Finite Group Databasebeta

← #34100101

Subgroups — #34100101

Order 20096.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#23758109C211yesmaximal, characteristic
2#23757291C211yesmaximal, characteristic, complemented
3#23757290C211yesmaximal, characteristic, complemented
4#23757270C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757239C211yesmaximal, characteristic, complemented
7#416×C157C211yesmaximal, characteristic, complemented
8#19097819—21no—
9#19097818—21no—
10#19097818C412yescomplemented
11#19097817—21no—
12#19097817C412yescomplemented
13#19097816—21no—
14#19097816—22no—
15C2×#14578954—22no—
16C2×#14578954C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18#41×D157—22no—
19#41×D157—22no—
20C2×#14578938C2211yescharacteristic, complemented
21#19097730C2211yescharacteristic, complemented
22#19097699—22no—
23#19097698—22no—
24#19097696—21no—
25#19097693—21no—
26#19097685—22no—
27#19097683—22no—
28C2×#41×C157C2211yescharacteristic
29C22×C8×C157C2211yescharacteristic, complemented
30#117×C157C2211yescharacteristic, complemented
31#103×C157—21no—
32#102×C157—21no—
33#14578955—22no—
34#14578955—22no—
35#14578954—24no—
36#14578954—24no—
37#14578954—22no—
38#14578954—22no—
39#14578954—21no—
40#14578954—21no—
41#14578954—21no—
42#14578954—21no—
43#14578954—21no—
44#14578954—21no—
45#14578954C2×C412yes—
46#14578954D412yes—
47#14578953—24no—
48#14578953—22no—
49#14578953—24no—
50#14578953—22no—
51C2×Dic314—22no—
52C2×Dic314C2×C411yescharacteristic
53C2×Dic314D411yescharacteristic
54Q8×D157—22no—
55Q8×D157—22no—
56C2×D628—22no—
57C2×D628C2×C411yescharacteristic
58C2×D628D411yescharacteristic
59C2×C22⋊D157—24no—
60C2×C22⋊D157—21no—
61C2×C22⋊D157D412yes—
62D4×D157—24no—
63D4×D157—22no—
64D4×D157—24no—
65D4×D157—22no—
66C2×C4×D157—24no—
67C2×C4×D157—21no—
68C2×C4×D157D412yes—
69#14578938—21no—
70#14578938—21no—
71C2×#11929434—21no—
72C4⋊Dic157—21no—
73C22⋊Dic157—21no—
74C4×Dic157—21no—
75#41×C157—21no—
76#41×C157—21no—
77#41×C157C2×C412yes—
78#41×C157D412yes—
79C2×Q8×C157D411yescharacteristic
80C2×D4×C157—21no—
81C2×D4×C157D411yescharacteristic
82C22×C4×C157—21no—
83C22×C4×C157C2311yescharacteristic
84C2×C8×C157—21no—
85C2×C8×C157—21no—
86C2×C8×C157—21no—
87C4⋊C4×C157—21no—
88C22⋊C4×C157—21no—
89C42×C157—21no—
90C22×D157—24no—
91C22×D157—21no—
92C22×D157C22⋊C412yes—
93Dic314—24no—
94Dic314—21no—
95Dic314—21no—
96Dic314—21no—
97Dic314C22⋊C412yes—
98D628—24no—
99D628—21no—
100D628—21no—
101D628—21no—
102D628C22⋊C412yes—
103C22⋊D157—24no—
104C22⋊D157—24no—
105C22⋊D157—22no—
106C22⋊D157—22no—
107C22⋊D157—22no—
108C22⋊D157—22no—
109C2×Dic157—24no—
110C2×Dic157—21no—
111C2×Dic157—41no—
112C2×Dic157C22⋊C412yes—
113C4×D157—24no—
114C4×D157—24no—
115C4×D157—22no—
116C4×D157—22no—
117C4×D157—22no—
118C4×D157—22no—
119#11929434—41no—
120C23×C157—21no—
121C23×C157C2×D411yescharacteristic
122Q8×C157—21no—
123Q8×C157C22⋊C412yes—
124D4×C157—22no—
125D4×C157—22no—
126D4×C157—21no—
127D4×C157C22⋊C412yes—
128C2×C4×C157—21no—
129C2×C4×C157—22no—
130C2×C4×C157—22no—
131C2×C4×C157—41no—
132C2×C4×C157C22×C411yescharacteristic
133C2×C4×C157C2×D411yescharacteristic
134C2×C4×C157C2×D411yescharacteristic
135C2×C4×C157C2×D411yescharacteristic
136C2×C4×C157C2×D411yescharacteristic
137C2×C4×C157C2×D411yescharacteristic
138C8×C157—41no—
139C22×C157—22no—
140C22×C157—41no—
141C22×C157—81no—
142C22×C157C2×C22⋊C411yescharacteristic
143C22×C157#12011yescharacteristic
144C22×C157#12011yescharacteristic
145C2×D157—22no—
146C2×D157—22no—
147C2×D157—42no—
148C2×D157—81no—
149C2×D157—84no—
150C4×C157—22no—
151C4×C157—81no—
152C4×C157C2×C22⋊C411yescharacteristic
153C4×C157C2×C22⋊C411yescharacteristic
154C4×C157#12011yescharacteristic
155C4×C157#12011yescharacteristic
156Dic157—22no—
157Dic157—22no—
158Dic157—81no—
159C2×C157—21no—
160C2×C157—21no—
161C2×C157—21no—
162C2×C157—42no—
163C2×C157#37911yescharacteristic
164D157—42no—
165D157—42no—
166C157#179411yescharacteristic, complemented, Sylow
167#1794—1571nomaximal, Sylow
168#585—1571no—
169#421—1571no—
170#421—1571no—
171#416—1571no—
172#416D15711yescharacteristic, complemented
173#410—1571no—
174#408—1571no—
1752−1+4—3141no—
1762−1+4—3141no—
1772−1+4—1572no—
178H(5,𝔽2)—3142no—
179H(5,𝔽2)—3141no—
180H(5,𝔽2)—3141no—
181H(5,𝔽2)—1572no—
182C2×#41—3142no—
183C2×#41—3142no—
184C2×#41—3142no—
185C2×#41—1571no—
186C2×#41—1571no—
187C2×#41C2×D15711yescharacteristic
188C2×M16—1571no—
189C22×C8C2×D15711yescharacteristic, complemented
190#117—1571no—
191#117C2×D15711yescharacteristic, complemented
192#104—3142no—
193#104—3142no—
194#103—21no—
195#103—3141no—
196#102—21no—
197#102—3141no—
198#99—3142no—
199C22⋊C8—3142no—
200#41—21no—
201#41—21no—
202#41—3144no—
203#41—3144no—
204#41—3142no—
205#41—3142no—
206#41—3144no—
207#41—3142no—
208#41—3142no—
209#41—3142no—
210#41—3142no—
211#41—3144no—
212#41—3141no—
213#41—3141no—
214#41—3141no—
215#41—3141no—
216#41—3141no—
217#41—3141no—
218#41—1572no—
219#41—1572no—
220#41C4×D15712yes—
221#41D62812yes—
222C2×Q8—3142no—
223C2×Q8—3142no—
224C2×Q8—3142no—
225C2×Q8—1571no—
226C2×Q8—1571no—
227C2×Q8C22⋊D15711yescharacteristic
228C2×D4—21no—
229C2×D4—3142no—
230C2×D4—3144no—
231C2×D4—3142no—
232C2×D4—3144no—
233C2×D4—3144no—
234C2×D4—3141no—
235C2×D4—3142no—
236C2×D4—1572no—
237C2×D4—1571no—
238C2×D4—1571no—
239C2×D4C22⋊D15711yescharacteristic
240C22×C4—21no—
241C22×C4—3144no—
242C22×C4—3141no—
243C22×C4—1572no—
244C22×C4C22×D15711yescharacteristic
245M16—3141no—
246M16—3141no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—3141no—
251C4⋊C4—21no—
252C4⋊C4—3141no—
253C22⋊C4—21no—
254C22⋊C4—3141no—
255C42—21no—
256C42—3141no—
257C23—21no—
258C23—3144no—
259C23—3141no—
260C23—1572no—
261C23D4×D15711yescharacteristic
262Q8—21no—
263Q8—3144no—
264Q8—3141no—
265Q8—3141no—
266Q8—3141no—
267Q8—1572no—
268Q8#1457893112yes—
269D4—22no—
270D4—22no—
271D4—21no—
272D4—3144no—
273D4—3144no—
274D4—3144no—
275D4—3142no—
276D4—3142no—
277D4—3142no—
278D4—3142no—
279D4—3141no—
280D4—3141no—
281D4—3141no—
282D4—1572no—
283D4#1457893112yes—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—3144no—
289C2×C4—3144no—
290C2×C4—3144no—
291C2×C4—3141no—
292C2×C4—3142no—
293C2×C4—3142no—
294C2×C4—3142no—
295C2×C4—3142no—
296C2×C4—6281no—
297C2×C4—1572no—
298C2×C4C2×C4×D15711yescharacteristic
299C2×C4D4×D15711yescharacteristic
300C2×C4D4×D15711yescharacteristic
301C2×C4D4×D15711yescharacteristic
302C2×C4C2×C22⋊D15711yescharacteristic
303C2×C4C2×D62811yescharacteristic
304C8—41no—
305C8—6281no—
306C22—22no—
307C22—41no—
308C22—81no—
309C22—3142no—
310C22—3142no—
311C22—6282no—
312C22—12561no—
313C22—12564no—
314C22C22⋊C4×D15711yescharacteristic
315C22#1909773911yescharacteristic
316C22#1909774111yescharacteristic
317C4—22no—
318C4—81no—
319C4—3142no—
320C4—3142no—
321C4—12561no—
322C4C22⋊C4×D15711yescharacteristic
323C4C2×#1457893111yescharacteristic
324C4#1909773911yescharacteristic
325C4#1909774111yescharacteristic
326C2—21no—
327C2—21no—
328C2—21no—
329C2—42no—
330C2—6282no—
331C2—6282no—
332C2#2375712811yescharacteristic
subgroupG/H
centerC4#19097741
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#416×C157C2
socleC2×C157#379
prime radical#23758109C2
cosocleC22×C4×C157C23
hypercenter#416D157
hypocenterC157#1794
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #23757291C2semidirect
3. #23757290C2semidirect
4. #23757270C2semidirect
5. #23757246C2semidirect
6. #23757239C2semidirect
7. #416×C157C2semidirect
10. #19097818C4semidirect
12. #19097817C4semidirect
20. C2×#14578938C22semidirect
21. #19097730C22semidirect
29. C22×C8×C157C22semidirect
30. #117×C157C22semidirect
166. C157#1794semidirect
172. #416D157semidirect
189. C22×C8C2×D157semidirect
191. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237572912
#237572902
#237572702
#237572462
#237572392
#416×C1572
#1794157