Finite Group Databasebeta

← #34113221

Subgroups — #34113221

Order 20096.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758009C211yesmaximal, characteristic, complemented
2#23758005C211yesmaximal, characteristic, complemented
3#23757661C211yesmaximal, characteristic, complemented
4#23757654C211yesmaximal, characteristic, complemented
5#23757493C211yesmaximal, characteristic, complemented
6#23757492C211yesmaximal, characteristic, complemented
7#23757490C211yesmaximal, characteristic, complemented
8#23757486C211yesmaximal, characteristic, complemented
9#23757466C211yesmaximal, characteristic, complemented
10#23757465C211yesmaximal, characteristic, complemented
11#23757401C211yesmaximal, characteristic, complemented
12#23757375C211yesmaximal, characteristic, complemented
13#23757374C211yesmaximal, characteristic, complemented
14D8×Dic157C211yesmaximal, characteristic, complemented
15#496×C157C211yesmaximal, characteristic, complemented
16C2×#14578944C2211yescharacteristic
17C2×#14578943C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578939C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21#19097767C2211yescharacteristic
22#19097760C2211yescharacteristic
23#19097758—21no—
24#19097758C2211yescharacteristic
25#19097758C2211yescharacteristic
26#19097757—21no—
27#19097755—21no—
28#19097753—21no—
29#19097752—21no—
30#19097752—21no—
31Q8×Dic157C2211yescharacteristic
32C4×Dic314C2211yescharacteristic
33C4×Dic314C2211yescharacteristic
34C4×D628C2211yescharacteristic
35D4×Dic157—21no—
36D4×Dic157C2211yescharacteristic
37D4×Dic157C2211yescharacteristic
38D4×Dic157C2211yescharacteristic
39#19097731—21no—
40#19097705C2211yescharacteristic
41C4⋊#11929434C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097700C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×SD16×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×D8×C157C2211yescharacteristic
55C2×D8×C157C2211yescharacteristic
56#127×C157C2211yescharacteristic
57#124×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59C2×Dic314D411yescharacteristic
60C2×Dic314C2311yescharacteristic
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C22×Dic157—21no—
65C22×Dic157—21no—
66C22×Dic157—21no—
67C2×C4×D157—21no—
68#14578944—41no—
69#14578943—22no—
70#14578942—41no—
71#14578939—41no—
72#14578939—41no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C4⋊Dic157—21no—
76C4⋊Dic157D411yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79#14578933—21no—
80#14578933D411yescharacteristic
81#14578933D411yescharacteristic
82C22⋊Dic157—21no—
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C4×Dic157C2311yescharacteristic
96C2×Q8×C157C2311yescharacteristic
97C2×D4×C157—21no—
98C2×D4×C157—21no—
99C2×D4×C157C2311yescharacteristic
100C2×D4×C157C2311yescharacteristic
101C2×D4×C157C2311yescharacteristic
102SD16×C157—22no—
103SD16×C157—41no—
104D8×C157—22no—
105D8×C157—41no—
106C2×C8×C157C2311yescharacteristic
107C2×C8×C157C2311yescharacteristic
108C22⋊C4×C157—21no—
109C22⋊C4×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112Dic314—21no—
113Dic314—21no—
114Dic314—41no—
115D628—21no—
116D628—21no—
117C22⋊D157—42no—
118C22⋊D157—81no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—81no—
123C2×Dic157—81no—
124C2×Dic157—81no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—81no—
130#11929434—21no—
131#11929434—21no—
132C23×C157—21no—
133C23×C157—21no—
134C23×C157—21no—
135Q8×C157—21no—
136Q8×C157—21no—
137D4×C157—21no—
138D4×C157—21no—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157—21no—
142D4×C157—21no—
143D4×C157—42no—
144D4×C157—81no—
145C2×C4×C157—21no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C8×C157—21no—
150C8×C157#4112yes—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157C22×D411yescharacteristic
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C4×C157—21no—
165C4×C157—21no—
166C4×C157—41no—
167C4×C157C2×#4111yescharacteristic
168C4×C157H(5,𝔽2)11yescharacteristic
169Dic157—21no—
170Dic157—21no—
171Dic157—21no—
172Dic157—21no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176C2×C157—81no—
177C2×C157—81no—
178C2×C157—81no—
179C2×C157#55011yescharacteristic
180C2×C157#57511yescharacteristic
181C2×C157#57611yescharacteristic
182D157—81no—
183C157#340311yescharacteristic, complemented, Sylow
184#3403—1571nomaximal, Sylow
185#559—1571no—
186#559—1571no—
187#496—1571no—
188#496D15711yescharacteristic, complemented
189#495—1571no—
190#464—1571no—
191#464—1571no—
192#463—1571no—
193#463—1571no—
194#459—1571no—
195#459—1571no—
196#446—1571no—
197#440—1571no—
198#440—1571no—
199C4×D8—1571no—
200C2×Q16—1571no—
201C2×SD16—1571no—
202C2×SD16—1571no—
203C2×SD16C2×D15711yescharacteristic
204C2×SD16C2×D15711yescharacteristic
205C2×D8—1571no—
206C2×D8—1571no—
207C2×D8C2×D15711yescharacteristic
208C2×D8C2×D15711yescharacteristic
209#127—1571no—
210#127C2×D15711yescharacteristic
211#124—3141no—
212#124—3141no—
213#124—1571no—
214#124—1571no—
215#124—1571no—
216#124C2×D15711yescharacteristic
217#123—3141no—
218#123—3141no—
219#123—3141no—
220#123—3141no—
221C4×Q8—1571no—
222C4×Q8—1571no—
223C4×Q8—1571no—
224C4×D4—3141no—
225C4×D4—3141no—
226C4×D4—1571no—
227C4×D4—1571no—
228C4×D4—1571no—
229C4×D4—1571no—
230#107—1571no—
231C4⋊C8—1571no—
232C4⋊C8—1571no—
233C4⋊C8—1571no—
234#103—1571no—
235#103—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#97—1571no—
241#97C2×D15711yescharacteristic
242C4×C8—1571no—
243C2×Q8—1571no—
244C2×Q8—1571no—
245C2×Q8C22×D15711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—3141no—
249C2×D4—3141no—
250C2×D4—1571no—
251C2×D4C22×D15711yescharacteristic
252C2×D4C22×D15711yescharacteristic
253C2×D4C22×D15711yescharacteristic
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—3141no—
257C22×C4—3141no—
258Q16—6281no—
259SD16—22no—
260SD16—41no—
261SD16—3142no—
262SD16—6281no—
263D8—22no—
264D8—41no—
265D8—6281no—
266D8—6281no—
267C2×C8—1571no—
268C2×C8—1571no—
269C2×C8C22×D15711yescharacteristic
270C2×C8C22×D15711yescharacteristic
271C4⋊C4—3141no—
272C4⋊C4—3141no—
273C4⋊C4—1571no—
274C4⋊C4—1571no—
275C4⋊C4—1571no—
276C4⋊C4—1571no—
277C4⋊C4—1571no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C42—3141no—
291C42—3141no—
292C42—1571no—
293C42—1571no—
294C42C22×D15711yescharacteristic
295C23—21no—
296C23—21no—
297C23—21no—
298C23—3141no—
299Q8—21no—
300Q8—21no—
301Q8—3141no—
302Q8—3141no—
303Q8—6281no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—81no—
312D4—3141no—
313D4—3141no—
314D4—6282no—
315D4—12561no—
316C2×C4—21no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—1571no—
321C2×C4—1571no—
322C2×C4—1571no—
323C2×C4—1571no—
324C2×C4—12561no—
325C2×C4—12561no—
326C2×C4—12561no—
327C2×C4—12561no—
328C2×C4D4×D15711yescharacteristic
329C2×C4D4×D15711yescharacteristic
330C2×C4C23×D15711yescharacteristic
331C8—21no—
332C8—3141no—
333C8—3141no—
334C8#1457895312yes—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22C2×D4×D15711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—3141no—
352C4—3141no—
353C4—3141no—
354C4—3141no—
355C4—6281no—
356C4—6281no—
357C4—6281no—
358C4C2×#1457895311yescharacteristic
359C4#1909781611yescharacteristic
360C2—81no—
361C2—81no—
362C2—81no—
363C2—12561no—
364C2#2375794311yescharacteristic
365C2#2375806911yescharacteristic
366C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC22×C157C22×D4
hypercenter#496D157
hypocenterC157#3403
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #23758009C2semidirect
2. #23758005C2semidirect
3. #23757661C2semidirect
4. #23757654C2semidirect
5. #23757493C2semidirect
6. #23757492C2semidirect
7. #23757490C2semidirect
8. #23757486C2semidirect
9. #23757466C2semidirect
10. #23757465C2semidirect
11. #23757401C2semidirect
12. #23757375C2semidirect
13. #23757374C2semidirect
14. D8×Dic157C2semidirect
15. #496×C157C2semidirect
183. C157#3403semidirect
188. #496D157semidirect

Maximal subgroups

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

subgroupindex
#237580092
#237580052
#237576612
#237576542
#237574932
#237574922
#237574902
#237574862
#237574662
#237574652
#237574012
#237573752
#237573742
D8×Dic1572
#496×C1572
#3403157