Finite Group Databasebeta

← #34102990

Subgroups — #34102990

Order 20096.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097727C211yesmaximal, characteristic
2#23757157C211yesmaximal, characteristic, complemented
3#23757157C211yesmaximal, characteristic, complemented
4#23757155C211yesmaximal, characteristic, complemented
5#23757155C211yesmaximal, characteristic, complemented
6#23757146C211yesmaximal, characteristic
7#23757145C211yesmaximal, characteristic
8#23757140C211yesmaximal, characteristic, complemented
9#23757137C211yesmaximal, characteristic, complemented
10#23757134C211yesmaximal, characteristic
11C4×#14578933C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic157C211yesmaximal, characteristic
13#23757119C211yesmaximal, characteristic, complemented
14C2×#19097678C211yesmaximal, characteristic
15C4×C22⋊C4×C157C211yesmaximal, characteristic
16C23×Dic157C2211yescharacteristic
17C23×Dic157C2211yescharacteristic
18#19097727C418yescomplemented
19C2×#14578933C2211yescharacteristic
20C2×#14578933C2211yescharacteristic
21C2×#14578933C2211yescharacteristic
22C2×#14578933C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×C22⋊Dic157C2211yescharacteristic
26C2×C22⋊Dic157C2211yescharacteristic
27C2×C4×Dic157C2211yescharacteristic
28C2×C4×Dic157C2211yescharacteristic
29C2×C4×Dic157C2211yescharacteristic
30C2×C4×Dic157C2211yescharacteristic
31#19097678C2211yescharacteristic
32#19097678C2211yescharacteristic
33#19097678C2211yescharacteristic
34#19097678C2211yescharacteristic
35#19097678C2211yescharacteristic
36#19097678C2211yescharacteristic
37#19097678C2211yescharacteristic
38#19097678C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45C23×C4×C157C2211yescharacteristic
46C2×C22⋊C4×C157C2211yescharacteristic
47C2×C22⋊C4×C157C2211yescharacteristic
48C2×C42×C157C2211yescharacteristic
49C2×C42×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52C22×Dic157—22no—
53C22×Dic157—22no—
54C22×Dic157—22no—
55C22×Dic157—22no—
56C22×Dic157—41no—
57C22×Dic157C2×C414yes—
58C22×Dic157C2311yescharacteristic
59C22×Dic157C2311yescharacteristic
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66#14578933—22no—
67#14578933—22no—
68#14578933—41no—
69#14578933—41no—
70#14578933C2×C414yescomplemented
71#14578933C2×C414yescomplemented
72C22⋊Dic157—22no—
73C22⋊Dic157C2×C414yes—
74C4×Dic157—22no—
75C4×Dic157—41no—
76C4×Dic157C2×C414yescomplemented
77C4×Dic157C2×C414yescomplemented
78C24×C157C2311yescharacteristic
79C22×C4×C157—22no—
80C22×C4×C157—22no—
81C22×C4×C157—41no—
82C22×C4×C157C2311yescharacteristic
83C22×C4×C157C2311yescharacteristic
84C22×C4×C157C2311yescharacteristic
85C22×C4×C157C2311yescharacteristic
86C22×C4×C157C2311yescharacteristic
87C22×C4×C157C2311yescharacteristic
88C22⋊C4×C157—22no—
89C22⋊C4×C157C2×C414yes—
90C42×C157—22no—
91C42×C157—41no—
92C2×Dic157—28no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—21no—
96C2×Dic157—21no—
97C2×Dic157—21no—
98C2×Dic157—21no—
99C2×Dic157—21no—
100C2×Dic157—21no—
101C2×Dic157—21no—
102C2×Dic157—21no—
103C2×Dic157—21no—
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—44no—
113C2×Dic157C22×C412yes—
114C2×Dic157C22×C412yes—
115C2×Dic157C22×C412yes—
116C2×Dic157C22×C412yes—
117C2×Dic157#4112yes—
118C23×C157—21no—
119C23×C157—21no—
120C23×C157—21no—
121C23×C157—21no—
122C23×C157C22×C412yes—
123C23×C157#4112yes—
124C23×C157#4112yes—
125C23×C157C2411yescharacteristic
126C2×C4×C157—21no—
127C2×C4×C157—21no—
128C2×C4×C157—21no—
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—44no—
142C2×C4×C157C22×C412yes—
143C2×C4×C157C22×C412yes—
144C2×C4×C157#4112yes—
145C22×C157—22no—
146C22×C157—22no—
147C22×C157—22no—
148C22×C157—22no—
149C22×C157—22no—
150C22×C157—22no—
151C22×C157#11714yes—
152C22×C157C23×C411yescharacteristic
153C22×C157C2×#4111yescharacteristic
154C22×C157C2×#4111yescharacteristic
155C22×C157C2×#4111yescharacteristic
156C22×C157C2×#4111yescharacteristic
157C22×C157H(5,𝔽2)11yescharacteristic
158C22×C1572−1+411yescharacteristic
159C4×C157—22no—
160C4×C157—22no—
161C4×C157—22no—
162C4×C157—41no—
163C4×C157—41no—
164C4×C157—41no—
165Dic157—22no—
166Dic157—22no—
167Dic157—22no—
168Dic157—22no—
169Dic157—22no—
170Dic157—41no—
171Dic157—41no—
172Dic157—41no—
173C2×C157—24no—
174C2×C157C2×#11711yescharacteristic
175C2×C157C4×#4111yescharacteristic
176C2×C157#52011yescharacteristic
177C2×C157#55111yescharacteristic
178C2×C157#55111yescharacteristic
179C2×C157#55211yescharacteristic
180C2×C157#55211yescharacteristic
181C157#234311yescharacteristic, complemented, Sylow
182#2343—1571nomaximal, Sylow
183C2×#117—1571no—
184#387—1571no—
185#387—1571no—
186#387—1571no—
187#387—1571no—
188#385—1571no—
189#385—1571no—
190#383—1571no—
191#382—1571no—
192#381—1571no—
193C4×C4⋊C4—1571no—
194C4×C22⋊C4—1571no—
195C4×C22⋊C4D15711yescharacteristic
196#376—1571no—
197C2×#95—1571no—
198C23×C4—1571no—
199C23×C4—1571no—
200C23×C4C2×D15711yescharacteristic
201#117—1578no—
202C2×C4⋊C4—1571no—
203C2×C4⋊C4—1571no—
204C2×C4⋊C4—1571no—
205C2×C4⋊C4—1571no—
206C2×C4⋊C4—1571no—
207C2×C4⋊C4—1571no—
208C2×C22⋊C4—1571no—
209C2×C22⋊C4—1571no—
210C2×C22⋊C4C2×D15711yescharacteristic
211C2×C22⋊C4C2×D15711yescharacteristic
212C2×C42—1571no—
213C2×C42—1571no—
214C2×C42—1571no—
215C2×C42—1571no—
216C2×C42C2×D15711yescharacteristic
217C2×C42C2×D15711yescharacteristic
218#95—1571no—
219#95—1571no—
220#95—1571no—
221#95—1571no—
222#95—1571no—
223#95—1571no—
224#95—1571no—
225#95—1571no—
226#95—1571no—
227#95—1571no—
228#95—1571no—
229#95—1571no—
230#95—1571no—
231#95—1571no—
232#95C2×D15711yescharacteristic
233#95C2×D15711yescharacteristic
234C24C22×D15711yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—3142no—
239C22×C4—3142no—
240C22×C4—3142no—
241C22×C4—3142no—
242C22×C4—6281no—
243C22×C4—1571no—
244C22×C4—1571no—
245C22×C4—1571no—
246C22×C4—1574no—
247C22×C4—1571no—
248C22×C4—1571no—
249C22×C4—1571no—
250C22×C4—1571no—
251C22×C4—1571no—
252C22×C4C22×D15711yescharacteristic
253C22×C4C22×D15711yescharacteristic
254C22×C4C22×D15711yescharacteristic
255C22×C4C22×D15711yescharacteristic
256C22×C4C22×D15711yescharacteristic
257C22×C4C22×D15711yescharacteristic
258C4⋊C4—3142no—
259C4⋊C4—3142no—
260C4⋊C4—6281no—
261C4⋊C4—6281no—
262C4⋊C4—1574no—
263C4⋊C4—1574no—
264C22⋊C4—22no—
265C22⋊C4—3142no—
266C22⋊C4—1574no—
267C22⋊C4C4×D15714yes—
268C42—22no—
269C42—41no—
270C42—3142no—
271C42—6281no—
272C42—1574no—
273C42—1574no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D15712yes—
279C23#1457895412yes—
280C23#1457895412yes—
281C23C23×D15711yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—3148no—
299C2×C4—3141no—
300C2×C4—3141no—
301C2×C4—3141no—
302C2×C4—3141no—
303C2×C4—3141no—
304C2×C4—3141no—
305C2×C4—3141no—
306C2×C4—3141no—
307C2×C4—3141no—
308C2×C4—3141no—
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—6284no—
319C2×C4—1572no—
320C2×C4—1572no—
321C2×C4—1572no—
322C2×C4—1572no—
323C2×C4—1572no—
324C2×C4C2×C4×D15712yes—
325C2×C4C2×C4×D15712yes—
326C2×C4#1457895312yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#1909772814yes—
334C22C22×C4×D15711yescharacteristic
335C22#41×D15711yescharacteristic
336C22C2×#1457895311yescharacteristic
337C22C2×#1457895411yescharacteristic
338C22C2×#1457895411yescharacteristic
339C22#1909781611yescharacteristic
340C22#1909781811yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3142no—
348C4—3142no—
349C4—3142no—
350C4—3142no—
351C4—3142no—
352C4—6281no—
353C4—6281no—
354C4—6281no—
355C2—24no—
356C2C2×#1909772811yescharacteristic
357C2C4×#1457895311yescharacteristic
358C2#2375775211yescharacteristic
359C2#2375794911yescharacteristic
360C2#2375795211yescharacteristic
361C2#2375795611yescharacteristic
362C2#2375796311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
FittingC4×C22⋊C4×C157C2
socle, cosocleC23×C157C24
prime radicalC24×C157C23
hypercenterC4×C22⋊C4D157
hypocenterC157#2343
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #23757157C2semidirect
3. #23757157C2semidirect
4. #23757155C2semidirect
5. #23757155C2semidirect
8. #23757140C2semidirect
9. #23757137C2semidirect
11. C4×#14578933C2semidirect
13. #23757119C2semidirect
18. #19097727C4semidirect
70. #14578933C2×C4semidirect
71. #14578933C2×C4semidirect
76. C4×Dic157C2×C4semidirect
77. C4×Dic157C2×C4semidirect
181. C157#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#190977272
#237571572
#237571552
#237571462
#237571452
#237571402
#237571372
#237571342
C4×#145789332
C22⋊C4×Dic1572
#237571192
C2×#190976782
C4×C22⋊C4×C1572
#2343157