Finite Group Databasebeta

← #28349124

Subgroups — #28349124

Order 13696.

362 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293360C211yesmaximal, characteristic, complemented
2#20293333C211yesmaximal, characteristic, complemented
3#20292984C211yesmaximal, characteristic, complemented
4#20292980C211yesmaximal, characteristic, complemented
5#20292972C211yesmaximal, characteristic, complemented
6#20292926C211yesmaximal, characteristic, complemented
7#20292921C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292841C211yesmaximal, characteristic, complemented
10#20292821C211yesmaximal, characteristic
11#20292751C211yesmaximal, characteristic, complemented
12#20292714C211yesmaximal, characteristic, complemented
13#20292709C211yesmaximal, characteristic, complemented
14C4×#13133803C211yesmaximal, characteristic, complemented
15#478×C107C211yesmaximal, characteristic, complemented
16C2×#13133807C2211yescharacteristic
17C2×D856C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812105C2211yescharacteristic
21#16812100C2211yescharacteristic
22#16812097C2211yescharacteristic
23#16812096C2211yescharacteristic
24#16812095—21no—
25#16812092—21no—
26#16812091—21no—
27#16812089—21no—
28#16812087—21no—
29#16812087—21no—
30C4×Dic214C2211yescharacteristic
31#16812074C2211yescharacteristic
32C4×D428C2211yescharacteristic
33#16812071C2211yescharacteristic
34#16812071C2211yescharacteristic
35#16812071C2211yescharacteristic
36C4×C22⋊D107—21no—
37C4⋊C4×D107—21no—
38#16812042C2211yescharacteristic
39Dic107⋊C8C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812032C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C4×C107⋊C8C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52#125×C107C2211yescharacteristic
53C4×D4×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×D428C2311yescharacteristic
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70#13133807—22no—
71D856—22no—
72#13133803—22no—
73C2×C107⋊C8C2311yescharacteristic
74C2×C107⋊C8C2311yescharacteristic
75C4⋊Dic107—21no—
76C4⋊Dic107D411yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C4×Dic107C2311yescharacteristic
94C2×D4×C107C2311yescharacteristic
95C22×C4×C107—21no—
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C4⋊C4×C107—21no—
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C22×D107—21no—
107C22×D107—21no—
108D4×C107—21no—
109D4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—81no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C22⋊D107—42no—
118C22⋊D107—81no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107C2×D411yescharacteristic
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125D428—21no—
126D428—21no—
127D428—21no—
128D428#4112yes—
129C4×D107—42no—
130C4×D107—81no—
131C4×D107—81no—
132C4×D107—81no—
133Dic214—22no—
134Dic214—21no—
135Dic214—21no—
136Dic214—81no—
137C8×C107—21no—
138C8×C107—21no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107C22×D411yescharacteristic
145C2×D107—22no—
146C2×D107—41no—
147C2×D107—41no—
148C2×D107—41no—
149C2×D107—41no—
150C2×D107—41no—
151C4×C107—21no—
152C4×C107—21no—
153C4×C107—41no—
154C4×C107—41no—
155C4×C107—41no—
156C4×C107C2×#4111yescharacteristic
157C4×C1072−1+411yescharacteristic
158Dic107—21no—
159Dic107—21no—
160Dic107—21no—
161Dic107—41no—
162Dic107—41no—
163Dic107#13512yes—
164C2×C107—81no—
165C2×C107#54811yescharacteristic
166C2×C107C2×#13511yescharacteristic
167C2×C107#57611yescharacteristic
168D107—42no—
169D107—81no—
170#3361—1071nomaximal, Sylow
171C107#336111yescharacteristic, complemented, Sylow
172#559—1071no—
173#554—1071no—
174#488—1071no—
175#488—1071no—
176#486—1071no—
177#478—1071no—
178#478D10711yescharacteristic, complemented
179#477—1071no—
180#463—1071no—
181#463—1071no—
182#459—1071no—
183C8×Q8—1071no—
184C4×SD16—1071no—
185C4×D8—1071no—
186C4×D8—1071no—
187C2×SD16—1071no—
188C2×D8—1071no—
189C2×D8—1071no—
190C4⋊Q8—1071no—
191#127—1071no—
192#125—1071no—
193#125C2×D10711yescharacteristic
194#124—1071no—
195#124—1071no—
196#124—2141no—
197#123—2141no—
198#123—2141no—
199C22⋊Q8—2141no—
200C22⋊Q8—2141no—
201C22⋊Q8—2141no—
202C4×Q8—1071no—
203C4×Q8—1071no—
204C4×D4—1071no—
205C4×D4—1071no—
206C4×D4—1071no—
207C4×D4—1071no—
208C4×D4—2141no—
209C4×D4C2×D10711yescharacteristic
210C2×C4⋊C4—2141no—
211#107—1071no—
212#107C2×D10711yescharacteristic
213#106C2×D10711yescharacteristic
214C4⋊C8—1071no—
215C4⋊C8—1071no—
216C4⋊C8C2×D10711yescharacteristic
217#103—1071no—
218#103—1071no—
219#103—1071no—
220#102—1071no—
221#102—1071no—
222#102—1071no—
223#102—1071no—
224#102—1071no—
225#102C2×D10711yescharacteristic
226#102C2×D10711yescharacteristic
227C4×C8—1071no—
228C4×C8—1071no—
229C4×C8—1071no—
230C2×Q8—1071no—
231C2×Q8—1071no—
232C2×Q8—2141no—
233C2×D4—1071no—
234C2×D4—1071no—
235C2×D4—2141no—
236C2×D4—2141no—
237C2×D4C22×D10711yescharacteristic
238C22×C4—21no—
239C22×C4—2141no—
240C22×C4—2141no—
241C22×C4—2141no—
242C22×C4—2141no—
243SD16—2142no—
244D8—2142no—
245D8—2142no—
246C2×C8—1071no—
247C2×C8—1071no—
248C2×C8C22×D10711yescharacteristic
249C2×C8C22×D10711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—1071no—
253C4⋊C4—1071no—
254C4⋊C4—1071no—
255C4⋊C4—1071no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4—2141no—
259C4⋊C4—2141no—
260C4⋊C4C22×D10711yescharacteristic
261C4⋊C4C22×D10711yescharacteristic
262C4⋊C4C22×D10711yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C42—1071no—
273C42—1071no—
274C42—2141no—
275C42C22×D10711yescharacteristic
276C23—21no—
277C23—2141no—
278C23—2141no—
279Q8—8561no—
280Q8—2142no—
281Q8—2141no—
282Q8—2141no—
283D4—21no—
284D4—21no—
285D4—8561no—
286D4—1072no—
287D4—4282no—
288D4—2141no—
289D4—2141no—
290D4—2141no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—81no—
295C2×C4—8561no—
296C2×C4—8561no—
297C2×C4—8561no—
298C2×C4—1071no—
299C2×C4—1071no—
300C2×C4—1071no—
301C2×C4—1071no—
302C2×C4—4282no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4D4×D10711yescharacteristic
306C2×C4D4×D10711yescharacteristic
307C2×C4C23×D10711yescharacteristic
308C8—21no—
309C8—21no—
310C8—2141no—
311C8—2141no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—4281no—
316C22—4281no—
317C22—4281no—
318C22—4281no—
319C22—4281no—
320C22—2142no—
321C22C2×D4×D10711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—1072no—
328C4—4281no—
329C4—4281no—
330C4—2141no—
331C4—2141no—
332C4—2141no—
333C4#41×D10711yescharacteristic
334C4#1681215711yescharacteristic
335C2—81no—
336C2—8561no—
337C2—4282no—
338C2#2029328711yescharacteristic
339C2#135×D10711yescharacteristic
340C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radical#20292821C2
hypercenter#478D107
hypocenterC107#3361
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20293360C2semidirect
2. #20293333C2semidirect
3. #20292984C2semidirect
4. #20292980C2semidirect
5. #20292972C2semidirect
6. #20292926C2semidirect
7. #20292921C2semidirect
8. #20292843C2semidirect
9. #20292841C2semidirect
11. #20292751C2semidirect
12. #20292714C2semidirect
13. #20292709C2semidirect
14. C4×#13133803C2semidirect
15. #478×C107C2semidirect
171. C107#3361semidirect
178. #478D107semidirect

Maximal subgroups

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

subgroupindex
#202933602
#202933332
#202929842
#202929802
#202929722
#202929262
#202929212
#202928432
#202928412
#202928212
#202927512
#202927142
#202927092
C4×#131338032
#478×C1072
#3361107