Finite Group Databasebeta

← #28347881

Subgroups — #28347881

Order 13696.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293248C211yesmaximal, characteristic
2#20292931C211yesmaximal, characteristic, complemented
3#20292924C211yesmaximal, characteristic, complemented
4#20292922C211yesmaximal, characteristic, complemented
5#20292912C211yesmaximal, characteristic, complemented
6#20292883C211yesmaximal, characteristic, complemented
7#20292875C211yesmaximal, characteristic
8#20292866C211yesmaximal, characteristic, complemented
9#20292858C211yesmaximal, characteristic
10#20292849C211yesmaximal, characteristic, complemented
11#20292849C211yesmaximal, characteristic, complemented
12#20292834C211yesmaximal, characteristic, complemented
13#20292829C211yesmaximal, characteristic, complemented
14#20292654C211yesmaximal, characteristic, complemented
15#541×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×#13133802C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22#16812109C2211yescharacteristic
23#16812104—21no—
24#16812097C2211yescharacteristic
25#16812095—21no—
26#16812091—21no—
27#16812088C2211yescharacteristic
28#16812087—21no—
29#16812083C2211yescharacteristic
30C4×Dic214C2211yescharacteristic
31C4×D428C2211yescharacteristic
32#16812042C2211yescharacteristic
33#16812042C2211yescharacteristic
34#16812040C2211yescharacteristic
35#16812040C2211yescharacteristic
36C4⋊(C107⋊C8)C2211yescharacteristic
37C4⋊(C107⋊C8)C2211yescharacteristic
38C4⋊(C107⋊C8)C2211yescharacteristic
39C4⋊(C107⋊C8)C2211yescharacteristic
40#16812033C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812033C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48C4⋊Q8×C107C2211yescharacteristic
49#126×C107—21no—
50#126×C107—21no—
51#125×C107—21no—
52#125×C107—21no—
53#125×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56C4×Q8×C107C2211yescharacteristic
57C4×Q8×C107C2211yescharacteristic
58#117×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C4×D107—21no—
64#13133808—41no—
65#13133808—41no—
66#13133805—41no—
67#13133805—41no—
68#13133802—41no—
69C2×C107⋊C8—41no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C22⋊Dic107—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84C4×Dic107—21no—
85C2×Q8×C107C2311yescharacteristic
86C2×Q8×C107C2311yescharacteristic
87C22×C4×C107C2311yescharacteristic
88C4⋊C4×C107—21no—
89C4⋊C4×C107—21no—
90C4⋊C4×C107—21no—
91C4⋊C4×C107—21no—
92C4⋊C4×C107—21no—
93C4⋊C4×C107—21no—
94C4⋊C4×C107—21no—
95C4⋊C4×C107—21no—
96C4⋊C4×C107D411yescharacteristic
97C4⋊C4×C107D411yescharacteristic
98C4⋊C4×C107C2311yescharacteristic
99C4⋊C4×C107C2311yescharacteristic
100C22⋊C4×C107—21no—
101C22⋊C4×C107—21no—
102C22⋊C4×C107D411yescharacteristic
103C22⋊C4×C107D411yescharacteristic
104C42×C107—21no—
105C42×C107—21no—
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107C2×D411yescharacteristic
109C22×D107—21no—
110Q8×C107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C22⋊D107—81no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130D428—21no—
131D428—21no—
132C4×D107—81no—
133Dic214—21no—
134Dic214—21no—
135Dic214—81no—
136C107⋊C8—21no—
137C107⋊C8—21no—
138C107⋊C8—21no—
139C107⋊C8—21no—
140C22×C107—21no—
141C22×C107—21no—
142C22×C107—21no—
143C22×C107C22×D411yescharacteristic
144C2×D107—41no—
145C2×D107—41no—
146C2×D107—41no—
147C4×C107—21no—
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—21no—
151C4×C107—21no—
152C4×C107—41no—
153C4×C107—41no—
154C4×C107—41no—
155C4×C107—41no—
156C4×C107H(5,𝔽2)11yescharacteristic
157C4×C1072−1+411yescharacteristic
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161C2×C107—41no—
162C2×C107#53711yescharacteristic
163C2×C107#57711yescharacteristic
164C2×C107#57811yescharacteristic
165D107—81no—
166#3258—1071nomaximal, Sylow
167C107#325811yescharacteristic, complemented, Sylow
168#542—1071no—
169#541D10711yescharacteristic, complemented
170#479—1071no—
171#478—1071no—
172#477—1071no—
173#475—1071no—
174#470—1071no—
175#468—1071no—
176#467—1071no—
177#465—1071no—
178#464—1071no—
179#464—1071no—
180#462—1071no—
181#461—1071no—
182#424—1071no—
183C2×Q16—1071no—
184C2×Q16—1071no—
185C2×SD16—1071no—
186C2×SD16—1071no—
187C2×M16—1071no—
188C22×C8—1071no—
189C4⋊Q8—1071no—
190C4⋊Q8C2×D10711yescharacteristic
191#126—21no—
192#126—21no—
193#126—2141no—
194#125—21no—
195#125—21no—
196#125C2×D10711yescharacteristic
197#124—1071no—
198#124—2141no—
199#123—2141no—
200C22⋊Q8—1071no—
201C22⋊Q8—2141no—
202C22⋊Q8C2×D10711yescharacteristic
203C22⋊Q8C2×D10711yescharacteristic
204#121—1071no—
205C4×Q8—1071no—
206C4×Q8C2×D10711yescharacteristic
207C4×Q8C2×D10711yescharacteristic
208C4×D4—1071no—
209#117C2×D10711yescharacteristic
210#107—1071no—
211#107—1071no—
212#106—1071no—
213#106—1071no—
214C4⋊C8—1071no—
215C4⋊C8—1071no—
216C4⋊C8—1071no—
217C4⋊C8—1071no—
218#103—1071no—
219#103—1071no—
220#103—1071no—
221#103—1071no—
222#103—1071no—
223#103—1071no—
224#102—1071no—
225#102—1071no—
226C2×Q8—1071no—
227C2×Q8—2141no—
228C2×Q8C22×D10711yescharacteristic
229C2×Q8C22×D10711yescharacteristic
230C2×D4—1071no—
231C2×D4—2141no—
232C22×C4—2141no—
233C22×C4C22×D10711yescharacteristic
234Q16—4281no—
235Q16—4281no—
236SD16—4281no—
237SD16—4281no—
238M16—4281no—
239C2×C8—1071no—
240C2×C8—1071no—
241C2×C8—1071no—
242C2×C8—1071no—
243C2×C8—4281no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—1071no—
253C4⋊C4—1071no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4C22⋊D10711yescharacteristic
258C4⋊C4C22⋊D10711yescharacteristic
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4C22⋊D10711yescharacteristic
269C22⋊C4C22⋊D10711yescharacteristic
270C42—21no—
271C42—21no—
272C42—2141no—
273C42C22×D10711yescharacteristic
274C42C22×D10711yescharacteristic
275C23—2141no—
276C23C2×C22⋊D10711yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—8561no—
282Q8—2141no—
283Q8—2141no—
284D4—8561no—
285D4—2141no—
286D4—2141no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—8561no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4C2×C22⋊D10711yescharacteristic
298C2×C4C2×C22⋊D10711yescharacteristic
299C2×C4C2×C22⋊D10711yescharacteristic
300C2×C4C2×C22⋊D10711yescharacteristic
301C2×C4C2×C22⋊D10711yescharacteristic
302C2×C4C23×D10711yescharacteristic
303C8—2141no—
304C8—2141no—
305C8—2141no—
306C8—2141no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—4281no—
311C22—4281no—
312C22—4281no—
313C22C22×C22⋊D10711yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—4281no—
324C4—4281no—
325C4—4281no—
326C4#1681215511yescharacteristic
327C4#1681215611yescharacteristic
328C2—41no—
329C2—8561no—
330C2#2029320511yescharacteristic
331C2#2029343511yescharacteristic
332C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#541×C107C2
socleC22×C107C22×D4
prime radical#16812083C22
hypercenter#541D107
hypocenterC107#3258
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20292931C2semidirect
3. #20292924C2semidirect
4. #20292922C2semidirect
5. #20292912C2semidirect
6. #20292883C2semidirect
8. #20292866C2semidirect
10. #20292849C2semidirect
11. #20292849C2semidirect
12. #20292834C2semidirect
13. #20292829C2semidirect
14. #20292654C2semidirect
15. #541×C107C2semidirect
167. C107#3258semidirect
169. #541D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202929312
#202929242
#202929222
#202929122
#202928832
#202928752
#202928662
#202928582
#202928492
#202928342
#202928292
#202926542
#541×C1072
#3258107