Finite Group Databasebeta

← #34101053

Subgroups — #34101053

Order 20096.

336 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#23757819C211yesmaximal, characteristic
2C2×#19097699C211yesmaximal, characteristic, complemented
3C2×#19097698C211yesmaximal, characteristic, complemented
4C2×#19097697C211yesmaximal, characteristic, complemented
5M16×Dic157C211yesmaximal, characteristic, complemented
6C4×#14578933C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097769C411yescharacteristic, complemented
11#19097767C411yescharacteristic, complemented
12#19097760C411yescharacteristic, complemented
13#19097758C411yescharacteristic, complemented
14#19097743—21no—
15#19097743—21no—
16C4×C22⋊D157—21no—
17C4×C22⋊D157—21no—
18C2×#14578933—21no—
19C2×C4×Dic157C2211yescharacteristic
20C2×C4×Dic157C2211yescharacteristic
21#19097699—22no—
22#19097698—22no—
23#19097697—22no—
24#19097681—21no—
25C8×Dic157—21no—
26#19097678—21no—
27C2×#41×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic
29C2×C42×C157C2211yescharacteristic
30#104×C157—22no—
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35C2×Dic314C2×C411yescharacteristic
36C2×D628C2×C411yescharacteristic
37C2×C22⋊D157—21no—
38C2×C22⋊D157—21no—
39C2×C22⋊D157—21no—
40C22×Dic157—21no—
41C22×Dic157D411yescharacteristic
42C22×Dic157D411yescharacteristic
43C2×C4×D157—21no—
44#14578938—21no—
45#14578938D412yescomplemented
46C2×#11929434—21no—
47#14578933—24no—
48#14578933—21no—
49#14578933—21no—
50C22⋊Dic157—21no—
51C22⋊Dic157—21no—
52#14578931—21no—
53#14578931—21no—
54C4×Dic157—21no—
55C4×Dic157—21no—
56C4×Dic157C2×C411yescharacteristic
57C4×Dic157C2×C411yescharacteristic
58C4×Dic157D411yescharacteristic
59C4×Dic157D411yescharacteristic
60#41×C157—21no—
61#41×C157—21no—
62#41×C157—21no—
63#41×C157—21no—
64C2×Q8×C157C2×C411yescharacteristic
65C2×D4×C157—21no—
66C2×D4×C157C2×C411yescharacteristic
67C22×C4×C157—21no—
68C22×C4×C157—21no—
69C22×C4×C157C2311yescharacteristic
70M16×C157—21no—
71M16×C157D412yescomplemented
72C2×C8×C157—21no—
73C42×C157—21no—
74C42×C157—21no—
75C22×D157—21no—
76Dic314—21no—
77Dic314—21no—
78D628—21no—
79D628—21no—
80C22⋊D157—42no—
81C22⋊D157—41no—
82C22⋊D157—41no—
83C22⋊D157—42no—
84C2×Dic157—22no—
85C2×Dic157—22no—
86C2×Dic157—21no—
87C2×Dic157—41no—
88C2×Dic157—41no—
89C2×Dic157—41no—
90C2×Dic157C22⋊C411yescharacteristic
91C2×Dic157C22⋊C411yescharacteristic
92C2×Dic157C22⋊C411yescharacteristic
93C2×Dic157C22⋊C411yescharacteristic
94C4×D157—41no—
95C4×D157—41no—
96#11929434—22no—
97C23×C157—21no—
98C23×C157C2×D411yescharacteristic
99Q8×C157—21no—
100Q8×C157—21no—
101D4×C157—21no—
102D4×C157—21no—
103D4×C157—41no—
104D4×C157—41no—
105C2×C4×C157—22no—
106C2×C4×C157—21no—
107C2×C4×C157—41no—
108C2×C4×C157—41no—
109C2×C4×C157—41no—
110C2×C4×C157—41no—
111C2×C4×C157C22×C411yescharacteristic
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157#4111yescharacteristic
116C2×C4×C157#4111yescharacteristic
117C8×C157—22no—
118C22×C157—21no—
119C22×C157—21no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157—41no—
123C22×C157C2×C22⋊C411yescharacteristic
124C22×C157#12411yescharacteristic
125C22×C157#12711yescharacteristic
126C2×D157—41no—
127C2×D157—41no—
128C2×D157—41no—
129C4×C157—42no—
130C4×C157—41no—
131C4×C157C4×D411yescharacteristic
132C4×C157C4×D411yescharacteristic
133C4×C157#12111yescharacteristic
134C4×C157#12111yescharacteristic
135Dic157—21no—
136Dic157—21no—
137Dic157—21no—
138Dic157—42no—
139Dic157—41no—
140Dic157#10412yescomplemented
141C2×C157—21no—
142C2×C157—21no—
143C2×C157—81no—
144C2×C157#39011yescharacteristic
145C2×C157C2×#10411yescharacteristic
146C2×C157#42111yescharacteristic
147D157—81no—
148C157#198011yescharacteristic, complemented, Sylow
149#1980—1571nomaximal, Sylow
150#532—1571no—
151C2×#104—1571no—
152C2×#104—1571no—
153C2×#104—1571no—
154C2×#104D15711yescharacteristic, complemented
155C4×M16—1571no—
156C4×C4⋊C4—1571no—
157C2×#41—1571no—
158C2×#41C2×D15711yescharacteristic
159C2×M16—1571no—
160C2×M16C2×D15711yescharacteristic
161C4⋊Q8—1571no—
162#127—1571no—
163#124—1571no—
164#124—1571no—
165#121—3141no—
166#121—3141no—
167C4×D4—3141no—
168C4×D4—3141no—
169C2×C4⋊C4—3141no—
170C2×C42—1571no—
171C2×C42—1571no—
172C2×C42C2×D15711yescharacteristic
173#104—22no—
174#104—3142no—
175#104—3142no—
176#104—3142no—
177#97—3141no—
178C4×C8—3141no—
179#95—3141no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—3141no—
185#41—3141no—
186#41—3141no—
187#41—3141no—
188C2×Q8—1571no—
189C2×Q8C4×D15711yescharacteristic
190C2×D4—21no—
191C2×D4—3141no—
192C2×D4—3141no—
193C2×D4—3141no—
194C2×D4—1571no—
195C2×D4C4×D15711yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—3141no—
199C22×C4—3141no—
200C22×C4—1571no—
201C22×C4—1571no—
202C22×C4C22×D15711yescharacteristic
203M16—21no—
204M16—3141no—
205M16—1572no—
206M16C22⋊D15712yescomplemented
207C2×C8—21no—
208C2×C8—3141no—
209C4⋊C4—3144no—
210C4⋊C4—3141no—
211C4⋊C4—3141no—
212C22⋊C4—3141no—
213C22⋊C4—3141no—
214C22⋊C4—3141no—
215C22⋊C4—3141no—
216C42—21no—
217C42—21no—
218C42—3141no—
219C42—3141no—
220C42—1571no—
221C42—1571no—
222C42—1571no—
223C42—1571no—
224C23—21no—
225C23—3141no—
226C23D4×D15711yescharacteristic
227Q8—21no—
228Q8—21no—
229Q8—3141no—
230Q8—3141no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—3141no—
236D4—3141no—
237D4—6282no—
238D4—6281no—
239D4—6281no—
240D4—6282no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—3142no—
248C2×C4—3142no—
249C2×C4—3141no—
250C2×C4—6281no—
251C2×C4—6281no—
252C2×C4—6281no—
253C2×C4—6281no—
254C2×C4—6281no—
255C2×C4—1571no—
256C2×C4—1571no—
257C2×C4—1571no—
258C2×C4—1571no—
259C2×C4C2×C4×D15711yescharacteristic
260C2×C4D4×D15711yescharacteristic
261C2×C4D4×D15711yescharacteristic
262C2×C4C2×C22⋊D15711yescharacteristic
263C2×C4#1457895311yescharacteristic
264C2×C4#1457895411yescharacteristic
265C8—22no—
266C8—3142no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—6281no—
273C22—6281no—
274C22—6281no—
275C22C22⋊C4×D15711yescharacteristic
276C22#1909775711yescharacteristic
277C22#1909776711yescharacteristic
278C4—42no—
279C4—41no—
280C4—3141no—
281C4—3141no—
282C4—3141no—
283C4—6282no—
284C4—6281no—
285C4—1572no—
286C4#1909773111yescharacteristic
287C4C4×C22⋊D15711yescharacteristic
288C4#1909774311yescharacteristic
289C4#1909774411yescharacteristic
290C2—21no—
291C2—21no—
292C2—81no—
293C2—12561no—
294C2#2375717311yescharacteristic
295C2#104×D15711yescharacteristic
296C2#2375728811yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757819C2
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1980
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#19097699C2semidirect
3. C2×#19097698C2semidirect
4. C2×#19097697C2semidirect
5. M16×Dic157C2semidirect
6. C4×#14578933C2semidirect
7. C2×#104×C157C2semidirect
10. #19097769C4semidirect
11. #19097767C4semidirect
12. #19097760C4semidirect
13. #19097758C4semidirect
45. #14578938D4semidirect
71. M16×C157D4semidirect
140. Dic157#104semidirect
148. C157#1980semidirect
154. C2×#104D157semidirect
206. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578192
C2×#190976992
C2×#190976982
C2×#190976972
M16×Dic1572
C4×#145789332
C2×#104×C1572
#1980157