Finite Group Databasebeta

← #28344744

Subgroups — #28344744

Order 13696.

1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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×C4×D107C211yesmaximal, characteristic, complemented
2C2×#16812084C212yesmaximal, complemented
3C2×#16812083C211yesmaximal, characteristic, complemented
4C2×#16812079C212yesmaximal, complemented
5#20292505C214yesmaximal, complemented
6#20292504C212yesmaximal, complemented
7#20292484C212yesmaximal, complemented
8#380×C107C211yesmaximal, characteristic, complemented
9C24×D107C2211yescharacteristic
10C22×D428C2214yes—
11C22×D428C2211yescharacteristic, complemented
12C22×C22⋊D107C2212yes—
13C23×Dic107C2211yescharacteristic
14C22×C4×D107—28no—
15C22×C4×D107—24no—
16C22×C4×D107C2212yes—
17C22×C4×D107C2212yes—
18#16812084—28no—
19#16812083—24no—
20#16812079—28no—
21C2×C4⋊Dic107C2211yescharacteristic, complemented
22C2×(C2×D107)⋊C4C2214yes—
23C2×(C2×D107)⋊C4C2214yescomplemented
24C2×(C2×D107)⋊C4C2212yescomplemented
25C107⋊#95C2214yescomplemented
26C23×C4×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2212yescomplemented
28C2×C22⋊C4×C107C2212yes—
29#95×C107C2212yescomplemented
30C23×D107—22no—
31C23×D107—22no—
32C23×D107—24no—
33C23×D107—24no—
34C23×D107D414yescomplemented
35C23×D107C2312yes—
36C23×D107C2312yes—
37C2×D428—28no—
38C2×D428—28no—
39C2×D428—28no—
40C2×D428—22no—
41C2×D428—24no—
42C2×C22⋊D107—24no—
43C2×C22⋊D107—24no—
44C2×C22⋊D107—24no—
45C22×Dic107—22no—
46C22×Dic107—24no—
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C2×C4×D107—232no—
50C2×C4×D107—24no—
51C2×C4×D107—24no—
52C2×C4×D107—24no—
53C2×C4×D107—24no—
54C2×C4×D107—24no—
55C2×C4×D107D418yescomplemented
56C4⋊Dic107—22no—
57(C2×D107)⋊C4—28no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—28no—
60C24×C107C2311yescharacteristic
61C22×C4×C107—24no—
62C22×C4×C107D414yescomplemented
63C22×C4×C107C2314yes—
64C22×C4×C107C2311yescharacteristic
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—24no—
67C22⋊C4×C107—24no—
68C23×C107—22no—
69C23×C107—22no—
70C23×C107C2×D412yes—
71C23×C107C2×D412yes—
72C23×C107#4112yes—
73C23×C107C2411yescharacteristic
74C22×D107—28no—
75C22×D107—216no—
76C22×D107—28no—
77C22×D107—28no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—22no—
81C22×D107—22no—
82C22×D107—216no—
83C22×D107—22no—
84C22×D107—24no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—24no—
88C22×D107C2×D414yes—
89C22×D107C2×D414yes—
90C22×D107#4114yes—
91C2×C4×C107—22no—
92C2×C4×C107—28no—
93C2×C4×C107—24no—
94C2×C4×C107—24no—
95C2×C4×C107—24no—
96C2×C4×C107C2×D414yes—
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D412yes—
99C22⋊D107—48no—
100C2×Dic107—28no—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107C2×D414yes—
107D428—416no—
108D428—44no—
109C4×D107—216no—
110C4×D107—216no—
111C22×C107—22no—
112C22×C107—24no—
113C22×C107—22no—
114C22×C107—24no—
115C22×C107#12114yescomplemented
116C22×C107C22×D412yes—
117C22×C107C22×D411yescharacteristic
118C22×C107C22×D411yescharacteristic
119C22×C107C2×#4112yes—
120C22×C107C2×#4111yescharacteristic
121C2×D107—232no—
122C2×D107—24no—
123C2×D107—24no—
124C2×D107—24no—
125C2×D107—24no—
126C2×D107—24no—
127C2×D107—24no—
128C2×D107—42no—
129C2×D107—44no—
130C2×D107—42no—
131C2×D107—42no—
132C2×D107—44no—
133C2×D107#12118yescomplemented
134C4×C107—22no—
135C4×C107—44no—
136C4×C107#12014yescomplemented
137Dic107—24no—
138Dic107—42no—
139C2×C107—24no—
140C2×C107C2×#12011yescharacteristic
141C2×C107C2×#12111yescharacteristic
142C2×C107C2×#12112yes—
143C2×C107#52511yescharacteristic
144C2×C107#52512yes—
145D107—28no—
146D107—82no—
147#2847—1071nomaximal, Sylow
148C107#284711yescharacteristic, complemented, Sylow
149C24×C4—1071no—
150C2×#121—1071no—
151C2×#121—1072no—
152C2×#120—1072no—
153#386—1072no—
154#386—1074no—
155#380—1072no—
156#380D10711yescharacteristic, complemented
157C25—1071no—
158C22×D4—1072no—
159C22×D4—1074no—
160C22×D4—1071no—
161C23×C4—1072no—
162C23×C4—1071no—
163C23×C4—1072no—
164C23×C4—2148no—
165C23×C4—2144no—
166C23×C4C2×D10711yescharacteristic
167#121—2148no—
168#121—2144no—
169#120—2148no—
170C2×C4⋊C4—1071no—
171C2×C4⋊C4C2×D10712yescomplemented
172C2×C22⋊C4—1074no—
173C2×C22⋊C4—1074no—
174C2×C22⋊C4—1072no—
175C2×C22⋊C4C2×D10712yes—
176#95—1074no—
177#95C2×D10712yescomplemented
178C24—1072no—
179C24—1074no—
180C24—1072no—
181C24—2144no—
182C24—2144no—
183C24—2142no—
184C24—2142no—
185C24C22×D10711yescharacteristic
186C2×D4—2148no—
187C2×D4—2148no—
188C2×D4—2148no—
189C2×D4—2144no—
190C2×D4—2144no—
191C2×D4—2144no—
192C2×D4—2144no—
193C2×D4—2142no—
194C22×C4—24no—
195C22×C4—1072no—
196C22×C4—1078no—
197C22×C4—1072no—
198C22×C4—2144no—
199C22×C4—2144no—
200C22×C4—21432no—
201C22×C4—2144no—
202C22×C4—2142no—
203C22×C4—2144no—
204C22×C4—2144no—
205C22×C4—2144no—
206C22×C4D42814yescomplemented
207C22×C4C22×D10714yes—
208C22×C4C22×D10711yescharacteristic
209C22×C4C22×D10711yescharacteristic
210C4⋊C4—24no—
211C4⋊C4—2142no—
212C22⋊C4—24no—
213C22⋊C4—2148no—
214C22⋊C4—2148no—
215C22⋊C4—2144no—
216C23—22no—
217C23—22no—
218C23—1074no—
219C23—1074no—
220C23—1074no—
221C23—21416no—
222C23—2148no—
223C23—2148no—
224C23—21416no—
225C23—2144no—
226C23—2142no—
227C23—2142no—
228C23—2144no—
229C23—2142no—
230C23—2148no—
231C23—2142no—
232C23—2142no—
233C23—2142no—
234C23—2142no—
235C23D4×D10712yes—
236C23C2×D42812yes—
237C23#1313381912yes—
238C23C23×D10711yescharacteristic
239D4—42816no—
240D4—4288no—
241D4—4284no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—1074no—
248C2×C4—21416no—
249C2×C4—2142no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4—2148no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—21416no—
256C2×C4D4×D10714yes—
257C2×C4C2×D42812yes—
258C2×C4C2×D42812yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—1078no—
264C22—4282no—
265C22—4282no—
266C22—4284no—
267C22—4282no—
268C22—4284no—
269C22—21432no—
270C22—2144no—
271C22—2144no—
272C22—2144no—
273C22—2144no—
274C22—2144no—
275C22—2144no—
276C22#1681208414yescomplemented
277C22C2×D4×D10711yescharacteristic
278C22C2×D4×D10712yes—
279C22C22×D42811yescharacteristic
280C22#41×D10712yes—
281C22C2×#1313381911yescharacteristic
282C4—22no—
283C4—44no—
284C4—4282no—
285C4—2144no—
286C4#1681207914yescomplemented
287C2—24no—
288C2—8562no—
289C2—2148no—
290C2C2×#1681207911yescharacteristic
291C2#121×D10712yes—
292C2C2×#1681208411yescharacteristic
293C2#2029313911yescharacteristic
294C2#2029314012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#380×C107C2
hypercenter#380D107
hypocenterC107#2847
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×C4×D107C2semidirect
2. C2×#16812084C2semidirect
3. C2×#16812083C2semidirect
4. C2×#16812079C2semidirect
5. #20292505C2semidirect
6. #20292504C2semidirect
7. #20292484C2semidirect
8. #380×C107C2semidirect
11. C22×D428C22semidirect
21. C2×C4⋊Dic107C22semidirect
23. C2×(C2×D107)⋊C4C22semidirect
24. C2×(C2×D107)⋊C4C22semidirect
25. C107⋊#95C22semidirect
27. C2×C4⋊C4×C107C22semidirect
29. #95×C107C22semidirect
34. C23×D107D4semidirect
55. C2×C4×D107D4semidirect
62. C22×C4×C107D4semidirect
115. C22×C107#121semidirect
133. C2×D107#121semidirect
136. C4×C107#120semidirect
148. C107#2847semidirect
156. #380D107semidirect
171. C2×C4⋊C4C2×D107semidirect
177. #95C2×D107semidirect
206. C22×C4D428semidirect
276. C22#16812084semidirect
286. C4#16812079semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1072
C2×#168120842
C2×#168120832
C2×#168120792
#202925052
#202925042
#202924842
#380×C1072
#2847107