Finite Group Databasebeta

← #34113201

Subgroups — #34113201

Order 20096.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758005C211yesmaximal, characteristic, complemented
2#23757997C211yesmaximal, characteristic, complemented
3#23757659C211yesmaximal, characteristic
4#23757658C211yesmaximal, characteristic, complemented
5#23757497C211yesmaximal, characteristic, complemented
6#23757497C211yesmaximal, characteristic, complemented
7#23757490C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757475C211yesmaximal, characteristic, complemented
10#23757471C211yesmaximal, characteristic, complemented
11#23757401C211yesmaximal, characteristic, complemented
12#23757386C211yesmaximal, characteristic, complemented
13#23757380C211yesmaximal, characteristic, complemented
14SD16×Dic157C211yesmaximal, characteristic, complemented
15#497×C157C211yesmaximal, characteristic, complemented
16C2×#14578941C2211yescharacteristic
17C2×#14578941C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19C2×D1256C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21#19097767C2211yescharacteristic
22#19097766—21no—
23#19097766—21no—
24#19097760C2211yescharacteristic
25#19097760C2211yescharacteristic
26#19097760C2211yescharacteristic
27#19097757—21no—
28#19097755—21no—
29#19097753—21no—
30#19097751—21no—
31Q8×Dic157C2211yescharacteristic
32Q8×Dic157C2211yescharacteristic
33Q8×Dic157C2211yescharacteristic
34C4×Dic314C2211yescharacteristic
35C4×D628C2211yescharacteristic
36C4×D628C2211yescharacteristic
37D4×Dic157C2211yescharacteristic
38#19097731—21no—
39#19097729—21no—
40#19097703C2211yescharacteristic
41C4⋊#11929434C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097700C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×Q16×C157C2211yescharacteristic
53C2×Q16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C4⋊Q8×C157C2211yescharacteristic
57#124×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59C2×Dic314D411yescharacteristic
60C2×D628C2311yescharacteristic
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C22×Dic157—21no—
65C2×C4×D157—21no—
66C2×C4×D157—21no—
67#14578941—41no—
68#14578941—41no—
69#14578941—41no—
70D1256—22no—
71#14578939—41no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C4⋊Dic157—21no—
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77C4⋊Dic157D411yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933D411yescharacteristic
81#14578933D411yescharacteristic
82C22⋊Dic157—21no—
83#14578931—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C4×Dic157C2311yescharacteristic
96C2×Q8×C157C2311yescharacteristic
97C2×Q8×C157C2311yescharacteristic
98C2×Q8×C157C2311yescharacteristic
99C2×D4×C157C2311yescharacteristic
100Q16×C157—22no—
101Q16×C157—41no—
102SD16×C157—22no—
103SD16×C157—41no—
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157—21no—
107C4⋊C4×C157—21no—
108C22⋊C4×C157—21no—
109C22⋊C4×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113Dic314—41no—
114D628—21no—
115D628—21no—
116D628—21no—
117D628—21no—
118C22⋊D157—42no—
119C22⋊D157—81no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—81no—
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—42no—
128C4×D157—81no—
129#11929434—21no—
130#11929434—21no—
131C23×C157—21no—
132Q8×C157—21no—
133Q8×C157—21no—
134Q8×C157—21no—
135Q8×C157—21no—
136Q8×C157—21no—
137Q8×C157—21no—
138D4×C157—21no—
139D4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157#4112yes—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157C22×D411yescharacteristic
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C4×C157—21no—
159C4×C157—21no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157—41no—
163C4×C157C2×#4111yescharacteristic
164C4×C157H(5,𝔽2)11yescharacteristic
165Dic157—21no—
166Dic157—21no—
167Dic157—21no—
168Dic157—21no—
169Dic157—41no—
170Dic157—41no—
171C2×C157—81no—
172C2×C157#55011yescharacteristic
173C2×C157#57511yescharacteristic
174C2×C157#57711yescharacteristic
175D157—81no—
176D157—81no—
177C157#340111yescharacteristic, complemented, Sylow
178#3401—1571nomaximal, Sylow
179#559—1571no—
180#556—1571no—
181#497D15711yescharacteristic, complemented
182#496—1571no—
183#495—1571no—
184#464—1571no—
185#464—1571no—
186#463—1571no—
187#463—1571no—
188#461—1571no—
189#460—1571no—
190#446—1571no—
191#442—1571no—
192#441—1571no—
193C4×SD16—1571no—
194C2×Q16C2×D15711yescharacteristic
195C2×Q16C2×D15711yescharacteristic
196C2×SD16—1571no—
197C2×SD16—1571no—
198C2×SD16—1571no—
199C2×SD16C2×D15711yescharacteristic
200C2×SD16C2×D15711yescharacteristic
201C2×D8—1571no—
202C2×D8—1571no—
203C4⋊Q8C2×D15711yescharacteristic
204#127—1571no—
205#126—3141no—
206#126—3141no—
207#124—3141no—
208#124—1571no—
209#124—1571no—
210#124—1571no—
211#124C2×D15711yescharacteristic
212#123—3141no—
213#123—3141no—
214C22⋊Q8—3141no—
215C4×Q8—1571no—
216C4×Q8—1571no—
217C4×Q8—1571no—
218C4×Q8—1571no—
219C4×D4—3141no—
220C4×D4—1571no—
221C4×D4—1571no—
222C4×D4—1571no—
223#117—3141no—
224#106—1571no—
225C4⋊C8—1571no—
226C4⋊C8—1571no—
227C4⋊C8—1571no—
228#103—1571no—
229#103—1571no—
230#103—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#97—1571no—
235#97C2×D15711yescharacteristic
236C4×C8—1571no—
237C2×Q8—1571no—
238C2×Q8C22×D15711yescharacteristic
239C2×Q8C22×D15711yescharacteristic
240C2×Q8C22×D15711yescharacteristic
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4—1571no—
245C2×D4C22×D15711yescharacteristic
246C22×C4—3141no—
247C22×C4—3141no—
248C22×C4—3141no—
249Q16—22no—
250Q16—41no—
251SD16—22no—
252SD16—41no—
253SD16—6281no—
254SD16—6281no—
255SD16—6281no—
256D8—3142no—
257D8—6281no—
258C2×C8—1571no—
259C2×C8—1571no—
260C2×C8C22×D15711yescharacteristic
261C2×C8C22×D15711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—1571no—
268C4⋊C4—1571no—
269C4⋊C4—1571no—
270C4⋊C4—1571no—
271C4⋊C4—1571no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C42—3141no—
284C42—3141no—
285C42—3141no—
286C42—1571no—
287C42—1571no—
288C42C22×D15711yescharacteristic
289C23—21no—
290C23—3141no—
291C23—3141no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—6281no—
299D4—21no—
300D4—21no—
301D4—3141no—
302D4—3141no—
303D4—3141no—
304D4—3141no—
305D4—6282no—
306D4—12561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—6282no—
313C2×C4—1571no—
314C2×C4—1571no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—12561no—
318C2×C4—12561no—
319C2×C4D4×D15711yescharacteristic
320C2×C4D4×D15711yescharacteristic
321C2×C4C23×D15711yescharacteristic
322C8—21no—
323C8—3141no—
324C8—3141no—
325C8#1457895312yes—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—6281no—
330C22—6281no—
331C22—6281no—
332C22—6281no—
333C22—6281no—
334C22—6281no—
335C22C2×D4×D15711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3141no—
342C4—3141no—
343C4—3141no—
344C4—3141no—
345C4—6281no—
346C4—6281no—
347C4C2×#1457895311yescharacteristic
348C4#1909781611yescharacteristic
349C2—81no—
350C2—12561no—
351C2—12561no—
352C2#2375794311yescharacteristic
353C2#2375807211yescharacteristic
354C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#497×C157C2
socleC22×C157C22×D4
prime radical#23757659C2
hypercenter#497D157
hypocenterC157#3401
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #23758005C2semidirect
2. #23757997C2semidirect
4. #23757658C2semidirect
5. #23757497C2semidirect
6. #23757497C2semidirect
7. #23757490C2semidirect
8. #23757485C2semidirect
9. #23757475C2semidirect
10. #23757471C2semidirect
11. #23757401C2semidirect
12. #23757386C2semidirect
13. #23757380C2semidirect
14. SD16×Dic157C2semidirect
15. #497×C157C2semidirect
177. C157#3401semidirect
181. #497D157semidirect

Maximal subgroups

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

subgroupindex
#237580052
#237579972
#237576592
#237576582
#237574972
#237574902
#237574852
#237574752
#237574712
#237574012
#237573862
#237573802
SD16×Dic1572
#497×C1572
#3401157