Finite Group Databasebeta

← #28339040

Subgroups — #28339040

Order 13696.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D107C211yesmaximal, characteristic
2C2×#16812067C211yesmaximal, characteristic, complemented
3C2×#16812066C211yesmaximal, characteristic, complemented
4#20292530C211yesmaximal, characteristic, complemented
5#20292529C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic, complemented
7#20292527C214yesmaximal, complemented
8#20292482C214yesmaximal, complemented
9#390×C107C211yesmaximal, characteristic, complemented
10C24×D107C2212yes—
11C22×D428C2211yescharacteristic
12C22×C22⋊D107C2212yes—
13C2×D4×D107—24no—
14C2×D4×D107—24no—
15C2×D4×D107C418yescomplemented
16C22×C4×D107C2211yescharacteristic
17#16812067—24no—
18#16812066—24no—
19C2×C4⋊Dic107C2211yescharacteristic, complemented
20C2×Dic107⋊C4C2212yescomplemented
21C2×C22⋊Dic107C2214yescomplemented
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2214yescomplemented
25C2×(C2×D107)⋊C4C2214yescomplemented
26C2×C4×Dic107C2212yescomplemented
27C2×C4×Dic107C2211yescharacteristic, complemented
28C22×D4×C107C2211yescharacteristic
29C2×C4⋊C4×C107C2211yescharacteristic, complemented
30C2×C22⋊C4×C107C2214yescomplemented
31C2×C42×C107C2211yescharacteristic, complemented
32C23×D107—24no—
33C23×D107—24no—
34C23×D107—44no—
35C23×D107—44no—
36C23×D107C2×C418yes—
37C23×D107C2312yes—
38C23×D107C2311yescharacteristic
39C2×D428—22no—
40C2×D428—22no—
41C2×D428C2×C414yes—
42C2×C22⋊D107—24no—
43C2×C22⋊D107—24no—
44C2×C22⋊D107C2×C418yes—
45D4×D107—232no—
46C22×Dic107C2312yes—
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C2×C4×D107C2×C414yes—
50C2×C4×D107D414yes—
51C2×C4×D107D414yes—
52C4⋊Dic107—22no—
53Dic107⋊C4—24no—
54C22⋊Dic107—44no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—24no—
57(C2×D107)⋊C4—44no—
58(C2×D107)⋊C4—44no—
59C4×Dic107—22no—
60C4×Dic107—24no—
61C24×C107C2312yes—
62C2×D4×C107—22no—
63C2×D4×C107—22no—
64C2×D4×C107C2×C414yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C22⋊C4×C107—44no—
70C42×C107—22no—
71C23×C107—22no—
72C23×C107—22no—
73C23×C107—24no—
74C23×C107—24no—
75C23×C107C22×C414yes—
76C23×C107C2411yescharacteristic
77C22×D107—22no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—22no—
81C22×D107—24no—
82C22×D107—24no—
83C22×D107—216no—
84C22×D107—416no—
85C22×D107—48no—
86C22×D107—416no—
87C22×D107—48no—
88C22×D107C22×C414yes—
89C22×D107C22×C412yes—
90C22×D107C2×D412yes—
91C22×D107C2×D412yes—
92D4×C107—28no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—24no—
97C2×C4×C107—22no—
98C2×C4×C107C22×C412yes—
99C2×C4×C107C2×D412yes—
100C2×C4×C107C2×D412yes—
101C22⋊D107—216no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—22no—
105C2×Dic107—24no—
106C2×Dic107—22no—
107C2×Dic107C22×C412yes—
108C2×Dic107C2×D412yes—
109C2×Dic107C2×D412yes—
110D428—28no—
111C4×D107C22⋊C4116yescomplemented
112C22×C107—24no—
113C22×C107—28no—
114C22×C107—42no—
115C22×C107—44no—
116C22×C107—42no—
117C22×C107C23×C411yescharacteristic
118C22×C107C22×D411yescharacteristic
119C22×C107C22×D411yescharacteristic
120C22×C107H(5,𝔽2)12yes—
121C22×C107H(5,𝔽2)12yes—
122C2×D107—24no—
123C2×D107—22no—
124C2×D107—24no—
125C2×D107—22no—
126C2×D107—24no—
127C2×D107—28no—
128C2×D107—432no—
129C2×D107—42no—
130C2×D107—44no—
131C2×D107—42no—
132C2×D107C2×C22⋊C414yes—
133C4×C107—42no—
134C4×C107—42no—
135C4×C107C2×C22⋊C414yes—
136Dic107—42no—
137Dic107—42no—
138Dic107C2×C22⋊C414yes—
139C2×C107—44no—
140C2×C107C22×C22⋊C411yescharacteristic
141C2×C107#51812yes—
142C2×C107#53511yescharacteristic
143C2×C107#53512yes—
144C2×C107#53511yescharacteristic
145D107—24no—
146D107—44no—
147#2309—1071nomaximal, Sylow
148C107#230911yescharacteristic, complemented, Sylow
149C23×D4—1071no—
150C2×#117—1071no—
151C2×#117—1071no—
152#390—1071no—
153#390—1071no—
154#390—1074no—
155#390—1071no—
156#390D10711yescharacteristic, complemented
157#379—1074no—
158C25—1072no—
159C22×D4—1072no—
160C22×D4—1078no—
161C22×D4—1071no—
162C22×D4—2144no—
163C22×D4—2144no—
164C22×D4C2×D10711yescharacteristic
165C23×C4—1071no—
166#117—2144no—
167#117—2144no—
168C2×C4⋊C4—1072no—
169C2×C4⋊C4—1071no—
170C2×C4⋊C4C2×D10711yescharacteristic, complemented
171C2×C22⋊C4—1074no—
172C2×C22⋊C4—1074no—
173C2×C22⋊C4—1074no—
174C2×C22⋊C4—1072no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4C2×D10714yescomplemented
177C2×C42—1072no—
178C2×C42—1071no—
179C2×C42C2×D10711yescharacteristic, complemented
180C24—1071no—
181C24—1078no—
182C24—1072no—
183C24—4284no—
184C24—4284no—
185C24—2144no—
186C24—2144no—
187C24C22×D10712yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—1078no—
191C2×D4—1074no—
192C2×D4—21432no—
193C2×D4—2144no—
194C2×D4—2144no—
195C2×D4—2142no—
196C2×D4—2142no—
197C2×D4C4×D10714yes—
198C22×C4—1071no—
199C22×C4—1074no—
200C22×C4—1074no—
201C22×C4—1074no—
202C22×C4—1072no—
203C22×C4—1072no—
204C22×C4C22×D10712yes—
205C22×C4C22×D10712yes—
206C22×C4C22×D10711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—2144no—
209C4⋊C4—2142no—
210C22⋊C4—44no—
211C22⋊C4—4284no—
212C22⋊C4—4284no—
213C22⋊C4—4284no—
214C22⋊C4—2144no—
215C22⋊C4—2144no—
216C42—22no—
217C42—2144no—
218C42—2142no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—1072no—
224C23—1072no—
225C23—1072no—
226C23—1074no—
227C23—42816no—
228C23—4288no—
229C23—42816no—
230C23—4288no—
231C23—21416no—
232C23—2142no—
233C23—2142no—
234C23—2144no—
235C23—2144no—
236C23—2142no—
237C23—2142no—
238C23C2×C4×D10714yes—
239C23C23×D10711yescharacteristic
240D4—28no—
241D4—21416no—
242D4—2148no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—1072no—
249C2×C4—1072no—
250C2×C4—1072no—
251C2×C4—10716no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4—2142no—
256C2×C4—2144no—
257C2×C4C2×C4×D10712yes—
258C2×C4D4×D10712yes—
259C2×C4D4×D10712yes—
260C22—24no—
261C22—28no—
262C22—42no—
263C22—44no—
264C22—42no—
265C22—1074no—
266C22—42832no—
267C22—4282no—
268C22—4284no—
269C22—4282no—
270C22—2144no—
271C22—2148no—
272C22—2142no—
273C22—2144no—
274C22—2142no—
275C22—2144no—
276C22C22×C4×D10711yescharacteristic
277C22C2×D4×D10711yescharacteristic
278C22C2×D4×D10711yescharacteristic
279C22#1681215412yes—
280C22#1681215512yes—
281C4—42no—
282C4—42no—
283C4—1074no—
284C4—4282no—
285C4—4282no—
286C4C22⋊C4×D10714yes—
287C2—44no—
288C2—4284no—
289C2—2144no—
290C2C2×C22⋊C4×D10711yescharacteristic
291C2#2029309912yes—
292C2#2029319011yescharacteristic
293C2#2029319312yes—
294C2#2029319511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#390×C107C2
socle, cosocleC23×C107C24
prime radicalC22×D4×D107C2
hypercenter#390D107
hypocenterC107#2309
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16812067C2semidirect
3. C2×#16812066C2semidirect
4. #20292530C2semidirect
5. #20292529C2semidirect
6. #20292528C2semidirect
7. #20292527C2semidirect
8. #20292482C2semidirect
9. #390×C107C2semidirect
15. C2×D4×D107C4semidirect
19. C2×C4⋊Dic107C22semidirect
20. C2×Dic107⋊C4C22semidirect
21. C2×C22⋊Dic107C22semidirect
24. C2×(C2×D107)⋊C4C22semidirect
25. C2×(C2×D107)⋊C4C22semidirect
26. C2×C4×Dic107C22semidirect
27. C2×C4×Dic107C22semidirect
29. C2×C4⋊C4×C107C22semidirect
30. C2×C22⋊C4×C107C22semidirect
31. C2×C42×C107C22semidirect
111. C4×D107C22⋊C4semidirect
148. C107#2309semidirect
156. #390D107semidirect
170. C2×C4⋊C4C2×D107semidirect
176. C2×C22⋊C4C2×D107semidirect
179. C2×C42C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168120672
C2×#168120662
#202925302
#202925292
#202925282
#202925272
#202924822
#390×C1072
#2309107