Finite Group Databasebeta

← #28346751

Subgroups — #28346751

Order 13696.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293228C211yesmaximal, characteristic
2#20293103C211yesmaximal, characteristic, complemented
3#20292954C211yesmaximal, characteristic, complemented
4#20292949C211yesmaximal, characteristic, complemented
5#20292936C211yesmaximal, characteristic, complemented
6#20292933C211yesmaximal, characteristic, complemented
7#20292826C211yesmaximal, characteristic, complemented
8#20292825C211yesmaximal, characteristic, complemented
9#20292767C211yesmaximal, characteristic, complemented
10#20292764C211yesmaximal, characteristic, complemented
11#20292729C211yesmaximal, characteristic, complemented
12C4×#13133805C211yesmaximal, characteristic, complemented
13C4×#13133806C211yesmaximal, characteristic, complemented
14#20292689C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20#16812105C2211yescharacteristic
21#16812100C2211yescharacteristic
22#16812092—21no—
23#16812091—21no—
24#16812084C2211yescharacteristic
25#16812083—21no—
26#16812082—21no—
27#16812080C2211yescharacteristic
28Q8×Dic107C2211yescharacteristic
29C4×Dic214—21no—
30C4×Dic214C2211yescharacteristic
31#16812074—21no—
32#16812074C2211yescharacteristic
33C4×D428C2211yescharacteristic
34#16812067—21no—
35#16812067C2211yescharacteristic
36#16812066—21no—
37#16812066C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812040C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C4×C107⋊C8C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61Q8×D107—22no—
62Q8×D107—22no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107C2311yescharacteristic
71#13133807—22no—
72#13133806—22no—
73#13133805—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
85C22⋊Dic107—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4D411yescharacteristic
90(C2×D107)⋊C4D411yescharacteristic
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99SD16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107—21no—
110C22×D107C2×D411yescharacteristic
111Q8×C107—21no—
112Q8×C107#4112yes—
113D4×C107—21no—
114D4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—81no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2411yescharacteristic
122C22⋊D107—81no—
123C22⋊D107—81no—
124C22⋊D107—81no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131D428—21no—
132D428—21no—
133D428—81no—
134C4×D107—24no—
135C4×D107—21no—
136C4×D107—21no—
137C4×D107—81no—
138Dic214—24no—
139Dic214—21no—
140Dic214#4112yes—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107C22×D411yescharacteristic
149C2×D107—21no—
150C2×D107—21no—
151C2×D107—21no—
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C4×C107—22no—
156C4×C107—21no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107C2×#4111yescharacteristic
161C4×C107C2×#4111yescharacteristic
162Dic107—21no—
163Dic107—22no—
164Dic107—21no—
165Dic107—21no—
166Dic107—41no—
167Dic107—41no—
168C2×C107—81no—
169C2×C107#52511yescharacteristic
170C2×C107#57711yescharacteristic
171C2×C107#57711yescharacteristic
172D107—41no—
173D107—81no—
174#3126—1071nomaximal, Sylow
175C107#312611yescharacteristic, complemented, Sylow
176#540—1071no—
177#519—1071no—
178#482—1071no—
179#482—1071no—
180#479—1071no—
181#479—1071no—
182#460—1071no—
183#460—1071no—
184#448—1071no—
185#448—1071no—
186#440—1071no—
187#440D10711yescharacteristic, complemented
188C4×SD16—1071no—
189C4×SD16—1071no—
190#433—1071no—
191C22×Q8—1071no—
192C2×SD16—1071no—
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×SD16C2×D10711yescharacteristic
196#127—1071no—
197#125—1071no—
198#123—2141no—
199#123—2141no—
200#121—1071no—
201#121—1071no—
202#121—2141no—
203#121—2141no—
204C4×Q8—1071no—
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×Q8—2141no—
208C4×Q8—2141no—
209C4×Q8C2×D10711yescharacteristic
210C4×D4—1071no—
211C4×D4C2×D10711yescharacteristic
212#117—1071no—
213#117—1071no—
214#117—2141no—
215#117—2141no—
216#107—1071no—
217#107C2×D10711yescharacteristic
218#106—1071no—
219#106—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222#103—1071no—
223#103—1071no—
224#103—1071no—
225#103C2×D10711yescharacteristic
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102C2×D10711yescharacteristic
230C22⋊C8—1071no—
231C22⋊C8—1071no—
232#97C2×D10711yescharacteristic
233C4×C8—1071no—
234C2×Q8—1071no—
235C2×Q8—2141no—
236C2×Q8—2142no—
237C2×Q8—2142no—
238C2×Q8C22×D10711yescharacteristic
239C2×D4—1071no—
240C2×D4—2141no—
241C2×D4—2141no—
242C2×D4—2141no—
243C2×D4—2141no—
244C2×D4C22×D10711yescharacteristic
245C22×C4—21no—
246C22×C4—1071no—
247C22×C4—2141no—
248C22×C4—2141no—
249SD16—22no—
250SD16—2142no—
251SD16—2142no—
252SD16—2142no—
253C2×C8—1071no—
254C2×C8—1071no—
255C2×C8C22×D10711yescharacteristic
256C2×C8C22×D10711yescharacteristic
257C4⋊C4—21no—
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⋊C4C22×D10711yescharacteristic
268C4⋊C4C22×D10711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1071no—
271C22⋊C4—1071no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C42—21no—
277C42—1071no—
278C42—2141no—
279C42—2141no—
280C42—2141no—
281C42—2141no—
282C42C22×D10711yescharacteristic
283C23—21no—
284C23—1071no—
285C23—2141no—
286Q8—21no—
287Q8—1072no—
288Q8—2144no—
289Q8—2141no—
290Q8#1313381812yes—
291D4—21no—
292D4—21no—
293D4—8561no—
294D4—8561no—
295D4—8561no—
296D4—8561no—
297D4—2141no—
298D4—2141no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—8561no—
304C2×C4—1071no—
305C2×C4—1071no—
306C2×C4—1071no—
307C2×C4—2144no—
308C2×C4—2141no—
309C2×C4—2141no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4D4×D10711yescharacteristic
314C2×C4D4×D10711yescharacteristic
315C2×C4C23×D10711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2141no—
319C8—2141no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—4281no—
324C22—4281no—
325C22—4281no—
326C22—2141no—
327C22—2141no—
328C22—2141no—
329C22C2×D4×D10711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—4281no—
336C4—4281no—
337C4—2141no—
338C4—2142no—
339C4—2141no—
340C4—2141no—
341C4#41×D10711yescharacteristic
342C4C2×#1313381811yescharacteristic
343C2—81no—
344C2—8561no—
345C2—4281no—
346C2#2029313611yescharacteristic
347C2#2029343611yescharacteristic
348C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radical#20293228C2
hypercenter#440D107
hypocenterC107#3126
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. #20293103C2semidirect
3. #20292954C2semidirect
4. #20292949C2semidirect
5. #20292936C2semidirect
6. #20292933C2semidirect
7. #20292826C2semidirect
8. #20292825C2semidirect
9. #20292767C2semidirect
10. #20292764C2semidirect
11. #20292729C2semidirect
12. C4×#13133805C2semidirect
13. C4×#13133806C2semidirect
14. #20292689C2semidirect
15. #440×C107C2semidirect
175. C107#3126semidirect
187. #440D107semidirect

Maximal subgroups

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

subgroupindex
#202932282
#202931032
#202929542
#202929492
#202929362
#202929332
#202928262
#202928252
#202927672
#202927642
#202927292
C4×#131338052
C4×#131338062
#202926892
#440×C1072
#3126107