Finite Group Databasebeta

← #28371311

Subgroups — #28371311

Order 13952.

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×D109C211yesmaximal, characteristic, complemented
2C2×#16812670C212yesmaximal, complemented
3C2×#16812667C212yesmaximal, complemented
4#20294906C212yesmaximal, complemented
5#20294902C214yesmaximal, complemented
6#20294886C211yesmaximal, characteristic, complemented
7#20294883C212yesmaximal, complemented
8C2×#121×C109C211yesmaximal, characteristic, complemented
9C24×D109C2211yescharacteristic
10C22×C22⋊D109C2214yes—
11C23×Dic109C2211yescharacteristic
12C22×C4×D109—28no—
13C22×C4×D109—24no—
14C22×C4×D109C2212yes—
15C22×C4×D109C2212yes—
16#16812670—28no—
17#16812667—28no—
18C2×C4⋊Dic109C2212yescomplemented
19C2×C22⋊Dic109C2212yes—
20C2×C22⋊Dic109C2214yescomplemented
21C2×#13182006C2214yes—
22#16812606C2214yescomplemented
23#16812606C2212yescomplemented
24C22×D4×C109C2212yes—
25C22×D4×C109C2211yescharacteristic, complemented
26C23×C4×C109C2211yescharacteristic
27#121×C109—24no—
28C2×C4⋊C4×C109C2211yescharacteristic, complemented
29C2×C22⋊C4×C109C2212yescomplemented
30C23×D109—22no—
31C23×D109—22no—
32C23×D109—24no—
33C23×D109—24no—
34C23×D109D414yescomplemented
35C23×D109C2312yes—
36C2×C22⋊D109—28no—
37C2×C22⋊D109—28no—
38C2×C22⋊D109—28no—
39C22×Dic109—22no—
40C22×Dic109—24no—
41C22×Dic109C2314yes—
42C22×Dic109C2312yes—
43C2×C4×D109—232no—
44C2×C4×D109—24no—
45C2×C4×D109—24no—
46C2×C4×D109—24no—
47C2×C4×D109—24no—
48C2×C4×D109—24no—
49C2×C4×D109D418yescomplemented
50C4⋊Dic109—24no—
51C22⋊Dic109—24no—
52C22⋊Dic109—28no—
53#13182006—28no—
54C24×C109C2312yes—
55C24×C109C2311yescharacteristic
56C2×D4×C109—24no—
57C2×D4×C109—24no—
58C2×D4×C109—24no—
59C2×D4×C109—22no—
60C2×D4×C109—24no—
61C22×C4×C109—24no—
62C22×C4×C109D414yescomplemented
63C22×C4×C109C2312yes—
64C22×C4×C109C2311yescharacteristic
65C22×C4×C109C2311yescharacteristic
66C4⋊C4×C109—22no—
67C22⋊C4×C109—24no—
68C23×C109—22no—
69C23×C109—22no—
70C23×C109—22no—
71C23×C109—24no—
72C23×C109—22no—
73C23×C109—22no—
74C23×C109—24no—
75C23×C109C2×D412yes—
76C23×C109C2×D412yes—
77C23×C109#4112yes—
78C23×C109C2411yescharacteristic
79C22×D109—28no—
80C22×D109—216no—
81C22×D109—28no—
82C22×D109—28no—
83C22×D109—22no—
84C22×D109—22no—
85C22×D109—22no—
86C22×D109—22no—
87C22×D109—216no—
88C22×D109C2×D414yes—
89C22×D109C2×D414yes—
90C22×D109#4114yes—
91D4×C109—48no—
92D4×C109—44no—
93C2×C4×C109—22no—
94C2×C4×C109—28no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109—22no—
98C2×C4×C109C2×D414yes—
99C2×C4×C109C2×D412yes—
100C2×C4×C109C2×D412yes—
101C22⋊D109—416no—
102C2×Dic109—28no—
103C2×Dic109—22no—
104C2×Dic109—22no—
105C2×Dic109—24no—
106C2×Dic109—24no—
107C2×Dic109—24no—
108C2×Dic109C2×D414yes—
109C4×D109—216no—
110C4×D109—216no—
111C22×C109—22no—
112C22×C109—24no—
113C22×C109—22no—
114C22×C109—24no—
115C22×C109—42no—
116C22×C109—44no—
117C22×C109—42no—
118C22×C109—42no—
119C22×C109—44no—
120C22×C109#12114yescomplemented
121C22×C109C22×D412yes—
122C22×C109C22×D411yescharacteristic
123C22×C109C22×D411yescharacteristic
124C22×C109C2×#4112yes—
125C22×C109C2×#4111yescharacteristic
126C2×D109—232no—
127C2×D109—24no—
128C2×D109—24no—
129C2×D109—24no—
130C2×D109—24no—
131C2×D109—24no—
132C2×D109—24no—
133C2×D109#12118yescomplemented
134C4×C109—22no—
135C4×C109—42no—
136C4×C109#12014yescomplemented
137Dic109—24no—
138Dic109—44no—
139C2×C109—24no—
140C2×C109—82no—
141C2×C109C2×#12011yescharacteristic
142C2×C109C2×#12112yes—
143C2×C109C2×#12111yescharacteristic
144C2×C109#52511yescharacteristic
145C2×C109#52512yes—
146D109—28no—
147#2847—1091nomaximal, Sylow
148C109#284711yescharacteristic, complemented, Sylow
149C24×C4—1091no—
150C2×#121—1092no—
151C2×#121D10911yescharacteristic, complemented
152C2×#120—1092no—
153#386—1092no—
154#386—1094no—
155#380—1091no—
156#380—1092no—
157C25—1091no—
158C22×D4—1094no—
159C22×D4C2×D10912yes—
160C22×D4C2×D10911yescharacteristic, complemented
161C23×C4—1092no—
162C23×C4—1091no—
163C23×C4—1092no—
164C23×C4—2188no—
165C23×C4—2184no—
166C23×C4C2×D10911yescharacteristic
167#121—24no—
168#121—2188no—
169#120—2188no—
170C2×C4⋊C4—1092no—
171C2×C4⋊C4C2×D10911yescharacteristic, complemented
172C2×C22⋊C4—1092no—
173C2×C22⋊C4—1094no—
174C2×C22⋊C4—1094no—
175C2×C22⋊C4C2×D10912yescomplemented
176#95—1094no—
177#95—1092no—
178C24—1092no—
179C24—1094no—
180C24—2184no—
181C24—2184no—
182C24—2182no—
183C24—2182no—
184C24C22×D10912yes—
185C24C22×D10911yescharacteristic
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—2188no—
192C2×D4—2188no—
193C2×D4—2188no—
194C22×C4—24no—
195C22×C4—1092no—
196C22×C4—1098no—
197C22×C4—1094no—
198C22×C4—2184no—
199C22×C4—2184no—
200C22×C4—21832no—
201C22×C4—2184no—
202C22×C4—2182no—
203C22×C4—2184no—
204C22×C4—2184no—
205C22×C4—2184no—
206C22×C4C22⋊D10914yescomplemented
207C22×C4C22×D10912yes—
208C22×C4C22×D10911yescharacteristic
209C22×C4C22×D10911yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—2184no—
212C22⋊C4—24no—
213C22⋊C4—2188no—
214C22⋊C4—2188no—
215C22⋊C4—2184no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—1094no—
224C23—1094no—
225C23—1094no—
226C23—21816no—
227C23—2188no—
228C23—2188no—
229C23—21816no—
230C23—2188no—
231C23—2182no—
232C23—2182no—
233C23—2182no—
234C23—2182no—
235C23D4×D10912yes—
236C23C2×C22⋊D10912yes—
237C23#1318202812yes—
238C23C23×D10911yescharacteristic
239D4—48no—
240D4—44no—
241D4—43616no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—1094no—
248C2×C4—21816no—
249C2×C4—2184no—
250C2×C4—2184no—
251C2×C4—2184no—
252C2×C4—2188no—
253C2×C4—2182no—
254C2×C4—2182no—
255C2×C4—21816no—
256C2×C4D4×D10914yes—
257C2×C4C2×C22⋊D10912yes—
258C2×C4C2×C22⋊D10912yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—1098no—
269C22—21832no—
270C22—2184no—
271C22—2184no—
272C22—2184no—
273C22—2184no—
274C22—2184no—
275C22—2184no—
276C22#1681267014yescomplemented
277C22C2×D4×D10912yes—
278C22C2×D4×D10911yescharacteristic
279C22C22×C22⋊D10911yescharacteristic
280C22#41×D10912yes—
281C22C2×#1318202811yescharacteristic
282C4—22no—
283C4—42no—
284C4—4364no—
285C4—2184no—
286C4#1681266714yescomplemented
287C2—24no—
288C2—82no—
289C2—2188no—
290C2C2×#1681266711yescharacteristic
291C2#121×D10912yes—
292C2C2×#1681267011yescharacteristic
293C2#2029553311yescharacteristic
294C2#2029553612yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
FittingC2×#121×C109C2
hypercenterC2×#121D109
hypocenterC109#2847
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×C4×D109C2semidirect
2. C2×#16812670C2semidirect
3. C2×#16812667C2semidirect
4. #20294906C2semidirect
5. #20294902C2semidirect
6. #20294886C2semidirect
7. #20294883C2semidirect
8. C2×#121×C109C2semidirect
18. C2×C4⋊Dic109C22semidirect
20. C2×C22⋊Dic109C22semidirect
22. #16812606C22semidirect
23. #16812606C22semidirect
25. C22×D4×C109C22semidirect
28. C2×C4⋊C4×C109C22semidirect
29. C2×C22⋊C4×C109C22semidirect
34. C23×D109D4semidirect
49. C2×C4×D109D4semidirect
62. C22×C4×C109D4semidirect
120. C22×C109#121semidirect
133. C2×D109#121semidirect
136. C4×C109#120semidirect
148. C109#2847semidirect
151. C2×#121D109semidirect
160. C22×D4C2×D109semidirect
171. C2×C4⋊C4C2×D109semidirect
175. C2×C22⋊C4C2×D109semidirect
206. C22×C4C22⋊D109semidirect
276. C22#16812670semidirect
286. C4#16812667semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1092
C2×#168126702
C2×#168126672
#202949062
#202949022
#202948862
#202948832
C2×#121×C1092
#2847109