Finite Group Databasebeta

← #49131191

Subgroups — #49131191

Order 36224.

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#31826972C211yesmaximal, characteristic
2C2×#22583568C211yesmaximal, characteristic, complemented
3#31826805C212yesmaximal, complemented
4#31826804C212yesmaximal, complemented
5#31826788C212yesmaximal, complemented
6#31826787C212yesmaximal, complemented
7#31826507C211yesmaximal, characteristic, complemented
8#31826505C211yesmaximal, characteristic, complemented
9#31826466C211yesmaximal, characteristic, complemented
10#31826428C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C22×C4×D283C2211yescharacteristic
13C2×#17985376C2211yescharacteristic
14#22583591—21no—
15#22583586C2211yescharacteristic
16#22583586C2211yescharacteristic
17#22583581C2211yescharacteristic
18#22583581C2211yescharacteristic
19#22583578—21no—
20C4×C22⋊D283—21no—
21C4×C22⋊D283—21no—
22#22583569C2211yescharacteristic
23#22583568—21no—
24#22583568—21no—
25#22583568C2212yes—
26#22583568C2212yes—
27C2×C4⋊Dic283C2211yescharacteristic
28C2×(C2×D283)⋊C4—21no—
29C2×C4×Dic283—21no—
30#22583544C2212yescomplemented
31#22583543C2212yescomplemented
32#22583542C2212yescomplemented
33#22583541C2212yescomplemented
34#22583534C2212yescomplemented
35#22583531C2212yescomplemented
36(C2×D283)⋊C8C2212yes—
37(C2×D283)⋊C8C2212yes—
38C2×#41×C283C2211yescharacteristic
39C2×M16×C283C2211yescharacteristic
40C2×C4⋊C4×C283C2211yescharacteristic
41#103×C283C2212yescomplemented
42#102×C283C2212yescomplemented
43C23×D283D411yescharacteristic
44C2×C22⋊D283—21no—
45C2×C22⋊D283—21no—
46C22×Dic283—21no—
47C22×Dic283D411yescharacteristic
48C2×C4×D283—22no—
49C2×C4×D283—22no—
50C2×C4×D283D411yescharacteristic
51C2×C4×D283D411yescharacteristic
52C2×C4×D283C2311yescharacteristic
53C2×C4×D283C2311yescharacteristic
54#17985376—22no—
55C2×C283⋊C8C2312yes—
56C4⋊Dic283—21no—
57C4⋊Dic283C2312yes—
58C4⋊Dic283C2311yescharacteristic
59C4⋊Dic283C2311yescharacteristic
60Dic283⋊C4—21no—
61Dic283⋊C4—21no—
62C22⋊Dic283—21no—
63C22⋊Dic283—21no—
64C22⋊Dic283—21no—
65(C2×D283)⋊C4—22no—
66(C2×D283)⋊C4—22no—
67(C2×D283)⋊C4—21no—
68(C2×D283)⋊C4—21no—
69C4×Dic283—22no—
70C4×Dic283—22no—
71C4×Dic283—21no—
72#41×C283—22no—
73#41×C283—22no—
74C2×Q8×C283C2311yescharacteristic
75C2×D4×C283—21no—
76C2×D4×C283C2311yescharacteristic
77C22×C4×C283—21no—
78C22×C4×C283—21no—
79C22×C4×C283C2311yescharacteristic
80M16×C283—22no—
81C2×C8×C283C2312yes—
82C4⋊C4×C283—21no—
83C4⋊C4×C283C2312yes—
84C22×D283—22no—
85C22×D283—41no—
86C22×D283—41no—
87C22×D283C2×D411yescharacteristic
88C22×D283C2×D411yescharacteristic
89C22⋊D283—42no—
90C22⋊D283—42no—
91C2×Dic283—22no—
92C2×Dic283—22no—
93C2×Dic283—21no—
94C2×Dic283—21no—
95C2×Dic283—42no—
96C2×Dic283C2×D411yescharacteristic
97C2×Dic283C2×D411yescharacteristic
98C4×D283—22no—
99C4×D283—22no—
100C4×D283—21no—
101C4×D283—21no—
102C4×D283—21no—
103C4×D283—21no—
104C283⋊C8—22no—
105C23×C283—21no—
106C23×C283C2×D411yescharacteristic
107Q8×C283—21no—
108Q8×C283—21no—
109D4×C283—21no—
110D4×C283—21no—
111D4×C283—42no—
112C2×C4×C283—22no—
113C2×C4×C283—21no—
114C2×C4×C283—42no—
115C2×C4×C283—42no—
116C2×C4×C283C2×D411yescharacteristic
117C2×C4×C283#4112yes—
118C2×C4×C283#4112yes—
119C2×C4×C283C2411yescharacteristic
120C8×C283—22no—
121C22×C283—21no—
122C22×C283—21no—
123C22×C283—41no—
124C22×C283—41no—
125C22×C283—41no—
126C22×C283#12312yes—
127C22×C283C22×D411yescharacteristic
128C2×D283—21no—
129C2×D283—21no—
130C2×D283—21no—
131C2×D283—21no—
132C2×D283—21no—
133C2×D283—21no—
134C2×D283—44no—
135C4×C283—41no—
136C4×C283—42no—
137C4×C283#12312yes—
138C4×C283C2×#4111yescharacteristic
139C4×C283C2×#4111yescharacteristic
140Dic283—21no—
141Dic283—21no—
142Dic283—42no—
143Dic283—41no—
144Dic283—41no—
145C2×C283—22no—
146C2×C283—81no—
147C2×C283C2×#12311yescharacteristic
148C2×C283#57511yescharacteristic
149C2×C283#57511yescharacteristic
150D283—41no—
151D283—41no—
152C283#309311yescharacteristic, complemented, Sylow
153#3093—2831nomaximal, Sylow
154#525—2831no—
155C2×#117—2831no—
156#485—2832no—
157#485—2832no—
158#482—2832no—
159#482—2832no—
160#428—2831no—
161#428—2831no—
162#417—2831no—
163#417D28311yescharacteristic, complemented
164#407—2831no—
165C2×#41C2×D28311yescharacteristic
166C23×C4—2831no—
167C2×M16—2831no—
168C2×M16C2×D28311yescharacteristic
169#123—5661no—
170C22⋊Q8—2831no—
171C22⋊Q8—2831no—
172#121—2831no—
173#121—2831no—
174#120—5661no—
175C4×D4—5661no—
176C4×D4—5661no—
177#117—2831no—
178#117—2832no—
179#117—2832no—
180#117—5661no—
181#117—5661no—
182C2×C4⋊C4—2831no—
183C2×C4⋊C4C2×D28311yescharacteristic
184C2×C22⋊C4—5661no—
185C2×C42—5661no—
186#107—2832no—
187#107—2832no—
188#106—2832no—
189#106—2832no—
190#103—2832no—
191#103C2×D28312yescomplemented
192#102—2832no—
193#102C2×D28312yescomplemented
194C22⋊C8—2832no—
195C22⋊C8—2832no—
196C24—2831no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D28311yescharacteristic
200C2×D4—21no—
201C2×D4—5661no—
202C2×D4—5661no—
203C2×D4C22×D28311yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2831no—
207C22×C4—2831no—
208C22×C4—2831no—
209C22×C4—2831no—
210C22×C4—2831no—
211C22×C4—5662no—
212C22×C4—5662no—
213C22×C4—5661no—
214C22×C4C22×D28311yescharacteristic
215M16—22no—
216M16—5662no—
217C2×C8—2832no—
218C2×C8C22×D28312yes—
219C4⋊C4—21no—
220C4⋊C4—2832no—
221C4⋊C4—2831no—
222C4⋊C4—2831no—
223C4⋊C4—5661no—
224C4⋊C4—5661no—
225C4⋊C4—5661no—
226C4⋊C4C22×D28312yes—
227C22⋊C4—5662no—
228C22⋊C4—5662no—
229C22⋊C4—5661no—
230C22⋊C4—5661no—
231C22⋊C4—5661no—
232C22⋊C4—5661no—
233C22⋊C4—5661no—
234C42—5662no—
235C42—5662no—
236C42—5661no—
237C23—21no—
238C23—2831no—
239C23—2831no—
240C23—5662no—
241C23—11321no—
242C23—11321no—
243C23D4×D28311yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—11322no—
250D4—11322no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—2831no—
256C2×C4—2831no—
257C2×C4—5662no—
258C2×C4—5662no—
259C2×C4—5662no—
260C2×C4—5661no—
261C2×C4—5661no—
262C2×C4—5661no—
263C2×C4—5661no—
264C2×C4—5662no—
265C2×C4—5661no—
266C2×C4—5661no—
267C2×C4—11322no—
268C2×C4D4×D28311yescharacteristic
269C2×C4#1798539112yes—
270C2×C4#1798539212yes—
271C2×C4C23×D28311yescharacteristic
272C8—22no—
273C8—5662no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—5661no—
280C22—5661no—
281C22—5661no—
282C22—5661no—
283C22—5661no—
284C22—5661no—
285C22—11324no—
286C22#2258359212yes—
287C22C2×D4×D28311yescharacteristic
288C4—41no—
289C4—42no—
290C4—5661no—
291C4—5661no—
292C4—11321no—
293C4—11322no—
294C4—11321no—
295C4#2258359212yes—
296C4C2×#1798539111yescharacteristic
297C4C2×#1798539211yescharacteristic
298C2—22no—
299C2—81no—
300C2—11321no—
301C2—11321no—
302C2C2×#2258359211yescharacteristic
303C2#3182726111yescharacteristic
304C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radical#31826972C2
hypercenter#417D283
hypocenterC283#3093
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#22583568C2semidirect
3. #31826805C2semidirect
4. #31826804C2semidirect
5. #31826788C2semidirect
6. #31826787C2semidirect
7. #31826507C2semidirect
8. #31826505C2semidirect
9. #31826466C2semidirect
10. #31826428C2semidirect
11. #417×C283C2semidirect
30. #22583544C22semidirect
31. #22583543C22semidirect
32. #22583542C22semidirect
33. #22583541C22semidirect
34. #22583534C22semidirect
35. #22583531C22semidirect
41. #103×C283C22semidirect
42. #102×C283C22semidirect
152. C283#3093semidirect
163. #417D283semidirect
191. #103C2×D283semidirect
193. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318269722
C2×#225835682
#318268052
#318268042
#318267882
#318267872
#318265072
#318265052
#318264662
#318264282
#417×C2832
#3093283