Finite Group Databasebeta

← #34102863

Subgroups — #34102863

Order 20096.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097749C211yesmaximal, characteristic
2#23757179C211yesmaximal, characteristic, complemented
3#23757177C211yesmaximal, characteristic, complemented
4#23757172C211yesmaximal, characteristic, complemented
5#23757170C211yesmaximal, characteristic, complemented
6#23757160C211yesmaximal, characteristic, complemented
7#23757155C211yesmaximal, characteristic, complemented
8#23757154C211yesmaximal, characteristic, complemented
9#23757147C211yesmaximal, characteristic
10#23757147C211yesmaximal, characteristic
11#23757130C211yesmaximal, characteristic
12C4×#14578931C211yesmaximal, characteristic
13#23757120C211yesmaximal, characteristic
14#95×D157C211yesmaximal, characteristic
15#387×C157C211yesmaximal, characteristic, complemented
16C22×Dic314C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20#19097749C418yescomplemented
21C2×C4⋊Dic157C2211yescharacteristic
22C2×C4⋊Dic157C2211yescharacteristic
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×#14578933C2211yescharacteristic
27C2×#14578933C2211yescharacteristic
28C2×#14578931C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×C4×Dic157C2211yescharacteristic
33C2×C4×Dic157C2211yescharacteristic
34C2×C4×Dic157C2211yescharacteristic
35#19097678C2211yescharacteristic
36#19097678C2211yescharacteristic
37#19097678C2211yescharacteristic
38#19097678C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45C2×C4⋊C4×C157C2211yescharacteristic
46C2×C4⋊C4×C157C2211yescharacteristic
47C2×C42×C157C2211yescharacteristic
48#95×C157C2211yescharacteristic
49#95×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52C23×D157C2311yescharacteristic
53C2×Dic314—41no—
54C2×Dic314—41no—
55C2×Dic314C2×C414yescomplemented
56C22×Dic157C2311yescharacteristic
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2311yescharacteristic
59C22×Dic157C2311yescharacteristic
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C2×C4×D157—22no—
64C2×C4×D157—41no—
65C2×C4×D157—41no—
66C2×C4×D157—41no—
67C2×C4×D157—41no—
68C2×C4×D157—41no—
69C2×C4×D157—41no—
70C2×C4×D157C2×C414yes—
71C2×C4×D157Q814yescomplemented
72C4⋊Dic157—41no—
73C4⋊Dic157—41no—
74C4⋊Dic157—41no—
75#14578933—41no—
76#14578933—41no—
77#14578933C2×C414yescomplemented
78#14578933C2×C414yescomplemented
79#14578931—22no—
80#14578931—22no—
81#14578931C2×C414yes—
82#14578931C2×C414yes—
83C4×Dic157—41no—
84C4×Dic157—41no—
85C4×Dic157—41no—
86C22×C4×C157C2311yescharacteristic
87C22×C4×C157C2311yescharacteristic
88C22×C4×C157C2311yescharacteristic
89C22×C4×C157C2311yescharacteristic
90C22×C4×C157C2311yescharacteristic
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C4⋊C4×C157—41no—
94C4⋊C4×C157C2×C414yescomplemented
95C42×C157—41no—
96C22×D157—21no—
97C22×D157—21no—
98C22×D157—21no—
99C22×D157—21no—
100C22×D157C22×C412yes—
101C22×D157C2×Q812yes—
102C22×D157#4112yes—
103Dic314—44no—
104C2×Dic157—21no—
105C2×Dic157—21no—
106C2×Dic157—21no—
107C2×Dic157—21no—
108C2×Dic157—21no—
109C2×Dic157—21no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—21no—
113C2×Dic157—21no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157C22×C412yes—
122C2×Dic157C22×C412yes—
123C2×Dic157C22×C412yes—
124C2×Dic157C2×Q812yes—
125C4×D157—44no—
126C4×D157—44no—
127C4×D157—44no—
128C23×C157C2411yescharacteristic
129C2×C4×C157—21no—
130C2×C4×C157—21no—
131C2×C4×C157—21no—
132C2×C4×C157—21no—
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157C22×C412yes—
147C2×C4×C157C22×C412yes—
148C2×C4×C157C22×C412yes—
149C2×C4×C157C2×Q812yes—
150C22×C157C23×C411yescharacteristic
151C22×C157C22×Q811yescharacteristic
152C22×C157C2×#4111yescharacteristic
153C22×C157H(5,𝔽2)11yescharacteristic
154C22×C157H(5,𝔽2)11yescharacteristic
155C22×C157H(5,𝔽2)11yescharacteristic
156C22×C1572−1+411yescharacteristic
157C2×D157—22no—
158C2×D157—22no—
159C2×D157—22no—
160C2×D157—22no—
161C2×D157—22no—
162C2×D157—22no—
163C2×D157C4×Q814yes—
164C4×C157—22no—
165C4×C157—22no—
166C4×C157—22no—
167C4×C157—22no—
168C4×C157—41no—
169C4×C157—41no—
170C4×C157—41no—
171Dic157—22no—
172Dic157—22no—
173Dic157—22no—
174Dic157—22no—
175Dic157—41no—
176Dic157—41no—
177Dic157—41no—
178C2×C157C2×C4×Q811yescharacteristic
179C2×C157#51811yescharacteristic
180C2×C157#52011yescharacteristic
181C2×C157#53811yescharacteristic
182C2×C157#53911yescharacteristic
183C2×C157#54311yescharacteristic
184C2×C157#54411yescharacteristic
185D157—24no—
186C157#232911yescharacteristic, complemented, Sylow
187#2329—1571nomaximal, Sylow
188C2×C22⋊Q8—1571no—
189#391—1571no—
190#391—1571no—
191#389—1571no—
192#389—1571no—
193#387—1571no—
194#387—1571no—
195#387—1571no—
196#387D15711yescharacteristic, complemented
197#385—1571no—
198#385—1571no—
199#380—1571no—
200C4×C22⋊C4—1571no—
201C4×C22⋊C4—1571no—
202C2×#95—1571no—
203C22×Q8—1571no—
204C23×C4—1571no—
205C23×C4—1571no—
206C23×C4—1571no—
207C22⋊Q8—1578no—
208C2×C4⋊C4—1571no—
209C2×C4⋊C4—1571no—
210C2×C4⋊C4—1571no—
211C2×C4⋊C4—1571no—
212C2×C4⋊C4—1571no—
213C2×C4⋊C4—1571no—
214C2×C4⋊C4—1571no—
215C2×C4⋊C4C2×D15711yescharacteristic
216C2×C4⋊C4C2×D15711yescharacteristic
217C2×C22⋊C4—1571no—
218C2×C22⋊C4—1571no—
219C2×C22⋊C4—1571no—
220C2×C22⋊C4—1571no—
221C2×C42—1571no—
222C2×C42—1571no—
223C2×C42—1571no—
224C2×C42C2×D15711yescharacteristic
225#95—1571no—
226#95—1571no—
227#95—1571no—
228#95—1571no—
229#95—1571no—
230#95—1571no—
231#95—1571no—
232#95—1571no—
233#95—1571no—
234#95—1571no—
235#95C2×D15711yescharacteristic
236#95C2×D15711yescharacteristic
237#95C2×D15711yescharacteristic
238#95C2×D15711yescharacteristic
239C24—1571no—
240C2×Q8—6281no—
241C2×Q8—6281no—
242C2×Q8—1574no—
243C22×C4—3142no—
244C22×C4—6281no—
245C22×C4—6281no—
246C22×C4—6281no—
247C22×C4—6281no—
248C22×C4—6281no—
249C22×C4—6281no—
250C22×C4—1571no—
251C22×C4—1571no—
252C22×C4—1571no—
253C22×C4—1571no—
254C22×C4—1571no—
255C22×C4—1571no—
256C22×C4—1571no—
257C22×C4—1574no—
258C22×C4—1574no—
259C22×C4C22×D15711yescharacteristic
260C22×C4C22×D15711yescharacteristic
261C22×C4C22×D15711yescharacteristic
262C22×C4C22×D15711yescharacteristic
263C22×C4C22×D15711yescharacteristic
264C22×C4C22×D15711yescharacteristic
265C22×C4C22×D15711yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—6281no—
268C4⋊C4—6281no—
269C4⋊C4—6281no—
270C4⋊C4—6281no—
271C4⋊C4—6281no—
272C4⋊C4—1574no—
273C4⋊C4—1574no—
274C4⋊C4C4×D15714yescomplemented
275C22⋊C4—3142no—
276C22⋊C4—3142no—
277C22⋊C4—1574no—
278C22⋊C4—1574no—
279C42—41no—
280C42—6281no—
281C42—6281no—
282C42—6281no—
283C23—3141no—
284C23—3141no—
285C23—3141no—
286C23—3141no—
287C23—1572no—
288C23—1572no—
289C23—1572no—
290C23C23×D15711yescharacteristic
291Q8—6284no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—6284no—
327C2×C4—6284no—
328C2×C4—6284no—
329C2×C4—1572no—
330C2×C4—1572no—
331C2×C4—1572no—
332C2×C4—1572no—
333C2×C4C2×C4×D15712yes—
334C2×C4C2×C4×D15712yes—
335C2×C4C2×C4×D15712yes—
336C2×C4Q8×D15712yes—
337C22—3142no—
338C22—3142no—
339C22—3142no—
340C22—3142no—
341C22—3142no—
342C22—3142no—
343C22—1574no—
344C22C22×C4×D15711yescharacteristic
345C22C2×Q8×D15711yescharacteristic
346C22#41×D15711yescharacteristic
347C22#1909781611yescharacteristic
348C22#1909781711yescharacteristic
349C22#1909781711yescharacteristic
350C22#1909781811yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3142no—
359C4—3142no—
360C4—3142no—
361C4—3142no—
362C4—6281no—
363C4—6281no—
364C4—6281no—
365C2—3144no—
366C2C4×Q8×D15711yescharacteristic
367C2#2375774511yescharacteristic
368C2#2375775311yescharacteristic
369C2#2375786211yescharacteristic
370C2#2375786611yescharacteristic
371C2#2375789811yescharacteristic
372C2#2375790311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#387×C157C2
socle, cosocleC23×C157C24
prime radicalC23×D157C23
hypercenter#387D157
hypocenterC157#2329
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #23757179C2semidirect
3. #23757177C2semidirect
4. #23757172C2semidirect
5. #23757170C2semidirect
6. #23757160C2semidirect
7. #23757155C2semidirect
8. #23757154C2semidirect
15. #387×C157C2semidirect
20. #19097749C4semidirect
55. C2×Dic314C2×C4semidirect
71. C2×C4×D157Q8semidirect
77. #14578933C2×C4semidirect
78. #14578933C2×C4semidirect
94. C4⋊C4×C157C2×C4semidirect
186. C157#2329semidirect
196. #387D157semidirect
274. C4⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977492
#237571792
#237571772
#237571722
#237571702
#237571602
#237571552
#237571542
#237571472
#237571302
C4×#145789312
#237571202
#95×D1572
#387×C1572
#2329157