Finite Group Databasebeta

← #28349143

Subgroups — #28349143

Order 13696.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293354C211yesmaximal, characteristic, complemented
2#20293346C211yesmaximal, characteristic
3#20292980C211yesmaximal, characteristic
4#20292976C211yesmaximal, characteristic, complemented
5#20292936C211yesmaximal, characteristic, complemented
6#20292931C211yesmaximal, characteristic, complemented
7#20292905C211yesmaximal, characteristic
8#20292848C211yesmaximal, characteristic
9#20292846C211yesmaximal, characteristic
10#20292836C211yesmaximal, characteristic, complemented
11C8×Dic214C211yesmaximal, characteristic, complemented
12C4×Dic428C211yesmaximal, characteristic, complemented
13#20292719C211yesmaximal, characteristic, complemented
14#20292718C211yesmaximal, characteristic
15#488×C107C211yesmaximal, characteristic
16C2×Dic428C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19#16812108—21no—
20#16812108C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812102—21no—
23#16812102—21no—
24#16812100C2211yescharacteristic
25#16812099—21no—
26#16812099—21no—
27#16812096C2211yescharacteristic
28#16812086—21no—
29Q8×Dic107C2211yescharacteristic
30C4×Dic214C2211yescharacteristic
31C4×Dic214C2211yescharacteristic
32C4×Dic214C2211yescharacteristic
33#16812074—21no—
34#16812074C2211yescharacteristic
35#16812074C2211yescharacteristic
36D4×Dic107C2211yescharacteristic
37#16812065—21no—
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812040C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812030C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52#125×C107C2211yescharacteristic
53#124×C107C2211yescharacteristic
54#103×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214D411yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×Dic214C2311yescharacteristic
63C22×Dic107—21no—
64Dic428—22no—
65#13133808—22no—
66#13133806—22no—
67C2×C107⋊C8C2311yescharacteristic
68C2×C107⋊C8C2311yescharacteristic
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107D411yescharacteristic
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74Dic107⋊C4—21no—
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4D411yescharacteristic
83Dic107⋊C4D411yescharacteristic
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C4×Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C4×Dic107C2311yescharacteristic
94C2×Q8×C107C2311yescharacteristic
95C2×D4×C107C2311yescharacteristic
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C4⋊C4×C107—21no—
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C22⋊C4×C107—21no—
103C22⋊C4×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106Q8×C107—21no—
107Q8×C107—21no—
108D4×C107—21no—
109D4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107C2×D411yescharacteristic
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2411yescharacteristic
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—42no—
121C2×Dic107C2×D411yescharacteristic
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125Dic214—21no—
126Dic214—22no—
127Dic214—21no—
128Dic214—21no—
129Dic214—42no—
130Dic214#4112yes—
131C8×C107—21no—
132C8×C107—21no—
133C107⋊C8—21no—
134C107⋊C8—21no—
135C22×C107—41no—
136C22×C107—41no—
137C22×C107—41no—
138C22×C107C22×D411yescharacteristic
139C4×C107—21no—
140C4×C107—21no—
141C4×C107—41no—
142C4×C107—41no—
143C4×C107—41no—
144C4×C107C2×#4111yescharacteristic
145C4×C1072−1+411yescharacteristic
146Dic107—22no—
147Dic107—21no—
148Dic107—21no—
149Dic107—21no—
150Dic107—41no—
151Dic107—41no—
152Dic107—41no—
153Dic107#13512yes—
154C2×C107—81no—
155C2×C107#54811yescharacteristic
156C2×C107C2×#13511yescharacteristic
157C2×C107#57811yescharacteristic
158#3364—1071nomaximal, Sylow
159C107#336411yescharacteristic, complemented, Sylow
160#557—1071no—
161#556—1071no—
162#488—1071no—
163#488D10711yescharacteristic
164#487—1071no—
165#479—1071no—
166#479—1071no—
167#474—1071no—
168#464—1071no—
169#464—1071no—
170#462—1071no—
171C8×Q8—1071no—
172C4×Q16—1071no—
173C4×Q16—1071no—
174C4×SD16—1071no—
175C2×Q16—1071no—
176C2×Q16—1071no—
177C2×SD16—1071no—
178C4⋊Q8—1071no—
179C4⋊Q8—1071no—
180C4⋊Q8—2141no—
181#126—2141no—
182#126—2141no—
183#125—1071no—
184#125—2141no—
185#125—2141no—
186#125C2×D10711yescharacteristic
187#124—1071no—
188#124C2×D10711yescharacteristic
189C22⋊Q8—2141no—
190C4×Q8—1071no—
191C4×Q8—1071no—
192C4×Q8—1071no—
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×Q8—1071no—
196C4×Q8—2141no—
197C4×D4—1071no—
198#117—2141no—
199#107—1071no—
200#107—1071no—
201#106—1071no—
202C4⋊C8—1071no—
203C4⋊C8—1071no—
204C4⋊C8—1071no—
205#103—1071no—
206#103—1071no—
207#103—1071no—
208#103—1071no—
209#103—1071no—
210#103C2×D10711yescharacteristic
211#103C2×D10711yescharacteristic
212#102—1071no—
213#102C2×D10711yescharacteristic
214#102C2×D10711yescharacteristic
215C4×C8—1071no—
216C4×C8—1071no—
217C4×C8C2×D10711yescharacteristic
218C2×Q8—1071no—
219C2×Q8—1071no—
220C2×Q8—1071no—
221C2×Q8—2141no—
222C2×Q8C22×D10711yescharacteristic
223C2×D4C22×D10711yescharacteristic
224C22×C4—2141no—
225Q16—2142no—
226Q16—2142no—
227SD16—2142no—
228C2×C8—1071no—
229C2×C8—1071no—
230C2×C8C22×D10711yescharacteristic
231C2×C8C22×D10711yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—1071no—
235C4⋊C4—1071no—
236C4⋊C4—1071no—
237C4⋊C4—1071no—
238C4⋊C4—1071no—
239C4⋊C4—2141no—
240C4⋊C4—2141no—
241C4⋊C4—2141no—
242C4⋊C4—2141no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4C22×D10711yescharacteristic
250C4⋊C4C22×D10711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C22⋊C4—2141no—
256C42—1071no—
257C42—1071no—
258C42—2141no—
259C42—2141no—
260C42—2141no—
261C42—2141no—
262C42—2141no—
263C42C22×D10711yescharacteristic
264C23—21no—
265Q8—21no—
266Q8—21no—
267Q8—1072no—
268Q8—4282no—
269Q8—2142no—
270Q8—2141no—
271Q8—2141no—
272Q8—2141no—
273D4—21no—
274D4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—1071no—
279C2×C4—1071no—
280C2×C4—1071no—
281C2×C4—1071no—
282C2×C4—4282no—
283C2×C4—2141no—
284C2×C4—2141no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4D4×D10711yescharacteristic
288C2×C4D4×D10711yescharacteristic
289C2×C4C23×D10711yescharacteristic
290C8—21no—
291C8—21no—
292C8—2141no—
293C8—2141no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22C2×D4×D10711yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—1072no—
304C4—4281no—
305C4—4281no—
306C4—4281no—
307C4—2141no—
308C4—2141no—
309C4—2141no—
310C4—2142no—
311C4#41×D10711yescharacteristic
312C4#1681215611yescharacteristic
313C2—81no—
314C2#2029328511yescharacteristic
315C2#135×D10711yescharacteristic
316C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#488×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×C107C23
hypercenter#488D107
hypocenterC107#3364
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20293354C2semidirect
4. #20292976C2semidirect
5. #20292936C2semidirect
6. #20292931C2semidirect
10. #20292836C2semidirect
11. C8×Dic214C2semidirect
12. C4×Dic428C2semidirect
13. #20292719C2semidirect
159. C107#3364semidirect

Maximal subgroups

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

subgroupindex
#202933542
#202933462
#202929802
#202929762
#202929362
#202929312
#202929052
#202928482
#202928462
#202928362
C8×Dic2142
C4×Dic4282
#202927192
#202927182
#488×C1072
#3364107