Finite Group Databasebeta

← #34109756

Subgroups — #34109756

Order 20096.

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

normal subgroups only

This group has 548 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×#19097799C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757785C211yesmaximal, characteristic, complemented
4C2×#19097746C211yesmaximal, characteristic, complemented
5#23757527C212yesmaximal, complemented
6#23757520C212yesmaximal, complemented
7#23757510C212yesmaximal, complemented
8#23757503C212yesmaximal, complemented
9#23757277C211yesmaximal, characteristic, complemented
10#23757272C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×D628—21no—
14C22×D628C2211yescharacteristic
15C22×C4×D157C2211yescharacteristic
16#19097799—24no—
17C2×#14578943C2212yescomplemented
18C2×D1256C2212yescomplemented
19C2×#14578938C2211yescharacteristic
20C2×#14578937C2212yescomplemented
21M16×D157—24no—
22C2×C8×D157C2212yescomplemented
23#19097755—21no—
24#19097751C2211yescharacteristic, complemented
25#19097751C2211yescharacteristic, complemented
26#19097746—21no—
27#19097746—21no—
28#19097746C2212yes—
29#19097746C2212yes—
30#19097746C2211yescharacteristic, complemented
31#19097746C2211yescharacteristic, complemented
32#19097741—21no—
33C4×D628—21no—
34C4×D628—21no—
35C2×#14578931—21no—
36#19097694C2212yescomplemented
37#19097691C2212yescomplemented
38#19097691C2212yescomplemented
39#19097691C2212yescomplemented
40C2×M16×C157C2211yescharacteristic
41#117×C157C2211yescharacteristic
42C2×C4⋊C4×C157C2211yescharacteristic
43#107×C157C2212yescomplemented
44#106×C157C2212yescomplemented
45C23×D157—21no—
46C23×D157D411yescharacteristic
47#14578954—22no—
48#14578954—22no—
49C2×Dic314C2311yescharacteristic
50C2×D628—21no—
51C2×D628—21no—
52C2×D628—22no—
53C2×D628—22no—
54C2×D628—22no—
55C2×D628—21no—
56C2×D628—22no—
57C2×D628—44no—
58C2×D628C2312yes—
59C2×D628C2311yescharacteristic
60C2×C22⋊D157—21no—
61C22×Dic157D411yescharacteristic
62C2×C4×D157—22no—
63C2×C4×D157—21no—
64C2×C4×D157—22no—
65C2×C4×D157D411yescharacteristic
66C2×C4×D157D411yescharacteristic
67C2×C4×D157C2311yescharacteristic
68C2×C4×D157C2311yescharacteristic
69#14578943—24no—
70D1256—24no—
71#14578938—22no—
72#14578937—24no—
73C2×#11929434C2312yes—
74C8×D157—24no—
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77#14578931—21no—
78#14578931—21no—
79#14578931—22no—
80#14578931—22no—
81#14578931—21no—
82#14578931—21no—
83C22×C4×C157—21no—
84C22×C4×C157C2311yescharacteristic
85M16×C157D414yescomplemented
86C2×C8×C157C2312yes—
87C4⋊C4×C157—21no—
88C4⋊C4×C157C2311yescharacteristic
89C4⋊C4×C157C2312yes—
90C4⋊C4×C157C2311yescharacteristic
91C22⋊C4×C157—21no—
92C42×C157—21no—
93C22×D157—22no—
94C22×D157—22no—
95C22×D157—21no—
96C22×D157—22no—
97C22×D157—41no—
98C22×D157—42no—
99C22×D157—42no—
100C22×D157—42no—
101C22×D157C2×D411yescharacteristic
102C22×D157C2×D411yescharacteristic
103Dic314—21no—
104Dic314—21no—
105D628—22no—
106D628—21no—
107D628—22no—
108D628—21no—
109D628—42no—
110D628—44no—
111D628—42no—
112D628—44no—
113D628—42no—
114C22⋊D157—42no—
115C2×Dic157—22no—
116C2×Dic157—21no—
117C2×Dic157C2×D411yescharacteristic
118C2×Dic157C2×D411yescharacteristic
119C4×D157—22no—
120C4×D157—22no—
121C4×D157—21no—
122C4×D157—21no—
123C4×D157—21no—
124C4×D157—21no—
125C4×D157—42no—
126#11929434—22no—
127C23×C157C2×D411yescharacteristic
128C2×C4×C157—22no—
129C2×C4×C157—21no—
130C2×C4×C157—21no—
131C2×C4×C157—42no—
132C2×C4×C157C2×D411yescharacteristic
133C2×C4×C157C2×D412yes—
134C2×C4×C157#4112yes—
135C2×C4×C157C2411yescharacteristic
136C8×C157C2×D414yescomplemented
137C22×C157—21no—
138C22×C157—21no—
139C22×C157#12112yes—
140C22×C157C22×D411yescharacteristic
141C2×D157—21no—
142C2×D157—21no—
143C2×D157—21no—
144C2×D157—21no—
145C2×D157—21no—
146C2×D157—44no—
147C2×D157—41no—
148C2×D157—42no—
149C2×D157—42no—
150C2×D157—41no—
151C2×D157—41no—
152C2×D157—42no—
153C2×D157—84no—
154C2×D157AGL(1,ℤ/8)12yescomplemented
155C4×C157—42no—
156C4×C157—41no—
157C4×C157—41no—
158C4×C157#12112yes—
159C4×C157C22×D411yescharacteristic
160C4×C157C2×#4111yescharacteristic
161Dic157—21no—
162Dic157—21no—
163Dic157—41no—
164C2×C157—22no—
165C2×C157C2×#12111yescharacteristic
166C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C157#57511yescharacteristic
168D157—22no—
169D157—41no—
170D157—81no—
171D157—82no—
172C157#305511yescharacteristic, complemented, Sylow
173#3055—1571nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1571no—
175C22×M16—1571no—
176#525—1571no—
177C2×#121—1571no—
178#469—1572no—
179#468—1572no—
180#466—1572no—
181#465—1572no—
182#428D15711yescharacteristic, complemented
183#418—1571no—
184#417—1571no—
185C2×#41—1571no—
186C22×D4—3141no—
187C22×D4—1571no—
188C23×C4—1571no—
189AGL(1,ℤ/8)—3144no—
190C2×SD16—1572no—
191C2×D8—1572no—
192C2×M16—3144no—
193C2×M16—1571no—
194C2×M16—1572no—
195C2×M16C2×D15711yescharacteristic
196C22×C8—1572no—
197#123—3141no—
198C22⋊Q8—1571no—
199C22⋊Q8—1571no—
200#121—3141no—
201#121—3141no—
202#121—1572no—
203#121—1572no—
204#121—1571no—
205#121—1571no—
206#120—3141no—
207C4×D4—3141no—
208C4×D4—3141no—
209#117C2×D15711yescharacteristic
210C2×C4⋊C4C2×D15711yescharacteristic
211C2×C22⋊C4—3141no—
212#107C2×D15712yescomplemented
213#106C2×D15712yescomplemented
214#103—1572no—
215#102—1572no—
216#102—1572no—
217#102—1572no—
218C24—3141no—
219C24—1571no—
220#41—3142no—
221#41—3142no—
222C2×Q8—1571no—
223C2×D4—3141no—
224C2×D4—3141no—
225C2×D4—3142no—
226C2×D4—3142no—
227C2×D4—3141no—
228C2×D4—3141no—
229C2×D4—3142no—
230C2×D4—3142no—
231C2×D4—6284no—
232C2×D4—1571no—
233C2×D4—1572no—
234C22×C4—21no—
235C22×C4—3142no—
236C22×C4—3142no—
237C22×C4—3141no—
238C22×C4—1571no—
239C22×C4—1571no—
240C22×C4—1571no—
241C22×C4—1571no—
242C22×C4—1571no—
243C22×C4C22×D15711yescharacteristic
244SD16—3144no—
245D8—3144no—
246M16—3142no—
247M16—3144no—
248M16D62814yescomplemented
249C2×C8—3144no—
250C2×C8—1572no—
251C2×C8C22×D15712yes—
252C4⋊C4—21no—
253C4⋊C4—3141no—
254C4⋊C4—3141no—
255C4⋊C4C22×D15711yescharacteristic
256C4⋊C4C22×D15711yescharacteristic
257C4⋊C4C22×D15712yes—
258C22⋊C4—21no—
259C22⋊C4—3141no—
260C22⋊C4—3141no—
261C22⋊C4—3142no—
262C22⋊C4—3142no—
263C22⋊C4—3141no—
264C22⋊C4—3141no—
265C42—21no—
266C23—3142no—
267C23—3142no—
268C23—3141no—
269C23—3142no—
270C23—6281no—
271C23—6282no—
272C23—6282no—
273C23—6282no—
274C23—1571no—
275C23—1571no—
276C23D4×D15711yescharacteristic
277Q8—3141no—
278Q8—3141no—
279D4—3141no—
280D4—3142no—
281D4—3142no—
282D4—3141no—
283D4—6284no—
284D4—6282no—
285D4—6284no—
286D4—6282no—
287D4—6282no—
288D4—6282no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—3142no—
294C2×C4—3142no—
295C2×C4—3142no—
296C2×C4—3141no—
297C2×C4—3141no—
298C2×C4—3141no—
299C2×C4—3141no—
300C2×C4—3141no—
301C2×C4—6282no—
302C2×C4—1571no—
303C2×C4—1571no—
304C2×C4D4×D15711yescharacteristic
305C2×C4C2×D62812yes—
306C2×C4#1457895512yes—
307C2×C4C23×D15711yescharacteristic
308C8—3142no—
309C8C2×D62814yescomplemented
310C22—21no—
311C22—21no—
312C22—3141no—
313C22—3141no—
314C22—3141no—
315C22—3141no—
316C22—3141no—
317C22—6284no—
318C22—6281no—
319C22—6282no—
320C22—6282no—
321C22—6281no—
322C22—6281no—
323C22—6282no—
324C22—1572no—
325C22—12564no—
326C22#1909774612yes—
327C22C2×D4×D15711yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—3141no—
332C4—3141no—
333C4—6281no—
334C4#1909774612yes—
335C4C22×D62811yescharacteristic
336C4C2×#1457895511yescharacteristic
337C2—22no—
338C2—3142no—
339C2—6281no—
340C2—12562no—
341C2—12561no—
342C2C2×#1909774611yescharacteristic
343C2AGL(1,ℤ/8)×D15711yescharacteristic
344C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
hypercenter#428D157
hypocenterC157#3055
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×#19097799C2semidirect
2. C2×M16×D157C2semidirect
3. #23757785C2semidirect
4. C2×#19097746C2semidirect
5. #23757527C2semidirect
6. #23757520C2semidirect
7. #23757510C2semidirect
8. #23757503C2semidirect
9. #23757277C2semidirect
10. #23757272C2semidirect
11. #428×C157C2semidirect
17. C2×#14578943C22semidirect
18. C2×D1256C22semidirect
20. C2×#14578937C22semidirect
22. C2×C8×D157C22semidirect
24. #19097751C22semidirect
25. #19097751C22semidirect
30. #19097746C22semidirect
31. #19097746C22semidirect
36. #19097694C22semidirect
37. #19097691C22semidirect
38. #19097691C22semidirect
39. #19097691C22semidirect
43. #107×C157C22semidirect
44. #106×C157C22semidirect
85. M16×C157D4semidirect
136. C8×C157C2×D4semidirect
154. C2×D157AGL(1,ℤ/8)semidirect
172. C157#3055semidirect
182. #428D157semidirect
212. #107C2×D157semidirect
213. #106C2×D157semidirect
248. M16D628semidirect
309. C8C2×D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×M16×D1572
#237577852
C2×#190977462
#237575272
#237575202
#237575102
#237575032
#237572772
#237572722
#428×C1572
#3055157