Finite Group Databasebeta

← #28347889

Subgroups — #28347889

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#20293239C211yesmaximal, characteristic, complemented
3#20292935C211yesmaximal, characteristic, complemented
4#20292928C211yesmaximal, characteristic, complemented
5#20292923C211yesmaximal, characteristic, complemented
6#20292913C211yesmaximal, characteristic, complemented
7#20292885C211yesmaximal, characteristic, complemented
8#20292873C211yesmaximal, characteristic
9#20292868C211yesmaximal, characteristic, complemented
10#20292856C211yesmaximal, characteristic
11#20292850C211yesmaximal, characteristic, complemented
12#20292850C211yesmaximal, characteristic, complemented
13#20292838C211yesmaximal, characteristic, complemented
14#20292833C211yesmaximal, characteristic, complemented
15#424×C107C211yesmaximal, characteristic, complemented
16C2×Dic428C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133807C2211yescharacteristic
20#16812109C2211yescharacteristic
21#16812109C2211yescharacteristic
22#16812104—21no—
23#16812102—21no—
24#16812102—21no—
25#16812101C2211yescharacteristic
26#16812099—21no—
27#16812099—21no—
28#16812097C2211yescharacteristic
29#16812095—21no—
30#16812091—21no—
31#16812088C2211yescharacteristic
32#16812088C2211yescharacteristic
33#16812088C2211yescharacteristic
34#16812087—21no—
35#16812083C2211yescharacteristic
36C4×Dic214C2211yescharacteristic
37C4×Dic214C2211yescharacteristic
38C4×Dic214C2211yescharacteristic
39C4×D428C2211yescharacteristic
40#16812043C2211yescharacteristic
41#16812043C2211yescharacteristic
42#16812041C2211yescharacteristic
43#16812041C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812034C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52C2×M16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54#117×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57C4⋊C8×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C2×D428C2311yescharacteristic
64C2×C22⋊D107—21no—
65C2×C4×D107—21no—
66Dic428—41no—
67Dic428—41no—
68#13133807—41no—
69#13133807—41no—
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107—21no—
95C22×C4×C107C2311yescharacteristic
96M16×C107—41no—
97C2×C8×C107—41no—
98C2×C8×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107D411yescharacteristic
103C4⋊C4×C107D411yescharacteristic
104C22⋊C4×C107D411yescharacteristic
105C22⋊C4×C107D411yescharacteristic
106C42×C107C2311yescharacteristic
107C42×C107C2311yescharacteristic
108C23×C107C2×D411yescharacteristic
109C22×D107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107C2×D411yescharacteristic
113C2×C4×C107C2×D411yescharacteristic
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2411yescharacteristic
118C22⋊D107—81no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126D428—21no—
127D428—21no—
128C4×D107—81no—
129Dic214—21no—
130Dic214—21no—
131Dic214—21no—
132Dic214—21no—
133Dic214—21no—
134Dic214—21no—
135Dic214—81no—
136C8×C107—21no—
137C8×C107—21no—
138C8×C107—21no—
139C8×C107—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×C107H(5,𝔽2)11yescharacteristic
153C4×C1072−1+411yescharacteristic
154Dic107—41no—
155Dic107—41no—
156Dic107—41no—
157Dic107—41no—
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#541—1071no—
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#424D10711yescharacteristic, complemented
183C2×Q16—1071no—
184C2×Q16—1071no—
185C2×SD16—1071no—
186C2×SD16—1071no—
187C2×M16C2×D10711yescharacteristic
188C22×C8C2×D10711yescharacteristic
189C4⋊Q8—1071no—
190C4⋊Q8—1071no—
191#126—2141no—
192#126—2141no—
193#126—2141no—
194#125—1071no—
195#125—2141no—
196#125—2141no—
197#124—1071no—
198#124—2141no—
199#123—2141no—
200C22⋊Q8—1071no—
201C22⋊Q8—1071no—
202C22⋊Q8—1071no—
203C22⋊Q8—2141no—
204#121—1071no—
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×Q8—1071no—
208C4×D4—1071no—
209#117C2×D10711yescharacteristic
210#107—1071no—
211#107—1071no—
212#106—1071no—
213#106—1071no—
214C4⋊C8C2×D10711yescharacteristic
215C4⋊C8C2×D10711yescharacteristic
216C4⋊C8C2×D10711yescharacteristic
217C4⋊C8C2×D10711yescharacteristic
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—1071no—
228C2×Q8—1071no—
229C2×Q8—2141no—
230C2×D4—1071no—
231C2×D4—2141no—
232C22×C4—2141no—
233C22×C4C22×D10711yescharacteristic
234Q16—4281no—
235Q16—4281no—
236SD16—4281no—
237SD16—4281no—
238M16—41no—
239C2×C8—41no—
240C2×C8C22×D10711yescharacteristic
241C2×C8C22×D10711yescharacteristic
242C2×C8C22×D10711yescharacteristic
243C2×C8C22×D10711yescharacteristic
244C4⋊C4—1071no—
245C4⋊C4—1071no—
246C4⋊C4—1071no—
247C4⋊C4—1071no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4D42811yescharacteristic
260C4⋊C4D42811yescharacteristic
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4D42811yescharacteristic
269C22⋊C4D42811yescharacteristic
270C42—2141no—
271C42—2141no—
272C42—2141no—
273C42C22×D10711yescharacteristic
274C42C22×D10711yescharacteristic
275C23—2141no—
276C23C2×D42811yescharacteristic
277Q8—8561no—
278Q8—2141no—
279Q8—2141no—
280Q8—2141no—
281Q8—2141no—
282Q8—2141no—
283Q8—2141no—
284D4—8561no—
285D4—2141no—
286D4—2141no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—8561no—
290C2×C4—2141no—
291C2×C4—2141no—
292C2×C4—2141no—
293C2×C4—2141no—
294C2×C4—2141no—
295C2×C4—2141no—
296C2×C4—2141no—
297C2×C4C2×D42811yescharacteristic
298C2×C4C2×D42811yescharacteristic
299C2×C4C2×D42811yescharacteristic
300C2×C4C2×D42811yescharacteristic
301C2×C4C2×D42811yescharacteristic
302C2×C4C23×D10711yescharacteristic
303C8—21no—
304C8—21no—
305C8—21no—
306C8—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—4281no—
311C22—4281no—
312C22—4281no—
313C22C22×D42811yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—4281no—
320C4—4281no—
321C4—4281no—
322C4—4281no—
323C4—4281no—
324C4—4281no—
325C4—4281no—
326C4#1681215411yescharacteristic
327C4#1681215711yescharacteristic
328C2—41no—
329C2—8561no—
330C2#2029320811yescharacteristic
331C2#2029343911yescharacteristic
332C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#424×C107C2
socleC22×C107C22×D4
prime radical#16812083C22
hypercenter#424D107
hypocenterC107#3258
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #20293239C2semidirect
3. #20292935C2semidirect
4. #20292928C2semidirect
5. #20292923C2semidirect
6. #20292913C2semidirect
7. #20292885C2semidirect
9. #20292868C2semidirect
11. #20292850C2semidirect
12. #20292850C2semidirect
13. #20292838C2semidirect
14. #20292833C2semidirect
15. #424×C107C2semidirect
167. C107#3258semidirect
182. #424D107semidirect

Maximal subgroups

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

subgroupindex
#202932482
#202932392
#202929352
#202929282
#202929232
#202929132
#202928852
#202928732
#202928682
#202928562
#202928502
#202928382
#202928332
#424×C1072
#3258107