Finite Group Databasebeta

← #28372962

Subgroups — #28372962

Order 13952.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20295533C211yesmaximal, characteristic
2C2×#16812657C211yesmaximal, characteristic, complemented
3#20295366C212yesmaximal, complemented
4#20295365C212yesmaximal, complemented
5#20295349C212yesmaximal, complemented
6#20295348C212yesmaximal, complemented
7#20295068C211yesmaximal, characteristic, complemented
8#20295066C211yesmaximal, characteristic, complemented
9#20295027C211yesmaximal, characteristic, complemented
10#20294989C211yesmaximal, characteristic, complemented
11#417×C109C211yesmaximal, characteristic, complemented
12C22×C4×D109C2211yescharacteristic
13C2×#13182013C2211yescharacteristic
14#16812680—21no—
15#16812675C2211yescharacteristic
16#16812675C2211yescharacteristic
17#16812670C2211yescharacteristic
18#16812670C2211yescharacteristic
19#16812667—21no—
20C4×C22⋊D109—21no—
21C4×C22⋊D109—21no—
22#16812658C2211yescharacteristic
23#16812657—21no—
24#16812657—21no—
25#16812657C2212yes—
26#16812657C2212yes—
27C2×C4⋊Dic109C2211yescharacteristic
28C2×#13182006—21no—
29C2×C4×Dic109—21no—
30#16812633C2212yescomplemented
31#16812632C2212yescomplemented
32#16812631C2212yescomplemented
33#16812630C2212yescomplemented
34#16812623C2212yescomplemented
35#16812620C2212yescomplemented
36#16812611C2212yes—
37#16812611C2212yes—
38C2×#41×C109C2211yescharacteristic
39C2×M16×C109C2211yescharacteristic
40C2×C4⋊C4×C109C2211yescharacteristic
41#103×C109C2212yescomplemented
42#102×C109C2212yescomplemented
43C23×D109D411yescharacteristic
44C2×C22⋊D109—21no—
45C2×C22⋊D109—21no—
46C22×Dic109—21no—
47C22×Dic109D411yescharacteristic
48C2×C4×D109—22no—
49C2×C4×D109—22no—
50C2×C4×D109D411yescharacteristic
51C2×C4×D109D411yescharacteristic
52C2×C4×D109C2311yescharacteristic
53C2×C4×D109C2311yescharacteristic
54#13182013—22no—
55C2×#11724784C2312yes—
56C4⋊Dic109—21no—
57C4⋊Dic109C2312yes—
58C4⋊Dic109C2311yescharacteristic
59C4⋊Dic109C2311yescharacteristic
60#13182008—21no—
61#13182008—21no—
62C22⋊Dic109—21no—
63C22⋊Dic109—21no—
64C22⋊Dic109—21no—
65#13182006—22no—
66#13182006—22no—
67#13182006—21no—
68#13182006—21no—
69C4×Dic109—21no—
70C4×Dic109—22no—
71C4×Dic109—22no—
72#41×C109—22no—
73#41×C109—22no—
74C2×Q8×C109C2311yescharacteristic
75C2×D4×C109—21no—
76C2×D4×C109C2311yescharacteristic
77C22×C4×C109—21no—
78C22×C4×C109—21no—
79C22×C4×C109C2311yescharacteristic
80M16×C109—22no—
81C2×C8×C109C2312yes—
82C4⋊C4×C109—21no—
83C4⋊C4×C109C2312yes—
84C23×C109—21no—
85C23×C109C2×D411yescharacteristic
86C22×D109—22no—
87C22×D109—41no—
88C22×D109—41no—
89C22×D109C2×D411yescharacteristic
90C22×D109C2×D411yescharacteristic
91Q8×C109—21no—
92Q8×C109—21no—
93D4×C109—21no—
94D4×C109—21no—
95D4×C109—42no—
96C2×C4×C109—22no—
97C2×C4×C109—21no—
98C2×C4×C109—42no—
99C2×C4×C109—42no—
100C2×C4×C109C2×D411yescharacteristic
101C2×C4×C109#4112yes—
102C2×C4×C109#4112yes—
103C2×C4×C109C2411yescharacteristic
104C22⋊D109—42no—
105C22⋊D109—42no—
106C2×Dic109—22no—
107C2×Dic109—22no—
108C2×Dic109—21no—
109C2×Dic109—21no—
110C2×Dic109—42no—
111C2×Dic109C2×D411yescharacteristic
112C2×Dic109C2×D411yescharacteristic
113C4×D109—22no—
114C4×D109—22no—
115C4×D109—21no—
116C4×D109—21no—
117C4×D109—21no—
118C4×D109—21no—
119C8×C109—22no—
120#11724784—22no—
121C22×C109—21no—
122C22×C109—21no—
123C22×C109—41no—
124C22×C109—41no—
125C22×C109—41no—
126C22×C109#12312yes—
127C22×C109C22×D411yescharacteristic
128C2×D109—21no—
129C2×D109—21no—
130C2×D109—21no—
131C2×D109—21no—
132C2×D109—21no—
133C2×D109—21no—
134C2×D109—44no—
135C4×C109—41no—
136C4×C109—42no—
137C4×C109#12312yes—
138C4×C109C2×#4111yescharacteristic
139C4×C109C2×#4111yescharacteristic
140Dic109—21no—
141Dic109—21no—
142Dic109—42no—
143Dic109—41no—
144Dic109—41no—
145C2×C109—22no—
146C2×C109—81no—
147C2×C109C2×#12311yescharacteristic
148C2×C109#57511yescharacteristic
149C2×C109#57511yescharacteristic
150D109—41no—
151D109—41no—
152#3093—1091nomaximal, Sylow
153C109#309311yescharacteristic, complemented, Sylow
154#525—1091no—
155C2×#117—1091no—
156#485—1092no—
157#485—1092no—
158#482—1092no—
159#482—1092no—
160#428—1091no—
161#428—1091no—
162#417—1091no—
163#417D10911yescharacteristic, complemented
164#407—1091no—
165C2×#41C2×D10911yescharacteristic
166C23×C4—1091no—
167C2×M16—1091no—
168C2×M16C2×D10911yescharacteristic
169#123—2181no—
170C22⋊Q8—1091no—
171C22⋊Q8—1091no—
172#121—1091no—
173#121—1091no—
174#120—2181no—
175C4×D4—2181no—
176C4×D4—2181no—
177#117—1091no—
178#117—1092no—
179#117—1092no—
180#117—2181no—
181#117—2181no—
182C2×C4⋊C4—1091no—
183C2×C4⋊C4C2×D10911yescharacteristic
184C2×C22⋊C4—2181no—
185C2×C42—2181no—
186#107—1092no—
187#107—1092no—
188#106—1092no—
189#106—1092no—
190#103—1092no—
191#103C2×D10912yescomplemented
192#102—1092no—
193#102C2×D10912yescomplemented
194C22⋊C8—1092no—
195C22⋊C8—1092no—
196C24—1091no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D10911yescharacteristic
200C2×D4—21no—
201C2×D4—2181no—
202C2×D4—2181no—
203C2×D4C22×D10911yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1091no—
207C22×C4—1091no—
208C22×C4—1091no—
209C22×C4—1091no—
210C22×C4—1091no—
211C22×C4—2181no—
212C22×C4—2182no—
213C22×C4—2182no—
214C22×C4C22×D10911yescharacteristic
215M16—22no—
216M16—2182no—
217C2×C8—1092no—
218C2×C8C22×D10912yes—
219C4⋊C4—21no—
220C4⋊C4—1092no—
221C4⋊C4—1091no—
222C4⋊C4—1091no—
223C4⋊C4—2181no—
224C4⋊C4—2181no—
225C4⋊C4—2181no—
226C4⋊C4C22×D10912yes—
227C22⋊C4—2182no—
228C22⋊C4—2182no—
229C22⋊C4—2181no—
230C22⋊C4—2181no—
231C22⋊C4—2181no—
232C22⋊C4—2181no—
233C22⋊C4—2181no—
234C42—2182no—
235C42—2182no—
236C42—2181no—
237C23—21no—
238C23—1091no—
239C23—1091no—
240C23—4361no—
241C23—4361no—
242C23—2182no—
243C23D4×D10911yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—4362no—
250D4—4362no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1091no—
256C2×C4—1091no—
257C2×C4—4362no—
258C2×C4—2182no—
259C2×C4—2181no—
260C2×C4—2181no—
261C2×C4—2182no—
262C2×C4—2182no—
263C2×C4—2182no—
264C2×C4—2181no—
265C2×C4—2181no—
266C2×C4—2181no—
267C2×C4—2181no—
268C2×C4D4×D10911yescharacteristic
269C2×C4#1318202812yes—
270C2×C4#1318202912yes—
271C2×C4C23×D10911yescharacteristic
272C8—22no—
273C8—2182no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—4364no—
280C22—2181no—
281C22—2181no—
282C22—2181no—
283C22—2181no—
284C22—2181no—
285C22—2181no—
286C22#1681268112yes—
287C22C2×D4×D10911yescharacteristic
288C4—41no—
289C4—42no—
290C4—4361no—
291C4—4362no—
292C4—4361no—
293C4—2181no—
294C4—2181no—
295C4#1681268112yes—
296C4C2×#1318202811yescharacteristic
297C4C2×#1318202911yescharacteristic
298C2—22no—
299C2—81no—
300C2—4361no—
301C2—4361no—
302C2C2×#1681268111yescharacteristic
303C2#2029582211yescharacteristic
304C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#417×C109C2
socleC22×C109C22×D4
prime radical#20295533C2
hypercenter#417D109
hypocenterC109#3093
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#16812657C2semidirect
3. #20295366C2semidirect
4. #20295365C2semidirect
5. #20295349C2semidirect
6. #20295348C2semidirect
7. #20295068C2semidirect
8. #20295066C2semidirect
9. #20295027C2semidirect
10. #20294989C2semidirect
11. #417×C109C2semidirect
30. #16812633C22semidirect
31. #16812632C22semidirect
32. #16812631C22semidirect
33. #16812630C22semidirect
34. #16812623C22semidirect
35. #16812620C22semidirect
41. #103×C109C22semidirect
42. #102×C109C22semidirect
153. C109#3093semidirect
163. #417D109semidirect
191. #103C2×D109semidirect
193. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955332
C2×#168126572
#202953662
#202953652
#202953492
#202953482
#202950682
#202950662
#202950272
#202949892
#417×C1092
#3093109