Finite Group Databasebeta

← #28348354

Subgroups — #28348354

Order 13696.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293319C211yesmaximal, characteristic, complemented
2#20293283C211yesmaximal, characteristic
3#20292980C211yesmaximal, characteristic, complemented
4#20292971C211yesmaximal, characteristic, complemented
5#20292938C211yesmaximal, characteristic, complemented
6#20292937C211yesmaximal, characteristic, complemented
7#20292867C211yesmaximal, characteristic, complemented
8#20292847C211yesmaximal, characteristic, complemented
9#20292786C211yesmaximal, characteristic, complemented
10#20292782C211yesmaximal, characteristic, complemented
11#20292719C211yesmaximal, characteristic, complemented
12C8×C22⋊D107C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14C2×#16812034C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812104—21no—
22#16812100—21no—
23#16812100C2211yescharacteristic
24#16812096C2211yescharacteristic
25#16812095—21no—
26#16812093—21no—
27#16812089C2211yescharacteristic
28#16812086—21no—
29#16812085C2211yescharacteristic
30#16812081—21no—
31Q8×Dic107C2211yescharacteristic
32#16812074C2211yescharacteristic
33D4×Dic107—21no—
34C4×C22⋊D107C2211yescharacteristic
35#16812067C2211yescharacteristic
36#16812065—21no—
37C2×C4⋊Dic107C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812042C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—22no—
60#13133817—41no—
61C2×Dic214—22no—
62C2×Dic214—21no—
63C2×Dic214—41no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107D411yescharacteristic
68C22×Dic107—21no—
69C22×Dic107—21no—
70C22×Dic107—21no—
71C2×C4×D107C2311yescharacteristic
72#13133808—22no—
73#13133808—41no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C8×D107—41no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107—21no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107D411yescharacteristic
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×D4×C107—21no—
100C2×D4×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102SD16×C107—41no—
103C2×C8×C107—41no—
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107—21no—
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C22⋊C4×C107—21no—
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107—21no—
116D4×C107—21no—
117D4×C107—21no—
118D4×C107—42no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C22⋊D107—22no—
127C22⋊D107—42no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—42no—
133C2×Dic107—42no—
134C2×Dic107—42no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138C4×D107—21no—
139C4×D107—21no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143Dic214—42no—
144Dic214#4112yes—
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#13512yes—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—21no—
159C4×C107—21no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107C2×#4111yescharacteristic
164C4×C107H(5,𝔽2)11yescharacteristic
165Dic107—22no—
166Dic107—21no—
167Dic107—21no—
168Dic107—21no—
169Dic107—41no—
170Dic107—41no—
171Dic107—41no—
172C2×C107—22no—
173C2×C107—81no—
174C2×C107#54611yescharacteristic
175C2×C107C2×#13511yescharacteristic
176C2×C107#57811yescharacteristic
177D107—41no—
178#3300—1071nomaximal, Sylow
179C107#330011yescharacteristic, complemented, Sylow
180#552—1071no—
181#548—1071no—
182#488—1071no—
183#485—1071no—
184#480—1071no—
185#479—1071no—
186#467—1071no—
187#465D10711yescharacteristic, complemented
188#464—1071no—
189#452—1071no—
190#451—1071no—
191C4×Q16—1071no—
192C8×D4—1071no—
193#416—1071no—
194C2×#103—1071no—
195C2×#41—1071no—
196C22×Q8—1071no—
197C2×Q16—1071no—
198C2×Q16—1071no—
199C2×SD16C2×D10711yescharacteristic
200C22×C8—1071no—
201C22×C8C2×D10711yescharacteristic
202#126—2141no—
203#125—1071no—
204#125—2141no—
205#124—1071no—
206#124—2141no—
207#123—2141no—
208C22⋊Q8—1071no—
209C22⋊Q8—1071no—
210C22⋊Q8—2141no—
211C22⋊Q8C2×D10711yescharacteristic
212#121—2141no—
213#121C2×D10711yescharacteristic
214C4×Q8—1071no—
215C4×Q8—1071no—
216C4×D4—1071no—
217C4×D4—2141no—
218#117—1071no—
219#117—2141no—
220C2×C4⋊C4—1071no—
221#107—1071no—
222#107—1071no—
223#106C2×D10711yescharacteristic
224C4⋊C8—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#103C2×D10711yescharacteristic
232#102—1071no—
233#102—1071no—
234#102C2×D10711yescharacteristic
235C22⋊C8—1071no—
236C22⋊C8—1071no—
237C4×C8—1071no—
238#41—4281no—
239#41—2142no—
240C2×Q8—1071no—
241C2×Q8—1071no—
242C2×Q8—4281no—
243C2×Q8—2142no—
244C2×Q8—2141no—
245C2×Q8C22×D10711yescharacteristic
246C2×D4—21no—
247C2×D4—1071no—
248C2×D4—2141no—
249C2×D4C22×D10711yescharacteristic
250C22×C4—1071no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4—2141no—
254C22×C4C22×D10711yescharacteristic
255Q16—4281no—
256Q16—2142no—
257SD16—41no—
258C2×C8—41no—
259C2×C8—1071no—
260C2×C8—1071no—
261C2×C8—4281no—
262C2×C8C22×D10711yescharacteristic
263C2×C8C22×D10711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1071no—
280C22⋊C4—1071no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—1071no—
287C42—2141no—
288C42—2141no—
289C42—2141no—
290C23—21no—
291C23—1071no—
292C23D4×D10711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1072no—
296Q8—4282no—
297Q8—2141no—
298Q8—2141no—
299Q8—2141no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—4282no—
304D4—2142no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—1071no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—4282no—
314C2×C4—4282no—
315C2×C4—4282no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4D4×D10711yescharacteristic
323C2×C4C23×D10711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2141no—
327C8—2141no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—2141no—
334C22—2141no—
335C22—2141no—
336C22#1681213012yes—
337C22C2×D4×D10711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—4281no—
343C4—4281no—
344C4—4281no—
345C4—2141no—
346C4—2141no—
347C4—2141no—
348C4—2142no—
349C4#41×D10711yescharacteristic
350C4#1681215411yescharacteristic
351C2—22no—
352C2—81no—
353C2—4281no—
354C2#2029326611yescharacteristic
355C2C2×#1681213011yescharacteristic
356C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
prime radical#20293283C2
hypercenter#465D107
hypocenterC107#3300
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20293319C2semidirect
3. #20292980C2semidirect
4. #20292971C2semidirect
5. #20292938C2semidirect
6. #20292937C2semidirect
7. #20292867C2semidirect
8. #20292847C2semidirect
9. #20292786C2semidirect
10. #20292782C2semidirect
11. #20292719C2semidirect
12. C8×C22⋊D107C2semidirect
13. #20292620C2semidirect
14. C2×#16812034C2semidirect
15. #465×C107C2semidirect
179. C107#3300semidirect
187. #465D107semidirect

Maximal subgroups

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

subgroupindex
#202933192
#202932832
#202929802
#202929712
#202929382
#202929372
#202928672
#202928472
#202927862
#202927822
#202927192
C8×C22⋊D1072
#202926202
C2×#168120342
#465×C1072
#3300107