Finite Group Databasebeta

← #28348342

Subgroups — #28348342

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#20293286C211yesmaximal, characteristic
2#20292984C211yesmaximal, characteristic, complemented
3#20292968C211yesmaximal, characteristic, complemented
4#20292940C211yesmaximal, characteristic, complemented
5#20292931C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292849C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292782C211yesmaximal, characteristic, complemented
11C4×#13133808C211yesmaximal, characteristic, complemented
12D4×C107⋊C8C211yesmaximal, characteristic, complemented
13#20292621C211yesmaximal, characteristic, complemented
14C2×#16812032C211yesmaximal, characteristic, complemented
15#552×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×#13133805C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23#16812097C2211yescharacteristic
24#16812095—21no—
25#16812088C2211yescharacteristic
26#16812087—21no—
27#16812083C2211yescharacteristic
28#16812081—21no—
29C4×Dic214C2211yescharacteristic
30(C22⋊D107)⋊C4—21no—
31#16812042C2211yescharacteristic
32#16812042C2211yescharacteristic
33#16812040C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35#16812033C2211yescharacteristic
36#16812032C2211yescharacteristic
37#16812032C2211yescharacteristic
38#16812032C2211yescharacteristic
39#16812032C2211yescharacteristic
40#16812032C2211yescharacteristic
41#16812032C2211yescharacteristic
42#16812029C2211yescharacteristic
43#16812029C2211yescharacteristic
44#16812029C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48#126×C107—21no—
49#125×C107—21no—
50#125×C107C2211yescharacteristic
51#123×C107—21no—
52C22⋊Q8×C107C2211yescharacteristic
53C22⋊Q8×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56#117×C107—21no—
57#117×C107C2211yescharacteristic
58C2×C4⋊C4×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61C2×Dic214—22no—
62C2×Dic214—21no—
63C2×Dic214—41no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71#13133808—22no—
72#13133808—41no—
73#13133805—41no—
74C2×C107⋊C8—41no—
75C2×C107⋊C8—41no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C4×Dic107—21no—
88C2×Q8×C107C2311yescharacteristic
89C2×D4×C107D411yescharacteristic
90C22×C4×C107—21no—
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C4⋊C4×C107—21no—
94C4⋊C4×C107—21no—
95C4⋊C4×C107—21no—
96C4⋊C4×C107—21no—
97C4⋊C4×C107—21no—
98C4⋊C4×C107—21no—
99C4⋊C4×C107D411yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107D411yescharacteristic
107C22⋊C4×C107D411yescharacteristic
108C42×C107—21no—
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107C2×D411yescharacteristic
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116D4×C107—22no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—42no—
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—42no—
132C22⋊D107—42no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—42no—
137D428—21no—
138D428—21no—
139C4×D107—42no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143Dic214—42no—
144Dic214#4112yes—
145C107⋊C8—21no—
146C107⋊C8—21no—
147C107⋊C8—21no—
148C107⋊C8—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107#13512yes—
155C22×C107C22×D411yescharacteristic
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C4×C107—21no—
160C4×C107—21no—
161C4×C107—21no—
162C4×C107—21no—
163C4×C107—41no—
164C4×C107—41no—
165C4×C107—41no—
166C4×C107—41no—
167C4×C107C2×#4111yescharacteristic
168C4×C107H(5,𝔽2)11yescharacteristic
169Dic107—22no—
170Dic107—41no—
171Dic107—41no—
172C2×C107—22no—
173C2×C107—41no—
174C2×C107#54611yescharacteristic
175C2×C107C2×#13511yescharacteristic
176C2×C107#57811yescharacteristic
177D107—81no—
178#3300—1071nomaximal, Sylow
179C107#330011yescharacteristic, complemented, Sylow
180#552D10711yescharacteristic, complemented
181#548—1071no—
182#488—1071no—
183#485—1071no—
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—21no—
203#125—21no—
204#125C2×D10711yescharacteristic
205#124—1071no—
206#124—2141no—
207#123—21no—
208C22⋊Q8—1071no—
209C22⋊Q8—2141no—
210C22⋊Q8C2×D10711yescharacteristic
211C22⋊Q8C2×D10711yescharacteristic
212#121—1071no—
213#121—2141no—
214C4×Q8—1071no—
215C4×Q8C2×D10711yescharacteristic
216C4×D4—2141no—
217C4×D4C2×D10711yescharacteristic
218#117—21no—
219#117C2×D10711yescharacteristic
220C2×C4⋊C4C2×D10711yescharacteristic
221#107—1071no—
222#107—1071no—
223#106—1071no—
224C4⋊C8—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#103—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
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—1071no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4C22⋊D10711yescharacteristic
250C22×C4—21no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4C22×D10711yescharacteristic
254C22×C4C22×D10711yescharacteristic
255Q16—4281no—
256Q16—2142no—
257SD16—4281no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8—1071no—
261C2×C8—1071no—
262C2×C8—4281no—
263C2×C8—4281no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—1071no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4C22⋊D10711yescharacteristic
274C4⋊C4C22×D10711yescharacteristic
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4C22⋊D10711yescharacteristic
285C22⋊C4C22⋊D10711yescharacteristic
286C42—21no—
287C42—21no—
288C42—2141no—
289C42C22×D10711yescharacteristic
290C23—2141no—
291C23C2×C22⋊D10711yescharacteristic
292C23C2×C22⋊D10711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1072no—
296Q8—4282no—
297Q8—2141no—
298Q8—2141no—
299Q8—2141no—
300D4—22no—
301D4—4282no—
302D4—4282no—
303D4—2141no—
304D4—2141no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—4282no—
315C2×C4—4282no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4C2×C22⋊D10711yescharacteristic
320C2×C4C2×C22⋊D10711yescharacteristic
321C2×C4C2×C22⋊D10711yescharacteristic
322C2×C4C2×C22⋊D10711yescharacteristic
323C2×C4C23×D10711yescharacteristic
324C8—2141no—
325C8—2141no—
326C8—2141no—
327C8—2141no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—4281no—
334C22—4281no—
335C22—4281no—
336C22#1681212912yes—
337C22C22×C22⋊D10711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4281no—
347C4—4281no—
348C4—2142no—
349C4#41×D10711yescharacteristic
350C4#1681215511yescharacteristic
351C2—22no—
352C2—41no—
353C2—8561no—
354C2#2029326511yescharacteristic
355C2C2×#1681212911yescharacteristic
356C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#552×C107C2
socleC22×C107C22×D4
prime radical#20293286C2
hypercenter#552D107
hypocenterC107#3300
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20292984C2semidirect
3. #20292968C2semidirect
4. #20292940C2semidirect
5. #20292931C2semidirect
6. #20292866C2semidirect
7. #20292858C2semidirect
8. #20292849C2semidirect
9. #20292787C2semidirect
10. #20292782C2semidirect
11. C4×#13133808C2semidirect
12. D4×C107⋊C8C2semidirect
13. #20292621C2semidirect
14. C2×#16812032C2semidirect
15. #552×C107C2semidirect
179. C107#3300semidirect
180. #552D107semidirect

Maximal subgroups

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

subgroupindex
#202932862
#202929842
#202929682
#202929402
#202929312
#202928662
#202928582
#202928492
#202927872
#202927822
C4×#131338082
D4×C107⋊C82
#202926212
C2×#168120322
#552×C1072
#3300107