Finite Group Databasebeta

← #34109611

Subgroups — #34109611

Order 20096.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19097795C212yesmaximal, complemented
2#127×D157C211yesmaximal, characteristic, complemented
3C4×#14578953C211yesmaximal, characteristic, complemented
4#23757474C211yesmaximal, characteristic, complemented
5#23757472C212yesmaximal, complemented
6#23757470C211yesmaximal, characteristic, complemented
7#23757468C211yesmaximal, characteristic, complemented
8#23757466C212yesmaximal, complemented
9#23757295C211yesmaximal, characteristic, complemented
10#23757276C212yesmaximal, complemented
11#459×C157C211yesmaximal, characteristic, complemented
12C2×#14578953C2211yescharacteristic
13C2×D4×D157—21no—
14C2×D4×D157—21no—
15C2×D4×D157C2212yes—
16#19097795—28no—
17C2×#14578943C2212yescomplemented
18C2×#14578942C2212yescomplemented
19C2×#14578939C2212yescomplemented
20C2×#14578937C2212yes—
21#19097768C2211yescharacteristic
22#19097767—21no—
23#19097767—21no—
24#19097767C2212yes—
25C4×Dic314C2211yescharacteristic, complemented
26#19097736C2211yescharacteristic, complemented
27D4×Dic157C2211yescharacteristic, complemented
28C4×C22⋊D157—21no—
29#19097731—21no—
30#19097729C2211yescharacteristic
31#19097727—21no—
32C2×C4×Dic157—21no—
33C42×D157C2211yescharacteristic
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097700C2212yescomplemented
36#19097693C2212yescomplemented
37#19097692C2212yescomplemented
38#19097691C2212yescomplemented
39C2×D8×C157C2212yescomplemented
40#127×C157C2211yescharacteristic
41C4×D4×C157C2211yescharacteristic, complemented
42C4⋊C8×C157C2211yescharacteristic, complemented
43#102×C157C2212yescomplemented
44C23×D157—22no—
45#14578953—22no—
46#14578953D414yescomplemented
47C2×Dic314C2311yescharacteristic
48C2×D628—21no—
49C2×D628—21no—
50C2×D628C2312yes—
51C2×C22⋊D157—21no—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54C2×C22⋊D157—22no—
55D4×D157—24no—
56D4×D157—24no—
57D4×D157—44no—
58D4×D157—44no—
59C22×Dic157—21no—
60C22×Dic157—21no—
61C2×C4×D157D411yescharacteristic
62C2×C4×D157D411yescharacteristic
63C2×C4×D157C2311yescharacteristic
64#14578943—24no—
65#14578942—24no—
66#14578939—24no—
67#14578937—24no—
68C2×#11929434C2312yes—
69C4⋊Dic157C2311yescharacteristic
70#14578933—21no—
71#14578933—21no—
72C22⋊Dic157—21no—
73#14578931—21no—
74C4×Dic157—21no—
75C4×Dic157—21no—
76C4×Dic157—21no—
77C4×Dic157D411yescharacteristic
78C4×Dic157D411yescharacteristic
79C4×Dic157C2311yescharacteristic
80C2×D4×C157—21no—
81C2×D4×C157—21no—
82C2×D4×C157C2312yes—
83C2×D4×C157C2311yescharacteristic
84C22×C4×C157—21no—
85D8×C157—24no—
86C2×C8×C157C2312yes—
87C4⋊C4×C157C2311yescharacteristic
88C22⋊C4×C157—21no—
89C42×C157C2311yescharacteristic
90C22×D157—22no—
91C22×D157—44no—
92C22×D157—44no—
93C22×D157—44no—
94C22×D157C2×D411yescharacteristic
95Dic314—21no—
96Dic314C2×D412yescomplemented
97D628—22no—
98D628—22no—
99D628—42no—
100D628—42no—
101C22⋊D157—24no—
102C22⋊D157—44no—
103C22⋊D157—44no—
104C22⋊D157—44no—
105C2×Dic157—21no—
106C2×Dic157—24no—
107C2×Dic157—21no—
108C2×Dic157—42no—
109C2×Dic157C2×D411yescharacteristic
110C2×Dic157C2×D411yescharacteristic
111C2×Dic157C2×D411yescharacteristic
112C4×D157—22no—
113C4×D157—22no—
114C4×D157C2×D412yes—
115C4×D157#4112yes—
116#11929434—22no—
117C23×C157—21no—
118C23×C157—22no—
119D4×C157—22no—
120D4×C157—21no—
121D4×C157—22no—
122D4×C157—42no—
123D4×C157—42no—
124D4×C157C2×D412yescomplemented
125C2×C4×C157—21no—
126C2×C4×C157—42no—
127C2×C4×C157C2×D411yescharacteristic
128C2×C4×C157C2×D411yescharacteristic
129C2×C4×C157C2411yescharacteristic
130C8×C157—22no—
131C22×C157—22no—
132C22×C157—41no—
133C22×C157—41no—
134C22×C157—42no—
135C22×C157—42no—
136C22×C157—42no—
137C22×C157C22×D411yescharacteristic
138C2×D157—21no—
139C2×D157—21no—
140C2×D157—42no—
141C2×D157—42no—
142C2×D157—42no—
143C2×D157—88no—
144C2×D157#12112yes—
145C4×C157—21no—
146C4×C157—41no—
147C4×C157AGL(1,ℤ/8)12yescomplemented
148C4×C157C22×D411yescharacteristic
149C4×C157C2×#4111yescharacteristic
150Dic157—22no—
151Dic157—21no—
152Dic157—41no—
153Dic157#12112yes—
154Dic157AGL(1,ℤ/8)12yescomplemented
155C2×C157—42no—
156C2×C157—82no—
157C2×C157C2×#12111yescharacteristic
158C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
160D157—22no—
161D157—82no—
162C157#303811yescharacteristic, complemented, Sylow
163#3038—1571nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1572no—
165C2×#127—1571no—
166C4×#41—1571no—
167#460—1571no—
168#460—1571no—
169#460—1572no—
170#459—1572no—
171#459—1571no—
172#459D15711yescharacteristic, complemented
173#423—1571no—
174#418—1572no—
175C2×#41—1571no—
176C22×D4—3141no—
177C22×D4—3141no—
178C22×D4—1572no—
179AGL(1,ℤ/8)—3148no—
180C2×SD16—1572no—
181C2×SD16—1572no—
182C2×D8—1572no—
183C2×D8C2×D15712yescomplemented
184C2×M16—1572no—
185#127—3141no—
186#127—3141no—
187#127—1572no—
188#127—1571no—
189#127C2×D15711yescharacteristic
190C4×Q8—1571no—
191C4×Q8—1571no—
192C4×D4—3141no—
193C4×D4—3141no—
194C4×D4—1571no—
195C4×D4C2×D15711yescharacteristic, complemented
196#117—3141no—
197#117—1571no—
198C2×C42—3141no—
199C2×C42—1571no—
200C4⋊C8—1572no—
201C4⋊C8—1571no—
202C4⋊C8C2×D15711yescharacteristic, complemented
203#102—1572no—
204#102—1572no—
205#102—1572no—
206#102C2×D15712yescomplemented
207C24—3142no—
208#41—3142no—
209#41—1574no—
210C2×Q8—1571no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—3141no—
214C2×D4—3144no—
215C2×D4—3144no—
216C2×D4—3142no—
217C2×D4—3142no—
218C2×D4—3142no—
219C2×D4—3141no—
220C2×D4—3141no—
221C2×D4—6284no—
222C2×D4—6284no—
223C2×D4—1572no—
224C2×D4C22×D15712yes—
225C2×D4C22×D15711yescharacteristic
226C22×C4—21no—
227C22×C4—3141no—
228C22×C4—3141no—
229C22×C4—1571no—
230C22×C4—1571no—
231C22×C4—1571no—
232SD16—3144no—
233SD16—3144no—
234D8—24no—
235D8—3144no—
236M16—3144no—
237C2×C8—1572no—
238C2×C8C22×D15712yes—
239C4⋊C4—3141no—
240C4⋊C4—3141no—
241C4⋊C4—1571no—
242C4⋊C4C22×D15711yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—3141no—
245C22⋊C4—3141no—
246C42—3141no—
247C42—3141no—
248C42—3141no—
249C42—1571no—
250C42—1571no—
251C42—1571no—
252C42C22×D15711yescharacteristic
253C23—21no—
254C23—22no—
255C23—3142no—
256C23—6284no—
257C23—6284no—
258C23—6284no—
259C23—1571no—
260Q8—3141no—
261Q8—1572no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—3144no—
268D4—3142no—
269D4—3142no—
270D4—6282no—
271D4—6284no—
272D4—6284no—
273D4—6284no—
274D4—6282no—
275D4D4×D15712yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—3142no—
279C2×C4—3141no—
280C2×C4—3144no—
281C2×C4—3142no—
282C2×C4—3141no—
283C2×C4—6282no—
284C2×C4—1571no—
285C2×C4—1571no—
286C2×C4—1571no—
287C2×C4—1572no—
288C2×C4—1572no—
289C2×C4D4×D15711yescharacteristic
290C2×C4D4×D15711yescharacteristic
291C2×C4C23×D15711yescharacteristic
292C8—22no—
293C8—3142no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—3141no—
301C22—3141no—
302C22—6282no—
303C22—6282no—
304C22—6282no—
305C22—1572no—
306C22—12568no—
307C22C2×D4×D15711yescharacteristic
308C4—21no—
309C4—41no—
310C4—3142no—
311C4—3141no—
312C4—6281no—
313C4—1572no—
314C4—1572no—
315C4#1909779512yescomplemented
316C4C2×D4×D15711yescharacteristic
317C4#41×D15711yescharacteristic
318C2—42no—
319C2—82no—
320C2—3142no—
321C2—12562no—
322C2#121×D15711yescharacteristic
323C2AGL(1,ℤ/8)×D15711yescharacteristic
324C2C2×#1909779511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3038
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. #127×D157C2semidirect
3. C4×#14578953C2semidirect
4. #23757474C2semidirect
5. #23757472C2semidirect
6. #23757470C2semidirect
7. #23757468C2semidirect
8. #23757466C2semidirect
9. #23757295C2semidirect
10. #23757276C2semidirect
11. #459×C157C2semidirect
17. C2×#14578943C22semidirect
18. C2×#14578942C22semidirect
19. C2×#14578939C22semidirect
25. C4×Dic314C22semidirect
26. #19097736C22semidirect
27. D4×Dic157C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097700C22semidirect
36. #19097693C22semidirect
37. #19097692C22semidirect
38. #19097691C22semidirect
39. C2×D8×C157C22semidirect
41. C4×D4×C157C22semidirect
42. C4⋊C8×C157C22semidirect
43. #102×C157C22semidirect
46. #14578953D4semidirect
96. Dic314C2×D4semidirect
124. D4×C157C2×D4semidirect
147. C4×C157AGL(1,ℤ/8)semidirect
154. Dic157AGL(1,ℤ/8)semidirect
162. C157#3038semidirect
172. #459D157semidirect
183. C2×D8C2×D157semidirect
195. C4×D4C2×D157semidirect
202. C4⋊C8C2×D157semidirect
206. #102C2×D157semidirect
275. D4D4×D157semidirect
315. C4#19097795semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
#127×D1572
C4×#145789532
#237574742
#237574722
#237574702
#237574682
#237574662
#237572952
#237572762
#459×C1572
#3038157