Finite Group Databasebeta

← #28349207

Subgroups — #28349207

Order 13696.

358 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#20293363C211yesmaximal, characteristic, complemented
2#20293336C211yesmaximal, characteristic, complemented
3#20292991C211yesmaximal, characteristic, complemented
4#20292985C211yesmaximal, characteristic, complemented
5#20292980C211yesmaximal, characteristic, complemented
6#20292935C211yesmaximal, characteristic, complemented
7#20292929C211yesmaximal, characteristic, complemented
8#20292905C211yesmaximal, characteristic, complemented
9#20292840C211yesmaximal, characteristic, complemented
10#20292840C211yesmaximal, characteristic, complemented
11#20292756C211yesmaximal, characteristic, complemented
12#20292739C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14D8×Dic107C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic
16C2×#13133806C2211yescharacteristic
17#16812108C2211yescharacteristic
18#16812101C2211yescharacteristic
19#16812100C2211yescharacteristic
20#16812096—21no—
21#16812096C2211yescharacteristic
22#16812096C2211yescharacteristic
23#16812090—21no—
24#16812090—21no—
25#16812088—21no—
26#16812086—21no—
27#16812086—21no—
28C4×Dic214C2211yescharacteristic
29C4×Dic214C2211yescharacteristic
30D4×Dic107—21no—
31D4×Dic107C2211yescharacteristic
32D4×Dic107C2211yescharacteristic
33D4×Dic107C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35C2×Dic107⋊C4—21no—
36#16812043C2211yescharacteristic
37#16812042C2211yescharacteristic
38#16812041C2211yescharacteristic
39Dic107⋊C8C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812020C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×D8×C107C2211yescharacteristic
53C2×D8×C107C2211yescharacteristic
54#127×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C22×Dic107—21no—
63C22×Dic107—21no—
64C22×Dic107—21no—
65C22×Dic107—21no—
66#13133806—22no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C4⋊Dic107—21no—
70C4⋊Dic107D411yescharacteristic
71C4⋊Dic107C2311yescharacteristic
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74Dic107⋊C4—21no—
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4D411yescharacteristic
80Dic107⋊C4D411yescharacteristic
81C22⋊Dic107—21no—
82C22⋊Dic107—21no—
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107C2311yescharacteristic
91C4×Dic107C2311yescharacteristic
92C2×D4×C107—21no—
93C2×D4×C107—21no—
94C2×D4×C107C2311yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C2×D4×C107C2311yescharacteristic
97C22×C4×C107—21no—
98D8×C107—22no—
99D8×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C23×C107—21no—
107C23×C107—21no—
108D4×C107—21no—
109D4×C107—21no—
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—21no—
113D4×C107—42no—
114D4×C107—81no—
115D4×C107#4112yes—
116C2×C4×C107—21no—
117C2×C4×C107—81no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2411yescharacteristic
121C2×Dic107—21no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107—42no—
126C2×Dic107—81no—
127C2×Dic107—81no—
128C2×Dic107—81no—
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133Dic214—21no—
134Dic214—21no—
135Dic214—41no—
136Dic214—81no—
137C8×C107—21no—
138C8×C107—21no—
139C107⋊C8—21no—
140C107⋊C8—21no—
141C22×C107—22no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107—41no—
145C22×C107—41no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107C22×D411yescharacteristic
151C4×C107—21no—
152C4×C107—21no—
153C4×C107—41no—
154C4×C107C2×#4111yescharacteristic
155C4×C1072−1+411yescharacteristic
156Dic107—21no—
157Dic107—21no—
158Dic107—21no—
159Dic107—21no—
160Dic107—41no—
161Dic107—41no—
162Dic107—41no—
163Dic107—41no—
164C2×C107—42no—
165C2×C107—81no—
166C2×C107—81no—
167C2×C107#54811yescharacteristic
168C2×C107#57511yescharacteristic
169C2×C107#57611yescharacteristic
170#3370—1071nomaximal, Sylow
171C107#337011yescharacteristic, complemented, Sylow
172#559—1071no—
173#554—1071no—
174#490—1071no—
175#489—1071no—
176#488—1071no—
177#479—1071no—
178#478—1071no—
179#474—1071no—
180#463—1071no—
181#463—1071no—
182#459D10711yescharacteristic
183#446—1071no—
184#442—1071no—
185#440—1071no—
186C4×D8—1071no—
187C2×SD16—1071no—
188C2×D8C2×D10711yescharacteristic
189C2×D8C2×D10711yescharacteristic
190C4⋊Q8—1071no—
191#127C2×D10711yescharacteristic
192#125—1071no—
193#125—1071no—
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#107—1071no—
213#106—1071no—
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
227#97—1071no—
228#97—1071no—
229C4×C8—1071no—
230C2×Q8—1071no—
231C2×Q8—1071no—
232C2×Q8—2141no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4C22×D10711yescharacteristic
236C2×D4C22×D10711yescharacteristic
237C2×D4C22×D10711yescharacteristic
238C22×C4—21no—
239C22×C4—2141no—
240C22×C4—2141no—
241C22×C4—2141no—
242C22×C4—2141no—
243SD16—2142no—
244D8—22no—
245D8—22no—
246C2×C8—1071no—
247C2×C8—1071no—
248C2×C8C22×D10711yescharacteristic
249C2×C8C22×D10711yescharacteristic
250C4⋊C4—1071no—
251C4⋊C4—1071no—
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⋊C4—2141no—
261C4⋊C4—2141no—
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—21no—
278C23—21no—
279Q8—8561no—
280Q8—4281no—
281Q8—2141no—
282Q8—2141no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—42no—
289D4—81no—
290D4#1313381712yes—
291C2×C4—21no—
292C2×C4—81no—
293C2×C4—8561no—
294C2×C4—8561no—
295C2×C4—8561no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—1071no—
299C2×C4—1071no—
300C2×C4—4282no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2141no—
305C2×C4D4×D10711yescharacteristic
306C2×C4D4×D10711yescharacteristic
307C2×C4C23×D10711yescharacteristic
308C8—21no—
309C8—21no—
310C8—2141no—
311C8—2141no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C2×D4×D10711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—4281no—
326C4—4281no—
327C4—4281no—
328C4—4281no—
329C4—2141no—
330C4—2141no—
331C4—2141no—
332C4—2141no—
333C4C2×#1313381711yescharacteristic
334C4#1681215611yescharacteristic
335C2—42no—
336C2—81no—
337C2—81no—
338C2#2029328311yescharacteristic
339C2#2029342311yescharacteristic
340C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3370
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20293363C2semidirect
2. #20293336C2semidirect
3. #20292991C2semidirect
4. #20292985C2semidirect
5. #20292980C2semidirect
6. #20292935C2semidirect
7. #20292929C2semidirect
8. #20292905C2semidirect
9. #20292840C2semidirect
10. #20292840C2semidirect
11. #20292756C2semidirect
12. #20292739C2semidirect
13. #20292727C2semidirect
14. D8×Dic107C2semidirect
171. C107#3370semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933362
#202929912
#202929852
#202929802
#202929352
#202929292
#202929052
#202928402
#202927562
#202927392
#202927272
D8×Dic1072
#459×C1072
#3370107