Finite Group Databasebeta

← #28344747

Subgroups — #28344747

Order 13696.

1068 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
1C23×C4×D107C211yesmaximal, characteristic
2C2×#16812087C212yesmaximal, complemented
3C2×#16812086C211yesmaximal, characteristic, complemented
4#20292501C212yesmaximal, complemented
5#20292500C211yesmaximal, characteristic, complemented
6#20292498C212yesmaximal, complemented
7#20292486C212yesmaximal, complemented
8#20292485C211yesmaximal, characteristic, complemented
9#20292482C211yesmaximal, characteristic
10#20292482C211yesmaximal, characteristic
11#385×C107C211yesmaximal, characteristic, complemented
12C24×D107C2211yescharacteristic
13C22×Dic214C2211yescharacteristic, complemented
14C23×Dic107C2211yescharacteristic
15C22×C4×D107—24no—
16C22×C4×D107—28no—
17C22×C4×D107C2212yes—
18C22×C4×D107C2212yes—
19#16812087—28no—
20#16812086—24no—
21C2×C4⋊Dic107C2211yescharacteristic, complemented
22C2×Dic107⋊C4C2211yescharacteristic, complemented
23C2×Dic107⋊C4C2211yescharacteristic, complemented
24C2×Dic107⋊C4C2212yescomplemented
25C2×Dic107⋊C4C2212yescomplemented
26C2×C22⋊Dic107C2211yescharacteristic
27C2×C22⋊Dic107C2211yescharacteristic
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yes—
31C2×(C2×D107)⋊C4C2212yes—
32C107⋊#95C2212yescomplemented
33C107⋊#95C2212yescomplemented
34C23×C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2212yescomplemented
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2211yescharacteristic
38#95×C107C2212yescomplemented
39C23×D107—22no—
40C23×D107—22no—
41C23×D107—24no—
42C23×D107—24no—
43C23×D107Q814yescomplemented
44C23×D107C2312yes—
45C2×Dic214—22no—
46C2×Dic214—24no—
47C22×Dic107—24no—
48C22×Dic107D414yes—
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2312yes—
51C22×Dic107C2311yescharacteristic
52C22×Dic107C2311yescharacteristic
53C22×Dic107C2311yescharacteristic
54C2×C4×D107—232no—
55C2×C4×D107—24no—
56C2×C4×D107—24no—
57C2×C4×D107—24no—
58C2×C4×D107—24no—
59C2×C4×D107—24no—
60C2×C4×D107D418yes—
61C4⋊Dic107—22no—
62Dic107⋊C4—24no—
63Dic107⋊C4—22no—
64Dic107⋊C4—22no—
65Dic107⋊C4—24no—
66C22⋊Dic107—22no—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—24no—
69(C2×D107)⋊C4—24no—
70(C2×D107)⋊C4—24no—
71(C2×D107)⋊C4—24no—
72C24×C107C2311yescharacteristic
73C22×C4×C107—22no—
74C22×C4×C107—24no—
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C22×C4×C107C2312yes—
78C22×C4×C107C2312yes—
79C4⋊C4×C107—24no—
80C22⋊C4×C107—22no—
81C22⋊C4×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107C2×D412yes—
85C23×C107C2×Q812yes—
86C23×C107#4112yes—
87C23×C107C2411yescharacteristic
88C22×D107—28no—
89C22×D107—216no—
90C22×D107—28no—
91C22×D107—28no—
92C22×D107—22no—
93C22×D107—22no—
94C22×D107—22no—
95C22×D107—22no—
96C22×D107—216no—
97C22×D107C2×D414yes—
98C22×D107C2×Q814yes—
99C22×D107#4114yes—
100C2×C4×C107—28no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107—21no—
105C2×C4×C107—22no—
106C2×C4×C107—22no—
107C2×C4×C107—22no—
108C2×C4×C107—22no—
109C2×C4×C107—22no—
110C2×C4×C107C2×D414yes—
111C2×Dic107—22no—
112C2×Dic107—28no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107—22no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107C2×D412yes—
121C2×Dic107C2×D412yes—
122C2×Dic107C2×D414yes—
123C4×D107—216no—
124C4×D107—216no—
125Dic214—44no—
126C22×C107—22no—
127C22×C107—24no—
128C22×C107—22no—
129C22×C107—24no—
130C22×C107C22⋊Q814yescomplemented
131C22×C107C22×D412yes—
132C22×C107C22×D411yescharacteristic
133C22×C107C22×Q811yescharacteristic
134C22×C107C2×#4112yes—
135C22×C107C2×#4111yescharacteristic
136C2×D107—232no—
137C2×D107—24no—
138C2×D107—24no—
139C2×D107—24no—
140C2×D107—24no—
141C2×D107—24no—
142C2×D107—24no—
143C2×D107C22⋊Q818yescomplemented
144C4×C107—24no—
145C4×C107—41no—
146C4×C107—41no—
147C4×C107—42no—
148Dic107—22no—
149Dic107—41no—
150Dic107—41no—
151Dic107—42no—
152Dic107#12014yes—
153C2×C107—24no—
154C2×C107C2×#12011yescharacteristic
155C2×C107C2×C22⋊Q812yes—
156C2×C107C2×C22⋊Q811yescharacteristic
157C2×C107#52511yescharacteristic
158C2×C107#52512yes—
159D107—28no—
160#2848—1071nomaximal, Sylow
161C107#284811yescharacteristic, complemented, Sylow
162C24×C4—1071no—
163C2×C22⋊Q8—1072no—
164C2×C22⋊Q8—1071no—
165#385—1071no—
166#385—1072no—
167#385—1072no—
168#385D10711yescharacteristic, complemented
169#380—1071no—
170#380—1072no—
171#379—1071no—
172#379—1071no—
173C25—1071no—
174C22×Q8—1071no—
175C23×C4—1071no—
176C23×C4—1072no—
177C23×C4—1072no—
178C23×C4—2144no—
179C23×C4—2148no—
180C23×C4C2×D10711yescharacteristic
181C22⋊Q8—2144no—
182C22⋊Q8—2148no—
183C2×C4⋊C4—1072no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1071no—
186C2×C4⋊C4—1072no—
187C2×C4⋊C4—1071no—
188C2×C4⋊C4C2×D10712yescomplemented
189C2×C22⋊C4—1071no—
190C2×C22⋊C4—1071no—
191C2×C22⋊C4—1072no—
192C2×C22⋊C4—1072no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4C2×D10711yescharacteristic
196C2×C22⋊C4C2×D10711yescharacteristic
197#95—1072no—
198#95—1072no—
199#95C2×D10712yescomplemented
200C24—1072no—
201C24—1074no—
202C24—2144no—
203C24—2144no—
204C24—2142no—
205C24—2142no—
206C24C22×D10711yescharacteristic
207C2×Q8—2142no—
208C2×Q8—2144no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—1071no—
212C22×C4—1074no—
213C22×C4—1071no—
214C22×C4—1078no—
215C22×C4—1071no—
216C22×C4—1072no—
217C22×C4—1071no—
218C22×C4—2144no—
219C22×C4—2144no—
220C22×C4—2144no—
221C22×C4—2144no—
222C22×C4—21432no—
223C22×C4—2144no—
224C22×C4—2144no—
225C22×C4C22×D10712yes—
226C22×C4C22×D10711yescharacteristic
227C22×C4C22×D10711yescharacteristic
228C22×C4C22×D10712yes—
229C4⋊C4—24no—
230C4⋊C4—2142no—
231C4⋊C4—2142no—
232C4⋊C4—2144no—
233C4⋊C4—2142no—
234C4⋊C4—2144no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—2144no—
238C22⋊C4—2144no—
239C22⋊C4—2144no—
240C22⋊C4—2144no—
241C22⋊C4—2142no—
242C22⋊C4—2142no—
243C23—22no—
244C23—22no—
245C23—1074no—
246C23—1074no—
247C23—1074no—
248C23—21416no—
249C23—2148no—
250C23—2148no—
251C23—21416no—
252C23—2148no—
253C23—2142no—
254C23—2142no—
255C23—2142no—
256C23—2142no—
257C23D4×D10712yes—
258C23Q8×D10712yes—
259C23#1313381812yes—
260C23C23×D10711yescharacteristic
261Q8—4284no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—1074no—
273C2×C4—1072no—
274C2×C4—1072no—
275C2×C4—21416no—
276C2×C4—21416no—
277C2×C4—2148no—
278C2×C4—2141no—
279C2×C4—2142no—
280C2×C4—2142no—
281C2×C4—2141no—
282C2×C4—2142no—
283C2×C4—2142no—
284C2×C4—2142no—
285C2×C4—2142no—
286C2×C4D4×D10714yes—
287C22—24no—
288C22—22no—
289C22—24no—
290C22—22no—
291C22—1078no—
292C22—21432no—
293C22—2144no—
294C22—2144no—
295C22—2144no—
296C22—2144no—
297C22—2144no—
298C22—2144no—
299C22#1681208714yescomplemented
300C22C2×D4×D10712yes—
301C22C2×D4×D10711yescharacteristic
302C22C2×Q8×D10711yescharacteristic
303C22#41×D10712yes—
304C22C2×#1313381811yescharacteristic
305C4—24no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—1074no—
310C4—4281no—
311C4—4282no—
312C4—4281no—
313C4—2142no—
314C2—24no—
315C2—2148no—
316C2#120×D10711yescharacteristic
317C2C22⋊Q8×D10712yes—
318C2C2×#1681208711yescharacteristic
319C2#2029313511yescharacteristic
320C2#2029313612yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radicalC24×D107C22
hypercenter#385D107
hypocenterC107#2848
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#16812087C2semidirect
3. C2×#16812086C2semidirect
4. #20292501C2semidirect
5. #20292500C2semidirect
6. #20292498C2semidirect
7. #20292486C2semidirect
8. #20292485C2semidirect
11. #385×C107C2semidirect
13. C22×Dic214C22semidirect
21. C2×C4⋊Dic107C22semidirect
22. C2×Dic107⋊C4C22semidirect
23. C2×Dic107⋊C4C22semidirect
24. C2×Dic107⋊C4C22semidirect
25. C2×Dic107⋊C4C22semidirect
32. C107⋊#95C22semidirect
33. C107⋊#95C22semidirect
35. C2×C4⋊C4×C107C22semidirect
38. #95×C107C22semidirect
43. C23×D107Q8semidirect
130. C22×C107C22⋊Q8semidirect
143. C2×D107C22⋊Q8semidirect
161. C107#2848semidirect
168. #385D107semidirect
188. C2×C4⋊C4C2×D107semidirect
199. #95C2×D107semidirect
299. C22#16812087semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1072
C2×#168120872
C2×#168120862
#202925012
#202925002
#202924982
#202924862
#202924852
#202924822
#385×C1072
#2848107