Finite Group Databasebeta

← #28340128

Subgroups — #28340128

Order 13696.

780 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16812087C212yesmaximal, complemented
2C2×C22⋊C4×D107C212yesmaximal
3#20292550C212yesmaximal, complemented
4#20292539C211yesmaximal, characteristic, complemented
5#20292501C212yesmaximal, complemented
6#20292498C212yesmaximal, complemented
7#20292484C212yesmaximal, complemented
8#20292482C211yesmaximal, characteristic
9#399×C107C211yesmaximal, characteristic, complemented
10C24×D107C2211yescharacteristic
11C22×Dic214C2211yescharacteristic, complemented
12C22×C4×D107C2212yes—
13C22×C4×D107C2212yes—
14#16812087—28no—
15C2×C4⋊Dic107C2212yescomplemented
16C2×Dic107⋊C4C2212yescomplemented
17C2×Dic107⋊C4C2212yescomplemented
18C2×C22⋊Dic107C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C2×(C2×D107)⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C22⋊C4×D107—28no—
25C107⋊#95C2211yescharacteristic, complemented
26C107⋊#95C2212yescomplemented
27C107⋊#95C2212yescomplemented
28C2×C4⋊C4×C107C2212yescomplemented
29C2×C4⋊C4×C107C2211yescharacteristic, complemented
30C2×C22⋊C4×C107C2212yes—
31C2×C22⋊C4×C107C2211yescharacteristic
32#95×C107C2211yescharacteristic, complemented
33C23×D107—24no—
34C23×D107—24no—
35C23×D107—41no—
36C23×D107—41no—
37C23×D107Q814yescomplemented
38C23×D107C2312yes—
39C2×Dic214—24no—
40C2×Dic214—41no—
41C22×Dic107C2312yes—
42C22×Dic107C2312yes—
43C22×Dic107C2312yes—
44C2×C4×D107—24no—
45C2×C4×D107—24no—
46C2×C4×D107—24no—
47C2×C4×D107—24no—
48C2×C4×D107—42no—
49C2×C4×D107—42no—
50C4⋊Dic107—42no—
51Dic107⋊C4—24no—
52Dic107⋊C4—24no—
53C22⋊Dic107—24no—
54C22⋊Dic107—41no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—24no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4D418yes—
59C24×C107C2311yescharacteristic
60C22×C4×C107C2311yescharacteristic
61C22×C4×C107C2311yescharacteristic
62C22×C4×C107C2312yes—
63C22×C4×C107C2312yes—
64C4⋊C4×C107—24no—
65C4⋊C4×C107—41no—
66C22⋊C4×C107—24no—
67C22⋊C4×C107—41no—
68C23×C107—22no—
69C23×C107—22no—
70C23×C107—21no—
71C23×C107—21no—
72C23×C107C2×Q812yes—
73C23×C107C2411yescharacteristic
74C22×D107—22no—
75C22×D107—22no—
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—216no—
79C22×D107—44no—
80C22×D107—44no—
81C22×D107—48no—
82C22×D107—44no—
83C22×D107C2×D414yes—
84C22×D107C2×Q814yes—
85C22×D107#4114yes—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107C2×D414yes—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107—22no—
100C2×Dic107—22no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107C2×D414yes—
105C4×D107—48no—
106C4×D107—48no—
107Dic214—44no—
108C22×C107—24no—
109C22×C107—22no—
110C22×C107—41no—
111C22×C107—42no—
112C22×C107—41no—
113C22×C107C22×D412yes—
114C22×C107C22×Q811yescharacteristic
115C22×C107C2×#4112yes—
116C22×C107H(5,𝔽2)11yescharacteristic
117C22×C107H(5,𝔽2)11yescharacteristic
118C2×D107—24no—
119C2×D107—24no—
120C2×D107—24no—
121C2×D107—24no—
122C2×D107—24no—
123C2×D107—24no—
124C2×D107—416no—
125C2×D107C22⋊Q818yescomplemented
126C4×C107—24no—
127C4×C107—41no—
128C4×C107—42no—
129C4×C107—41no—
130Dic107—24no—
131Dic107—42no—
132Dic107—42no—
133C2×C107—42no—
134C2×C107C2×C22⋊Q812yes—
135C2×C107#54311yescharacteristic
136C2×C107D4211yescharacteristic
137C2×C107#54612yes—
138C2×C107#55111yescharacteristic
139D107—28no—
140#2437—1071nomaximal, Sylow
141C107#243711yescharacteristic, complemented, Sylow
142C2×C22⋊Q8—1072no—
143C22×C22⋊C4—1072no—
144#399D10711yescharacteristic, complemented
145#396—1072no—
146#393—1071no—
147#385—1072no—
148#385—1072no—
149#380—1072no—
150#379—1071no—
151C25—1071no—
152C22×Q8—1071no—
153C23×C4—1072no—
154C23×C4—1072no—
155C22⋊Q8—2148no—
156C2×C4⋊C4—1072no—
157C2×C4⋊C4—1072no—
158C2×C4⋊C4—1072no—
159C2×C4⋊C4C2×D10712yescomplemented
160C2×C4⋊C4C2×D10711yescharacteristic, complemented
161C2×C22⋊C4—1072no—
162C2×C22⋊C4—1071no—
163C2×C22⋊C4—1072no—
164C2×C22⋊C4—1072no—
165C2×C22⋊C4—1072no—
166C2×C22⋊C4—1072no—
167C2×C22⋊C4—2148no—
168C2×C22⋊C4C2×D10712yes—
169C2×C22⋊C4C2×D10711yescharacteristic
170#95—1071no—
171#95—1072no—
172#95—1072no—
173#95C2×D10711yescharacteristic, complemented
174C24—1072no—
175C24—1074no—
176C24—4281no—
177C24—4281no—
178C24—2144no—
179C24—2144no—
180C24C22×D10711yescharacteristic
181C2×Q8—4281no—
182C2×Q8—2144no—
183C22×C4—1072no—
184C22×C4—1072no—
185C22×C4—1072no—
186C22×C4—4282no—
187C22×C4—4282no—
188C22×C4—2144no—
189C22×C4—2144no—
190C22×C4—2144no—
191C22×C4—2144no—
192C22×C4C22×D10711yescharacteristic
193C22×C4C22×D10711yescharacteristic
194C22×C4C22×D10712yes—
195C22×C4C22×D10712yes—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—4282no—
199C4⋊C4—2144no—
200C4⋊C4—2144no—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—1078no—
204C22⋊C4—4281no—
205C22⋊C4—2144no—
206C22⋊C4—2144no—
207C22⋊C4—2144no—
208C22⋊C4—2144no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—1074no—
214C23—1074no—
215C23—1074no—
216C23—4284no—
217C23—4284no—
218C23—4288no—
219C23—4284no—
220C23—21416no—
221C23—2142no—
222C23—2142no—
223C23—2142no—
224C23—2142no—
225C23Q8×D10712yes—
226C23C23×D10711yescharacteristic
227Q8—4284no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—1074no—
238C2×C4—4288no—
239C2×C4—4288no—
240C2×C4—2142no—
241C2×C4—2142no—
242C2×C4—2142no—
243C2×C4—2142no—
244C2×C4—2142no—
245C2×C4—2142no—
246C2×C4—2142no—
247C2×C4—2142no—
248C2×C4D4×D10714yes—
249C22—24no—
250C22—22no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—1078no—
255C22—42816no—
256C22—2144no—
257C22—2144no—
258C22—2144no—
259C22—2144no—
260C22—2144no—
261C22—2144no—
262C22C2×D4×D10712yes—
263C22C2×Q8×D10711yescharacteristic
264C22#41×D10712yes—
265C22#1681215411yescharacteristic
266C22#1681215511yescharacteristic
267C4—24no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—4282no—
272C4—4282no—
273C4—2144no—
274C2—42no—
275C2—2148no—
276C2C22⋊Q8×D10712yes—
277C2#2029325211yescharacteristic
278C2#2029326211yescharacteristic
279C2#2029326912yes—
280C2#2029330511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#399×C107C2
prime radicalC24×D107C22
hypercenter#399D107
hypocenterC107#2437
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16812087C2semidirect
3. #20292550C2semidirect
4. #20292539C2semidirect
5. #20292501C2semidirect
6. #20292498C2semidirect
7. #20292484C2semidirect
9. #399×C107C2semidirect
11. C22×Dic214C22semidirect
15. C2×C4⋊Dic107C22semidirect
16. C2×Dic107⋊C4C22semidirect
17. C2×Dic107⋊C4C22semidirect
25. C107⋊#95C22semidirect
26. C107⋊#95C22semidirect
27. C107⋊#95C22semidirect
28. C2×C4⋊C4×C107C22semidirect
29. C2×C4⋊C4×C107C22semidirect
32. #95×C107C22semidirect
37. C23×D107Q8semidirect
125. C2×D107C22⋊Q8semidirect
141. C107#2437semidirect
144. #399D107semidirect
159. C2×C4⋊C4C2×D107semidirect
160. C2×C4⋊C4C2×D107semidirect
173. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120872
C2×C22⋊C4×D1072
#202925502
#202925392
#202925012
#202924982
#202924842
#202924822
#399×C1072
#2437107