Finite Group Databasebeta

← #28349196

Subgroups — #28349196

Order 13696.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic214C211yesmaximal, characteristic, complemented
2#20293346C211yesmaximal, characteristic
3#20292986C211yesmaximal, characteristic
4#20292985C211yesmaximal, characteristic
5#20292976C211yesmaximal, characteristic, complemented
6#20292915C211yesmaximal, characteristic
7#20292913C211yesmaximal, characteristic, complemented
8#20292910C211yesmaximal, characteristic, complemented
9#20292847C211yesmaximal, characteristic
10#20292835C211yesmaximal, characteristic, complemented
11#20292756C211yesmaximal, characteristic, complemented
12#20292734C211yesmaximal, characteristic, complemented
13#20292732C211yesmaximal, characteristic, complemented
14SD16×Dic107C211yesmaximal, characteristic
15#464×C107C211yesmaximal, characteristic
16C2×#13133808C2211yescharacteristic
17#16812109—21no—
18#16812109C2211yescharacteristic
19#16812108—21no—
20#16812108—21no—
21#16812108C2211yescharacteristic
22#16812108C2211yescharacteristic
23#16812107C2211yescharacteristic
24#16812102—21no—
25#16812102—21no—
26#16812096C2211yescharacteristic
27#16812086—21no—
28Q8×Dic107C2211yescharacteristic
29Q8×Dic107C2211yescharacteristic
30Q8×Dic107C2211yescharacteristic
31C4×Dic214—21no—
32C4×Dic214C2211yescharacteristic
33C4×Dic214C2211yescharacteristic
34D4×Dic107C2211yescharacteristic
35#16812065—21no—
36#16812041C2211yescharacteristic
37#16812041C2211yescharacteristic
38#16812040C2211yescharacteristic
39Dic107⋊C8C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812020C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54#124×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214D411yescharacteristic
63C2×Dic214C2311yescharacteristic
64C22×Dic107—21no—
65#13133808—22no—
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C4⋊Dic107—21no—
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107D411yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C4×Dic107—21no—
87C4×Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107—21no—
90C4×Dic107C2311yescharacteristic
91C4×Dic107C2311yescharacteristic
92C2×Q8×C107C2311yescharacteristic
93C2×Q8×C107C2311yescharacteristic
94C2×D4×C107C2311yescharacteristic
95Q16×C107—22no—
96SD16×C107—22no—
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C42×C107—21no—
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106Q8×C107—21no—
107Q8×C107—21no—
108Q8×C107—21no—
109Q8×C107#4112yes—
110D4×C107—21no—
111D4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2411yescharacteristic
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107—21no—
122C2×Dic107—42no—
123C2×Dic107C2×D411yescharacteristic
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127Dic214—22no—
128Dic214—21no—
129Dic214—21no—
130Dic214—81no—
131Dic214—81no—
132Dic214—81no—
133C8×C107—21no—
134C8×C107—21no—
135C107⋊C8—21no—
136C107⋊C8—21no—
137C22×C107—41no—
138C22×C107—41no—
139C22×C107—41no—
140C22×C107C22×D411yescharacteristic
141C4×C107—22no—
142C4×C107—21no—
143C4×C107—21no—
144C4×C107—41no—
145C4×C107—41no—
146C4×C107C2×#4111yescharacteristic
147C4×C1072−1+411yescharacteristic
148Dic107—21no—
149Dic107—21no—
150Dic107—21no—
151Dic107—41no—
152Dic107—41no—
153Dic107—41no—
154Dic107—41no—
155Dic107#13712yes—
156C2×C107—81no—
157C2×C107#54811yescharacteristic
158C2×C107C2×#13711yescharacteristic
159C2×C107#57711yescharacteristic
160#3369—1071nomaximal, Sylow
161C107#336911yescharacteristic, complemented, Sylow
162Q82—1071no—
163#556—1071no—
164#489—1071no—
165#489—1071no—
166#487—1071no—
167#476—1071no—
168#475—1071no—
169#475—1071no—
170#464—1071no—
171#464D10711yescharacteristic
172#462—1071no—
173#446—1071no—
174#441—1071no—
175#441—1071no—
176C4×SD16—1071no—
177C2×Q16—1071no—
178C2×Q16C2×D10711yescharacteristic
179C2×SD16C2×D10711yescharacteristic
180C4⋊Q8—1071no—
181C4⋊Q8—1071no—
182C4⋊Q8—1071no—
183C4⋊Q8—1071no—
184C4⋊Q8—2141no—
185C4⋊Q8—2141no—
186C4⋊Q8—2141no—
187#126—2141no—
188#126—2141no—
189#124—1071no—
190#124C2×D10711yescharacteristic
191C22⋊Q8—2141no—
192C4×Q8—1071no—
193C4×Q8—1071no—
194C4×Q8—1071no—
195C4×Q8—1071no—
196C4×Q8—1071no—
197C4×Q8—2141no—
198C4×Q8C2×D10711yescharacteristic
199C4×D4—1071no—
200#117—2141no—
201#106—1071no—
202#106—1071no—
203#106—1071no—
204C4⋊C8—1071no—
205C4⋊C8—1071no—
206C4⋊C8C2×D10711yescharacteristic
207#103—1071no—
208#103—1071no—
209#103—1071no—
210#103—1071no—
211#103—1071no—
212#103—1071no—
213#103C2×D10711yescharacteristic
214#102—1071no—
215#102—1071no—
216#102C2×D10711yescharacteristic
217#97—1071no—
218#97—1071no—
219C4×C8—1071no—
220C2×Q8—1071no—
221C2×Q8—1071no—
222C2×Q8—2141no—
223C2×Q8—2141no—
224C2×Q8—2141no—
225C2×Q8C22×D10711yescharacteristic
226C2×Q8C22×D10711yescharacteristic
227C2×D4C22×D10711yescharacteristic
228C22×C4—2141no—
229Q16—22no—
230Q16—2142no—
231SD16—22no—
232C2×C8—1071no—
233C2×C8—1071no—
234C2×C8C22×D10711yescharacteristic
235C2×C8C22×D10711yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—1071no—
238C4⋊C4—1071no—
239C4⋊C4—1071no—
240C4⋊C4—1071no—
241C4⋊C4—1071no—
242C4⋊C4—1071no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4C22×D10711yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—2141no—
256C22⋊C4—2141no—
257C22⋊C4—2141no—
258C42—21no—
259C42—1071no—
260C42—1071no—
261C42—2141no—
262C42—2141no—
263C42—2141no—
264C42—2141no—
265C42C22×D10711yescharacteristic
266C23—21no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—8561no—
271Q8—8561no—
272Q8—8561no—
273Q8—2142no—
274Q8—2141no—
275Q8—2141no—
276Q8#1313381712yes—
277D4—21no—
278D4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—1071no—
283C2×C4—1071no—
284C2×C4—1071no—
285C2×C4—1071no—
286C2×C4—4282no—
287C2×C4—2141no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4D4×D10711yescharacteristic
292C2×C4D4×D10711yescharacteristic
293C2×C4C23×D10711yescharacteristic
294C8—21no—
295C8—21no—
296C8—2141no—
297C8—2141no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D10711yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—1072no—
308C4—4281no—
309C4—4281no—
310C4—4281no—
311C4—4281no—
312C4—2141no—
313C4—2141no—
314C4—2141no—
315C4C2×#1313381711yescharacteristic
316C4#1681215611yescharacteristic
317C2—81no—
318C2#2029328311yescharacteristic
319C2#137×D10711yescharacteristic
320C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#464×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×C107C23
hypercenter#464D107
hypocenterC107#3369
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. Q8×Dic214C2semidirect
5. #20292976C2semidirect
7. #20292913C2semidirect
8. #20292910C2semidirect
10. #20292835C2semidirect
11. #20292756C2semidirect
12. #20292734C2semidirect
13. #20292732C2semidirect
161. C107#3369semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2142
#202933462
#202929862
#202929852
#202929762
#202929152
#202929132
#202929102
#202928472
#202928352
#202927562
#202927342
#202927322
SD16×Dic1072
#464×C1072
#3369107