Finite Group Databasebeta

← #28347888

Subgroups — #28347888

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#20293244C211yesmaximal, characteristic
2#20293237C211yesmaximal, characteristic, complemented
3#20292933C211yesmaximal, characteristic, complemented
4#20292929C211yesmaximal, characteristic, complemented
5#20292921C211yesmaximal, characteristic, complemented
6#20292910C211yesmaximal, characteristic, complemented
7#20292885C211yesmaximal, characteristic, complemented
8#20292866C211yesmaximal, characteristic, complemented
9#20292854C211yesmaximal, characteristic
10#20292848C211yesmaximal, characteristic, complemented
11#20292847C211yesmaximal, characteristic, complemented
12#20292836C211yesmaximal, characteristic, complemented
13#20292830C211yesmaximal, characteristic, complemented
14#20292655C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic
16C2×Dic428C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812102—21no—
23#16812102—21no—
24#16812102—21no—
25#16812100—21no—
26#16812100C2211yescharacteristic
27#16812099—21no—
28#16812096—21no—
29#16812096C2211yescharacteristic
30#16812090—21no—
31#16812088C2211yescharacteristic
32#16812088C2211yescharacteristic
33#16812086—21no—
34Q8×Dic107C2211yescharacteristic
35#16812074C2211yescharacteristic
36#16812074C2211yescharacteristic
37D4×Dic107C2211yescharacteristic
38#16812068C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812041C2211yescharacteristic
41#16812040C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812033C2211yescharacteristic
49#16812032C2211yescharacteristic
50#16812032C2211yescharacteristic
51#16812030C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C22×Dic107—21no—
63Dic428—41no—
64#13133808—41no—
65#13133806—41no—
66C2×C107⋊C8—41no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107D411yescharacteristic
72C4⋊Dic107D411yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107D411yescharacteristic
89C22⋊Dic107D411yescharacteristic
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107C2311yescharacteristic
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×D4×C107—21no—
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99SD16×C107—41no—
100M16×C107—41no—
101C2×C8×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C4⋊C4×C107—21no—
104C4⋊C4×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C22⋊C4×C107—21no—
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110Q8×C107—21no—
111Q8×C107—21no—
112D4×C107—21no—
113D4×C107—21no—
114D4×C107—81no—
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×C107C2411yescharacteristic
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—81no—
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131Dic214—21no—
132Dic214—21no—
133Dic214—21no—
134Dic214—21no—
135Dic214—81no—
136C8×C107—21no—
137C8×C107—21no—
138C107⋊C8—21no—
139C107⋊C8—21no—
140C22×C107—21no—
141C22×C107—21no—
142C22×C107—21no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107C22×D411yescharacteristic
147C4×C107—21no—
148C4×C107—41no—
149C4×C107—41no—
150C4×C107—41no—
151C4×C107H(5,𝔽2)11yescharacteristic
152C4×C1072−1+411yescharacteristic
153Dic107—21no—
154Dic107—21no—
155Dic107—21no—
156Dic107—21no—
157Dic107—41no—
158Dic107—41no—
159Dic107—41no—
160Dic107—41no—
161C2×C107—41no—
162C2×C107—81no—
163C2×C107#53711yescharacteristic
164C2×C107#57711yescharacteristic
165C2×C107#57811yescharacteristic
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#468D10711yescharacteristic
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×SD16C2×D10711yescharacteristic
187C2×M16C2×D10711yescharacteristic
188C22×C8—1071no—
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—2141no—
203C22⋊Q8C2×D10711yescharacteristic
204#121C2×D10711yescharacteristic
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×Q8—1071no—
208C4×D4—1071no—
209#117—1071no—
210#107—1071no—
211#107C2×D10711yescharacteristic
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#103C2×D10711yescharacteristic
224#102—1071no—
225#102C2×D10711yescharacteristic
226C2×Q8—1071no—
227C2×Q8—1071no—
228C2×Q8—2141no—
229C2×Q8C22×D10711yescharacteristic
230C2×D4—21no—
231C2×D4C22×D10711yescharacteristic
232C22×C4—2141no—
233C22×C4C22×D10711yescharacteristic
234Q16—4281no—
235Q16—4281no—
236SD16—41no—
237SD16—4281no—
238M16—41no—
239C2×C8—1071no—
240C2×C8—1071no—
241C2×C8—4281no—
242C2×C8C22×D10711yescharacteristic
243C2×C8C22×D10711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—1071no—
246C4⋊C4—1071no—
247C4⋊C4—1071no—
248C4⋊C4—1071no—
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⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—1071no—
264C22⋊C4—1071no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C42—1071no—
271C42—1071no—
272C42—2141no—
273C42—2141no—
274C42—2141no—
275C23—21no—
276C23D4×D10711yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—8561no—
280Q8—2141no—
281Q8—2141no—
282Q8—2141no—
283Q8—2141no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—8561no—
293C2×C4—1071no—
294C2×C4—1071no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—2141no—
298C2×C4—2141no—
299C2×C4—2141no—
300C2×C4—2141no—
301C2×C4D4×D10711yescharacteristic
302C2×C4C23×D10711yescharacteristic
303C8—21no—
304C8—21no—
305C8—2141no—
306C8—2141no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C2×D4×D10711yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—4281no—
319C4—4281no—
320C4—4281no—
321C4—4281no—
322C4—2141no—
323C4—2141no—
324C4—2141no—
325C4—2141no—
326C4#1681215411yescharacteristic
327C4#1681215611yescharacteristic
328C2—41no—
329C2—81no—
330C2#2029320411yescharacteristic
331C2#2029343711yescharacteristic
332C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
prime radical#121×C107C22
hypercenter#468D107
hypocenterC107#3258
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #20293237C2semidirect
3. #20292933C2semidirect
4. #20292929C2semidirect
5. #20292921C2semidirect
6. #20292910C2semidirect
7. #20292885C2semidirect
8. #20292866C2semidirect
10. #20292848C2semidirect
11. #20292847C2semidirect
12. #20292836C2semidirect
13. #20292830C2semidirect
14. #20292655C2semidirect
167. C107#3258semidirect

Maximal subgroups

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

subgroupindex
#202932442
#202932372
#202929332
#202929292
#202929212
#202929102
#202928852
#202928662
#202928542
#202928482
#202928472
#202928362
#202928302
#202926552
#468×C1072
#3258107