Finite Group Databasebeta

← #28349536

Subgroups — #28349536

Order 13696.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293350C211yesmaximal, characteristic, complemented
2#20293022C211yesmaximal, characteristic, complemented
3#20293019C211yesmaximal, characteristic, complemented
4#20293009C211yesmaximal, characteristic
5#20292849C211yesmaximal, characteristic, complemented
6#20292846C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic
8#20292839C211yesmaximal, characteristic
9#20292837C211yesmaximal, characteristic, complemented
10#20292834C211yesmaximal, characteristic, complemented
11#20292758C211yesmaximal, characteristic, complemented
12#20292727C211yesmaximal, characteristic, complemented
13#20292725C211yesmaximal, characteristic, complemented
14C4×#13133808C211yesmaximal, characteristic, complemented
15#556×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×#13133806C2211yescharacteristic
21C2×#13133806C2211yescharacteristic
22C2×#13133805C2211yescharacteristic
23C2×#13133805C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25#16812109C2211yescharacteristic
26#16812104—21no—
27#16812104—21no—
28#16812097C2211yescharacteristic
29#16812097C2211yescharacteristic
30#16812085—21no—
31C4×Dic214C2211yescharacteristic
32C4×Dic214C2211yescharacteristic
33C4×D428C2211yescharacteristic
34#16812067—21no—
35#16812042C2211yescharacteristic
36C4⋊(C107⋊C8)C2211yescharacteristic
37C4⋊(C107⋊C8)C2211yescharacteristic
38C4⋊(C107⋊C8)C2211yescharacteristic
39#16812033C2211yescharacteristic
40#16812033C2211yescharacteristic
41#16812032C2211yescharacteristic
42#16812032C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812029C2211yescharacteristic
45#16812020C2211yescharacteristic
46#16812020C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48C4⋊Q8×C107C2211yescharacteristic
49#126×C107—21no—
50#126×C107—21no—
51#124×C107C2211yescharacteristic
52#124×C107C2211yescharacteristic
53C22⋊Q8×C107—21no—
54C4×Q8×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4×Q8×C107C2211yescharacteristic
57C4×D4×C107C2211yescharacteristic
58#117×C107—21no—
59C2×Dic214C2311yescharacteristic
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C4×D107—21no—
63#13133808—22no—
64#13133808—41no—
65#13133808—41no—
66#13133808—41no—
67#13133806—22no—
68#13133806—41no—
69#13133805—22no—
70#13133805—41no—
71#13133803—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86C4×Dic107—21no—
87C4×Dic107—21no—
88C2×Q8×C107D411yescharacteristic
89C2×Q8×C107C2311yescharacteristic
90C2×Q8×C107C2311yescharacteristic
91C2×D4×C107C2311yescharacteristic
92C22×C4×C107—21no—
93C4⋊C4×C107—21no—
94C4⋊C4×C107—21no—
95C4⋊C4×C107—21no—
96C4⋊C4×C107—21no—
97C4⋊C4×C107D411yescharacteristic
98C4⋊C4×C107D411yescharacteristic
99C4⋊C4×C107D411yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C42×C107—21no—
107C42×C107—21no—
108C42×C107C2311yescharacteristic
109C42×C107C2311yescharacteristic
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C22×D107—21no—
113Q8×C107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107—41no—
118D4×C107—21no—
119D4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—42no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134D428—21no—
135D428—21no—
136C4×D107—42no—
137Dic214—21no—
138Dic214—21no—
139Dic214—21no—
140Dic214—21no—
141C107⋊C8—21no—
142C107⋊C8—21no—
143C107⋊C8—21no—
144C107⋊C8#4112yes—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107C22×D411yescharacteristic
149C2×D107—41no—
150C2×D107—41no—
151C2×D107—41no—
152C4×C107—21no—
153C4×C107—21no—
154C4×C107—21no—
155C4×C107—21no—
156C4×C107—21no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107C2×#4111yescharacteristic
162C4×C107H(5,𝔽2)11yescharacteristic
163Dic107—41no—
164Dic107—41no—
165Dic107—41no—
166C2×C107—81no—
167C2×C107#55011yescharacteristic
168C2×C107#57511yescharacteristic
169C2×C107#57811yescharacteristic
170D107—81no—
171#3402—1071nomaximal, Sylow
172C107#340211yescharacteristic, complemented, Sylow
173#556—1071no—
174#556D10711yescharacteristic, complemented
175#497—1071no—
176#497—1071no—
177#495—1071no—
178#464—1071no—
179#464—1071no—
180#463—1071no—
181#463—1071no—
182#462—1071no—
183#462—1071no—
184#446—1071no—
185#440—1071no—
186#440—1071no—
187C4×Q16—1071no—
188C2×Q16—1071no—
189C2×Q16—1071no—
190C2×Q16—1071no—
191C2×Q16—1071no—
192C2×SD16—1071no—
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×SD16—1071no—
196C2×D8—1071no—
197C4⋊Q8—1071no—
198C4⋊Q8C2×D10711yescharacteristic
199#126—21no—
200#126—21no—
201#126—2141no—
202#126—2141no—
203#124—1071no—
204#124—1071no—
205#124C2×D10711yescharacteristic
206#124C2×D10711yescharacteristic
207C22⋊Q8—21no—
208C22⋊Q8—2141no—
209C4×Q8—1071no—
210C4×Q8—1071no—
211C4×Q8C2×D10711yescharacteristic
212C4×Q8C2×D10711yescharacteristic
213C4×Q8C2×D10711yescharacteristic
214C4×D4—1071no—
215C4×D4C2×D10711yescharacteristic
216#117—21no—
217#117—2141no—
218#107—1071no—
219C4⋊C8—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222#103—1071no—
223#103—1071no—
224#103—1071no—
225#103—1071no—
226#102—1071no—
227#102—1071no—
228#97—1071no—
229#97—1071no—
230C4×C8—1071no—
231C2×Q8—1071no—
232C2×Q8—1071no—
233C2×Q8C22⋊D10711yescharacteristic
234C2×Q8C22×D10711yescharacteristic
235C2×Q8C22×D10711yescharacteristic
236C2×D4—1071no—
237C2×D4C22×D10711yescharacteristic
238C22×C4—21no—
239C22×C4—2141no—
240Q16—4281no—
241Q16—4281no—
242Q16—4281no—
243Q16—2142no—
244SD16—4281no—
245SD16—4281no—
246SD16—2142no—
247SD16—2142no—
248D8—4281no—
249C2×C8—1071no—
250C2×C8—1071no—
251C2×C8—1071no—
252C2×C8—1071no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—1071no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4C22⋊D10711yescharacteristic
263C4⋊C4C22⋊D10711yescharacteristic
264C4⋊C4C22⋊D10711yescharacteristic
265C4⋊C4C22×D10711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C42—21no—
277C42—21no—
278C42—2141no—
279C42—2141no—
280C42C22×D10711yescharacteristic
281C42C22×D10711yescharacteristic
282C42C22×D10711yescharacteristic
283C23—21no—
284C23—2141no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—2141no—
291Q8—2141no—
292Q8—2141no—
293Q8—2141no—
294D4—21no—
295D4—21no—
296D4—2141no—
297D4—2141no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—4282no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4—2141no—
306C2×C4C2×C22⋊D10711yescharacteristic
307C2×C4C2×C22⋊D10711yescharacteristic
308C2×C4C2×C22⋊D10711yescharacteristic
309C2×C4C2×C22⋊D10711yescharacteristic
310C2×C4C2×C22⋊D10711yescharacteristic
311C2×C4C2×C22⋊D10711yescharacteristic
312C2×C4C23×D10711yescharacteristic
313C8—1072no—
314C8—2141no—
315C8—2141no—
316C8—2141no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—4281no—
321C22—4281no—
322C22—4281no—
323C22C22×C22⋊D10711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4281no—
334C4—4281no—
335C4—4281no—
336C4#41×D10711yescharacteristic
337C4#1681215511yescharacteristic
338C2—81no—
339C2—8561no—
340C2#2029329611yescharacteristic
341C2#2029342511yescharacteristic
342C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#556×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#556D107
hypocenterC107#3402
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20293350C2semidirect
2. #20293022C2semidirect
3. #20293019C2semidirect
5. #20292849C2semidirect
6. #20292846C2semidirect
9. #20292837C2semidirect
10. #20292834C2semidirect
11. #20292758C2semidirect
12. #20292727C2semidirect
13. #20292725C2semidirect
14. C4×#13133808C2semidirect
15. #556×C107C2semidirect
172. C107#3402semidirect
174. #556D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202930222
#202930192
#202930092
#202928492
#202928462
#202928422
#202928392
#202928372
#202928342
#202927582
#202927272
#202927252
C4×#131338082
#556×C1072
#3402107