Finite Group Databasebeta

← #28339442

Subgroups — #28339442

Order 13696.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812077C211yesmaximal, characteristic
2C2×(C22⋊D107)⋊C4C212yesmaximal, complemented
3C2×C22⋊C4×D107C212yesmaximal, complemented
4#20292515C212yesmaximal, complemented
5#20292507C211yesmaximal, characteristic, complemented
6#20292503C212yesmaximal, complemented
7#20292498C212yesmaximal, complemented
8C4×(C2×D107)⋊C4C212yesmaximal, complemented
9#390×C107C211yesmaximal, characteristic, complemented
10C24×D107C2211yescharacteristic
11C22×C22⋊D107C2212yes—
12C23×Dic107C2211yescharacteristic, complemented
13C22×C4×D107C2212yes—
14C22×C4×D107C2212yes—
15#16812077C418yescomplemented
16(C22⋊D107)⋊C4—28no—
17C2×Dic107⋊C4C2212yescomplemented
18C2×C22⋊Dic107C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C2×C22⋊Dic107C2211yescharacteristic, complemented
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C22⋊C4×D107—28no—
25C2×C4×Dic107C2212yescomplemented
26C107⋊#95C2212yescomplemented
27C107⋊#95C2212yescomplemented
28C107⋊#95C2212yescomplemented
29C22×D4×C107C2211yescharacteristic
30C2×C4⋊C4×C107C2211yescharacteristic, complemented
31C2×C22⋊C4×C107C2212yes—
32C2×C22⋊C4×C107C2212yescomplemented
33C2×C42×C107C2211yescharacteristic, complemented
34C23×D107—24no—
35C23×D107—24no—
36C23×D107—41no—
37C23×D107—41no—
38C23×D107C2×C414yescomplemented
39C23×D107C2312yes—
40C2×C22⋊D107—24no—
41C2×C22⋊D107—24no—
42C2×C22⋊D107C2×C418yescomplemented
43C22×Dic107—24no—
44C22×Dic107—41no—
45C22×Dic107C2312yes—
46C22×Dic107C2312yes—
47C22×Dic107C2312yes—
48C2×C4×D107—24no—
49C2×C4×D107—24no—
50C2×C4×D107—24no—
51C2×C4×D107—24no—
52C2×C4×D107—42no—
53C2×C4×D107—42no—
54Dic107⋊C4—24no—
55C22⋊Dic107—24no—
56C22⋊Dic107—41no—
57C22⋊Dic107C2×C414yescomplemented
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4C2×C418yescomplemented
60(C2×D107)⋊C4D418yescomplemented
61C4×Dic107—24no—
62C24×C107C2311yescharacteristic
63C24×C107C2311yescharacteristic
64C2×D4×C107—41no—
65C2×D4×C107—41no—
66C2×D4×C107C2×C414yescomplemented
67C22×C4×C107C2312yes—
68C22×C4×C107C2311yescharacteristic
69C22×C4×C107C2312yes—
70C4⋊C4×C107—41no—
71C22⋊C4×C107—24no—
72C22⋊C4×C107—24no—
73C42×C107—41no—
74C23×C107—22no—
75C23×C107—22no—
76C23×C107—22no—
77C23×C107—22no—
78C23×C107—21no—
79C23×C107—21no—
80C23×C107—21no—
81C23×C107—21no—
82C23×C107C22×C412yes—
83C23×C107C22×C412yes—
84C23×C107C2411yescharacteristic
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—22no—
89C22×D107—216no—
90C22×D107—44no—
91C22×D107—44no—
92C22×D107—48no—
93C22×D107—44no—
94C22×D107C22×C414yes—
95C22×D107C2×D414yes—
96C22×D107#4114yes—
97D4×C107—44no—
98C2×C4×C107—22no—
99C2×C4×C107—21no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—22no—
105C2×C4×C107C22×C412yes—
106C2×C4×C107C2×D414yes—
107C22⋊D107—216no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—44no—
116C2×Dic107C22×C414yes—
117C2×Dic107C2×D414yes—
118C4×D107—48no—
119C4×D107—48no—
120C22×C107—24no—
121C22×C107—24no—
122C22×C107—22no—
123C22×C107—22no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—41no—
129C22×C107—41no—
130C22×C107C23×C411yescharacteristic
131C22×C107C22×D412yes—
132C22×C107C2×#4112yes—
133C22×C107H(5,𝔽2)11yescharacteristic
134C22×C107H(5,𝔽2)11yescharacteristic
135C2×D107—24no—
136C2×D107—24no—
137C2×D107—24no—
138C2×D107—24no—
139C2×D107—24no—
140C2×D107—24no—
141C2×D107—416no—
142C2×D107C4×D418yescomplemented
143C4×C107—22no—
144C4×C107—24no—
145C4×C107—42no—
146Dic107—24no—
147Dic107—24no—
148Dic107—42no—
149C2×C107—42no—
150C2×C107—42no—
151C2×C107C2×C4×D412yes—
152C2×C107#51811yescharacteristic
153C2×C107D4211yescharacteristic
154C2×C107#54612yes—
155C2×C107#55111yescharacteristic
156D107—28no—
157#2358—1071nomaximal, Sylow
158C107#235811yescharacteristic, complemented, Sylow
159C2×#120—1071no—
160C2×C4×D4—1072no—
161C22×C22⋊C4—1072no—
162#390D10711yescharacteristic, complemented
163#388—1072no—
164#386—1071no—
165#386—1072no—
166#385—1072no—
167C4×C22⋊C4—1072no—
168C25—1071no—
169C22×D4—1072no—
170C22×D4C2×D10711yescharacteristic
171C23×C4—1071no—
172C23×C4—1072no—
173C23×C4—1072no—
174#120—1078no—
175C4×D4—2148no—
176C2×C4⋊C4—1072no—
177C2×C4⋊C4C2×D10711yescharacteristic, complemented
178C2×C22⋊C4—1071no—
179C2×C22⋊C4—1072no—
180C2×C22⋊C4—1072no—
181C2×C22⋊C4—1071no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4—1072no—
184C2×C22⋊C4—2148no—
185C2×C22⋊C4C2×D10712yes—
186C2×C22⋊C4C2×D10712yescomplemented
187C2×C42—1072no—
188C2×C42C2×D10711yescharacteristic, complemented
189#95—1072no—
190#95—1072no—
191#95—1072no—
192C24—1072no—
193C24—1074no—
194C24—4281no—
195C24—4281no—
196C24—2144no—
197C24—2144no—
198C24C22×D10711yescharacteristic
199C24C22×D10711yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—1078no—
203C2×D4—2144no—
204C2×D4—2144no—
205C2×D4C4×D10714yescomplemented
206C22×C4—1072no—
207C22×C4—1072no—
208C22×C4—1072no—
209C22×C4—4282no—
210C22×C4—4282no—
211C22×C4—4281no—
212C22×C4—2144no—
213C22×C4—2144no—
214C22×C4—2144no—
215C22×C4—2144no—
216C22×C4—2144no—
217C22×C4C22×D10712yes—
218C22×C4C22×D10711yescharacteristic
219C22×C4C22×D10712yes—
220C4⋊C4—41no—
221C4⋊C4—2144no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—1074no—
225C22⋊C4—1078no—
226C22⋊C4—1078no—
227C22⋊C4—4281no—
228C22⋊C4—2144no—
229C22⋊C4—2144no—
230C42—41no—
231C42—2144no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—1074no—
241C23—1074no—
242C23—1074no—
243C23—4284no—
244C23—4284no—
245C23—4288no—
246C23—4284no—
247C23—21416no—
248C23—2142no—
249C23—2142no—
250C23—2142no—
251C23—2142no—
252C23C2×C4×D10712yes—
253C23C2×C4×D10712yes—
254C23C23×D10711yescharacteristic
255D4—44no—
256D4—21416no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—1074no—
265C2×C4—1074no—
266C2×C4—4288no—
267C2×C4—4288no—
268C2×C4—4284no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4—2142no—
272C2×C4—2142no—
273C2×C4—2142no—
274C2×C4—2142no—
275C2×C4—2142no—
276C2×C4C2×C4×D10712yes—
277C2×C4D4×D10714yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—1078no—
289C22—42816no—
290C22—2144no—
291C22—2144no—
292C22—2144no—
293C22—2144no—
294C22—2144no—
295C22—2144no—
296C22C22×C4×D10711yescharacteristic
297C22C2×D4×D10712yes—
298C22#41×D10712yes—
299C22#1681215411yescharacteristic
300C22#1681215411yescharacteristic
301C4—24no—
302C4—22no—
303C4—42no—
304C4—4282no—
305C4—2144no—
306C4—2144no—
307C2—42no—
308C2—42no—
309C2—2148no—
310C2C4×D4×D10712yes—
311C2#2029310011yescharacteristic
312C2#2029326211yescharacteristic
313C2#2029326612yes—
314C2#2029330711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#390×C107C2
socle, cosocleC23×C107C24
prime radicalC2×#16812077C2
hypercenter#390D107
hypocenterC107#2358
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D107)⋊C4C2semidirect
3. C2×C22⋊C4×D107C2semidirect
4. #20292515C2semidirect
5. #20292507C2semidirect
6. #20292503C2semidirect
7. #20292498C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. #390×C107C2semidirect
12. C23×Dic107C22semidirect
15. #16812077C4semidirect
17. C2×Dic107⋊C4C22semidirect
20. C2×C22⋊Dic107C22semidirect
25. C2×C4×Dic107C22semidirect
26. C107⋊#95C22semidirect
27. C107⋊#95C22semidirect
28. C107⋊#95C22semidirect
30. C2×C4⋊C4×C107C22semidirect
32. C2×C22⋊C4×C107C22semidirect
33. C2×C42×C107C22semidirect
38. C23×D107C2×C4semidirect
42. C2×C22⋊D107C2×C4semidirect
57. C22⋊Dic107C2×C4semidirect
59. (C2×D107)⋊C4C2×C4semidirect
60. (C2×D107)⋊C4D4semidirect
66. C2×D4×C107C2×C4semidirect
142. C2×D107C4×D4semidirect
158. C107#2358semidirect
162. #390D107semidirect
177. C2×C4⋊C4C2×D107semidirect
186. C2×C22⋊C4C2×D107semidirect
188. C2×C42C2×D107semidirect
205. C2×D4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120772
C2×(C22⋊D107)⋊C42
C2×C22⋊C4×D1072
#202925152
#202925072
#202925032
#202924982
C4×(C2×D107)⋊C42
#390×C1072
#2358107