Finite Group Databasebeta

← #28345290

Subgroups — #28345290

Order 13696.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic
2#20293106C211yesmaximal, characteristic, complemented
3#20293099C211yesmaximal, characteristic, complemented
4#20292736C211yesmaximal, characteristic, complemented
5#20292729C211yesmaximal, characteristic, complemented
6#20292724C211yesmaximal, characteristic, complemented
7#20292714C211yesmaximal, characteristic, complemented
8C4×#13133806C211yesmaximal, characteristic, complemented
9D8×Dic107C211yesmaximal, characteristic, complemented
10C4×#13133803C211yesmaximal, characteristic, complemented
11#20292666C211yesmaximal, characteristic, complemented
12#20292620C211yesmaximal, characteristic, complemented
13#102×D107C211yesmaximal, characteristic, complemented
14#20292584C211yesmaximal, characteristic, complemented
15#442×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18#16812133C418yescomplemented
19C2×#13133807C2211yescharacteristic
20C2×#13133806C2211yescharacteristic
21C2×#13133803C2211yescharacteristic
22C2×#13133801C2211yescharacteristic
23C2×C8×D107C2211yescharacteristic
24C4×Dic214C2211yescharacteristic, complemented
25#16812074C2211yescharacteristic, complemented
26C4×D428C2211yescharacteristic
27D4×Dic107C2211yescharacteristic
28D4×Dic107C2211yescharacteristic, complemented
29#16812071C2211yescharacteristic
30C4×C22⋊D107—21no—
31C4×C22⋊D107—21no—
32(C22⋊D107)⋊C4—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067C2211yescharacteristic
35#16812066C2211yescharacteristic
36#16812065—21no—
37C2×Dic107⋊C4—21no—
38C4⋊C4×D107C2211yescharacteristic
39C2×(C2×D107)⋊C4—21no—
40C22⋊C4×D107—21no—
41#16812043C2211yescharacteristic, complemented
42#16812042C2211yescharacteristic, complemented
43#16812040C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812032C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic
50#16812019C2211yescharacteristic, complemented
51C4×C107⋊C8C2211yescharacteristic, complemented
52C8×Dic107C2211yescharacteristic, complemented
53C2×D8×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic, complemented
56#106×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic, complemented
59#97×C107C2211yescharacteristic, complemented
60C23×D107—21no—
61#13133817—22no—
62#13133817C2×C414yescomplemented
63C2×Dic214C2311yescharacteristic
64C2×D428C2311yescharacteristic
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67D4×D107—22no—
68D4×D107C2×C414yescomplemented
69C22×Dic107—21no—
70C22×Dic107—21no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107C2311yescharacteristic
76#13133807C2×C414yescomplemented
77#13133806C2×C414yescomplemented
78#13133803C2×C414yescomplemented
79#13133801C2×C414yescomplemented
80C2×C107⋊C8C2311yescharacteristic
81C2×C107⋊C8C2311yescharacteristic
82C8×D107—22no—
83C4⋊Dic107C2311yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4D411yescharacteristic
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4D411yescharacteristic
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107—21no—
100C4×Dic107C2311yescharacteristic
101C2×D4×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104C22×C4×C107—21no—
105D8×C107C2×C414yescomplemented
106C2×C8×C107C2311yescharacteristic
107C2×C8×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C22⋊C4×C107—21no—
112C42×C107C2311yescharacteristic
113C23×C107—21no—
114C23×C107—21no—
115C22×D107—21no—
116C22×D107—24no—
117C22×D107—81no—
118C22×D107—81no—
119C22×D107C2×D411yescharacteristic
120D4×C107—21no—
121D4×C107—21no—
122D4×C107C22×C412yes—
123D4×C107C22×C412yes—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—81no—
127C2×C4×C107—81no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—24no—
132C22⋊D107—24no—
133C2×Dic107—21no—
134C2×Dic107—24no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—81no—
138C2×Dic107—81no—
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141C2×Dic107C2×D411yescharacteristic
142D428—21no—
143D428C22×C412yes—
144C4×D107—81no—
145C4×D107—81no—
146C4×D107—81no—
147C4×D107C22×C412yes—
148C4×D107#4112yes—
149Dic214—21no—
150Dic214C22×C412yes—
151C8×C107—21no—
152C8×C107C22×C412yes—
153C107⋊C8—21no—
154C107⋊C8C22×C412yes—
155C22×C107—22no—
156C22×C107—22no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—22no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—84no—
168C2×D107C4×D412yes—
169C4×C107—21no—
170C4×C107—21no—
171C4×C107—41no—
172C4×C107—41no—
173C4×C107C23×C411yescharacteristic
174C4×C107C2×#4111yescharacteristic
175Dic107—22no—
176Dic107—21no—
177Dic107—21no—
178Dic107—41no—
179Dic107—41no—
180Dic107C4×D412yes—
181C2×C107—42no—
182C2×C107—42no—
183C2×C107C2×C4×D411yescharacteristic
184C2×C107#57611yescharacteristic
185C2×C107#57711yescharacteristic
186D107—22no—
187D107—42no—
188#2946—1071nomaximal, Sylow
189C107#294611yescharacteristic, complemented, Sylow
190C2×AGL(1,ℤ/8)—1071no—
191#520—1071no—
192#518—1071no—
193#442—1071no—
194#442D10711yescharacteristic, complemented
195#440—1071no—
196#440—1071no—
197C4×SD16—1071no—
198C4×SD16—1071no—
199C4×D8—1071no—
200C4×D8—1071no—
201#428—1071no—
202#416—1071no—
203C2×#102—1071no—
204#405—1071no—
205C2×#41—1071no—
206C22×D4—1071no—
207AGL(1,ℤ/8)—1078no—
208C2×SD16—1071no—
209C2×SD16—1071no—
210C2×D8—1071no—
211C2×D8C2×D10711yescharacteristic
212C2×M16—1071no—
213C22×C8—1071no—
214C4×Q8—1071no—
215C4×Q8—1071no—
216C4×D4—1071no—
217C4×D4—1071no—
218C4×D4—1071no—
219C4×D4—1071no—
220C4×D4—2141no—
221C4×D4—2141no—
222C4×D4—2141no—
223C4×D4—2141no—
224C4×D4C2×D10711yescharacteristic
225C4×D4C2×D10711yescharacteristic, complemented
226#117—1071no—
227#117—1071no—
228#117—2141no—
229C2×C4⋊C4—1071no—
230C2×C4⋊C4—2141no—
231C2×C22⋊C4—2141no—
232C2×C22⋊C4—2141no—
233#107—1071no—
234#107—1071no—
235#106—1071no—
236#106C2×D10711yescharacteristic, complemented
237#103—1071no—
238#103—1071no—
239#102—1071no—
240#102—1071no—
241#102—1071no—
242#102—1071no—
243#102C2×D10711yescharacteristic
244#102C2×D10711yescharacteristic, complemented
245#97—1071no—
246#97C2×D10711yescharacteristic, complemented
247C4×C8—1071no—
248C4×C8—1071no—
249C24—2141no—
250#41—1074no—
251#41—2142no—
252C2×Q8—1071no—
253C2×D4—1074no—
254C2×D4—1071no—
255C2×D4—2142no—
256C2×D4—2141no—
257C2×D4—2141no—
258C2×D4C22×D10711yescharacteristic
259C2×D4C22×D10711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—1071no—
263C22×C4—2141no—
264C22×C4—2141no—
265C22×C4—2141no—
266C22×C4—2141no—
267C22×C4—2141no—
268C22×C4—2141no—
269SD16—1074no—
270SD16—1074no—
271D8—1074no—
272D8C4×D10714yescomplemented
273M16—1074no—
274C2×C8—1071no—
275C2×C8—1071no—
276C2×C8—2142no—
277C2×C8C22×D10711yescharacteristic
278C2×C8C22×D10711yescharacteristic
279C4⋊C4—1071no—
280C4⋊C4—1071no—
281C4⋊C4—1071no—
282C4⋊C4—1071no—
283C4⋊C4—2141no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4C22×D10711yescharacteristic
287C4⋊C4C22×D10711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—1071no—
291C22⋊C4—1071no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C42—1071no—
298C42—2141no—
299C42—2141no—
300C42—2141no—
301C42C22×D10711yescharacteristic
302C23—21no—
303C23—21no—
304C23—8561no—
305C23—8561no—
306C23—1071no—
307C23—2144no—
308C23—2141no—
309Q8—1072no—
310Q8—2141no—
311D4—21no—
312D4—21no—
313D4—1072no—
314D4—2144no—
315D4—2144no—
316D4—2141no—
317D4C2×C4×D10712yes—
318D4C2×C4×D10712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—8561no—
324C2×C4—8561no—
325C2×C4—8561no—
326C2×C4—8561no—
327C2×C4—8561no—
328C2×C4—1071no—
329C2×C4—1072no—
330C2×C4—1072no—
331C2×C4—1071no—
332C2×C4—1071no—
333C2×C4—2144no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4D4×D10711yescharacteristic
338C2×C4D4×D10711yescharacteristic
339C2×C4C23×D10711yescharacteristic
340C8—21no—
341C8—1072no—
342C8—2141no—
343C8C2×C4×D10712yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—8564no—
351C22—1072no—
352C22—4281no—
353C22—4281no—
354C22—2142no—
355C22—2141no—
356C22—2141no—
357C22C2×D4×D10711yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—1072no—
363C4—4281no—
364C4—4281no—
365C4—2141no—
366C4—2141no—
367C4—2142no—
368C4C22×C4×D10711yescharacteristic
369C4#41×D10711yescharacteristic
370C2—42no—
371C2—42no—
372C2—4282no—
373C2—2142no—
374C2C4×D4×D10711yescharacteristic
375C2#2029343111yescharacteristic
376C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#442×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812133C2
cosocleC2×C4×C107C24
hypercenter#442D107
hypocenterC107#2946
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #20293106C2semidirect
3. #20293099C2semidirect
4. #20292736C2semidirect
5. #20292729C2semidirect
6. #20292724C2semidirect
7. #20292714C2semidirect
8. C4×#13133806C2semidirect
9. D8×Dic107C2semidirect
10. C4×#13133803C2semidirect
11. #20292666C2semidirect
12. #20292620C2semidirect
13. #102×D107C2semidirect
14. #20292584C2semidirect
15. #442×C107C2semidirect
18. #16812133C4semidirect
24. C4×Dic214C22semidirect
25. #16812074C22semidirect
28. D4×Dic107C22semidirect
41. #16812043C22semidirect
42. #16812042C22semidirect
43. #16812040C22semidirect
44. #16812034C22semidirect
45. #16812032C22semidirect
48. #16812030C22semidirect
50. #16812019C22semidirect
51. C4×C107⋊C8C22semidirect
52. C8×Dic107C22semidirect
55. C4×D4×C107C22semidirect
56. #106×C107C22semidirect
58. #102×C107C22semidirect
59. #97×C107C22semidirect
62. #13133817C2×C4semidirect
68. D4×D107C2×C4semidirect
76. #13133807C2×C4semidirect
77. #13133806C2×C4semidirect
78. #13133803C2×C4semidirect
79. #13133801C2×C4semidirect
105. D8×C107C2×C4semidirect
189. C107#2946semidirect
194. #442D107semidirect
225. C4×D4C2×D107semidirect
236. #106C2×D107semidirect
244. #102C2×D107semidirect
246. #97C2×D107semidirect
272. D8C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202931062
#202930992
#202927362
#202927292
#202927242
#202927142
C4×#131338062
D8×Dic1072
C4×#131338032
#202926662
#202926202
#102×D1072
#202925842
#442×C1072
#2946107