no—
65C2×C4×D107—21no—
66C2×C4×D107—21no—
67C2×C4×D107D411yescharacteristic
68C2×C4×D107D411yescharacteristic
69C2×C4×D107C2311yescharacteristic
70C2×C4×D107C2311yescharacteristic
71C2×C107⋊C8—22no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94#41×C107—22no—
95#41×C107—22no—
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C22×C4×C107—21no—
100C22×C4×C107C2311yescharacteristic
101C2×C8×C107—22no—
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C42×C107—21no—
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—22no—
111C22×D107—22no—
112C22×D107—41no—
113C22×D107C2×D411yescharacteristic
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—21no—
118D4×C107—21no—
119D4×C107—42no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—42no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107#4112yes—
126C2×C4×C107#4112yes—
127C2×C4×C107C2411yescharacteristic
128C22⋊D107—42no—
129C22⋊D107—42no—
130C2×Dic107—22no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C4×D107—22no—
138C4×D107—22no—
139C4×D107—21no—
140C4×D107—21no—
141C4×D107—21no—
142C4×D107—21no—
143C4×D107—42no—
144C4×D107—42no—
145C8×C107—21no—
146C8×C107—21no—
147C107⋊C8—21no—
148C107⋊C8—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107#12312yes—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—21no—
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—44no—
162C2×D107#13512yes—
163C4×C107—41no—
164C4×C107—41no—
165C4×C107—41no—
166C4×C107#12312yes—
167C4×C107C2×#4111yescharacteristic
168C4×C107C2×#4111yescharacteristic
169Dic107—21no—
170Dic107—21no—
171Dic107—41no—
172Dic107—41no—
173Dic107—41no—
174Dic107—41no—
175C2×C107—22no—
176C2×C107—81no—
177C2×C107C2×#12311yescharacteristic
178C2×C107C2×#13511yescharacteristic
179C2×C107#57511yescharacteristic
180D107—22no—
181D107—41no—
182#3091—1071nomaximal, Sylow
183C107#309111yescharacteristic, complemented, Sylow
184#525—1071no—
185C2×#117—1071no—
186#485—1071no—
187#485—1071no—
188#484—1071no—
189#483—1071no—
190#482—1071no—
191#482—1071no—
192#481—1071no—
193#480—1071no—
194#427—1071no—
195#427—1071no—
196#416—1071no—
197#416D10711yescharacteristic, complemented
198C2×C22⋊C8—1071no—
199C2×#41C2×D10711yescharacteristic
200C23×C4—1071no—
201C22×C8—1071no—
202C22×C8C2×D10711yescharacteristic
203#123—2141no—
204C22⋊Q8—1071no—
205C22⋊Q8—1071no—
206#121—1071no—
207#121—1071no—
208#120—2141no—
209C4×D4—2141no—
210C4×D4—2141no—
211#117—1071no—
212#117—1071no—
213#117—1071no—
214#117—1071no—
215#117—2141no—
216#117—2141no—
217#117C2×D10711yescharacteristic
218C2×C4⋊C4—1071no—
219C2×C4⋊C4—1071no—
220C2×C22⋊C4—2141no—
221C2×C42—2141no—
222#107—1071no—
223#107—1071no—
224#107—1071no—
225#107—1071no—
226#106—1071no—
227#106—1071no—
228#106—1071no—
229#106—1071no—
230#103—1071no—
231#103—1071no—
232#103C2×D10711yescharacteristic, complemented
233#103C2×D10711yescharacteristic, complemented
234#102—1071no—
235#102—1071no—
236#102C2×D10711yescharacteristic, complemented
237#102C2×D10711yescharacteristic, complemented
238C22⋊C8—1071no—
239C22⋊C8—1071no—
240C22⋊C8—1071no—
241C22⋊C8—1071no—
242C24—1071no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D10711yescharacteristic
246C2×D4—21no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4C22×D10711yescharacteristic
250C22×C4—21no—
251C22×C4—1071no—
252C22×C4—1071no—
253C22×C4—1071no—
254C22×C4—1071no—
255C22×C4—1071no—
256C22×C4—2141no—
257C22×C4—2141no—
258C22×C4—2142no—
259C22×C4—2142no—
260C22×C4C22×D10711yescharacteristic
261C2×C8—22no—
262C2×C8—1071no—
263C2×C8—1071no—
264C2×C8—2142no—
265C2×C8C22×D10711yescharacteristic
266C2×C8C22×D10711yescharacteristic
267C4⋊C4—1071no—
268C4⋊C4—1071no—
269C4⋊C4—1071no—
270C4⋊C4—1071no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—21no—
287C42—2141no—
288C42—2141no—
289C42—2141no—
290C42—2141no—
291C23—21no—
292C23—1071no—
293C23—1071no—
294C23—4281no—
295C23—2142no—
296C23—2142no—
297C23D4×D10711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—4282no—
304D4—4282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—1071no—
310C2×C4—1071no—
311C2×C4—4282no—
312C2×C4—4282no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2142no—
318C2×C4—2142no—
319C2×C4—2142no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4D4×D10711yescharacteristic
325C2×C4#1313381712yes—
326C2×C4#1313381812yes—
327C2×C4C23×D10711yescharacteristic
328C8—21no—
329C8—21no—
330C8—2141no—
331C8—2141no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—1072no—
338C22—4284no—
339C22—2141no—
340C22—2141no—
341C22—2141no—
342C22—2141no—
343C22—2141no—
344C22#1681209112yes—
345C22C2×D4×D10711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—4281no—
350C4—4281no—
351C4—4281no—
352C4—4281no—
353C4—2141no—
354C4—2141no—
355C4#1681209112yes—
356C4C2×#1313381711yescharacteristic
357C4C2×#1313381811yescharacteristic
358C2—22no—
359C2—81no—
360C2—4281no—
361C2—2142no—
362C2C2×#1681209111yescharacteristic
363C2#135×D10711yescharacteristic
364C2#2029342411yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radical#20293134C2
hypercenter#416D107
hypocenterC107#3091
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D107C2semidirect
3. #20292967C2semidirect
4. #20292966C2semidirect
5. #20292961C2semidirect
6. #20292956C2semidirect
7. #20292950C2semidirect
8. #20292949C2semidirect
9. #20292944C2semidirect
10. #20292939C2semidirect
11. #20292665C2semidirect
12. #20292663C2semidirect
13. #20292623C2semidirect
14. C2×(C2×D107)⋊C8C2semidirect
15. #416×C107C2semidirect
36. #16812043C22semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. #16812041C22semidirect
42. #16812040C22semidirect
43. #16812040C22semidirect
44. #16812033C22semidirect
45. #16812033C22semidirect
46. #16812030C22semidirect
47. #16812030C22semidirect
55. #103×C107C22semidirect
56. #103×C107C22semidirect
57. #102×C107C22semidirect
58. #102×C107C22semidirect
183. C107#3091semidirect
197. #416D107semidirect
232. #103C2×D107semidirect
233. #103C2×D107semidirect
236. #102C2×D107semidirect
237. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931342
#117×D1072
#202929672
#202929662
#202929612
#202929562
#202929502
#202929492
#202929442
#202929392
#202926652
#202926632
#202926232
C2×(C2×D107)⋊C82
#416×C1072
#3091107