Finite Group Databasebeta

← #28339818

Subgroups — #28339818

Order 13696.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C23×D428C211yesmaximal, characteristic, complemented
2C2×#16812084C212yesmaximal, complemented
3C2×#16812079C212yesmaximal, complemented
4C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
5#20292545C212yesmaximal, complemented
6#20292537C212yesmaximal, complemented
7#20292528C212yesmaximal, complemented
8#20292482C212yesmaximal, complemented
9#380×C107C211yesmaximal, characteristic, complemented
10C24×D107C2212yes—
11C22×D428—28no—
12C22×D428—24no—
13C22×D428C2212yes—
14C22×D428C2212yes—
15C22×D428C2214yescomplemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic, complemented
18#16812084—28no—
19#16812079—28no—
20C4×C22⋊D107—24no—
21C2×Dic107⋊C4C2211yescharacteristic, complemented
22C2×C22⋊Dic107C2211yescharacteristic
23C2×(C2×D107)⋊C4C2214yescomplemented
24C2×(C2×D107)⋊C4C2214yescomplemented
25C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
26C2×(C2×D107)⋊C4C2214yescomplemented
27C2×C4×Dic107C2211yescharacteristic, complemented
28C23×C4×C107C2211yescharacteristic
29C2×C4⋊C4×C107C2212yescomplemented
30C2×C22⋊C4×C107C2212yes—
31#95×C107C2212yescomplemented
32C23×D107—24no—
33C23×D107—28no—
34C23×D107—24no—
35C23×D107—24no—
36C23×D107—44no—
37C23×D107C2314yes—
38C23×D107C2311yescharacteristic
39C2×D428—232no—
40C2×D428—28no—
41C2×D428—24no—
42C2×D428—24no—
43C2×D428—24no—
44C2×D428—28no—
45C2×D428—28no—
46C2×D428—44no—
47C2×D428D418yescomplemented
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107D414yescomplemented
51C22×Dic107C2311yescharacteristic
52C22×Dic107C2311yescharacteristic
53C22×Dic107C2311yescharacteristic
54C2×C4×D107—22no—
55C2×C4×D107—24no—
56Dic107⋊C4—22no—
57C22⋊Dic107—22no—
58(C2×D107)⋊C4—28no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—28no—
61(C2×D107)⋊C4—44no—
62C4×Dic107—22no—
63C24×C107C2311yescharacteristic
64C22×C4×C107—24no—
65C22×C4×C107D414yescomplemented
66C22×C4×C107C2314yes—
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2311yescharacteristic
69C4⋊C4×C107—24no—
70C22⋊C4×C107—24no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107C2×D412yes—
74C23×C107C2×D412yes—
75C23×C107#4112yes—
76C23×C107C2411yescharacteristic
77C22×D107—216no—
78C22×D107—21no—
79C22×D107—24no—
80C22×D107—22no—
81C22×D107—24no—
82C22×D107—24no—
83C22×D107—21no—
84C22×D107—24no—
85C22×D107—24no—
86C22×D107—22no—
87C22×D107—24no—
88C22×D107—416no—
89C22×D107—416no—
90C22×D107—48no—
91C22×D107—48no—
92C22×D107C2×D418yes—
93C22×D107C2×D412yescomplemented
94C2×C4×C107—22no—
95C2×C4×C107—28no—
96C2×C4×C107—24no—
97C2×C4×C107—24no—
98C2×C4×C107—24no—
99C2×C4×C107C2×D414yes—
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D412yes—
102C22⋊D107—28no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107—22no—
107C2×Dic107—21no—
108C2×Dic107—22no—
109C2×Dic107C2×D412yescomplemented
110D428—216no—
111D428—216no—
112D428—416no—
113C4×D107—44no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—24no—
117C22×C107—24no—
118C22×C107#12114yescomplemented
119C22×C107C22×D412yes—
120C22×C107C22×D411yescharacteristic
121C22×C107C22×D411yescharacteristic
122C22×C107C2×#4111yescharacteristic
123C22×C107H(5,𝔽2)12yes—
124C2×D107—28no—
125C2×D107—22no—
126C2×D107—24no—
127C2×D107—28no—
128C2×D107—28no—
129C2×D107—432no—
130C2×D107—41no—
131C2×D107—44no—
132C2×D107—41no—
133C2×D107—44no—
134C2×D107—44no—
135C2×D107—42no—
136C2×D107—44no—
137C4×C107—22no—
138C4×C107—44no—
139C4×C107#12014yescomplemented
140Dic107—22no—
141Dic107—41no—
142Dic107—41no—
143C2×C107—24no—
144C2×C107C2×#12011yescharacteristic
145C2×C107C2×#12111yescharacteristic
146C2×C107#53511yescharacteristic
147C2×C107D4212yes—
148C2×C107#54612yes—
149D107—42no—
150D107—48no—
151#2408—1071nomaximal, Sylow
152C107#240811yescharacteristic, complemented, Sylow
153C23×D4—1071no—
154C2×#121—1072no—
155C2×#120—1072no—
156C2×C4×D4—1071no—
157#394—1072no—
158#392—1072no—
159#390—1072no—
160#380D10711yescharacteristic, complemented
161#379—1072no—
162C25—1072no—
163C22×D4—1071no—
164C22×D4—1072no—
165C22×D4—1074no—
166C22×D4—1072no—
167C22×D4—2148no—
168C22×D4—2144no—
169C23×C4—1071no—
170C23×C4C2×D10711yescharacteristic
171#121—2148no—
172#120—2148no—
173C4×D4—2144no—
174C2×C4⋊C4—1071no—
175C2×C4⋊C4C2×D10712yescomplemented
176C2×C22⋊C4—1074no—
177C2×C22⋊C4—1071no—
178C2×C22⋊C4—1071no—
179C2×C22⋊C4—1074no—
180C2×C22⋊C4—1074no—
181C2×C22⋊C4C2×D10712yes—
182C2×C42—1071no—
183#95C2×D10712yescomplemented
184C24—1074no—
185C24—1071no—
186C24—4284no—
187C24—2148no—
188C24—2144no—
189C24—2144no—
190C24—2144no—
191C24C22×D10711yescharacteristic
192C2×D4—1074no—
193C2×D4—1078no—
194C2×D4—4284no—
195C2×D4—2148no—
196C2×D4—2148no—
197C2×D4—2148no—
198C2×D4—21432no—
199C2×D4—2142no—
200C2×D4—2142no—
201C2×D4—2144no—
202C2×D4—2144no—
203C2×D4—2144no—
204C22×C4—24no—
205C22×C4—1071no—
206C22×C4—1071no—
207C22×C4—1071no—
208C22×C4—2144no—
209C22×C4—2142no—
210C22×C4D42814yescomplemented
211C22×C4C22×D10714yes—
212C22×C4C22×D10711yescharacteristic
213C22×C4C22×D10711yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—2142no—
216C22⋊C4—24no—
217C22⋊C4—4284no—
218C22⋊C4—2148no—
219C22⋊C4—2148no—
220C22⋊C4—2142no—
221C22⋊C4—2142no—
222C42—2142no—
223C23—22no—
224C23—22no—
225C23—1078no—
226C23—1072no—
227C23—4288no—
228C23—42816no—
229C23—42816no—
230C23—4288no—
231C23—21416no—
232C23—2144no—
233C23—2142no—
234C23—2144no—
235C23—2141no—
236C23—2144no—
237C23—2141no—
238C23—2144no—
239C23—2144no—
240C23—2142no—
241C23—2144no—
242C23D4×D10712yes—
243C23C2×D42812yes—
244C23#1313381912yes—
245C23C23×D10711yescharacteristic
246D4—42816no—
247D4—21416no—
248D4—2148no—
249D4—21416no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—1072no—
256C2×C4—4284no—
257C2×C4—2142no—
258C2×C4—2141no—
259C2×C4—2141no—
260C2×C4—2142no—
261C2×C4—2141no—
262C2×C4—2141no—
263C2×C4D4×D10714yes—
264C2×C4C2×D42812yes—
265C2×C4C2×D42812yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—42832no—
271C22—4284no—
272C22—4284no—
273C22—4281no—
274C22—4284no—
275C22—4284no—
276C22—4282no—
277C22—4281no—
278C22—2142no—
279C22—2144no—
280C22—2148no—
281C22—2148no—
282C22—2148no—
283C22#1681208414yescomplemented
284C22C2×D4×D10711yescharacteristic
285C22C2×D4×D10712yes—
286C22C22×D42811yescharacteristic
287C22C2×#1313381911yescharacteristic
288C22#1681215512yes—
289C4—22no—
290C4—44no—
291C4—4281no—
292C4—4281no—
293C4—2142no—
294C4#1681207914yescomplemented
295C2—24no—
296C2—4288no—
297C2—4282no—
298C2C2×#1681207911yescharacteristic
299C2C2×#1681208411yescharacteristic
300C2#2029319611yescharacteristic
301C2#2029326112yes—
302C2#2029326812yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#380×C107C2
hypercenter#380D107
hypocenterC107#2408
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D428C2semidirect
2. C2×#16812084C2semidirect
3. C2×#16812079C2semidirect
4. C2×C4×C22⋊D107C2semidirect
5. #20292545C2semidirect
6. #20292537C2semidirect
7. #20292528C2semidirect
8. #20292482C2semidirect
9. #380×C107C2semidirect
15. C22×D428C22semidirect
17. C22×C4×D107C22semidirect
21. C2×Dic107⋊C4C22semidirect
23. C2×(C2×D107)⋊C4C22semidirect
24. C2×(C2×D107)⋊C4C22semidirect
25. C2×(C2×D107)⋊C4C22semidirect
26. C2×(C2×D107)⋊C4C22semidirect
27. C2×C4×Dic107C22semidirect
29. C2×C4⋊C4×C107C22semidirect
31. #95×C107C22semidirect
47. C2×D428D4semidirect
50. C2×C22⋊D107D4semidirect
65. C22×C4×C107D4semidirect
93. C22×D107C2×D4semidirect
109. C2×Dic107C2×D4semidirect
118. C22×C107#121semidirect
139. C4×C107#120semidirect
152. C107#2408semidirect
160. #380D107semidirect
175. C2×C4⋊C4C2×D107semidirect
183. #95C2×D107semidirect
210. C22×C4D428semidirect
283. C22#16812084semidirect
294. C4#16812079semidirect

Maximal subgroups

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

subgroupindex
C23×D4282
C2×#168120842
C2×#168120792
C2×C4×C22⋊D1072
#202925452
#202925372
#202925282
#202924822
#380×C1072
#2408107