Finite Group Databasebeta

← #34110332

Subgroups — #34110332

Order 20096.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757877C211yesmaximal, characteristic, complemented
2#23757752C211yesmaximal, characteristic, complemented
3#23757590C211yesmaximal, characteristic, complemented
4#23757585C211yesmaximal, characteristic, complemented
5#23757581C211yesmaximal, characteristic, complemented
6#23757573C211yesmaximal, characteristic, complemented
7#23757470C211yesmaximal, characteristic, complemented
8#23757415C211yesmaximal, characteristic, complemented
9#23757415C211yesmaximal, characteristic, complemented
10#23757385C211yesmaximal, characteristic, complemented
11#23757370C211yesmaximal, characteristic, complemented
12#23757364C211yesmaximal, characteristic, complemented
13D8×Dic157C211yesmaximal, characteristic, complemented
14#23757333C211yesmaximal, characteristic, complemented
15#459×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578942C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19#19097763C2211yescharacteristic
20#19097752—21no—
21#19097752—21no—
22#19097743—21no—
23#19097743—21no—
24#19097742C2211yescharacteristic
25#19097742C2211yescharacteristic
26C4×Dic314C2211yescharacteristic
27#19097736C2211yescharacteristic
28D4×Dic157C2211yescharacteristic
29D4×Dic157C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31C4×C22⋊D157—21no—
32#19097731—21no—
33#19097728C2211yescharacteristic
34#19097727—21no—
35C2×#14578933—21no—
36C4⋊C4×D157C2211yescharacteristic
37#19097705C2211yescharacteristic
38#19097704C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41C4⋊#11929434C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097692C2211yescharacteristic
45#19097692C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097683C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×D8×C157C2211yescharacteristic
53C2×D8×C157C2211yescharacteristic
54#127×C157C2211yescharacteristic
55C4×D4×C157C2211yescharacteristic
56C4⋊C8×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—22no—
61C2×Dic314C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65C22×Dic157—21no—
66C22×Dic157—21no—
67C22×Dic157—21no—
68C22×Dic157—21no—
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578942—22no—
72#14578942—22no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C4⋊Dic157C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78#14578933—21no—
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84#14578933D411yescharacteristic
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90#14578931D411yescharacteristic
91#14578931D411yescharacteristic
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C2×D4×C157—21no—
96C2×D4×C157—21no—
97C2×D4×C157C2311yescharacteristic
98C2×D4×C157C2311yescharacteristic
99C2×D4×C157C2311yescharacteristic
100C22×C4×C157—21no—
101D8×C157—22no—
102D8×C157—22no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C42×C157C2311yescharacteristic
108C22×D157C2×D411yescharacteristic
109Dic314—21no—
110Dic314#4112yes—
111C22⋊D157—24no—
112C22⋊D157—81no—
113C22⋊D157—81no—
114C2×Dic157—24no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—81no—
120C2×Dic157—81no—
121C2×Dic157—81no—
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C4×D157—21no—
126C4×D157—21no—
127C4×D157—81no—
128#11929434—21no—
129#11929434—21no—
130C23×C157—21no—
131C23×C157—21no—
132C23×C157—21no—
133D4×C157—21no—
134D4×C157—21no—
135D4×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157—81no—
139D4×C157—81no—
140D4×C157#4112yes—
141C2×C4×C157—21no—
142C2×C4×C157—81no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157—21no—
148C22×C157—22no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157C22×D411yescharacteristic
158C2×D157—21no—
159C2×D157—21no—
160C2×D157—21no—
161C4×C157—21no—
162C4×C157—21no—
163C4×C157—41no—
164C4×C157C2×#4111yescharacteristic
165C4×C157C2×#4111yescharacteristic
166Dic157—21no—
167Dic157—22no—
168Dic157—21no—
169Dic157—21no—
170Dic157—41no—
171Dic157—41no—
172Dic157—41no—
173C2×C157—42no—
174C2×C157—81no—
175C2×C157—81no—
176C2×C157#52511yescharacteristic
177C2×C157#57611yescharacteristic
178C2×C157#57711yescharacteristic
179D157—41no—
180C157#311911yescharacteristic, complemented, Sylow
181#3119—1571nomaximal, Sylow
182#540—1571no—
183#520—1571no—
184#481—1571no—
185#480—1571no—
186#479—1571no—
187#478—1571no—
188#460—1571no—
189#459D15711yescharacteristic, complemented
190#449—1571no—
191#449—1571no—
192#442—1571no—
193#440—1571no—
194C4×SD16—1571no—
195C4×D8—1571no—
196#433—1571no—
197C2×#41—1571no—
198C2×SD16—1571no—
199C2×SD16—1571no—
200C2×D8C2×D15711yescharacteristic
201C2×D8C2×D15711yescharacteristic
202#127C2×D15711yescharacteristic
203#125—1571no—
204#123—3141no—
205#123—3141no—
206#121—3141no—
207#121—3141no—
208#121—1571no—
209#121—1571no—
210C4×Q8—1571no—
211C4×Q8—1571no—
212C4×D4—3141no—
213C4×D4—3141no—
214C4×D4—1571no—
215C4×D4—1571no—
216C4×D4—1571no—
217C4×D4C2×D15711yescharacteristic
218#117—3141no—
219#117—1571no—
220C2×C4⋊C4—3141no—
221C2×C4⋊C4—1571no—
222#107—1571no—
223#107—1571no—
224#106—1571no—
225#106—1571no—
226C4⋊C8—1571no—
227C4⋊C8C2×D15711yescharacteristic
228#103—1571no—
229#103—1571no—
230#102—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102C2×D15711yescharacteristic
235#102C2×D15711yescharacteristic
236C22⋊C8—1571no—
237C22⋊C8—1571no—
238#97—1571no—
239C4×C8—1571no—
240#41—3142no—
241#41—3142no—
242C2×Q8—1571no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—3141no—
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4C22×D15711yescharacteristic
249C2×D4C22×D15711yescharacteristic
250C2×D4C22×D15711yescharacteristic
251C22×C4—21no—
252C22×C4—3141no—
253C22×C4—3141no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—3141no—
257C22×C4—1571no—
258SD16—3142no—
259SD16—3142no—
260D8—22no—
261D8—22no—
262C2×C8—1571no—
263C2×C8—1571no—
264C2×C8C22×D15711yescharacteristic
265C2×C8C22×D15711yescharacteristic
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—1571no—
272C4⋊C4—1571no—
273C4⋊C4—1571no—
274C4⋊C4—1571no—
275C4⋊C4—1571no—
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—1571no—
284C22⋊C4—1571no—
285C42—3141no—
286C42—3141no—
287C42—1571no—
288C42C22×D15711yescharacteristic
289C23—21no—
290C23—21no—
291C23—21no—
292C23—1571no—
293Q8—3141no—
294Q8—1572no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—3144no—
303D4—12561no—
304D4—12561no—
305D4#1457895312yes—
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—3144no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1571no—
318C2×C4—12561no—
319C2×C4—12561no—
320C2×C4—12561no—
321C2×C4—12561no—
322C2×C4D4×D15711yescharacteristic
323C2×C4D4×D15711yescharacteristic
324C2×C4C23×D15711yescharacteristic
325C8—21no—
326C8—21no—
327C8—3141no—
328C8—3141no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3141no—
339C22—3141no—
340C22—3141no—
341C22C2×D4×D15711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—3141no—
346C4—3141no—
347C4—3142no—
348C4—3141no—
349C4—6281no—
350C4—6281no—
351C4—6281no—
352C4#41×D15711yescharacteristic
353C4C2×#1457895311yescharacteristic
354C2—42no—
355C2—81no—
356C2—81no—
357C2—6281no—
358C2#2375778311yescharacteristic
359C2#2375807411yescharacteristic
360C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3119
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #23757877C2semidirect
2. #23757752C2semidirect
3. #23757590C2semidirect
4. #23757585C2semidirect
5. #23757581C2semidirect
6. #23757573C2semidirect
7. #23757470C2semidirect
8. #23757415C2semidirect
9. #23757415C2semidirect
10. #23757385C2semidirect
11. #23757370C2semidirect
12. #23757364C2semidirect
13. D8×Dic157C2semidirect
14. #23757333C2semidirect
15. #459×C157C2semidirect
180. C157#3119semidirect
189. #459D157semidirect

Maximal subgroups

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

subgroupindex
#237578772
#237577522
#237575902
#237575852
#237575812
#237575732
#237574702
#237574152
#237573852
#237573702
#237573642
D8×Dic1572
#237573332
#459×C1572
#3119157