Finite Group Databasebeta

← #28349083

Subgroups — #28349083

Order 13696.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293340C211yesmaximal, characteristic
2#20293332C211yesmaximal, characteristic, complemented
3#20293031C211yesmaximal, characteristic, complemented
4#20292971C211yesmaximal, characteristic, complemented
5#20292968C211yesmaximal, characteristic, complemented
6#20292942C211yesmaximal, characteristic, complemented
7#20292940C211yesmaximal, characteristic, complemented
8#20292921C211yesmaximal, characteristic, complemented
9#20292916C211yesmaximal, characteristic, complemented
10#20292874C211yesmaximal, characteristic, complemented
11#20292873C211yesmaximal, characteristic, complemented
12#20292714C211yesmaximal, characteristic, complemented
13#20292703C211yesmaximal, characteristic, complemented
14#20292666C211yesmaximal, characteristic, complemented
15#428×C107C211yesmaximal, characteristic, complemented
16C2×#13133807C2211yescharacteristic
17C2×#13133801C2211yescharacteristic
18#16812108C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812099—21no—
21#16812097—21no—
22#16812095—21no—
23#16812089C2211yescharacteristic
24#16812088—21no—
25#16812088C2211yescharacteristic
26#16812087—21no—
27#16812084C2211yescharacteristic
28#16812083C2211yescharacteristic
29#16812082—21no—
30#16812074C2211yescharacteristic
31#16812071C2211yescharacteristic
32C4×C22⋊D107—21no—
33C4×C22⋊D107C2211yescharacteristic
34#16812067C2211yescharacteristic
35#16812066—21no—
36C4⋊C4×D107C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812040C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812032C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×M16×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54C2×C4⋊C4×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#106×C107C2211yescharacteristic
58#106×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214C2311yescharacteristic
62C2×D428—21no—
63C2×D428C2311yescharacteristic
64C2×C22⋊D107—21no—
65C2×C22⋊D107D411yescharacteristic
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107C2311yescharacteristic
69#13133807—22no—
70#13133801—22no—
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82C22⋊Dic107—21no—
83C22⋊Dic107D411yescharacteristic
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4D411yescharacteristic
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C22×C4×C107—21no—
94C22×C4×C107C2311yescharacteristic
95M16×C107—22no—
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C4⋊C4×C107—21no—
99C4⋊C4×C107C2311yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C42×C107—21no—
105C23×C107C2×D411yescharacteristic
106C22×D107—21no—
107C22×D107C2×D411yescharacteristic
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—81no—
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C22⋊D107—22no—
118C22⋊D107—81no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125D428—21no—
126D428—21no—
127D428—81no—
128C4×D107—21no—
129C4×D107—21no—
130C4×D107—42no—
131C4×D107—81no—
132Dic214—21no—
133Dic214—21no—
134Dic214—81no—
135Dic214—81no—
136C8×C107—21no—
137C8×C107#4112yes—
138C107⋊C8—21no—
139C107⋊C8—21no—
140C22×C107—21no—
141C22×C107—21no—
142C22×C107—21no—
143C22×C107C22×D411yescharacteristic
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—21no—
147C2×D107—41no—
148C2×D107—41no—
149C2×D107—41no—
150C4×C107—21no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107—41no—
154C4×C107—41no—
155C4×C107C2×#4111yescharacteristic
156C4×C1072−1+411yescharacteristic
157Dic107—21no—
158Dic107—21no—
159Dic107—41no—
160Dic107—41no—
161Dic107—41no—
162Dic107#13712yes—
163C2×C107—41no—
164C2×C107#54711yescharacteristic
165C2×C107C2×#13711yescharacteristic
166C2×C107#57611yescharacteristic
167D107—41no—
168D107—81no—
169#3357—1071nomaximal, Sylow
170C107#335711yescharacteristic, complemented, Sylow
171#555—1071no—
172#554—1071no—
173#500—1071no—
174#485—1071no—
175#485—1071no—
176#480—1071no—
177#480—1071no—
178#477—1071no—
179#476—1071no—
180#468—1071no—
181#468—1071no—
182C4×SD16—1071no—
183#436—1071no—
184#428—1071no—
185#428D10711yescharacteristic, complemented
186C2×SD16—1071no—
187C2×M16—1071no—
188C2×M16C2×D10711yescharacteristic
189C4⋊Q8—1071no—
190C4⋊Q8—1071no—
191#125—2141no—
192#124—2141no—
193#124—2141no—
194C22⋊Q8—1071no—
195C22⋊Q8—1071no—
196C22⋊Q8—2141no—
197C22⋊Q8—2141no—
198#121—1071no—
199#121—1071no—
200#121—2141no—
201C4×Q8—1071no—
202C4×D4—1071no—
203C4×D4—1071no—
204C4×D4—2141no—
205#117—1071no—
206#117—2141no—
207#117C2×D10711yescharacteristic
208C2×C4⋊C4—1071no—
209C2×C4⋊C4C2×D10711yescharacteristic
210#107—1071no—
211#107—1071no—
212#107—1071no—
213#107—1071no—
214#107C2×D10711yescharacteristic
215#107C2×D10711yescharacteristic
216#106—1071no—
217#106C2×D10711yescharacteristic
218#106C2×D10711yescharacteristic
219C4⋊C8—1071no—
220#103—1071no—
221#103—1071no—
222#103—1071no—
223#102—1071no—
224#102—1071no—
225#102—1071no—
226C22⋊C8—1071no—
227C22⋊C8—1071no—
228C4×C8—1071no—
229C2×Q8—1071no—
230C2×Q8—2141no—
231C2×Q8—2141no—
232C2×D4—1071no—
233C2×D4—1071no—
234C2×D4—2141no—
235C2×D4—2141no—
236C22×C4—21no—
237C22×C4—1071no—
238C22×C4—2141no—
239C22×C4—2141no—
240C22×C4C22×D10711yescharacteristic
241SD16—2142no—
242M16—22no—
243M16—2142no—
244C2×C8—1071no—
245C2×C8—1071no—
246C2×C8C22×D10711yescharacteristic
247C2×C8C22×D10711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—1071no—
250C4⋊C4—1071no—
251C4⋊C4—1071no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4—2141no—
255C4⋊C4—2141no—
256C4⋊C4—2141no—
257C4⋊C4—2141no—
258C4⋊C4C22×D10711yescharacteristic
259C4⋊C4C22×D10711yescharacteristic
260C4⋊C4C22×D10711yescharacteristic
261C4⋊C4C22×D10711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—1071no—
264C22⋊C4—1071no—
265C22⋊C4—2141no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C42—21no—
272C42—1071no—
273C42—2141no—
274C42—2141no—
275C23—1071no—
276C23—2141no—
277C23D4×D10711yescharacteristic
278Q8—8561no—
279Q8—8561no—
280Q8—2141no—
281Q8—2141no—
282D4—8561no—
283D4—8561no—
284D4—2142no—
285D4—2141no—
286D4—2141no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—81no—
294C2×C4—8561no—
295C2×C4—1071no—
296C2×C4—1071no—
297C2×C4—1071no—
298C2×C4—4282no—
299C2×C4—2141no—
300C2×C4—2141no—
301C2×C4—2141no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4D4×D10711yescharacteristic
305C2×C4C23×D10711yescharacteristic
306C8—21no—
307C8—2141no—
308C8—2141no—
309C8#1313381812yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—4281no—
314C22—4281no—
315C22—4281no—
316C22—2141no—
317C22—2141no—
318C22—2141no—
319C22C2×D4×D10711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—1072no—
326C4—4281no—
327C4—4281no—
328C4—4281no—
329C4—2141no—
330C4—2141no—
331C4C2×#1313381811yescharacteristic
332C4#1681215711yescharacteristic
333C2—41no—
334C2—8561no—
335C2—4281no—
336C2#2029327811yescharacteristic
337C2#137×D10711yescharacteristic
338C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radical#20293340C2
hypercenter#428D107
hypocenterC107#3357
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #20293332C2semidirect
3. #20293031C2semidirect
4. #20292971C2semidirect
5. #20292968C2semidirect
6. #20292942C2semidirect
7. #20292940C2semidirect
8. #20292921C2semidirect
9. #20292916C2semidirect
10. #20292874C2semidirect
11. #20292873C2semidirect
12. #20292714C2semidirect
13. #20292703C2semidirect
14. #20292666C2semidirect
15. #428×C107C2semidirect
170. C107#3357semidirect
185. #428D107semidirect

Maximal subgroups

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

subgroupindex
#202933402
#202933322
#202930312
#202929712
#202929682
#202929422
#202929402
#202929212
#202929162
#202928742
#202928732
#202927142
#202927032
#202926662
#428×C1072
#3357107