Finite Group Databasebeta

← #34109767

Subgroups — #34109767

Order 20096.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19097804C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757785C211yesmaximal, characteristic
4C2×#19097751C211yesmaximal, characteristic, complemented
5#23757532C212yesmaximal, complemented
6#23757520C212yesmaximal, complemented
7#23757515C212yesmaximal, complemented
8#23757503C212yesmaximal, complemented
9#23757281C211yesmaximal, characteristic, complemented
10#23757272C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×Dic314C2211yescharacteristic
14C22×C4×D157C2211yescharacteristic
15#19097804—24no—
16C2×Dic628C2212yescomplemented
17C2×#14578943C2212yescomplemented
18C2×#14578938C2211yescharacteristic
19C2×#14578937C2212yes—
20M16×D157—24no—
21C2×C8×D157C2212yes—
22#19097755—21no—
23#19097751—21no—
24#19097751—21no—
25#19097751C2212yes—
26#19097751C2211yescharacteristic
27#19097751C2212yes—
28#19097751C2211yescharacteristic
29#19097746C2211yescharacteristic
30#19097746C2211yescharacteristic
31#19097741—21no—
32C4×D628—21no—
33C4×D628—21no—
34C2×C4⋊Dic157—21no—
35C2×#14578931—21no—
36#19097694C2212yescomplemented
37#19097694C2212yescomplemented
38#19097694C2212yescomplemented
39#19097691C2212yescomplemented
40C2×M16×C157C2211yescharacteristic
41#117×C157C2211yescharacteristic
42C2×C4⋊C4×C157C2211yescharacteristic
43#107×C157C2212yescomplemented
44#106×C157C2212yescomplemented
45C23×D157D411yescharacteristic
46#14578954—22no—
47#14578954—22no—
48C2×Dic314—21no—
49C2×Dic314—22no—
50C2×Dic314—22no—
51C2×Dic314C2312yes—
52C2×Dic314C2311yescharacteristic
53C2×D628—21no—
54C2×D628—21no—
55C2×D628C2311yescharacteristic
56C2×C22⋊D157—21no—
57C22×Dic157—21no—
58C22×Dic157D411yescharacteristic
59C2×C4×D157—22no—
60C2×C4×D157—21no—
61C2×C4×D157—22no—
62C2×C4×D157D411yescharacteristic
63C2×C4×D157D411yescharacteristic
64C2×C4×D157C2311yescharacteristic
65C2×C4×D157C2311yescharacteristic
66Dic628—24no—
67#14578943—24no—
68#14578938—22no—
69#14578937—24no—
70C2×#11929434C2312yes—
71C8×D157—24no—
72C4⋊Dic157—22no—
73C4⋊Dic157—22no—
74C4⋊Dic157—21no—
75C4⋊Dic157—21no—
76#14578931—22no—
77#14578931—22no—
78#14578931—21no—
79#14578931—21no—
80#14578931—21no—
81#14578931—21no—
82C22×C4×C157—21no—
83C22×C4×C157C2311yescharacteristic
84M16×C157D414yescomplemented
85C2×C8×C157C2312yes—
86C4⋊C4×C157—21no—
87C4⋊C4×C157C2311yescharacteristic
88C4⋊C4×C157C2312yes—
89C4⋊C4×C157C2311yescharacteristic
90C22⋊C4×C157—21no—
91C42×C157—21no—
92C22×D157—22no—
93C22×D157—22no—
94C22×D157—21no—
95C22×D157—41no—
96C22×D157C2×D411yescharacteristic
97C22×D157C2×D411yescharacteristic
98Dic314—21no—
99Dic314—22no—
100Dic314—22no—
101Dic314—21no—
102Dic314—42no—
103D628—21no—
104D628—21no—
105D628—42no—
106D628—42no—
107C22⋊D157—42no—
108C2×Dic157—22no—
109C2×Dic157—21no—
110C2×Dic157—22no—
111C2×Dic157—42no—
112C2×Dic157C2×D411yescharacteristic
113C2×Dic157C2×D411yescharacteristic
114C4×D157—22no—
115C4×D157—22no—
116C4×D157—21no—
117C4×D157—21no—
118C4×D157—21no—
119C4×D157—21no—
120C4×D157—42no—
121#11929434—22no—
122C23×C157C2×D411yescharacteristic
123C2×C4×C157—21no—
124C2×C4×C157—21no—
125C2×C4×C157—22no—
126C2×C4×C157—42no—
127C2×C4×C157C2×D411yescharacteristic
128C2×C4×C157C2×D412yes—
129C2×C4×C157#4112yes—
130C2×C4×C157C2411yescharacteristic
131C8×C157C2×D414yes—
132C22×C157—21no—
133C22×C157—21no—
134C22×C157#12112yes—
135C22×C157C22×D411yescharacteristic
136C2×D157—21no—
137C2×D157—21no—
138C2×D157—21no—
139C2×D157—21no—
140C2×D157—21no—
141C2×D157—44no—
142C2×D157—41no—
143C2×D157—41no—
144C2×D157—41no—
145C2×D157#13712yescomplemented
146C4×C157—41no—
147C4×C157—42no—
148C4×C157—41no—
149C4×C157#12112yes—
150C4×C157C22×D411yescharacteristic
151C4×C157C2×#4111yescharacteristic
152Dic157—21no—
153Dic157—21no—
154Dic157—41no—
155Dic157—42no—
156C2×C157—22no—
157C2×C157C2×#12111yescharacteristic
158C2×C157C2×#13711yescharacteristic
159C2×C157#57511yescharacteristic
160D157—22no—
161D157—41no—
162D157—81no—
163C157#305611yescharacteristic, complemented, Sylow
164#3056—1571nomaximal, Sylow
165C2×#137—1571no—
166C22×M16—1571no—
167#525—1571no—
168C2×C22⋊Q8—1571no—
169#470—1572no—
170#468—1572no—
171#467—1572no—
172#465—1572no—
173#428D15711yescharacteristic, complemented
174#419—1571no—
175#417—1571no—
176C2×#41—1571no—
177C22×Q8—1571no—
178C23×C4—1571no—
179#137—3144no—
180C2×Q16—1572no—
181C2×SD16—1572no—
182C2×M16—3144no—
183C2×M16—1571no—
184C2×M16—1572no—
185C2×M16C2×D15711yescharacteristic
186C22×C8—1572no—
187#123—3141no—
188C22⋊Q8—3141no—
189C22⋊Q8—3141no—
190C22⋊Q8—1572no—
191C22⋊Q8—1572no—
192C22⋊Q8—1571no—
193C22⋊Q8—1571no—
194#121—1571no—
195#121—1571no—
196#120—3141no—
197C4×D4—3141no—
198C4×D4—3141no—
199#117C2×D15711yescharacteristic
200C2×C4⋊C4—3141no—
201C2×C4⋊C4C2×D15711yescharacteristic
202C2×C22⋊C4—3141no—
203#107C2×D15712yescomplemented
204#106C2×D15712yescomplemented
205#103—1572no—
206#103—1572no—
207#103—1572no—
208#102—1572no—
209C24—1571no—
210#41—3142no—
211#41—3142no—
212C2×Q8—3141no—
213C2×Q8—3142no—
214C2×Q8—3142no—
215C2×Q8—1572no—
216C2×Q8—1571no—
217C2×D4—3141no—
218C2×D4—3141no—
219C2×D4—3141no—
220C2×D4—1571no—
221C22×C4—21no—
222C22×C4—3142no—
223C22×C4—3142no—
224C22×C4—3141no—
225C22×C4—3141no—
226C22×C4—1571no—
227C22×C4—1571no—
228C22×C4—1571no—
229C22×C4—1571no—
230C22×C4—1571no—
231C22×C4C22×D15711yescharacteristic
232Q16—3144no—
233SD16—3144no—
234M16—3142no—
235M16—3144no—
236M16D62814yescomplemented
237C2×C8—3144no—
238C2×C8—1572no—
239C2×C8C22×D15712yes—
240C4⋊C4—21no—
241C4⋊C4—3142no—
242C4⋊C4—3142no—
243C4⋊C4—3141no—
244C4⋊C4—3141no—
245C4⋊C4C22×D15712yes—
246C4⋊C4C22×D15711yescharacteristic
247C4⋊C4C22×D15711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—3142no—
250C22⋊C4—3142no—
251C22⋊C4—3141no—
252C22⋊C4—3141no—
253C22⋊C4—3141no—
254C22⋊C4—3141no—
255C42—21no—
256C23—3142no—
257C23—3142no—
258C23—3141no—
259C23—6281no—
260C23—1571no—
261C23—1571no—
262C23D4×D15711yescharacteristic
263Q8—3142no—
264Q8—3141no—
265Q8—3142no—
266Q8—3141no—
267Q8—6282no—
268D4—3141no—
269D4—3141no—
270D4—6282no—
271D4—6282no—
272D4—6282no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3142no—
278C2×C4—3142no—
279C2×C4—3142no—
280C2×C4—3141no—
281C2×C4—3141no—
282C2×C4—3141no—
283C2×C4—3141no—
284C2×C4—3141no—
285C2×C4—3142no—
286C2×C4—6282no—
287C2×C4—6282no—
288C2×C4—1571no—
289C2×C4—1571no—
290C2×C4D4×D15711yescharacteristic
291C2×C4C2×D62812yes—
292C2×C4#1457895512yes—
293C2×C4C23×D15711yescharacteristic
294C8—3142no—
295C8C2×D62814yes—
296C22—21no—
297C22—21no—
298C22—3141no—
299C22—3141no—
300C22—3141no—
301C22—3141no—
302C22—3141no—
303C22—6284no—
304C22—6281no—
305C22—6281no—
306C22—6281no—
307C22—1572no—
308C22#1909774612yes—
309C22C2×D4×D15711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3141no—
314C4—3141no—
315C4—6281no—
316C4—6282no—
317C4#1909774612yes—
318C4C22×D62811yescharacteristic
319C4C2×#1457895511yescharacteristic
320C2—22no—
321C2—3142no—
322C2—6281no—
323C2—12561no—
324C2C2×#1909774611yescharacteristic
325C2#137×D15711yescharacteristic
326C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radical#23757785C2
hypercenter#428D157
hypocenterC157#3056
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19097804C2semidirect
2. C2×M16×D157C2semidirect
4. C2×#19097751C2semidirect
5. #23757532C2semidirect
6. #23757520C2semidirect
7. #23757515C2semidirect
8. #23757503C2semidirect
9. #23757281C2semidirect
10. #23757272C2semidirect
11. #428×C157C2semidirect
16. C2×Dic628C22semidirect
17. C2×#14578943C22semidirect
36. #19097694C22semidirect
37. #19097694C22semidirect
38. #19097694C22semidirect
39. #19097691C22semidirect
43. #107×C157C22semidirect
44. #106×C157C22semidirect
84. M16×C157D4semidirect
145. C2×D157#137semidirect
163. C157#3056semidirect
173. #428D157semidirect
203. #107C2×D157semidirect
204. #106C2×D157semidirect
236. M16D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C2×M16×D1572
#237577852
C2×#190977512
#237575322
#237575202
#237575152
#237575032
#237572812
#237572722
#428×C1572
#3056157