Finite Group Databasebeta

← #34109391

Subgroups — #34109391

Order 20096.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14578953C211yesmaximal, characteristic
2C2×#19097800C212yesmaximal, complemented
3C22⋊Q8×D157C211yesmaximal, characteristic, complemented
4#23757436C212yesmaximal, complemented
5#23757432C212yesmaximal, complemented
6#23757431C211yesmaximal, characteristic, complemented
7#23757428C211yesmaximal, characteristic, complemented
8#23757424C211yesmaximal, characteristic, complemented
9#23757271C212yesmaximal, complemented
10#23757235C211yesmaximal, characteristic, complemented
11#450×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—28no—
13C2×#14578953—21no—
14C2×#14578953—21no—
15C2×#14578953C2212yes—
16C2×#14578953C2212yes—
17C22×Dic314C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic, complemented
19C22×C22⋊D157—21no—
20C23×Dic157—21no—
21C22×C4×D157C2211yescharacteristic
22#19097800—28no—
23C2×Dic628C2212yescomplemented
24C2×#14578944C2212yescomplemented
25C2×#14578942C2212yescomplemented
26C2×#14578937C2212yescomplemented
27#19097751C2211yescharacteristic, complemented
28#19097750C2211yescharacteristic
29#19097749—21no—
30#19097748—21no—
31#19097747C2211yescharacteristic, complemented
32C4⋊C4×D157—21no—
33C4⋊C4×D157C2211yescharacteristic, complemented
34C22⋊C4×D157—21no—
35#19097696C2212yescomplemented
36#19097694C2212yescomplemented
37#19097692C2212yescomplemented
38C22⋊#11929434C2211yescharacteristic
39#19097683C2212yescomplemented
40C22×D4×C157C2211yescharacteristic
41C2×SD16×C157C2212yescomplemented
42C22⋊Q8×C157C2211yescharacteristic
43#102×C157C2212yescomplemented
44C22⋊C8×C157C2211yescharacteristic
45C23×D157D411yescharacteristic
46#14578953—28no—
47#14578953—28no—
48#14578953—24no—
49#14578953—24no—
50#14578953—24no—
51#14578953D418yes—
52C2×Dic314—24no—
53C2×Dic314—21no—
54C2×Dic314—21no—
55C2×Dic314C2311yescharacteristic
56C2×Dic314C2312yes—
57Q8×D157—24no—
58C2×C22⋊D157—28no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157—22no—
61C22×Dic157—28no—
62C22×Dic157—22no—
63C22×Dic157—22no—
64C22×Dic157—21no—
65C22×Dic157D411yescharacteristic
66C2×C4×D157—24no—
67C2×C4×D157—21no—
68C2×C4×D157—21no—
69C2×C4×D157D411yescharacteristic
70C2×C4×D157D411yescharacteristic
71C2×C4×D157C2311yescharacteristic
72C2×C4×D157C2311yescharacteristic
73Dic628—24no—
74#14578944—24no—
75#14578942—24no—
76#14578937—24no—
77C2×#11929434C2312yes—
78C4⋊Dic157—21no—
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86C24×C157—21no—
87C2×Q8×C157C2311yescharacteristic
88C2×D4×C157—24no—
89C2×D4×C157—21no—
90C2×D4×C157C2312yes—
91C22×C4×C157C2311yescharacteristic
92SD16×C157—24no—
93C2×C8×C157C2312yes—
94C4⋊C4×C157—21no—
95C4⋊C4×C157C2311yescharacteristic
96C22⋊C4×C157—21no—
97C22×D157—22no—
98C22×D157—22no—
99C22×D157—41no—
100C22×D157C2×D411yescharacteristic
101C22×D157C2×D411yescharacteristic
102Dic314—24no—
103Dic314—22no—
104Dic314—22no—
105Dic314—42no—
106Dic314C2×D414yes—
107C22⋊D157—28no—
108C22⋊D157—28no—
109C2×Dic157—24no—
110C2×Dic157—28no—
111C2×Dic157—22no—
112C2×Dic157—22no—
113C2×Dic157—28no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157C2×D411yescharacteristic
117C2×Dic157C2×D411yescharacteristic
118C4×D157—22no—
119C4×D157—22no—
120C4×D157—22no—
121C4×D157—42no—
122C4×D157—42no—
123C4×D157C2×D414yes—
124#11929434—22no—
125C23×C157—24no—
126C23×C157—22no—
127C23×C157—42no—
128C23×C157—42no—
129C23×C157C2×D411yescharacteristic
130Q8×C157—22no—
131D4×C157—24no—
132D4×C157—22no—
133D4×C157C2×D414yes—
134C2×C4×C157—22no—
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157C2×D411yescharacteristic
138C2×C4×C157C2411yescharacteristic
139C8×C157—22no—
140C22×C157—21no—
141C22×C157—21no—
142C22×C157—24no—
143C22×C157—48no—
144C22×C157—42no—
145C22×C157—42no—
146C22×C157#13712yescomplemented
147C22×C157C22×D411yescharacteristic
148C2×D157—21no—
149C2×D157—21no—
150C2×D157—21no—
151C2×D157—21no—
152C2×D157—21no—
153C2×D157—44no—
154C2×D157#12012yes—
155C4×C157—21no—
156C4×C157—41no—
157C4×C157—41no—
158C4×C157C22×D412yes—
159Dic157—21no—
160Dic157—24no—
161Dic157—41no—
162Dic157—41no—
163Dic157#12012yes—
164C2×C157—22no—
165C2×C157—44no—
166C2×C157C2×#12011yescharacteristic
167C2×C157C2×#13711yescharacteristic
168C2×C157#57511yescharacteristic
169D157—22no—
170D157—41no—
171C157#300711yescharacteristic, complemented, Sylow
172#3007—1571nomaximal, Sylow
173C22×#41—1571no—
174C2×#137—1572no—
175C2×C22⋊Q8—1571no—
176#452—1572no—
177#452—1572no—
178#451—1571no—
179#451—1571no—
180#450—1571no—
181#450D15711yescharacteristic, complemented
182#417—1572no—
183#407—1571no—
184C2×#41—3148no—
185C2×#41—3141no—
186C2×#41—3141no—
187C2×#41—1572no—
188C2×#41—1572no—
189C22×Q8—1571no—
190C22×Q8—1571no—
191C22×D4—3141no—
192C22×D4C2×D15711yescharacteristic
193C23×C4—3141no—
194C23×C4—1571no—
195#137—3148no—
196C2×Q16—1572no—
197C2×Q16—1572no—
198C2×SD16—1572no—
199C2×SD16C2×D15712yescomplemented
200C2×M16—1572no—
201C22⋊Q8—3141no—
202C22⋊Q8—3141no—
203C22⋊Q8—1571no—
204C22⋊Q8—1571no—
205C22⋊Q8—1571no—
206C22⋊Q8C2×D15711yescharacteristic
207C2×C4⋊C4—3141no—
208C2×C4⋊C4—1571no—
209C2×C22⋊C4—3141no—
210#103—1572no—
211#103—1572no—
212#102—1572no—
213#102C2×D15712yescomplemented
214C22⋊C8—1571no—
215C22⋊C8—1572no—
216C22⋊C8C2×D15711yescharacteristic
217C24—21no—
218C24—1571no—
219#41—3148no—
220#41—3148no—
221#41—3144no—
222#41—3144no—
223#41—3144no—
224#41—1578no—
225C2×Q8—3144no—
226C2×Q8—3141no—
227C2×Q8—3141no—
228C2×Q8—3144no—
229C2×Q8—1571no—
230C2×Q8—1572no—
231C2×Q8C22×D15711yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—3148no—
235C2×D4—3142no—
236C2×D4—3142no—
237C2×D4C22×D15712yes—
238C22×C4—3144no—
239C22×C4—3141no—
240C22×C4—3141no—
241C22×C4—3148no—
242C22×C4—3141no—
243C22×C4—3142no—
244C22×C4—3142no—
245C22×C4—1571no—
246C22×C4—1571no—
247C22×C4—1571no—
248C22×C4—1571no—
249C22×C4—1571no—
250C22×C4C22×D15711yescharacteristic
251Q16—3144no—
252Q16—3144no—
253SD16—24no—
254SD16—3144no—
255M16—3144no—
256C2×C8—1572no—
257C2×C8C22×D15712yes—
258C4⋊C4—21no—
259C4⋊C4—3141no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—1571no—
264C4⋊C4C22×D15711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3141no—
267C22⋊C4—3141no—
268C22⋊C4—3141no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—3142no—
274C23—3142no—
275C23—6281no—
276C23—1571no—
277C23—1571no—
278C23D4×D15711yescharacteristic
279Q8—22no—
280Q8—3144no—
281Q8—3142no—
282Q8—3142no—
283Q8—6282no—
284Q8—1574no—
285D4—24no—
286D4—22no—
287D4—3148no—
288D4—3148no—
289D4D4×D15714yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—3142no—
294C2×C4—3142no—
295C2×C4—3144no—
296C2×C4—3148no—
297C2×C4—3142no—
298C2×C4—3142no—
299C2×C4—3148no—
300C2×C4—3142no—
301C2×C4—3141no—
302C2×C4—3141no—
303C2×C4—6282no—
304C2×C4—6282no—
305C2×C4—1571no—
306C2×C4—1571no—
307C2×C4—1574no—
308C2×C4D4×D15711yescharacteristic
309C2×C4C23×D15711yescharacteristic
310C8—22no—
311C8—3142no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—3141no—
319C22—3141no—
320C22—3141no—
321C22—3141no—
322C22—3141no—
323C22—6284no—
324C22—1572no—
325C22#1909780012yescomplemented
326C22C2×D4×D15711yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—3141no—
331C4—3144no—
332C4—6281no—
333C4—6281no—
334C4—1572no—
335C4C2×D4×D15712yes—
336C2—22no—
337C2—44no—
338C2—3142no—
339C2—6281no—
340C2#120×D15711yescharacteristic
341C2C2×#1909780011yescharacteristic
342C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalC22×#14578953C2
hypercenter#450D157
hypocenterC157#3007
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#19097800C2semidirect
3. C22⋊Q8×D157C2semidirect
4. #23757436C2semidirect
5. #23757432C2semidirect
6. #23757431C2semidirect
7. #23757428C2semidirect
8. #23757424C2semidirect
9. #23757271C2semidirect
10. #23757235C2semidirect
11. #450×C157C2semidirect
18. C2×Q8×D157C22semidirect
23. C2×Dic628C22semidirect
24. C2×#14578944C22semidirect
25. C2×#14578942C22semidirect
26. C2×#14578937C22semidirect
27. #19097751C22semidirect
31. #19097747C22semidirect
33. C4⋊C4×D157C22semidirect
35. #19097696C22semidirect
36. #19097694C22semidirect
37. #19097692C22semidirect
39. #19097683C22semidirect
41. C2×SD16×C157C22semidirect
43. #102×C157C22semidirect
146. C22×C157#137semidirect
171. C157#3007semidirect
181. #450D157semidirect
199. C2×SD16C2×D157semidirect
213. #102C2×D157semidirect
325. C22#19097800semidirect

Maximal subgroups

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

subgroupindex
C22×#145789532
C2×#190978002
C22⋊Q8×D1572
#237574362
#237574322
#237574312
#237574282
#237574242
#237572712
#237572352
#450×C1572
#3007157