Finite Group Databasebeta

← #28347905

Subgroups — #28347905

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#20293235C211yesmaximal, characteristic, complemented
2#20293231C211yesmaximal, characteristic
3#20292935C211yesmaximal, characteristic, complemented
4#20292934C211yesmaximal, characteristic, complemented
5#20292929C211yesmaximal, characteristic, complemented
6#20292925C211yesmaximal, characteristic, complemented
7#20292872C211yesmaximal, characteristic, complemented
8#20292870C211yesmaximal, characteristic, complemented
9#20292855C211yesmaximal, characteristic, complemented
10#20292853C211yesmaximal, characteristic, complemented
11#20292830C211yesmaximal, characteristic, complemented
12#20292830C211yesmaximal, characteristic, complemented
13#20292824C211yesmaximal, characteristic, complemented
14#20292652C211yesmaximal, characteristic, complemented
15#460×C107C211yesmaximal, characteristic, complemented
16C2×#13133806C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×C8×D107C2211yescharacteristic
20#16812107C2211yescharacteristic
21#16812104—21no—
22#16812103—21no—
23#16812101—21no—
24#16812101C2211yescharacteristic
25#16812100C2211yescharacteristic
26#16812099—21no—
27#16812090—21no—
28#16812090—21no—
29#16812089C2211yescharacteristic
30#16812085C2211yescharacteristic
31#16812081—21no—
32#16812081—21no—
33#16812080C2211yescharacteristic
34#16812080C2211yescharacteristic
35C4×Dic214C2211yescharacteristic
36D4×Dic107C2211yescharacteristic
37D4×Dic107C2211yescharacteristic
38#16812066C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812040C2211yescharacteristic
43C4⋊(C107⋊C8)C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812033C2211yescharacteristic
49#16812033C2211yescharacteristic
50#16812030C2211yescharacteristic
51#16812030C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54#127×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C2×C22⋊D107—21no—
61C2×C22⋊D107—21no—
62C22×Dic107—21no—
63C22×Dic107—21no—
64C2×C4×D107C2311yescharacteristic
65#13133806—41no—
66#13133806—41no—
67#13133801—41no—
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2311yescharacteristic
70C8×D107—41no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92(C2×D107)⋊C4D411yescharacteristic
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×D4×C107—21no—
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C2×D4×C107C2311yescharacteristic
100SD16×C107—41no—
101SD16×C107—41no—
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107C2311yescharacteristic
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107—21no—
110C22×D107C2×D411yescharacteristic
111Q8×C107—21no—
112Q8×C107—21no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—21no—
116D4×C107—21no—
117D4×C107—81no—
118D4×C107—81no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C22⋊D107—81no—
125C22⋊D107—81no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—81no—
131C2×Dic107—81no—
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C4×D107—21no—
136C4×D107—21no—
137Dic214—21no—
138Dic214—21no—
139C8×C107—21no—
140C8×C107—21no—
141C107⋊C8—21no—
142C107⋊C8—21no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107C22×D411yescharacteristic
150C2×D107—21no—
151C2×D107—21no—
152C2×D107—21no—
153C4×C107—21no—
154C4×C107—21no—
155C4×C107—41no—
156C4×C107—41no—
157C4×C107H(5,𝔽2)11yescharacteristic
158C4×C1072−1+411yescharacteristic
159Dic107—21no—
160Dic107—21no—
161Dic107—21no—
162Dic107—41no—
163Dic107—41no—
164Dic107—41no—
165Dic107—41no—
166C2×C107—81no—
167C2×C107—81no—
168C2×C107#53711yescharacteristic
169C2×C107#57711yescharacteristic
170C2×C107#57711yescharacteristic
171D107—41no—
172#3260—1071nomaximal, Sylow
173C107#326011yescharacteristic, complemented, Sylow
174#541—1071no—
175#540—1071no—
176#479—1071no—
177#479—1071no—
178#478—1071no—
179#478—1071no—
180#468—1071no—
181#468—1071no—
182#465—1071no—
183#465—1071no—
184#461—1071no—
185#461—1071no—
186#460—1071no—
187#460D10711yescharacteristic, complemented
188#424—1071no—
189C2×SD16—1071no—
190C2×SD16—1071no—
191C2×SD16C2×D10711yescharacteristic
192C2×SD16C2×D10711yescharacteristic
193C2×M16—1071no—
194C22×C8—1071no—
195C4⋊Q8—1071no—
196#127C2×D10711yescharacteristic
197#126—2141no—
198#126—2141no—
199#125—1071no—
200#125—1071no—
201#125—2141no—
202#125—2141no—
203#123—2141no—
204#123—2141no—
205C22⋊Q8—1071no—
206C22⋊Q8—1071no—
207#121—1071no—
208#121—1071no—
209#121—2141no—
210#121—2141no—
211C4×Q8—1071no—
212C4×Q8C2×D10711yescharacteristic
213C4×D4—1071no—
214C4×D4—1071no—
215#117—1071no—
216#107—1071no—
217#107—1071no—
218#106—1071no—
219#106—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223C4⋊C8C2×D10711yescharacteristic
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#102—1071no—
229#102—1071no—
230#102C2×D10711yescharacteristic
231#102C2×D10711yescharacteristic
232C2×Q8—1071no—
233C2×Q8C22×D10711yescharacteristic
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—2141no—
237C2×D4—2141no—
238C2×D4C22×D10711yescharacteristic
239C2×D4C22×D10711yescharacteristic
240C22×C4—1071no—
241C22×C4—2141no—
242C22×C4—2141no—
243SD16—41no—
244SD16—41no—
245SD16—4281no—
246SD16—4281no—
247M16—4281no—
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—1071no—
259C4⋊C4—2141no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4C22×D10711yescharacteristic
269C22⋊C4—1071no—
270C22⋊C4—1071no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C42—21no—
278C42—1071no—
279C42—2141no—
280C42C22×D10711yescharacteristic
281C23—21no—
282C23—21no—
283C23—1071no—
284Q8—21no—
285Q8—21no—
286Q8—2141no—
287Q8—2141no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—81no—
294D4—8561no—
295D4—8561no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—8561no—
299C2×C4—8561no—
300C2×C4—1071no—
301C2×C4—1071no—
302C2×C4—1071no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4—2141no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4D4×D10711yescharacteristic
310C2×C4D4×D10711yescharacteristic
311C2×C4C23×D10711yescharacteristic
312C8—21no—
313C8—21no—
314C8—2141no—
315C8—2141no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22C2×D4×D10711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—4281no—
331C4—4281no—
332C4—4281no—
333C4—4281no—
334C4—2141no—
335C4—2141no—
336C4—2141no—
337C4#1681215411yescharacteristic
338C4#1681215611yescharacteristic
339C2—81no—
340C2—81no—
341C2—4281no—
342C2#2029320611yescharacteristic
343C2#2029343311yescharacteristic
344C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293231C2
hypercenter#460D107
hypocenterC107#3260
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #20293235C2semidirect
3. #20292935C2semidirect
4. #20292934C2semidirect
5. #20292929C2semidirect
6. #20292925C2semidirect
7. #20292872C2semidirect
8. #20292870C2semidirect
9. #20292855C2semidirect
10. #20292853C2semidirect
11. #20292830C2semidirect
12. #20292830C2semidirect
13. #20292824C2semidirect
14. #20292652C2semidirect
15. #460×C107C2semidirect
173. C107#3260semidirect
187. #460D107semidirect

Maximal subgroups

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

subgroupindex
#202932352
#202932312
#202929352
#202929342
#202929292
#202929252
#202928722
#202928702
#202928552
#202928532
#202928302
#202928242
#202926522
#460×C1072
#3260107