Finite Group Databasebeta

← #34103122

Subgroups — #34103122

Order 20096.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097752C211yesmaximal, characteristic
2C2×#19097727C211yesmaximal, characteristic, complemented
3#23757168C211yesmaximal, characteristic, complemented
4#23757167C211yesmaximal, characteristic, complemented
5#23757161C211yesmaximal, characteristic, complemented
6#23757157C211yesmaximal, characteristic, complemented
7#23757154C211yesmaximal, characteristic, complemented
8#23757153C211yesmaximal, characteristic
9#23757146C211yesmaximal, characteristic, complemented
10#23757146C211yesmaximal, characteristic, complemented
11C4×#14578933C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic157C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic157C211yesmaximal, characteristic, complemented
14C2×#19097678C211yesmaximal, characteristic, complemented
15C2×C4×D4×C157C211yesmaximal, characteristic
16C23×Dic157C2211yescharacteristic
17C23×Dic157C2211yescharacteristic
18#19097752C418yescomplemented
19#19097727—24no—
20C2×#14578933C2211yescharacteristic
21C2×#14578933C2211yescharacteristic
22C2×#14578933C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×C22⋊Dic157C2211yescharacteristic
27C2×C22⋊Dic157C2211yescharacteristic
28C2×C22⋊Dic157C2211yescharacteristic
29C2×C22⋊Dic157C2211yescharacteristic
30C2×C22⋊Dic157C2211yescharacteristic
31C2×C22⋊Dic157C2211yescharacteristic
32C2×C4×Dic157C2211yescharacteristic
33C2×C4×Dic157C2211yescharacteristic
34C2×C4×Dic157C2211yescharacteristic
35C2×C4×Dic157C2211yescharacteristic
36#19097678C2211yescharacteristic
37#19097678C2211yescharacteristic
38#19097678C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45#19097678C2211yescharacteristic
46C22×D4×C157C2211yescharacteristic
47C23×C4×C157C2211yescharacteristic
48C23×C4×C157C2211yescharacteristic
49C4×D4×C157—24no—
50C2×C4⋊C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157C2211yescharacteristic
52C2×C22⋊C4×C157C2211yescharacteristic
53C2×C42×C157C2211yescharacteristic
54C22×Dic157—22no—
55C22×Dic157—22no—
56C22×Dic157—22no—
57C22×Dic157—22no—
58C22×Dic157—41no—
59C22×Dic157C2×C414yescomplemented
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68#14578933—22no—
69#14578933—22no—
70#14578933—22no—
71#14578933—41no—
72#14578933C2×C414yescomplemented
73#14578933C2×C414yescomplemented
74C22⋊Dic157—22no—
75C22⋊Dic157—22no—
76C22⋊Dic157—41no—
77C22⋊Dic157C2×C414yescomplemented
78C22⋊Dic157C2×C414yescomplemented
79C22⋊Dic157C2×C414yescomplemented
80C4×Dic157—22no—
81C4×Dic157—22no—
82C4×Dic157—22no—
83C4×Dic157—41no—
84C24×C157C2311yescharacteristic
85C24×C157C2311yescharacteristic
86C2×D4×C157—22no—
87C2×D4×C157—22no—
88C2×D4×C157C2×C414yes—
89C22×C4×C157—22no—
90C22×C4×C157—22no—
91C22×C4×C157—22no—
92C22×C4×C157—22no—
93C22×C4×C157—41no—
94C22×C4×C157—41no—
95C22×C4×C157C2311yescharacteristic
96C22×C4×C157C2311yescharacteristic
97C22×C4×C157C2311yescharacteristic
98C22×C4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C4⋊C4×C157—22no—
101C22⋊C4×C157—22no—
102C22⋊C4×C157D414yes—
103C42×C157—22no—
104C2×Dic157—28no—
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×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—44no—
125C2×Dic157C22×C412yes—
126C2×Dic157C22×C412yes—
127C2×Dic157C22×C412yes—
128C2×Dic157C22×C412yes—
129C2×Dic157#4112yes—
130C23×C157—21no—
131C23×C157—21no—
132C23×C157—21no—
133C23×C157—21no—
134C23×C157—21no—
135C23×C157—21no—
136C23×C157—21no—
137C23×C157—21no—
138C23×C157—21no—
139C23×C157—21no—
140C23×C157C22×C412yes—
141C23×C157C22×C412yes—
142C23×C157C2×D412yes—
143C23×C157#4112yes—
144C23×C157C2411yescharacteristic
145D4×C157—28no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157—21no—
154C2×C4×C157—21no—
155C2×C4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—44no—
158C2×C4×C157—44no—
159C2×C4×C157C22×C412yes—
160C2×C4×C157C2×D412yes—
161C2×C4×C157C2×D412yes—
162C2×C4×C157#4112yes—
163C22×C157—22no—
164C22×C157—22no—
165C22×C157—22no—
166C22×C157—22no—
167C22×C157—22no—
168C22×C157—22no—
169C22×C157—22no—
170C22×C157—22no—
171C22×C157—22no—
172C22×C157—41no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157C4×D414yes—
177C22×C157C23×C411yescharacteristic
178C22×C157C22×D411yescharacteristic
179C22×C157C2×#4111yescharacteristic
180C22×C157C2×#4111yescharacteristic
181C22×C157C2×#4111yescharacteristic
182C22×C157H(5,𝔽2)11yescharacteristic
183C22×C1572−1+411yescharacteristic
184C4×C157—22no—
185C4×C157—22no—
186C4×C157—22no—
187C4×C157—22no—
188C4×C157—41no—
189Dic157—22no—
190Dic157—22no—
191Dic157—22no—
192Dic157—22no—
193Dic157—22no—
194Dic157—41no—
195Dic157—41no—
196Dic157—41no—
197C2×C157—24no—
198C2×C157—42no—
199C2×C157C2×C4×D411yescharacteristic
200C2×C157C4×#4111yescharacteristic
201C2×C157#52011yescharacteristic
202C2×C157#54611yescharacteristic
203C2×C157#54711yescharacteristic
204C2×C157#55111yescharacteristic
205C2×C157#55211yescharacteristic
206C157#235911yescharacteristic, complemented, Sylow
207#2359—1571nomaximal, Sylow
208C2×#123—1571no—
209C2×C4×D4D15711yescharacteristic
210C2×#117—1571no—
211#389—1571no—
212#388—1571no—
213#388—1571no—
214#387—1571no—
215#387—1571no—
216#386—1571no—
217#385—1571no—
218#385—1571no—
219C4×C4⋊C4—1571no—
220C4×C22⋊C4—1571no—
221C4×C22⋊C4—1571no—
222C2×#95—1571no—
223C22×D4C2×D15711yescharacteristic
224C23×C4—1571no—
225C23×C4—1571no—
226C23×C4C2×D15711yescharacteristic
227C23×C4C2×D15711yescharacteristic
228#123—1578no—
229C4×D4—24no—
230#117—3144no—
231C2×C4⋊C4—1571no—
232C2×C4⋊C4—1571no—
233C2×C4⋊C4—1571no—
234C2×C4⋊C4—1571no—
235C2×C4⋊C4—1571no—
236C2×C4⋊C4—1571no—
237C2×C4⋊C4C2×D15711yescharacteristic
238C2×C22⋊C4—1571no—
239C2×C22⋊C4—1571no—
240C2×C22⋊C4—1571no—
241C2×C22⋊C4—1571no—
242C2×C22⋊C4—1571no—
243C2×C22⋊C4—1571no—
244C2×C22⋊C4C2×D15711yescharacteristic
245C2×C22⋊C4C2×D15711yescharacteristic
246C2×C42—1571no—
247C2×C42—1571no—
248C2×C42—1571no—
249C2×C42—1571no—
250C2×C42C2×D15711yescharacteristic
251#95—1571no—
252#95—1571no—
253#95—1571no—
254#95—1571no—
255#95—1571no—
256#95—1571no—
257#95—1571no—
258#95—1571no—
259#95—1571no—
260#95—1571no—
261C24C22×D15711yescharacteristic
262C24C22×D15711yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C4×D15714yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—3142no—
273C22×C4—3142no—
274C22×C4—3142no—
275C22×C4—3142no—
276C22×C4—6281no—
277C22×C4—1571no—
278C22×C4—1571no—
279C22×C4—1571no—
280C22×C4—1574no—
281C22×C4—1571no—
282C22×C4—1571no—
283C22×C4—1571no—
284C22×C4—1571no—
285C22×C4—1571no—
286C22×C4C22×D15711yescharacteristic
287C22×C4C22×D15711yescharacteristic
288C22×C4C22×D15711yescharacteristic
289C22×C4C22×D15711yescharacteristic
290C22×C4C22×D15711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—3142no—
293C4⋊C4—3142no—
294C4⋊C4—3142no—
295C4⋊C4—6281no—
296C4⋊C4—1574no—
297C4⋊C4—1574no—
298C22⋊C4—22no—
299C22⋊C4—3142no—
300C22⋊C4—3142no—
301C22⋊C4—6281no—
302C22⋊C4—1574no—
303C22⋊C4—1574no—
304C22⋊C4—1574no—
305C22⋊C4C22⋊D15714yes—
306C42—22no—
307C42—3142no—
308C42—3142no—
309C42—3142no—
310C42—6281no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×D15712yes—
322C23C2×C4×D15712yes—
323C23C2×C22⋊D15712yes—
324C23#1457895412yes—
325C23C23×D15711yescharacteristic
326D4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—3148no—
341C2×C4—3141no—
342C2×C4—3141no—
343C2×C4—3141no—
344C2×C4—3141no—
345C2×C4—3141no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—3141no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—3141no—
360C2×C4—6284no—
361C2×C4—1572no—
362C2×C4—1572no—
363C2×C4—1572no—
364C2×C4—1572no—
365C2×C4—1572no—
366C2×C4C2×C4×D15712yes—
367C2×C4C2×C22⋊D15712yes—
368C2×C4C2×C22⋊D15712yes—
369C2×C4#1457895312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C4×C22⋊D15714yes—
384C22C22×C4×D15711yescharacteristic
385C22C22×C22⋊D15711yescharacteristic
386C22#41×D15711yescharacteristic
387C22C2×#1457895311yescharacteristic
388C22C2×#1457895411yescharacteristic
389C22#1909781611yescharacteristic
390C22#1909781811yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—3142no—
397C4—3142no—
398C4—3142no—
399C4—3142no—
400C4—3142no—
401C4—6281no—
402C4—6281no—
403C4—6281no—
404C2—24no—
405C2—42no—
406C2C2×C4×C22⋊D15711yescharacteristic
407C2C4×#1457895311yescharacteristic
408C2#2375775211yescharacteristic
409C2#2375791711yescharacteristic
410C2#2375792211yescharacteristic
411C2#2375795211yescharacteristic
412C2#2375796311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
FittingC2×C4×D4×C157C2
socle, cosocleC23×C157C24
prime radicalC22×D4×C157C22
hypercenterC2×C4×D4D157
hypocenterC157#2359
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#19097727C2semidirect
3. #23757168C2semidirect
4. #23757167C2semidirect
5. #23757161C2semidirect
6. #23757157C2semidirect
7. #23757154C2semidirect
9. #23757146C2semidirect
10. #23757146C2semidirect
11. C4×#14578933C2semidirect
12. C22⋊C4×Dic157C2semidirect
13. C4×C22⋊Dic157C2semidirect
14. C2×#19097678C2semidirect
18. #19097752C4semidirect
59. C22×Dic157C2×C4semidirect
72. #14578933C2×C4semidirect
73. #14578933C2×C4semidirect
77. C22⋊Dic157C2×C4semidirect
78. C22⋊Dic157C2×C4semidirect
79. C22⋊Dic157C2×C4semidirect
206. C157#2359semidirect

Maximal subgroups

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

subgroupindex
C2×#190977522
C2×#190977272
#237571682
#237571672
#237571612
#237571572
#237571542
#237571532
#237571462
C4×#145789332
C22⋊C4×Dic1572
C4×C22⋊Dic1572
C2×#190976782
C2×C4×D4×C1572
#2359157