Finite Group Databasebeta

← #34102572

Subgroups — #34102572

Order 20096.

696 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×#19097731C211yesmaximal, characteristic
2C2×#19097729C211yesmaximal, characteristic, complemented
3C22×C4×Dic157C211yesmaximal, characteristic, complemented
4#23757165C211yesmaximal, characteristic, complemented
5#23757163C211yesmaximal, characteristic, complemented
6#23757160C211yesmaximal, characteristic, complemented
7#23757154C211yesmaximal, characteristic, complemented
8#23757152C211yesmaximal, characteristic
9#23757150C211yesmaximal, characteristic
10#23757148C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic157C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic157C211yesmaximal, characteristic, complemented
13#23757120C211yesmaximal, characteristic, complemented
14#23757118C211yesmaximal, characteristic, complemented
15#385×C157C211yesmaximal, characteristic, complemented
16C22×C22⋊D157C2211yescharacteristic
17C23×Dic157C2211yescharacteristic
18C23×Dic157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20#19097731C418yescomplemented
21#19097729—24no—
22C2×C4⋊Dic157C2211yescharacteristic
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×C22⋊Dic157C2211yescharacteristic
26C2×#14578931C2211yescharacteristic
27C2×#14578931C2211yescharacteristic
28C2×#14578931C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×C4×Dic157—24no—
32C2×C4×Dic157C2211yescharacteristic
33C2×C4×Dic157C2211yescharacteristic
34C2×C4×Dic157C2211yescharacteristic
35C2×C4×Dic157C2211yescharacteristic
36C2×C4×Dic157C2211yescharacteristic
37C2×C4×Dic157C2211yescharacteristic
38C2×C4×Dic157C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45#19097678C2211yescharacteristic
46#19097678C2211yescharacteristic
47C23×C4×C157C2211yescharacteristic
48C2×C4⋊C4×C157C2211yescharacteristic
49C2×C4⋊C4×C157C2211yescharacteristic
50C2×C22⋊C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157C2211yescharacteristic
52#95×C157C2211yescharacteristic
53#95×C157C2211yescharacteristic
54C23×D157C2311yescharacteristic
55C2×C22⋊D157—22no—
56C2×C22⋊D157—22no—
57C2×C22⋊D157C2×C414yes—
58C22×Dic157—22no—
59C22×Dic157—22no—
60C22×Dic157—22no—
61C22×Dic157—22no—
62C22×Dic157—22no—
63C22×Dic157C2×C414yes—
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C22×Dic157C2311yescharacteristic
69C22×Dic157C2311yescharacteristic
70C22×Dic157C2311yescharacteristic
71C2×C4×D157—22no—
72C2×C4×D157—22no—
73C2×C4×D157C2×C414yescomplemented
74C4⋊Dic157—22no—
75C4⋊Dic157—22no—
76#14578933C2×C414yescomplemented
77C22⋊Dic157—22no—
78#14578931—22no—
79#14578931—22no—
80#14578931—22no—
81#14578931—22no—
82#14578931C2×C414yescomplemented
83C4×Dic157—22no—
84C4×Dic157—22no—
85C4×Dic157—22no—
86C4×Dic157—22no—
87C4×Dic157—22no—
88C4×Dic157C2×C414yescomplemented
89C4×Dic157D414yescomplemented
90C24×C157C2311yescharacteristic
91C22×C4×C157—22no—
92C22×C4×C157—22no—
93C22×C4×C157—22no—
94C22×C4×C157C2311yescharacteristic
95C22×C4×C157C2311yescharacteristic
96C22×C4×C157C2311yescharacteristic
97C22×C4×C157C2311yescharacteristic
98C22×C4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C4⋊C4×C157—22no—
101C4⋊C4×C157—22no—
102C22⋊C4×C157—22no—
103C22⋊C4×C157C2×C414yes—
104C22×D157—21no—
105C22×D157—21no—
106C22×D157—21no—
107C22×D157—21no—
108C22×D157—21no—
109C22×D157—21no—
110C22×D157C22×C412yes—
111C22⋊D157—28no—
112C2×Dic157—28no—
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—28no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—21no—
128C2×Dic157C22×C412yes—
129C2×Dic157C22×C412yes—
130C2×Dic157C22×C412yes—
131C2×Dic157C2×D412yes—
132C2×Dic157C2×D412yes—
133C2×Dic157#4112yes—
134C2×Dic157#4112yes—
135C4×D157—28no—
136C23×C157—21no—
137C23×C157—21no—
138C23×C157—21no—
139C23×C157—21no—
140C23×C157C22×C412yes—
141C23×C157#4112yes—
142C23×C157#4112yes—
143C23×C157C2411yescharacteristic
144C2×C4×C157—28no—
145C2×C4×C157—21no—
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—21no—
158C2×C4×C157—21no—
159C2×C4×C157C22×C412yes—
160C2×C4×C157C22×C412yes—
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#11714yescomplemented
170C22×C157C23×C411yescharacteristic
171C22×C157C22×D411yescharacteristic
172C22×C157C2×#4111yescharacteristic
173C22×C157C2×#4111yescharacteristic
174C22×C157C2×#4111yescharacteristic
175C22×C157C2×#4111yescharacteristic
176C22×C157C2×#4111yescharacteristic
177C2×D157—22no—
178C2×D157—22no—
179C2×D157—22no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C4×C157—22no—
185C4×C157—22no—
186C4×C157—22no—
187C4×C157—22no—
188C4×C157—41no—
189C4×C157—41no—
190Dic157—22no—
191Dic157—22no—
192Dic157—22no—
193Dic157—22no—
194Dic157—41no—
195Dic157—41no—
196Dic157C4×D414yes—
197C2×C157—24no—
198C2×C157C2×#11711yescharacteristic
199C2×C157C2×C4×D411yescharacteristic
200C2×C157C4×#4111yescharacteristic
201C2×C157#52511yescharacteristic
202C2×C157#52511yescharacteristic
203C2×C157#52911yescharacteristic
204C2×C157#52911yescharacteristic
205D157—42no—
206C157#229011yescharacteristic, complemented, Sylow
207#2290—1571nomaximal, Sylow
208C2×C4×D4—1571no—
209C2×#117—1571no—
210C22×C42—1571no—
211#388—1571no—
212#388—1571no—
213#387—1571no—
214#387—1571no—
215#386—1571no—
216#386—1571no—
217#385—1571no—
218#385D15711yescharacteristic, complemented
219C4×C4⋊C4—1571no—
220C4×C22⋊C4—1571no—
221C4×C22⋊C4—1571no—
222#376—1571no—
223C22×D4—1571no—
224C23×C4—1571no—
225C23×C4—1571no—
226C23×C4—1571no—
227C23×C4C2×D15711yescharacteristic
228C4×D4—1578no—
229#117—3144no—
230C2×C4⋊C4—1571no—
231C2×C4⋊C4—1571no—
232C2×C4⋊C4—1571no—
233C2×C4⋊C4C2×D15711yescharacteristic
234C2×C4⋊C4C2×D15711yescharacteristic
235C2×C22⋊C4—1571no—
236C2×C22⋊C4—1571no—
237C2×C22⋊C4—1571no—
238C2×C22⋊C4—1571no—
239C2×C22⋊C4—1571no—
240C2×C22⋊C4—1571no—
241C2×C22⋊C4C2×D15711yescharacteristic
242C2×C22⋊C4C2×D15711yescharacteristic
243C2×C42—3144no—
244C2×C42—1571no—
245C2×C42—1571no—
246C2×C42—1571no—
247C2×C42—1571no—
248C2×C42—1571no—
249C2×C42—1571no—
250C2×C42—1571no—
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#95C2×D15711yescharacteristic
260#95C2×D15711yescharacteristic
261C24—1571no—
262C24C22×D15711yescharacteristic
263C2×D4—3142no—
264C2×D4—3142no—
265C2×D4—1574no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—3142no—
270C22×C4—3142no—
271C22×C4—3142no—
272C22×C4—3142no—
273C22×C4—3142no—
274C22×C4—3142no—
275C22×C4—3142no—
276C22×C4—1571no—
277C22×C4—1571no—
278C22×C4—1574no—
279C22×C4—1571no—
280C22×C4—1571no—
281C22×C4—1571no—
282C22×C4—1571no—
283C22×C4—1574no—
284C22×C4—1571no—
285C22×C4C22×D15711yescharacteristic
286C22×C4C22×D15711yescharacteristic
287C22×C4C22×D15711yescharacteristic
288C22×C4C22×D15711yescharacteristic
289C22×C4C22×D15711yescharacteristic
290C22×C4C22×D15711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—3142no—
294C4⋊C4—3142no—
295C4⋊C4—1574no—
296C22⋊C4—22no—
297C22⋊C4—3142no—
298C22⋊C4—3142no—
299C22⋊C4—3142no—
300C22⋊C4—3142no—
301C22⋊C4—3142no—
302C22⋊C4—1574no—
303C22⋊C4C4×D15714yes—
304C42—3142no—
305C42—3142no—
306C42—3142no—
307C42—3142no—
308C42—3142no—
309C42—1574no—
310C42—1574no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—3141no—
316C23—3141no—
317C23—3141no—
318C23—3141no—
319C23—3141no—
320C23—3141no—
321C23—1572no—
322C23C2×C4×D15712yes—
323C23#1457895312yes—
324C23#1457895512yes—
325C23C23×D15711yescharacteristic
326D4—3148no—
327C2×C4—28no—
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—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3148no—
343C2×C4—3148no—
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—3148no—
353C2×C4—3141no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—1572no—
360C2×C4—1572no—
361C2×C4—1572no—
362C2×C4—1572no—
363C2×C4—1572no—
364C2×C4—1572no—
365C2×C4—1572no—
366C2×C4C2×C4×D15712yes—
367C2×C4C2×C4×D15712yes—
368C2×C4D4×D15712yes—
369C2×C4#1457895312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—3142no—
377C22—3142no—
378C22—3142no—
379C22—6281no—
380C22—6281no—
381C22—6281no—
382C22—6281no—
383C22#1909772914yescomplemented
384C22C22×C4×D15711yescharacteristic
385C22C2×D4×D15711yescharacteristic
386C22#41×D15711yescharacteristic
387C22#41×D15711yescharacteristic
388C22C2×#1457895311yescharacteristic
389C22C2×#1457895311yescharacteristic
390C22C2×#1457895511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—3142no—
398C4—3142no—
399C4—3142no—
400C4—3142no—
401C4—6281no—
402C4—6281no—
403C4—1574no—
404C2—24no—
405C2—6282no—
406C2C2×#1909772911yescharacteristic
407C2C4×D4×D15711yescharacteristic
408C2C4×#1457895311yescharacteristic
409C2#2375777911yescharacteristic
410C2#2375778611yescharacteristic
411C2#2375780011yescharacteristic
412C2#2375780911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#385×C157C2
socle, cosocleC23×C157C24
prime radicalC22×C22⋊D157C22
hypercenter#385D157
hypocenterC157#2290
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#19097729C2semidirect
3. C22×C4×Dic157C2semidirect
4. #23757165C2semidirect
5. #23757163C2semidirect
6. #23757160C2semidirect
7. #23757154C2semidirect
10. #23757148C2semidirect
11. C4⋊C4×Dic157C2semidirect
12. C22⋊C4×Dic157C2semidirect
13. #23757120C2semidirect
14. #23757118C2semidirect
15. #385×C157C2semidirect
20. #19097731C4semidirect
73. C2×C4×D157C2×C4semidirect
76. #14578933C2×C4semidirect
82. #14578931C2×C4semidirect
88. C4×Dic157C2×C4semidirect
89. C4×Dic157D4semidirect
169. C22×C157#117semidirect
206. C157#2290semidirect
218. #385D157semidirect
383. C22#19097729semidirect

Maximal subgroups

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

subgroupindex
C2×#190977312
C2×#190977292
C22×C4×Dic1572
#237571652
#237571632
#237571602
#237571542
#237571522
#237571502
#237571482
C4⋊C4×Dic1572
C22⋊C4×Dic1572
#237571202
#237571182
#385×C1572
#2290157