Finite Group Databasebeta

← #28348351

Subgroups — #28348351

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#20293316C211yesmaximal, characteristic, complemented
2#20293287C211yesmaximal, characteristic
3#20292983C211yesmaximal, characteristic, complemented
4#20292942C211yesmaximal, characteristic, complemented
5#20292937C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292857C211yesmaximal, characteristic, complemented
8#20292852C211yesmaximal, characteristic, complemented
9#20292791C211yesmaximal, characteristic, complemented
10#20292784C211yesmaximal, characteristic, complemented
11#20292721C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13#20292621C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#485×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×Dic428C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812102—21no—
24#16812101—21no—
25#16812100C2211yescharacteristic
26#16812098C2211yescharacteristic
27#16812095—21no—
28#16812091—21no—
29#16812089C2211yescharacteristic
30#16812088C2211yescharacteristic
31#16812087—21no—
32#16812084C2211yescharacteristic
33#16812082—21no—
34#16812074C2211yescharacteristic
35#16812074C2211yescharacteristic
36#16812071—21no—
37C4×C22⋊D107C2211yescharacteristic
38#16812066—21no—
39C4⋊C4×D107C2211yescharacteristic
40#16812042C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C22⋊Q8×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63C2×Dic214C2311yescharacteristic
64Q8×D107—22no—
65Q8×D107—41no—
66C2×D428—21no—
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107D411yescharacteristic
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74Dic428—22no—
75#13133808—41no—
76#13133807—41no—
77C2×C107⋊C8—41no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—41no—
81C4⋊Dic107—21no—
82C4⋊Dic107—21no—
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91C22⋊Dic107D411yescharacteristic
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107C2311yescharacteristic
100C2×Q8×C107C2311yescharacteristic
101C22×C4×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107C2311yescharacteristic
106C4⋊C4×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C22⋊C4×C107—21no—
109C22⋊C4×C107—21no—
110C42×C107—21no—
111C23×C107C2×D411yescharacteristic
112C22×D107—21no—
113C22×D107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C22⋊D107—22no—
125C22⋊D107—42no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132D428—21no—
133D428—21no—
134D428—42no—
135C4×D107—21no—
136C4×D107—21no—
137C4×D107—42no—
138C4×D107—42no—
139C4×D107—42no—
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—21no—
152C22×C107C22×D411yescharacteristic
153C2×D107—21no—
154C2×D107—21no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107#13512yes—
159C4×C107—21no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107—41no—
164C4×C107C2×#4111yescharacteristic
165C4×C107H(5,𝔽2)11yescharacteristic
166Dic107—22no—
167Dic107—21no—
168Dic107—21no—
169Dic107—21no—
170Dic107—41no—
171Dic107—41no—
172C2×C107—41no—
173C2×C107#54611yescharacteristic
174C2×C107C2×#13511yescharacteristic
175C2×C107#57811yescharacteristic
176D107—22no—
177D107—81no—
178#3300—1071nomaximal, Sylow
179C107#330011yescharacteristic, complemented, Sylow
180#552—1071no—
181#548—1071no—
182#488—1071no—
183#485D10711yescharacteristic, complemented
184#480—1071no—
185#479—1071no—
186#467—1071no—
187#465—1071no—
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×SD16—1071no—
200C22×C8—1071no—
201C22×C8—1071no—
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—1071no—
213#121—2141no—
214C4×Q8—1071no—
215C4×Q8—1071no—
216C4×D4—1071no—
217C4×D4—2141no—
218#117—2141no—
219#117C2×D10711yescharacteristic
220C2×C4⋊C4—1071no—
221#107—1071no—
222#107C2×D10711yescharacteristic
223#106C2×D10711yescharacteristic
224C4⋊C8—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103C2×D10711yescharacteristic
231#103C2×D10711yescharacteristic
232#102—1071no—
233#102—1071no—
234#102—1071no—
235C22⋊C8—1071no—
236C22⋊C8C2×D10711yescharacteristic
237C4×C8—1071no—
238#41—4281no—
239#41—2142no—
240C2×Q8—1071no—
241C2×Q8—1071no—
242C2×Q8—4281no—
243C2×Q8—2141no—
244C2×Q8—2142no—
245C2×Q8C22×D10711yescharacteristic
246C2×D4—1071no—
247C2×D4—1071no—
248C2×D4—2141no—
249C2×D4—2141no—
250C22×C4—1071no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4—2141no—
254C22×C4C22×D10711yescharacteristic
255Q16—4281no—
256Q16—2142no—
257SD16—4281no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8—4281no—
261C2×C8—4281no—
262C2×C8C22×D10711yescharacteristic
263C2×C8C22×D10711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4C22×D10711yescharacteristic
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—21no—
287C42—1071no—
288C42—2141no—
289C42—2141no—
290C23—1071no—
291C23—2141no—
292C23D4×D10711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1072no—
296Q8—4282no—
297Q8—2141no—
298Q8—2141no—
299Q8—2141no—
300D4—4282no—
301D4—4282no—
302D4—2142no—
303D4—2141no—
304D4—2141no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—1071no—
314C2×C4—4282no—
315C2×C4—4282no—
316C2×C4—4282no—
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—21no—
331C22—1072no—
332C22—4281no—
333C22—4281no—
334C22—4281no—
335C22—2141no—
336C22—2141no—
337C22C2×D4×D10711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4281no—
344C4—4281no—
345C4—2141no—
346C4—2141no—
347C4—2141no—
348C4—2142no—
349C4#41×D10711yescharacteristic
350C4#1681215511yescharacteristic
351C2—41no—
352C2—8561no—
353C2—2142no—
354C2#2029326911yescharacteristic
355C2#135×D10711yescharacteristic
356C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#485×C107C2
socleC22×C107C22×D4
prime radical#20293287C2
hypercenter#485D107
hypocenterC107#3300
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #20293316C2semidirect
3. #20292983C2semidirect
4. #20292942C2semidirect
5. #20292937C2semidirect
6. #20292866C2semidirect
7. #20292857C2semidirect
8. #20292852C2semidirect
9. #20292791C2semidirect
10. #20292784C2semidirect
11. #20292721C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. #20292621C2semidirect
14. #103×D107C2semidirect
15. #485×C107C2semidirect
179. C107#3300semidirect
183. #485D107semidirect

Maximal subgroups

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

subgroupindex
#202933162
#202932872
#202929832
#202929422
#202929372
#202928662
#202928572
#202928522
#202927912
#202927842
#202927212
(C22⋊D107)⋊C82
#202926212
#103×D1072
#485×C1072
#3300107