Finite Group Databasebeta

← #34109415

Subgroups — #34109415

Order 20096.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19097817C211yesmaximal, characteristic
2C2×#19097804C211yesmaximal, characteristic, complemented
3C22×#14578943C211yesmaximal, characteristic, complemented
4#23757844C211yesmaximal, characteristic, complemented
5#23757437C214yesmaximal, complemented
6#23757426C214yesmaximal, complemented
7#23757279C211yesmaximal, characteristic, complemented
8#23757270C211yesmaximal, characteristic, complemented
9#408×C157C211yesmaximal, characteristic, complemented
10#19097817—24no—
11#19097817—24no—
12C2×#14578955—21no—
13C2×#14578954C2212yes—
14C2×#14578954C2211yescharacteristic
15C22×Dic314C2211yescharacteristic, complemented
16C22×D628C2212yes—
17C22×D628C2211yescharacteristic
18C2×D4×D157—22no—
19C2×D4×D157—21no—
20#19097804—24no—
21C2×Dic628C2212yescomplemented
22C2×#14578943—24no—
23C2×#14578943C2212yescomplemented
24C2×#14578943C2214yescomplemented
25#19097769—21no—
26#19097758—21no—
27#19097752—22no—
28#19097750C2214yescomplemented
29#19097730C2211yescharacteristic, complemented
30#19097696C2212yescomplemented
31#19097693C2212yescomplemented
32#19097693C2214yescomplemented
33C2×#41×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic, complemented
35C22×C8×C157C2211yescharacteristic, complemented
36C22⋊C8×C157C2214yescomplemented
37C23×D157—22no—
38C23×D157—21no—
39#14578955—28no—
40#14578954—24no—
41#14578954—24no—
42#14578954—22no—
43#14578954D414yes—
44C2×Dic314—21no—
45C2×Dic314—22no—
46C2×Dic314—22no—
47C2×Dic314C2312yes—
48C2×Dic314C2311yescharacteristic
49C2×D628—24no—
50C2×D628—24no—
51C2×D628—22no—
52C2×D628—21no—
53C2×D628—22no—
54C2×D628D414yes—
55C2×D628C2311yescharacteristic
56C2×D628C2312yes—
57C2×C22⋊D157—22no—
58C2×C22⋊D157—21no—
59D4×D157—216no—
60D4×D157—28no—
61C22×Dic157—21no—
62C2×C4×D157—22no—
63C2×C4×D157—21no—
64Dic628—24no—
65#14578943—28no—
66#14578943—24no—
67C4⋊Dic157C2312yes—
68#14578933—24no—
69C22⋊Dic157—21no—
70C22⋊Dic157—24no—
71C4×Dic157—21no—
72#41×C157—22no—
73#41×C157—22no—
74C2×Q8×C157D411yescharacteristic
75C2×D4×C157D412yes—
76C2×D4×C157D411yescharacteristic
77C22×C4×C157C2312yes—
78C22×C4×C157C2311yescharacteristic
79M16×C157—22no—
80C2×C8×C157—22no—
81C2×C8×C157C2312yes—
82C2×C8×C157C2312yes—
83C22×D157—22no—
84C22×D157—21no—
85C22×D157—28no—
86C22×D157—24no—
87C22×D157—82no—
88C22×D157—84no—
89Dic314—21no—
90Dic314—22no—
91Dic314—22no—
92Dic314—42no—
93Dic314C2×D412yes—
94D628—28no—
95D628—24no—
96D628—22no—
97D628—21no—
98D628C2×D412yes—
99D628C2×D414yes—
100C22⋊D157—28no—
101C22⋊D157—24no—
102C2×Dic157—21no—
103C2×Dic157—22no—
104C2×Dic157—22no—
105C2×Dic157—42no—
106C4×D157—28no—
107C4×D157—24no—
108C23×C157C2×D412yes—
109C23×C157C2×D411yescharacteristic
110Q8×C157—22no—
111D4×C157—24no—
112D4×C157—22no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2411yescharacteristic
120C8×C157—22no—
121C8×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—24no—
125C22×C157#12012yes—
126C22×C157C22×D411yescharacteristic
127C2×D157—24no—
128C2×D157—22no—
129C2×D157—44no—
130C2×D157—42no—
131C2×D157—84no—
132C2×D157—88no—
133C4×C157—22no—
134C4×C157#12012yes—
135C4×C157C22×D411yescharacteristic
136C4×C157C22×D411yescharacteristic
137Dic157—22no—
138Dic157—42no—
139Dic157—42no—
140C2×C157—22no—
141C2×C157—42no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57712yes—
144D157—44no—
145D157—42no—
146C157#300911yescharacteristic, complemented, Sylow
147#3009—1571nomaximal, Sylow
148C2×H(5,𝔽2)—1571no—
149C2×#137—1571no—
150C22×SD16—1571no—
151#536—1571no—
152#452—1574no—
153#450—1574no—
154#418—1571no—
155#416—1571no—
156#408D15711yescharacteristic, complemented
157H(5,𝔽2)—3144no—
158H(5,𝔽2)—3144no—
159C2×#41—3141no—
160C2×#41—1572no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C22×Q8—1571no—
164C22×D4—3142no—
165C22×D4—3141no—
166C22×D4—1572no—
167C22×D4—1571no—
168#137—3144no—
169C2×Q16—1572no—
170C2×SD16—3144no—
171C2×SD16—1574no—
172C2×SD16—1572no—
173C2×M16C2×D15711yescharacteristic, complemented
174C22×C8C2×D15711yescharacteristic, complemented
175C4⋊Q8—3141no—
176#124—3141no—
177#123—3142no—
178C22⋊Q8—1574no—
179#117—1571no—
180#103—1572no—
181#102—1572no—
182#102—1574no—
183C22⋊C8C2×D15714yescomplemented
184C24—3142no—
185C24—3141no—
186#41—22no—
187#41—22no—
188#41—3144no—
189#41—3144no—
190#41—3148no—
191#41—3142no—
192#41—1574no—
193C2×Q8—3141no—
194C2×Q8—3142no—
195C2×Q8—3142no—
196C2×Q8—1572no—
197C2×Q8—1571no—
198C2×Q8D62811yescharacteristic
199C2×D4—3144no—
200C2×D4—3144no—
201C2×D4—31416no—
202C2×D4—3148no—
203C2×D4—3142no—
204C2×D4—3141no—
205C2×D4—3142no—
206C2×D4—3142no—
207C2×D4—3141no—
208C2×D4—1574no—
209C2×D4—1571no—
210C2×D4—1572no—
211C2×D4D62812yes—
212C2×D4D62811yescharacteristic
213C22×C4—3142no—
214C22×C4—3141no—
215C22×C4—3141no—
216C22×C4C22×D15712yes—
217C22×C4C22×D15711yescharacteristic
218Q16—3144no—
219SD16—3148no—
220SD16—3144no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D15712yes—
224C2×C8C22×D15712yes—
225C4⋊C4—3144no—
226C4⋊C4—1572no—
227C22⋊C4—3141no—
228C22⋊C4—3144no—
229C42—3141no—
230C23—3148no—
231C23—3142no—
232C23—3141no—
233C23—3144no—
234C23—12562no—
235C23—12564no—
236C23C2×D62812yes—
237C23C2×D62811yescharacteristic
238Q8—22no—
239Q8—3141no—
240Q8—3142no—
241Q8—3142no—
242Q8—6282no—
243Q8—1572no—
244D4—24no—
245D4—22no—
246D4—3148no—
247D4—3148no—
248D4—3144no—
249D4—3144no—
250D4—3142no—
251D4—3141no—
252D4—1572no—
253D4—1574no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3148no—
257C2×C4—3141no—
258C2×C4—3144no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—6282no—
262C2×C4D4×D15712yes—
263C2×C4D4×D15712yes—
264C2×C4C2×D62812yes—
265C2×C4C2×D62811yescharacteristic
266C2×C4C23×D15711yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—3144no—
273C22—3142no—
274C22—6284no—
275C22—6282no—
276C22—12564no—
277C22—12568no—
278C22#1909774112yes—
279C22C22×D62811yescharacteristic
280C4—22no—
281C4—3142no—
282C4—6282no—
283C4—6282no—
284C4#1909774112yes—
285C4C2×D4×D15711yescharacteristic
286C4C2×D4×D15711yescharacteristic
287C2—22no—
288C2—42no—
289C2—6284no—
290C2—6282no—
291C2C2×#1909774111yescharacteristic
292C2#2375808512yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#408×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097817C2
hypercenter#408D157
hypocenterC157#3009
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#19097804C2semidirect
3. C22×#14578943C2semidirect
4. #23757844C2semidirect
5. #23757437C2semidirect
6. #23757426C2semidirect
7. #23757279C2semidirect
8. #23757270C2semidirect
9. #408×C157C2semidirect
15. C22×Dic314C22semidirect
21. C2×Dic628C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578943C22semidirect
28. #19097750C22semidirect
29. #19097730C22semidirect
30. #19097696C22semidirect
31. #19097693C22semidirect
32. #19097693C22semidirect
34. C2×M16×C157C22semidirect
35. C22×C8×C157C22semidirect
36. C22⋊C8×C157C22semidirect
146. C157#3009semidirect
156. #408D157semidirect
173. C2×M16C2×D157semidirect
174. C22×C8C2×D157semidirect
183. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190978042
C22×#145789432
#237578442
#237574372
#237574262
#237572792
#237572702
#408×C1572
#3009157