Finite Group Databasebeta

← #28346707

Subgroups — #28346707

Order 13696.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293237C211yesmaximal, characteristic, complemented
2#20293102C211yesmaximal, characteristic
3#20292969C211yesmaximal, characteristic, complemented
4#20292965C211yesmaximal, characteristic, complemented
5#20292929C211yesmaximal, characteristic, complemented
6#20292925C211yesmaximal, characteristic, complemented
7#20292830C211yesmaximal, characteristic, complemented
8#20292830C211yesmaximal, characteristic, complemented
9#20292778C211yesmaximal, characteristic
10#20292728C211yesmaximal, characteristic, complemented
11#20292724C211yesmaximal, characteristic, complemented
12#20292718C211yesmaximal, characteristic, complemented
13SD16×Dic107C211yesmaximal, characteristic, complemented
14#20292685C211yesmaximal, characteristic, complemented
15#450×C107C211yesmaximal, characteristic
16C2×#13133806C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18#16812107C2211yescharacteristic, complemented
19#16812102—21no—
20#16812102—21no—
21#16812100—21no—
22#16812100C2211yescharacteristic, complemented
23#16812099—21no—
24#16812088C2211yescharacteristic
25Q8×Dic107C2211yescharacteristic, complemented
26#16812074C2211yescharacteristic, complemented
27D4×Dic107—21no—
28D4×Dic107—21no—
29D4×Dic107C2211yescharacteristic
30D4×Dic107C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32D4×Dic107C2211yescharacteristic
33#16812068C2211yescharacteristic
34#16812068C2211yescharacteristic
35C2×C22⋊Dic107—21no—
36C2×C22⋊Dic107—21no—
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812041C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812033C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22×D4×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C22×Dic107—21no—
61C22×Dic107—21no—
62C22×Dic107—21no—
63C22×Dic107—21no—
64#13133806—22no—
65#13133806—22no—
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C4⋊Dic107—21no—
69C4⋊Dic107—21no—
70C4⋊Dic107D411yescharacteristic
71C4⋊Dic107D411yescharacteristic
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80C22⋊Dic107—21no—
81C22⋊Dic107—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107D411yescharacteristic
87C22⋊Dic107D411yescharacteristic
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107C2311yescharacteristic
92C4×Dic107C2311yescharacteristic
93C24×C107—21no—
94C2×Q8×C107C2311yescharacteristic
95C2×D4×C107—22no—
96C2×D4×C107—22no—
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C2×D4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101SD16×C107—22no—
102SD16×C107—22no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C23×C107—24no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107—81no—
112C23×C107—81no—
113C23×C107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107—21no—
116D4×C107—24no—
117D4×C107—21no—
118D4×C107—21no—
119D4×C107#4112yes—
120D4×C107#4112yes—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—81no—
132C2×Dic107—81no—
133C2×Dic107—81no—
134C2×Dic107—81no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138C2×Dic107C2×D411yescharacteristic
139Dic214—21no—
140Dic214—21no—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—22no—
147C22×C107—22no—
148C22×C107—21no—
149C22×C107—21no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—84no—
155C22×C107C22×D411yescharacteristic
156C4×C107—21no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107C2×#4111yescharacteristic
160C4×C107C2×#4111yescharacteristic
161Dic107—21no—
162Dic107—21no—
163Dic107—21no—
164Dic107—21no—
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168Dic107—41no—
169C2×C107—41no—
170C2×C107—42no—
171C2×C107—42no—
172C2×C107#52511yescharacteristic
173C2×C107#57711yescharacteristic
174C2×C107#57711yescharacteristic
175#3123—1071nomaximal, Sylow
176C107#312311yescharacteristic, complemented, Sylow
177#541—1071no—
178#518—1071no—
179#485—1071no—
180#485—1071no—
181#478—1071no—
182#478—1071no—
183#461—1071no—
184#461—1071no—
185#450—1071no—
186#450D10711yescharacteristic
187#440—1071no—
188#440—1071no—
189C4×SD16—1071no—
190C4×SD16—1071no—
191#433—1071no—
192C22×D4C2×D10711yescharacteristic
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×SD16C2×D10711yescharacteristic
196C2×SD16C2×D10711yescharacteristic
197C4⋊Q8—1071no—
198#126—2141no—
199#126—2141no—
200#125—1071no—
201#125—2141no—
202#125—2141no—
203C22⋊Q8—1071no—
204C22⋊Q8C2×D10711yescharacteristic
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×D4—1071no—
208C4×D4—1071no—
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—2141no—
212C4×D4—2141no—
213#117—1071no—
214#117—1071no—
215C2×C22⋊C4—2141no—
216C2×C22⋊C4—2141no—
217#107—1071no—
218#107—1071no—
219#106—1071no—
220#106—1071no—
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223#103—1071no—
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#102—1071no—
228#102—1071no—
229#102C2×D10711yescharacteristic
230#102C2×D10711yescharacteristic
231C22⋊C8—1071no—
232C22⋊C8C2×D10711yescharacteristic
233#97—1071no—
234C4×C8—1071no—
235C24—21no—
236C2×Q8—1071no—
237C2×Q8C22×D10711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D10711yescharacteristic
242C2×D4C22×D10711yescharacteristic
243C22×C4—2141no—
244C22×C4—2141no—
245C22×C4—2141no—
246C22×C4—2141no—
247C22×C4C22×D10711yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—2142no—
251SD16—2142no—
252C2×C8—1071no—
253C2×C8—1071no—
254C2×C8C22×D10711yescharacteristic
255C2×C8C22×D10711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4C22×D10711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1071no—
272C22⋊C4—1071no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C42—1071no—
280C42—1071no—
281C42—2141no—
282C42—2141no—
283C42—2141no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D10711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—2141no—
293Q8—2141no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1313381712yes—
298D4#1313381712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—8561no—
304C2×C4—8561no—
305C2×C4—8561no—
306C2×C4—8561no—
307C2×C4—1071no—
308C2×C4—1071no—
309C2×C4—1071no—
310C2×C4—1071no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2141no—
320C8—2141no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D10711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—4281no—
336C4—4281no—
337C4—4281no—
338C4—4281no—
339C4—2141no—
340C4—2141no—
341C4—2141no—
342C4—2141no—
343C4C2×#1313381711yescharacteristic
344C4C2×#1313381711yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2029313311yescharacteristic
349C2#2029343311yescharacteristic
350C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×C107C22
hypercenter#450D107
hypocenterC107#3123
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20293237C2semidirect
3. #20292969C2semidirect
4. #20292965C2semidirect
5. #20292929C2semidirect
6. #20292925C2semidirect
7. #20292830C2semidirect
8. #20292830C2semidirect
10. #20292728C2semidirect
11. #20292724C2semidirect
12. #20292718C2semidirect
13. SD16×Dic107C2semidirect
14. #20292685C2semidirect
18. #16812107C22semidirect
22. #16812100C22semidirect
25. Q8×Dic107C22semidirect
26. #16812074C22semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812041C22semidirect
40. #16812040C22semidirect
41. Dic107⋊C8C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812034C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812033C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
176. C107#3123semidirect

Maximal subgroups

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

subgroupindex
#202932372
#202931022
#202929692
#202929652
#202929292
#202929252
#202928302
#202927782
#202927282
#202927242
#202927182
SD16×Dic1072
#202926852
#450×C1072
#3123107