Finite Group Databasebeta

← #34109769

Subgroups — #34109769

Order 20096.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19097801C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757780C211yesmaximal, characteristic
4C2×#19097751C211yesmaximal, characteristic, complemented
5#23757530C212yesmaximal, complemented
6#23757518C212yesmaximal, complemented
7#23757513C212yesmaximal, complemented
8#23757501C212yesmaximal, complemented
9#23757313C211yesmaximal, characteristic, complemented
10#23757283C211yesmaximal, characteristic, complemented
11#417×C157C211yesmaximal, characteristic, complemented
12C22×Dic314C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14#19097801—24no—
15C2×#14578944C2212yescomplemented
16C2×#14578942C2212yescomplemented
17C2×#14578938C2211yescharacteristic
18C2×#14578937C2212yes—
19M16×D157—24no—
20C2×C8×D157C2212yes—
21#19097752—21no—
22#19097751—21no—
23#19097751—21no—
24#19097751C2212yes—
25#19097751C2212yes—
26#19097747C2211yescharacteristic
27#19097747C2211yescharacteristic
28#19097742C2211yescharacteristic
29#19097742C2211yescharacteristic
30#19097739—21no—
31C4×C22⋊D157—21no—
32C4×C22⋊D157—21no—
33#19097730C2211yescharacteristic
34C2×C4⋊Dic157—21no—
35C2×#14578931—21no—
36#19097704C2212yescomplemented
37#19097702C2212yescomplemented
38#19097696C2212yescomplemented
39#19097696C2212yescomplemented
40C2×#41×C157C2211yescharacteristic
41C2×M16×C157C2211yescharacteristic
42C2×C4⋊C4×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44#102×C157C2212yescomplemented
45C23×D157D411yescharacteristic
46C2×Dic314—21no—
47C2×Dic314—22no—
48C2×Dic314—22no—
49C2×Dic314C2312yes—
50C2×C22⋊D157—21no—
51C2×C22⋊D157—21no—
52C22×Dic157—21no—
53C22×Dic157D411yescharacteristic
54C2×C4×D157—22no—
55C2×C4×D157—22no—
56C2×C4×D157D411yescharacteristic
57C2×C4×D157D411yescharacteristic
58C2×C4×D157C2311yescharacteristic
59C2×C4×D157C2311yescharacteristic
60#14578944—24no—
61#14578942—24no—
62#14578938D414yescomplemented
63#14578937—24no—
64C2×#11929434C2312yes—
65C8×D157—24no—
66C4⋊Dic157—22no—
67C4⋊Dic157—22no—
68C4⋊Dic157C2311yescharacteristic
69C4⋊Dic157C2311yescharacteristic
70#14578933—21no—
71#14578933—21no—
72C22⋊Dic157—21no—
73C22⋊Dic157—21no—
74C22⋊Dic157—21no—
75#14578931—22no—
76#14578931—22no—
77#14578931—21no—
78#14578931—21no—
79C4×Dic157—21no—
80#41×C157—22no—
81#41×C157—22no—
82C2×Q8×C157C2311yescharacteristic
83C2×D4×C157—21no—
84C2×D4×C157C2311yescharacteristic
85C22×C4×C157—21no—
86C22×C4×C157—21no—
87C22×C4×C157C2311yescharacteristic
88M16×C157—22no—
89C2×C8×C157C2312yes—
90C4⋊C4×C157—21no—
91C4⋊C4×C157C2312yes—
92C22×D157—22no—
93C22×D157—22no—
94C22×D157—41no—
95C22×D157C2×D411yescharacteristic
96C22×D157C2×D411yescharacteristic
97Dic314—22no—
98Dic314—22no—
99Dic314—42no—
100C22⋊D157—42no—
101C22⋊D157—42no—
102C2×Dic157—22no—
103C2×Dic157—22no—
104C2×Dic157—21no—
105C2×Dic157—21no—
106C2×Dic157—42no—
107C2×Dic157C2×D411yescharacteristic
108C2×Dic157C2×D411yescharacteristic
109C4×D157—22no—
110C4×D157—22no—
111C4×D157—21no—
112C4×D157—21no—
113C4×D157—21no—
114C4×D157—21no—
115#11929434C2×D414yes—
116C23×C157—21no—
117C23×C157C2×D411yescharacteristic
118Q8×C157—21no—
119Q8×C157—21no—
120D4×C157—21no—
121D4×C157—21no—
122D4×C157—42no—
123C2×C4×C157—22no—
124C2×C4×C157—21no—
125C2×C4×C157—42no—
126C2×C4×C157—42no—
127C2×C4×C157C2×D411yescharacteristic
128C2×C4×C157C2×D412yes—
129C2×C4×C157#4112yes—
130C2×C4×C157C2411yescharacteristic
131C8×C157—22no—
132C22×C157—21no—
133C22×C157—21no—
134C22×C157—41no—
135C22×C157—41no—
136C22×C157—41no—
137C22×C157#12112yes—
138C22×C157C22×D411yescharacteristic
139C2×D157—21no—
140C2×D157—21no—
141C2×D157—21no—
142C2×D157—21no—
143C2×D157—21no—
144C2×D157—44no—
145C2×D157#13712yescomplemented
146C4×C157—41no—
147C4×C157—42no—
148C4×C157#12112yes—
149C4×C157C22×D411yescharacteristic
150C4×C157C2×#4111yescharacteristic
151Dic157—21no—
152Dic157—21no—
153Dic157—42no—
154Dic157—41no—
155Dic157—41no—
156C2×C157—22no—
157C2×C157—81no—
158C2×C157C2×#12111yescharacteristic
159C2×C157C2×#13711yescharacteristic
160C2×C157#57511yescharacteristic
161D157—22no—
162D157—41no—
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#428—1571no—
174#419—1571no—
175#417D15711yescharacteristic, complemented
176C2×#41C2×D15711yescharacteristic
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—1571no—
191C22⋊Q8—1571no—
192C22⋊Q8—1572no—
193C22⋊Q8—1572no—
194#121—1571no—
195#121—1571no—
196#120—3141no—
197C4×D4—3141no—
198C4×D4—3141no—
199#117—1571no—
200C2×C4⋊C4—3141no—
201C2×C4⋊C4C2×D15711yescharacteristic
202C2×C22⋊C4—3141no—
203#107—1572no—
204#106—1572no—
205#103—1572no—
206#103—1572no—
207#103C2×D15712yescomplemented
208#102C2×D15712yescomplemented
209C24—1571no—
210#41—22no—
211#41—22no—
212C2×Q8—3141no—
213C2×Q8—3142no—
214C2×Q8—3142no—
215C2×Q8—1572no—
216C2×Q8C22×D15711yescharacteristic
217C2×D4—21no—
218C2×D4—3141no—
219C2×D4—3141no—
220C2×D4C22×D15711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—3142no—
224C22×C4—3142no—
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—22no—
235M16—3144no—
236M16—1574no—
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⋊C4—1571no—
246C4⋊C4—1571no—
247C4⋊C4C22×D15712yes—
248C22⋊C4—3142no—
249C22⋊C4—3142no—
250C22⋊C4—3141no—
251C22⋊C4—3141no—
252C22⋊C4—3141no—
253C22⋊C4—3141no—
254C22⋊C4—3141no—
255C42—3141no—
256C23—21no—
257C23—3142no—
258C23—3142no—
259C23—6281no—
260C23—1571no—
261C23—1571no—
262C23D4×D15711yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—3142no—
266Q8—3142no—
267Q8—6282no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—6282no—
272D4—6282no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
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—3141no—
286C2×C4—3142no—
287C2×C4—6282no—
288C2×C4—1571no—
289C2×C4—1571no—
290C2×C4D4×D15711yescharacteristic
291C2×C4D4×D15712yes—
292C2×C4#1457895412yes—
293C2×C4C23×D15711yescharacteristic
294C8—22no—
295C8—1574no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—3141no—
302C22—3141no—
303C22—3141no—
304C22—3141no—
305C22—3141no—
306C22—6284no—
307C22—1572no—
308C22#1909774412yes—
309C22C2×D4×D15711yescharacteristic
310C4—41no—
311C4—42no—
312C4—3141no—
313C4—3141no—
314C4—6282no—
315C4—6281no—
316C4—6281no—
317C4#1909774412yes—
318C4C2×D4×D15711yescharacteristic
319C4C2×#1457895411yescharacteristic
320C2—22no—
321C2—81no—
322C2—3142no—
323C2—6281no—
324C2C2×#1909774411yescharacteristic
325C2#137×D15711yescharacteristic
326C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
prime radical#23757780C2
hypercenter#417D157
hypocenterC157#3056
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19097801C2semidirect
2. C2×M16×D157C2semidirect
4. C2×#19097751C2semidirect
5. #23757530C2semidirect
6. #23757518C2semidirect
7. #23757513C2semidirect
8. #23757501C2semidirect
9. #23757313C2semidirect
10. #23757283C2semidirect
11. #417×C157C2semidirect
15. C2×#14578944C22semidirect
16. C2×#14578942C22semidirect
36. #19097704C22semidirect
37. #19097702C22semidirect
38. #19097696C22semidirect
39. #19097696C22semidirect
43. #103×C157C22semidirect
44. #102×C157C22semidirect
62. #14578938D4semidirect
145. C2×D157#137semidirect
163. C157#3056semidirect
175. #417D157semidirect
207. #103C2×D157semidirect
208. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C2×M16×D1572
#237577802
C2×#190977512
#237575302
#237575182
#237575132
#237575012
#237573132
#237572832
#417×C1572
#3056157