Finite Group Databasebeta

← #28366169

Subgroups — #28366169

Order 13952.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16812667C211yesmaximal, characteristic
2#20294926C212yesmaximal, complemented
3#20294917C212yesmaximal, complemented
4#20294916C212yesmaximal, complemented
5#20294914C212yesmaximal, complemented
6#20294906C211yesmaximal, characteristic, complemented
7#20294903C212yesmaximal, complemented
8#20294881C212yesmaximal, complemented
9#390×C109C211yesmaximal, characteristic, complemented
10C24×D109C2211yescharacteristic
11C22×C22⋊D109C2212yes—
12C23×Dic109C2211yescharacteristic, complemented
13#16812667C418yescomplemented
14C2×#13182008C2212yescomplemented
15C2×C22⋊Dic109C2212yes—
16C2×C22⋊Dic109C2211yescharacteristic
17C2×C22⋊Dic109C2211yescharacteristic, complemented
18C2×#13182006C2212yes—
19C2×#13182006C2212yes—
20C2×#13182006C2212yes—
21C2×#13182006C2212yes—
22C2×#13182006C2212yes—
23C2×C4×Dic109C2212yescomplemented
24#16812606C2212yescomplemented
25#16812606C2212yescomplemented
26#16812606C2212yescomplemented
27C22×D4×C109C2211yescharacteristic
28C2×C4⋊C4×C109C2211yescharacteristic, complemented
29C2×C22⋊C4×C109C2212yes—
30C2×C22⋊C4×C109C2212yescomplemented
31C2×C42×C109C2211yescharacteristic, complemented
32C23×D109—41no—
33C23×D109—41no—
34C23×D109—42no—
35C23×D109—42no—
36C23×D109C2×C414yescomplemented
37C23×D109C2312yes—
38C2×C22⋊D109—42no—
39C2×C22⋊D109—42no—
40C2×C22⋊D109C2×C418yes—
41C22×Dic109—41no—
42C22×Dic109—42no—
43C22×Dic109C2312yes—
44C22×Dic109C2312yes—
45C22×Dic109C2312yes—
46#13182008—42no—
47C22⋊Dic109—42no—
48C22⋊Dic109—41no—
49C22⋊Dic109C2×C414yescomplemented
50#13182006—42no—
51#13182006—42no—
52#13182006—42no—
53#13182006—42no—
54#13182006C2×C418yescomplemented
55C4×Dic109—42no—
56C24×C109C2311yescharacteristic
57C24×C109C2311yescharacteristic
58C2×D4×C109—41no—
59C2×D4×C109—41no—
60C2×D4×C109C2×C414yes—
61C22×C4×C109C2312yes—
62C22×C4×C109C2312yes—
63C22×C4×C109C2311yescharacteristic
64C4⋊C4×C109—41no—
65C22⋊C4×C109—42no—
66C22⋊C4×C109—42no—
67C42×C109—41no—
68C23×C109—22no—
69C23×C109—22no—
70C23×C109—21no—
71C23×C109—21no—
72C23×C109—21no—
73C23×C109—21no—
74C23×C109—22no—
75C23×C109—22no—
76C23×C109C22×C412yes—
77C23×C109C22×C412yes—
78C23×C109C2411yescharacteristic
79C22×D109—22no—
80C22×D109—22no—
81C22×D109—22no—
82C22×D109—22no—
83C22×D109—22no—
84C22×D109—22no—
85C22×D109—48no—
86C22×D109—44no—
87C22×D109—44no—
88C22×D109—48no—
89C22×D109—44no—
90C22×D109C22×C414yes—
91D4×C109—44no—
92C2×C4×C109—21no—
93C2×C4×C109—21no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109—22no—
98C2×C4×C109—22no—
99C2×C4×C109—22no—
100C2×C4×C109C22×C412yes—
101C22⋊D109—48no—
102C2×Dic109—22no—
103C2×Dic109—22no—
104C2×Dic109—22no—
105C2×Dic109—22no—
106C2×Dic109—22no—
107C2×Dic109—22no—
108C2×Dic109—22no—
109C2×Dic109—22no—
110C2×Dic109—44no—
111C2×Dic109C22×C414yes—
112C22×C109—24no—
113C22×C109—22no—
114C22×C109—22no—
115C22×C109—24no—
116C22×C109—41no—
117C22×C109—42no—
118C22×C109—41no—
119C22×C109—42no—
120C22×C109—41no—
121C22×C109—41no—
122C22×C109C23×C411yescharacteristic
123C22×C109H(5,𝔽2)12yes—
124C22×C109H(5,𝔽2)11yescharacteristic
125C22×C109H(5,𝔽2)11yescharacteristic
126C22×C109H(5,𝔽2)12yes—
127C2×D109—24no—
128C2×D109—24no—
129C2×D109—24no—
130C2×D109—416no—
131C2×D109—42no—
132C2×D109—42no—
133C2×D109—42no—
134C2×D109—42no—
135C4×C109—22no—
136C4×C109—42no—
137C4×C109—42no—
138Dic109—24no—
139Dic109—42no—
140Dic109—42no—
141C2×C109—42no—
142C2×C109—42no—
143C2×C109#51812yes—
144C2×C109#51811yescharacteristic
145C2×C109#56011yescharacteristic
146C2×C109#56012yes—
147C2×C109H(3,𝔽4)11yescharacteristic
148D109—44no—
149#2375—1091nomaximal, Sylow
150C109#237511yescharacteristic, complemented, Sylow
151C2×#120—1091no—
152#390—1092no—
153#390D10911yescharacteristic, complemented
154#388—1092no—
155#388—1092no—
156#388—1092no—
157#386—1091no—
158#386—1092no—
159#379—1092no—
160C25—1091no—
161C22×D4—1092no—
162C22×D4C2×D10911yescharacteristic
163C23×C4—1091no—
164#120—1098no—
165C2×C4⋊C4—1092no—
166C2×C4⋊C4C2×D10911yescharacteristic, complemented
167C2×C22⋊C4—1091no—
168C2×C22⋊C4—1092no—
169C2×C22⋊C4—1092no—
170C2×C22⋊C4—1091no—
171C2×C22⋊C4—1092no—
172C2×C22⋊C4—1092no—
173C2×C22⋊C4—1092no—
174C2×C22⋊C4—1092no—
175C2×C22⋊C4C2×D10912yes—
176C2×C22⋊C4C2×D10912yescomplemented
177C2×C42—1092no—
178C2×C42C2×D10911yescharacteristic, complemented
179#95—1092no—
180#95—1092no—
181#95—1092no—
182C24—1094no—
183C24—1092no—
184C24—4362no—
185C24—4361no—
186C24—4361no—
187C24—4362no—
188C24C22×D10911yescharacteristic
189C24C22×D10911yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—1098no—
193C2×D4—4362no—
194C2×D4—4362no—
195C2×D4C4×D10914yes—
196C22×C4—1092no—
197C22×C4—1092no—
198C22×C4—1092no—
199C22×C4—4361no—
200C22×C4—4362no—
201C22×C4C22×D10912yes—
202C22×C4C22×D10912yes—
203C22×C4C22×D10911yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—4362no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—1094no—
209C22⋊C4—1098no—
210C22⋊C4—4362no—
211C22⋊C4—4362no—
212C22⋊C4—4362no—
213C22⋊C4—4362no—
214C22⋊C4—4361no—
215C22⋊C4—4362no—
216C42—41no—
217C42—4362no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—1094no—
227C23—4368no—
228C23—4364no—
229C23—4364no—
230C23—4368no—
231C23—4364no—
232C23—2182no—
233C23—2182no—
234C23—2182no—
235C23—2182no—
236C23—2182no—
237C23—2182no—
238C23C2×C4×D10912yes—
239C23C2×C4×D10912yes—
240C23C23×D10911yescharacteristic
241D4—44no—
242D4—4368no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—1094no—
252C2×C4—4364no—
253C2×C4—2182no—
254C2×C4—2182no—
255C2×C4—2182no—
256C2×C4—2182no—
257C2×C4—2182no—
258C2×C4—2182no—
259C2×C4—2182no—
260C2×C4—2182no—
261C2×C4C2×C4×D10912yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—43616no—
273C22—4362no—
274C22—4362no—
275C22—4362no—
276C22—4362no—
277C22—2184no—
278C22—2184no—
279C22—2184no—
280C22C22×C4×D10911yescharacteristic
281C22#1681274412yes—
282C22#1681274411yescharacteristic
283C22#1681274411yescharacteristic
284C22#1681274512yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—4362no—
289C4—4362no—
290C4—2184no—
291C2—42no—
292C2—42no—
293C2—4364no—
294C2#2029549812yes—
295C2#2029549911yescharacteristic
296C2#2029576311yescharacteristic
297C2#2029576412yes—
298C2#2029576811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#390×C109C2
socle, cosocleC23×C109C24
prime radicalC2×#16812667C2
hypercenter#390D109
hypocenterC109#2375
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20294926C2semidirect
3. #20294917C2semidirect
4. #20294916C2semidirect
5. #20294914C2semidirect
6. #20294906C2semidirect
7. #20294903C2semidirect
8. #20294881C2semidirect
9. #390×C109C2semidirect
12. C23×Dic109C22semidirect
13. #16812667C4semidirect
14. C2×#13182008C22semidirect
17. C2×C22⋊Dic109C22semidirect
23. C2×C4×Dic109C22semidirect
24. #16812606C22semidirect
25. #16812606C22semidirect
26. #16812606C22semidirect
28. C2×C4⋊C4×C109C22semidirect
30. C2×C22⋊C4×C109C22semidirect
31. C2×C42×C109C22semidirect
36. C23×D109C2×C4semidirect
49. C22⋊Dic109C2×C4semidirect
54. #13182006C2×C4semidirect
150. C109#2375semidirect
153. #390D109semidirect
166. C2×C4⋊C4C2×D109semidirect
176. C2×C22⋊C4C2×D109semidirect
178. C2×C42C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126672
#202949262
#202949172
#202949162
#202949142
#202949062
#202949032
#202948812
#390×C1092
#2375109