Finite Group Databasebeta

← #28346296

Subgroups — #28346296

Order 13696.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293254C211yesmaximal, characteristic
2#20293103C211yesmaximal, characteristic
3#20292937C211yesmaximal, characteristic, complemented
4#20292935C211yesmaximal, characteristic, complemented
5#20292934C211yesmaximal, characteristic, complemented
6#20292932C211yesmaximal, characteristic, complemented
7#20292920C211yesmaximal, characteristic, complemented
8#20292919C211yesmaximal, characteristic, complemented
9#20292911C211yesmaximal, characteristic, complemented
10#20292666C211yesmaximal, characteristic
11#20292662C211yesmaximal, characteristic
12#20292652C211yesmaximal, characteristic
13#20292634C211yesmaximal, characteristic
14#103×D107C211yesmaximal, characteristic
15#475×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×#13133801C2211yescharacteristic
18C2×C8×D107C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812107—21no—
21#16812101C2211yescharacteristic
22#16812100C2211yescharacteristic
23#16812099—21no—
24#16812087—21no—
25#16812085—21no—
26Q8×Dic107C2211yescharacteristic
27C4×Dic214—21no—
28C4×Dic214C2211yescharacteristic
29#16812074—21no—
30#16812074C2211yescharacteristic
31#16812067—21no—
32#16812067C2211yescharacteristic
33#16812067C2211yescharacteristic
34#16812066—21no—
35#16812066C2211yescharacteristic
36C4⋊C4×D107C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812040C2211yescharacteristic
43C4⋊(C107⋊C8)C2211yescharacteristic
44Dic107⋊C8C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812033C2211yescharacteristic
49#16812033C2211yescharacteristic
50#16812032C2211yescharacteristic
51#16812032C2211yescharacteristic
52C4⋊Q8×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic
54#106×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214C2311yescharacteristic
62Q8×D107—22no—
63Q8×D107Q814yescomplemented
64C2×C4×D107—21no—
65C2×C4×D107—21no—
66C2×C4×D107C2311yescharacteristic
67#13133801—22no—
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2311yescharacteristic
70C8×D107—22no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4D411yescharacteristic
88(C2×D107)⋊C4D411yescharacteristic
89C4×Dic107—21no—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107—21no—
96C2×Q8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107—21no—
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C42×C107—21no—
105C42×C107C2311yescharacteristic
106C22×D107C2×D411yescharacteristic
107Q8×C107—21no—
108Q8×C107—81no—
109Q8×C107C2×Q812yes—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2411yescharacteristic
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121C2×Dic107C2×D411yescharacteristic
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124C4×D107—24no—
125C4×D107—81no—
126C4×D107C2×Q812yes—
127C4×D107#4112yes—
128Dic214—21no—
129Dic214—24no—
130Dic214—81no—
131Dic214C2×Q812yes—
132C8×C107—21no—
133C8×C107—21no—
134C107⋊C8—21no—
135C107⋊C8—21no—
136C22×C107C22×D411yescharacteristic
137C2×D107—21no—
138C2×D107—21no—
139C2×D107C22⋊Q812yes—
140C4×C107—22no—
141C4×C107—21no—
142C4×C107—21no—
143C4×C107—41no—
144C4×C107—41no—
145C4×C107—41no—
146C4×C107C22×Q811yescharacteristic
147C4×C107C2×#4111yescharacteristic
148Dic107—22no—
149Dic107—21no—
150Dic107—21no—
151Dic107—41no—
152Dic107—41no—
153Dic107—41no—
154Dic107C22⋊Q812yes—
155C2×C107C2×C22⋊Q811yescharacteristic
156C2×C107#57711yescharacteristic
157C2×C107#57811yescharacteristic
158D107—22no—
159#3082—1071nomaximal, Sylow
160C107#308211yescharacteristic, complemented, Sylow
161#544—1071no—
162#519—1071no—
163#479—1071no—
164#479—1071no—
165#479—1071no—
166#479—1071no—
167#477—1071no—
168#477—1071no—
169#475—1071no—
170#475D10711yescharacteristic, complemented
171#428—1071no—
172#427—1071no—
173#424—1071no—
174#419—1071no—
175C2×#103—1071no—
176C22×Q8—1071no—
177C2×M16—1071no—
178C22×C8—1071no—
179C4⋊Q8—1071no—
180C4⋊Q8—2141no—
181C4⋊Q8C2×D10711yescharacteristic
182#125—1071no—
183#125—1071no—
184#125—2141no—
185C22⋊Q8—2141no—
186C22⋊Q8—2141no—
187C4×Q8—1071no—
188C4×Q8—1071no—
189C4×Q8—1071no—
190C4×Q8—2141no—
191C4×Q8—2141no—
192C4×Q8C2×D10711yescharacteristic
193#117—1071no—
194#117—1071no—
195#117—1071no—
196#117—2141no—
197#117—2141no—
198C2×C4⋊C4—1071no—
199#107—1071no—
200#107—1071no—
201#107—1071no—
202#107—1071no—
203#106—1071no—
204#106—1071no—
205#106C2×D10711yescharacteristic
206#106C2×D10711yescharacteristic
207C4⋊C8—1071no—
208C4⋊C8—1071no—
209C4⋊C8—1071no—
210C4⋊C8C2×D10711yescharacteristic
211#103—1071no—
212#103—1071no—
213#103—1071no—
214#103—1071no—
215#103—1071no—
216#103—1071no—
217#103C2×D10711yescharacteristic
218#103C2×D10711yescharacteristic
219C2×Q8—21no—
220C2×Q8—1074no—
221C2×Q8—1071no—
222C2×Q8—2141no—
223C2×Q8—2141no—
224C2×Q8—2142no—
225C2×Q8C22×D10711yescharacteristic
226C22×C4—1071no—
227C22×C4—2141no—
228C22×C4—2141no—
229M16—2142no—
230C2×C8—1071no—
231C2×C8—1071no—
232C2×C8—2142no—
233C2×C8C22×D10711yescharacteristic
234C2×C8C22×D10711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—1071no—
238C4⋊C4—1071no—
239C4⋊C4—1071no—
240C4⋊C4—1071no—
241C4⋊C4—1071no—
242C4⋊C4—2141no—
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⋊C4C22×D10711yescharacteristic
252C4⋊C4C22×D10711yescharacteristic
253C4⋊C4C22×D10711yescharacteristic
254C22⋊C4—1071no—
255C22⋊C4—1071no—
256C22⋊C4—2141no—
257C22⋊C4—2141no—
258C42—21no—
259C42—1071no—
260C42—2141no—
261C42—2141no—
262C42—2141no—
263C42—2141no—
264C42—2141no—
265C42C22×D10711yescharacteristic
266C23—1071no—
267Q8—21no—
268Q8—81no—
269Q8—8561no—
270Q8—1072no—
271Q8—2144no—
272Q8—2141no—
273Q8Q8×D10712yes—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—8561no—
279C2×C4—1071no—
280C2×C4—1072no—
281C2×C4—1072no—
282C2×C4—1071no—
283C2×C4—1071no—
284C2×C4—2144no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4—2141no—
289C2×C4D4×D10711yescharacteristic
290C2×C4D4×D10711yescharacteristic
291C2×C4C23×D10711yescharacteristic
292C8—21no—
293C8—21no—
294C8—2141no—
295C8—2141no—
296C22—1072no—
297C22—2141no—
298C22—2141no—
299C22C2×D4×D10711yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—1072no—
307C4—4281no—
308C4—4281no—
309C4—4281no—
310C4—2142no—
311C4—2141no—
312C4—2141no—
313C4C2×Q8×D10711yescharacteristic
314C4#41×D10711yescharacteristic
315C2—2142no—
316C2C22⋊Q8×D10711yescharacteristic
317C2#2029343311yescharacteristic
318C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#475×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#475D107
hypocenterC107#3082
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #20292937C2semidirect
4. #20292935C2semidirect
5. #20292934C2semidirect
6. #20292932C2semidirect
7. #20292920C2semidirect
8. #20292919C2semidirect
9. #20292911C2semidirect
15. #475×C107C2semidirect
63. Q8×D107Q8semidirect
160. C107#3082semidirect
170. #475D107semidirect

Maximal subgroups

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

subgroupindex
#202932542
#202931032
#202929372
#202929352
#202929342
#202929322
#202929202
#202929192
#202929112
#202926662
#202926622
#202926522
#202926342
#103×D1072
#475×C1072
#3082107