Finite Group Databasebeta

← #34109671

Subgroups — #34109671

Order 20096.

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

normal subgroups only

This group has 504 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
1C2×#19097797C211yesmaximal, characteristic, complemented
2C2×#19097791C211yesmaximal, characteristic, complemented
3#23757753C211yesmaximal, characteristic, complemented
4#23757492C212yesmaximal, complemented
5#23757485C212yesmaximal, complemented
6#23757470C212yesmaximal, complemented
7#23757465C212yesmaximal, complemented
8#23757300C211yesmaximal, characteristic, complemented
9#23757274C211yesmaximal, characteristic, complemented
10C2×#19097692C211yesmaximal, characteristic, complemented
11#535×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13#19097797—24no—
14#19097791—24no—
15C2×#14578944C2211yescharacteristic, complemented
16C2×#14578942C2212yes—
17C2×#14578942C2211yescharacteristic, complemented
18C2×#14578941C2211yescharacteristic, complemented
19C2×#14578939C2212yes—
20C2×#14578939C2211yescharacteristic, complemented
21C2×#14578938C2211yescharacteristic
22C22×#11929434C2211yescharacteristic
23C4×Dic314C2212yescomplemented
24C4×D628C2212yescomplemented
25#19097731—22no—
26#19097729—21no—
27C2×C4⋊Dic157C2211yescharacteristic
28C4⋊C4×D157—21no—
29C4⋊#11929434C2212yescomplemented
30C4⋊#11929434C2212yescomplemented
31#19097695C2212yescomplemented
32#19097692C2212yes—
33#19097692C2212yes—
34#19097692C2212yescomplemented
35C2×#41×C157C2211yescharacteristic
36C22×D4×C157C2211yescharacteristic
37#127×C157C2212yes—
38#124×C157C2212yes—
39#121×C157—21no—
40#121×C157—21no—
41#120×C157—22no—
42#117×C157C2211yescharacteristic
43#14578954—22no—
44#14578954D414yescomplemented
45C2×Dic314C2311yescharacteristic
46C2×D628C2311yescharacteristic
47C2×C22⋊D157—21no—
48C22×Dic157—21no—
49C2×C4×D157—21no—
50C2×C4×D157—22no—
51#14578944—22no—
52#14578942—24no—
53#14578942—22no—
54#14578941—22no—
55#14578939—24no—
56#14578939—22no—
57#14578938—22no—
58C2×#11929434—22no—
59C2×#11929434C2311yescharacteristic
60C2×#11929434C2311yescharacteristic
61C2×#11929434C2312yes—
62C4⋊Dic157—21no—
63C4⋊Dic157C2312yes—
64#14578933—22no—
65#14578931—22no—
66C4×Dic157—22no—
67C24×C157—21no—
68#41×C157—22no—
69#41×C157—22no—
70C2×Q8×C157C2311yescharacteristic
71C2×D4×C157—21no—
72C2×D4×C157—22no—
73C2×D4×C157—21no—
74C2×D4×C157—22no—
75C2×D4×C157—22no—
76C2×D4×C157—22no—
77C2×D4×C157C2312yes—
78C2×D4×C157C2311yescharacteristic
79C22×C4×C157—21no—
80C22×C4×C157C2311yescharacteristic
81C4⋊C4×C157D411yescharacteristic
82C4⋊C4×C157D411yescharacteristic
83C22⋊C4×C157—22no—
84C22⋊C4×C157—22no—
85C22⋊C4×C157D412yes—
86C42×C157C2312yes—
87C22×D157—21no—
88Dic314—21no—
89Dic314C2×D412yes—
90D628—21no—
91D628C2×D412yes—
92C22⋊D157—24no—
93C2×Dic157—21no—
94C2×Dic157—22no—
95C2×Dic157—81no—
96C4×D157—24no—
97C4×D157—82no—
98#11929434—21no—
99#11929434—21no—
100#11929434—22no—
101C23×C157—22no—
102C23×C157—21no—
103C23×C157—42no—
104C23×C157—81no—
105C23×C157—81no—
106C23×C157C2×D411yescharacteristic
107Q8×C157—21no—
108Q8×C157—21no—
109D4×C157—22no—
110D4×C157—21no—
111D4×C157—22no—
112D4×C157—21no—
113D4×C157—42no—
114D4×C157—42no—
115D4×C157—82no—
116D4×C157—82no—
117C2×C4×C157—21no—
118C2×C4×C157—42no—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D411yescharacteristic
123C2×C4×C157#4112yes—
124C2×C4×C157C2411yescharacteristic
125C22×C157—21no—
126C22×C157—21no—
127C22×C157—41no—
128C22×C157—42no—
129C22×C157—41no—
130C22×C157—42no—
131C22×C157—41no—
132C22×C157—42no—
133C22×C157—84no—
134C22×C157#12112yes—
135C22×C157C22×D411yescharacteristic
136C2×D157—22no—
137C2×D157—41no—
138C2×D157—41no—
139C4×C157—22no—
140C4×C157—22no—
141C4×C157—41no—
142C4×C157#12112yes—
143C4×C157C22×D411yescharacteristic
144C4×C157C2×#4111yescharacteristic
145Dic157—22no—
146Dic157—42no—
147C2×C157—22no—
148C2×C157—81no—
149C2×C157—82no—
150C2×C157C2×#12111yescharacteristic
151C2×C157#57611yescharacteristic
152C2×C157#57711yescharacteristic
153D157—42no—
154C157#304511yescharacteristic, complemented, Sylow
155#3045—1571nomaximal, Sylow
156C2×AGL(1,ℤ/8)—1571no—
157C2×#135—1571no—
158#535D15711yescharacteristic, complemented
159#520—1571no—
160#464—1572no—
161#463—1572no—
162#460—1572no—
163#459—1572no—
164#424—1571no—
165#417—1571no—
166C2×#102—1571no—
167C2×#41—1571no—
168C2×#41C2×D15711yescharacteristic
169C22×D4C2×D15711yescharacteristic
170AGL(1,ℤ/8)—3144no—
171#135—3144no—
172C2×Q16—1571no—
173C2×SD16—1571no—
174C2×SD16—1571no—
175C2×SD16—1572no—
176C2×D8—1571no—
177C2×D8—1572no—
178C2×M16—1571no—
179C22×C8—1571no—
180#127C2×D15712yes—
181#124C2×D15712yes—
182#121—21no—
183#121—21no—
184#120—22no—
185C4×Q8—1572no—
186C4×D4—3142no—
187C4×D4—1572no—
188#117—3141no—
189#117C2×D15711yescharacteristic
190C2×C4⋊C4—3141no—
191C2×C4⋊C4—1571no—
192C4⋊C8—1572no—
193C4⋊C8—1572no—
194#103—1572no—
195#102—1572no—
196#102—1572no—
197#102—1572no—
198C24—21no—
199#41—22no—
200#41—22no—
201#41—3142no—
202#41—1574no—
203C2×Q8—1571no—
204C2×Q8C22×D15711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—3141no—
212C2×D4—1571no—
213C2×D4C22×D15712yes—
214C2×D4C22×D15711yescharacteristic
215C22×C4—21no—
216C22×C4—3141no—
217C22×C4—3141no—
218C22×C4—3142no—
219C22×C4C22×D15711yescharacteristic
220Q16—3142no—
221SD16—3142no—
222SD16—3144no—
223SD16—3142no—
224D8—3142no—
225D8—3144no—
226M16—3142no—
227C2×C8—3142no—
228C2×C8—1571no—
229C2×C8—1571no—
230C2×C8—1572no—
231C4⋊C4—3141no—
232C4⋊C4—3142no—
233C4⋊C4—1572no—
234C4⋊C4C22⋊D15711yescharacteristic
235C4⋊C4C22⋊D15711yescharacteristic
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—3142no—
239C22⋊C4C22⋊D15712yes—
240C42—3142no—
241C42C22×D15712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—3141no—
248C23C2×C22⋊D15711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—3141no—
252Q8—1572no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—42no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—3144no—
262D4—3141no—
263D4—1572no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—3141no—
267C2×C4—3144no—
268C2×C4—3142no—
269C2×C4—12562no—
270C2×C4—12561no—
271C2×C4D4×D15712yes—
272C2×C4C2×C22⋊D15712yes—
273C2×C4C2×C22⋊D15712yes—
274C2×C4C2×C22⋊D15711yescharacteristic
275C2×C4#1457895312yes—
276C2×C4C23×D15711yescharacteristic
277C8—3141no—
278C8—3141no—
279C8—3142no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—84no—
289C22—3142no—
290C22—6281no—
291C22—6281no—
292C22#1909774212yes—
293C22C22×C22⋊D15711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—3142no—
298C4—6282no—
299C4#1909774212yes—
300C4C2×D4×D15711yescharacteristic
301C4C2×#1457895311yescharacteristic
302C2—22no—
303C2—82no—
304C2—81no—
305C2—6282no—
306C2C2×#1909774211yescharacteristic
307C2#2375807611yescharacteristic
308C2#2375808411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#535×C157C2
socleC22×C157C22×D4
hypercenter#535D157
hypocenterC157#3045
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. C2×#19097791C2semidirect
3. #23757753C2semidirect
4. #23757492C2semidirect
5. #23757485C2semidirect
6. #23757470C2semidirect
7. #23757465C2semidirect
8. #23757300C2semidirect
9. #23757274C2semidirect
10. C2×#19097692C2semidirect
11. #535×C157C2semidirect
15. C2×#14578944C22semidirect
17. C2×#14578942C22semidirect
18. C2×#14578941C22semidirect
20. C2×#14578939C22semidirect
23. C4×Dic314C22semidirect
24. C4×D628C22semidirect
29. C4⋊#11929434C22semidirect
30. C4⋊#11929434C22semidirect
31. #19097695C22semidirect
34. #19097692C22semidirect
44. #14578954D4semidirect
154. C157#3045semidirect
158. #535D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
C2×#190977912
#237577532
#237574922
#237574852
#237574702
#237574652
#237573002
#237572742
C2×#190976922
#535×C1572
#3045157