Finite Group Databasebeta

← #28372953

Subgroups — #28372953

Order 13952.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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×C4⋊C4×D109C211yesmaximal, characteristic, complemented
3#20295360C212yesmaximal, complemented
4#20295355C212yesmaximal, complemented
5#20295343C212yesmaximal, complemented
6#20295338C212yesmaximal, complemented
7#20295068C211yesmaximal, characteristic, complemented
8#20295066C211yesmaximal, characteristic, complemented
9#20295027C211yesmaximal, characteristic, complemented
10#20294989C211yesmaximal, characteristic, complemented
11#417×C109C211yesmaximal, characteristic, complemented
12C22×C4×D109—21no—
13C22×C4×D109C2211yescharacteristic
14C2×#13182013C2211yescharacteristic
15#16812680—21no—
16#16812675C2211yescharacteristic
17#16812675C2211yescharacteristic
18#16812670C2211yescharacteristic
19#16812670C2211yescharacteristic
20#16812667—21no—
21C4×C22⋊D109—21no—
22C4×C22⋊D109—21no—
23#16812658C2211yescharacteristic
24C2×C4⋊Dic109C2211yescharacteristic
25C2×#13182008—21no—
26C4⋊C4×D109—21no—
27C4⋊C4×D109—21no—
28C4⋊C4×D109C2212yes—
29C4⋊C4×D109C2212yes—
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×D109—22no—
51C2×C4×D109—22no—
52C2×C4×D109—44no—
53C2×C4×D109D411yescharacteristic
54C2×C4×D109D411yescharacteristic
55C2×C4×D109C2311yescharacteristic
56C2×C4×D109C2311yescharacteristic
57#13182013—22no—
58C2×#11724784C2312yes—
59C4⋊Dic109—21no—
60C4⋊Dic109C2312yes—
61C4⋊Dic109C2311yescharacteristic
62C4⋊Dic109C2311yescharacteristic
63#13182008—22no—
64#13182008—22no—
65#13182008—21no—
66#13182008—21no—
67C22⋊Dic109—21no—
68C22⋊Dic109—21no—
69C22⋊Dic109—21no—
70#13182006—21no—
71#13182006—21no—
72C4×Dic109—21no—
73#41×C109—22no—
74#41×C109—22no—
75C2×Q8×C109C2311yescharacteristic
76C2×D4×C109—21no—
77C2×D4×C109C2311yescharacteristic
78C22×C4×C109—21no—
79C22×C4×C109—21no—
80C22×C4×C109C2311yescharacteristic
81M16×C109—22no—
82C2×C8×C109C2312yes—
83C4⋊C4×C109—21no—
84C4⋊C4×C109C2312yes—
85C23×C109—21no—
86C23×C109C2×D411yescharacteristic
87C22×D109—22no—
88C22×D109—22no—
89C22×D109—22no—
90C22×D109C2×D411yescharacteristic
91C22×D109C2×D411yescharacteristic
92Q8×C109—21no—
93Q8×C109—21no—
94D4×C109—21no—
95D4×C109—21no—
96D4×C109—42no—
97C2×C4×C109—22no—
98C2×C4×C109—21no—
99C2×C4×C109—42no—
100C2×C4×C109—42no—
101C2×C4×C109C2×D411yescharacteristic
102C2×C4×C109#4112yes—
103C2×C4×C109#4112yes—
104C2×C4×C109C2411yescharacteristic
105C22⋊D109—42no—
106C22⋊D109—42no—
107C2×Dic109—22no—
108C2×Dic109—21no—
109C2×Dic109—21no—
110C2×Dic109—22no—
111C2×Dic109—42no—
112C2×Dic109C2×D411yescharacteristic
113C2×Dic109C2×D411yescharacteristic
114C4×D109—22no—
115C4×D109—22no—
116C4×D109—21no—
117C4×D109—21no—
118C4×D109—21no—
119C4×D109—21no—
120C4×D109—44no—
121C4×D109—44no—
122C8×C109—22no—
123#11724784—22no—
124C22×C109—21no—
125C22×C109—21no—
126C22×C109—41no—
127C22×C109—41no—
128C22×C109—41no—
129C22×C109#12312yes—
130C22×C109C22×D411yescharacteristic
131C2×D109—28no—
132C2×D109—21no—
133C2×D109—21no—
134C2×D109—21no—
135C2×D109—21no—
136C2×D109AGL(1,ℤ/8)12yes—
137C2×D109#13712yes—
138C4×C109—41no—
139C4×C109—42no—
140C4×C109#12312yes—
141C4×C109C2×#4111yescharacteristic
142C4×C109C2×#4111yescharacteristic
143Dic109—21no—
144Dic109—21no—
145Dic109—42no—
146Dic109—41no—
147Dic109—41no—
148C2×C109—22no—
149C2×C109—81no—
150C2×C109C2×#12311yescharacteristic
151C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C109C2×#13711yescharacteristic
153D109—22no—
154D109—22no—
155#3092—1091nomaximal, Sylow
156C109#309211yescharacteristic, complemented, Sylow
157#525—1091no—
158C22×C4⋊C4—1091no—
159#484—1092no—
160#483—1092no—
161#481—1092no—
162#480—1092no—
163#428—1091no—
164#428—1091no—
165#417—1091no—
166#417D10911yescharacteristic, complemented
167#407—1091no—
168C2×#41C2×D10911yescharacteristic
169C23×C4—1091no—
170C23×C4—2181no—
171C2×M16—1091no—
172C2×M16C2×D10911yescharacteristic
173#123—2181no—
174C22⋊Q8—1091no—
175C22⋊Q8—1091no—
176#121—1091no—
177#121—1091no—
178#120—2181no—
179C4×D4—2181no—
180C4×D4—2181no—
181#117—1091no—
182C2×C4⋊C4—1091no—
183C2×C4⋊C4—1092no—
184C2×C4⋊C4—1092no—
185C2×C4⋊C4—2181no—
186C2×C4⋊C4—2181no—
187C2×C4⋊C4—2181no—
188C2×C4⋊C4C2×D10911yescharacteristic
189#107—1092no—
190#107—1092no—
191#106—1092no—
192#106—1092no—
193#103—1092no—
194#103C2×D10912yescomplemented
195#102—1092no—
196#102C2×D10912yescomplemented
197C22⋊C8—1092no—
198C22⋊C8—1092no—
199C24—1091no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D10911yescharacteristic
203C2×D4—21no—
204C2×D4—2181no—
205C2×D4—2181no—
206C2×D4C22×D10911yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—1091no—
210C22×C4—1091no—
211C22×C4—1091no—
212C22×C4—1091no—
213C22×C4—1091no—
214C22×C4—4364no—
215C22×C4—2182no—
216C22×C4—2182no—
217C22×C4—2181no—
218C22×C4—2182no—
219C22×C4—2182no—
220C22×C4C22×D10911yescharacteristic
221M16—22no—
222M16—2182no—
223C2×C8—1092no—
224C2×C8C22×D10912yes—
225C4⋊C4—21no—
226C4⋊C4—1092no—
227C4⋊C4—1091no—
228C4⋊C4—1091no—
229C4⋊C4—2182no—
230C4⋊C4—2182no—
231C4⋊C4—2181no—
232C4⋊C4—2181no—
233C4⋊C4—2181no—
234C4⋊C4C22×D10912yes—
235C22⋊C4—2181no—
236C22⋊C4—2181no—
237C22⋊C4—2181no—
238C22⋊C4—2181no—
239C22⋊C4—2181no—
240C42—2181no—
241C23—21no—
242C23—1091no—
243C23—1091no—
244C23—2182no—
245C23—2182no—
246C23—2182no—
247C23D4×D10911yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—4362no—
254D4—4362no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1091no—
260C2×C4—1091no—
261C2×C4—4364no—
262C2×C4—4364no—
263C2×C4—4362no—
264C2×C4—2181no—
265C2×C4—2181no—
266C2×C4—2182no—
267C2×C4—2182no—
268C2×C4—2182no—
269C2×C4—2182no—
270C2×C4—2181no—
271C2×C4—2181no—
272C2×C4—2181no—
273C2×C4—2181no—
274C2×C4D4×D10911yescharacteristic
275C2×C4#1318202812yes—
276C2×C4#1318202912yes—
277C2×C4C23×D10911yescharacteristic
278C8—22no—
279C8—2182no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—1092no—
286C22—1092no—
287C22—2188no—
288C22—2181no—
289C22—2181no—
290C22—2181no—
291C22—2181no—
292C22#1681268112yes—
293C22C2×D4×D10911yescharacteristic
294C4—41no—
295C4—42no—
296C4—4362no—
297C4—4361no—
298C4—4361no—
299C4—2181no—
300C4—2181no—
301C4#1681268112yes—
302C4C2×#1318202811yescharacteristic
303C4C2×#1318202911yescharacteristic
304C2—22no—
305C2—81no—
306C2—2182no—
307C2—2182no—
308C2C2×#1681268111yescharacteristic
309C2AGL(1,ℤ/8)×D10911yescharacteristic
310C2#137×D10911yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#417×C109C2
socleC22×C109C22×D4
prime radical#20295533C2
hypercenter#417D109
hypocenterC109#3092
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×C4⋊C4×D109C2semidirect
3. #20295360C2semidirect
4. #20295355C2semidirect
5. #20295343C2semidirect
6. #20295338C2semidirect
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
156. C109#3092semidirect
166. #417D109semidirect
194. #103C2×D109semidirect
196. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955332
C2×C4⋊C4×D1092
#202953602
#202953552
#202953432
#202953382
#202950682
#202950662
#202950272
#202949892
#417×C1092
#3092109