Finite Group Databasebeta

← #28347918

Subgroups — #28347918

Order 13696.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293250C211yesmaximal, characteristic, complemented
2#20293244C211yesmaximal, characteristic, complemented
3#20292921C211yesmaximal, characteristic, complemented
4#20292920C211yesmaximal, characteristic, complemented
5#20292906C211yesmaximal, characteristic, complemented
6#20292905C211yesmaximal, characteristic, complemented
7#20292873C211yesmaximal, characteristic, complemented
8#20292866C211yesmaximal, characteristic, complemented
9#20292861C211yesmaximal, characteristic
10#20292852C211yesmaximal, characteristic, complemented
11#20292847C211yesmaximal, characteristic, complemented
12#20292844C211yesmaximal, characteristic, complemented
13#20292841C211yesmaximal, characteristic, complemented
14#20292655C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812104—21no—
23#16812102—21no—
24#16812098C2211yescharacteristic
25#16812096—21no—
26#16812096C2211yescharacteristic
27#16812095—21no—
28#16812091—21no—
29#16812090—21no—
30#16812089—21no—
31#16812088C2211yescharacteristic
32#16812086—21no—
33#16812083C2211yescharacteristic
34Q8×Dic107C2211yescharacteristic
35#16812074C2211yescharacteristic
36D4×Dic107C2211yescharacteristic
37#16812071C2211yescharacteristic
38#16812068C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812042C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812030C2211yescharacteristic
51#16812029C2211yescharacteristic
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×Dic214—21no—
61C2×Dic214C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C22×Dic107—21no—
65C2×C4×D107—21no—
66#13133808—41no—
67#13133807—41no—
68#13133803—41no—
69C2×C107⋊C8—41no—
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C4⋊Dic107—21no—
73C4⋊Dic107D411yescharacteristic
74C4⋊Dic107D411yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81C22⋊Dic107—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107D411yescharacteristic
86C22⋊Dic107D411yescharacteristic
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—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
110C22×D107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—81no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2411yescharacteristic
123C22⋊D107—81no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—81no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132D428—21no—
133D428—21no—
134C4×D107—81no—
135Dic214—21no—
136Dic214—21no—
137Dic214—81no—
138Dic214—81no—
139C8×C107—21no—
140C8×C107—21no—
141C107⋊C8—21no—
142C107⋊C8—21no—
143C22×C107—21no—
144C22×C107—21no—
145C22×C107—21no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107C22×D411yescharacteristic
150C2×D107—41no—
151C2×D107—41no—
152C2×D107—41no—
153C4×C107—21no—
154C4×C107—41no—
155C4×C107—41no—
156C4×C107—41no—
157C4×C107H(5,𝔽2)11yescharacteristic
158C4×C1072−1+411yescharacteristic
159Dic107—21no—
160Dic107—21no—
161Dic107—21no—
162Dic107—21no—
163Dic107—41no—
164Dic107—41no—
165Dic107—41no—
166C2×C107—41no—
167C2×C107—81no—
168C2×C107#53711yescharacteristic
169C2×C107#57611yescharacteristic
170C2×C107#57811yescharacteristic
171D107—81no—
172#3262—1071nomaximal, Sylow
173C107#326211yescharacteristic, complemented, Sylow
174#542—1071no—
175#542—1071no—
176#477—1071no—
177#477—1071no—
178#474—1071no—
179#474—1071no—
180#468—1071no—
181#468D10711yescharacteristic, complemented
182#467—1071no—
183#466—1071no—
184#464—1071no—
185#464—1071no—
186#463—1071no—
187#463—1071no—
188#424—1071no—
189C2×Q16—1071no—
190C2×SD16—1071no—
191C2×SD16C2×D10711yescharacteristic
192C2×D8—1071no—
193C2×M16C2×D10711yescharacteristic
194C22×C8—1071no—
195C4⋊Q8—1071no—
196C4⋊Q8—1071no—
197#126—2141no—
198#126—2141no—
199#124—1071no—
200#124—1071no—
201#124—2141no—
202#124—2141no—
203#123—2141no—
204#123—2141no—
205C22⋊Q8—1071no—
206C22⋊Q8—2141no—
207C22⋊Q8—2141no—
208C22⋊Q8C2×D10711yescharacteristic
209#121—1071no—
210#121C2×D10711yescharacteristic
211C4×Q8—1071no—
212C4×Q8—1071no—
213C4×D4—1071no—
214C4×D4—1071no—
215#117—1071no—
216#107—1071no—
217#107—1071no—
218#107—1071no—
219#107C2×D10711yescharacteristic
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223C4⋊C8—1071no—
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103C2×D10711yescharacteristic
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102C2×D10711yescharacteristic
232C2×Q8—1071no—
233C2×Q8—2141no—
234C2×Q8—2141no—
235C2×Q8C22×D10711yescharacteristic
236C2×D4—21no—
237C2×D4—1071no—
238C2×D4—2141no—
239C2×D4C22×D10711yescharacteristic
240C22×C4—2141no—
241C22×C4—2141no—
242C22×C4C22×D10711yescharacteristic
243Q16—4281no—
244SD16—41no—
245SD16—4281no—
246D8—4281no—
247M16—41no—
248C2×C8—1071no—
249C2×C8—1071no—
250C2×C8—4281no—
251C2×C8C22×D10711yescharacteristic
252C2×C8C22×D10711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—1071no—
255C4⋊C4—1071no—
256C4⋊C4—1071no—
257C4⋊C4—1071no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4C22×D10711yescharacteristic
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—1071no—
268C22⋊C4—1071no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C42—1071no—
278C42—1071no—
279C42—2141no—
280C42—2141no—
281C23—21no—
282C23—2141no—
283C23D4×D10711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—8561no—
287Q8—8561no—
288Q8—2141no—
289Q8—2141no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—8561no—
294D4—2141no—
295D4—2141no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—8561no—
302C2×C4—8561no—
303C2×C4—1071no—
304C2×C4—1071no—
305C2×C4—1071no—
306C2×C4—1071no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2141no—
310C2×C4D4×D10711yescharacteristic
311C2×C4C23×D10711yescharacteristic
312C8—21no—
313C8—21no—
314C8—2141no—
315C8—2141no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—4281no—
323C22—4281no—
324C22—4281no—
325C22C2×D4×D10711yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—4281no—
331C4—4281no—
332C4—4281no—
333C4—2141no—
334C4—2141no—
335C4—2141no—
336C4—2141no—
337C4#1681215411yescharacteristic
338C4#1681215611yescharacteristic
339C2—41no—
340C2—81no—
341C2—8561no—
342C2#2029320411yescharacteristic
343C2#2029343111yescharacteristic
344C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
prime radical#20292861C2
hypercenter#468D107
hypocenterC107#3262
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
1. #20293250C2semidirect
2. #20293244C2semidirect
3. #20292921C2semidirect
4. #20292920C2semidirect
5. #20292906C2semidirect
6. #20292905C2semidirect
7. #20292873C2semidirect
8. #20292866C2semidirect
10. #20292852C2semidirect
11. #20292847C2semidirect
12. #20292844C2semidirect
13. #20292841C2semidirect
14. #20292655C2semidirect
15. #468×C107C2semidirect
173. C107#3262semidirect
181. #468D107semidirect

Maximal subgroups

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

subgroupindex
#202932502
#202932442
#202929212
#202929202
#202929062
#202929052
#202928732
#202928662
#202928612
#202928522
#202928472
#202928442
#202928412
#202926552
#468×C1072
#3262107