Finite Group Databasebeta

← #34110336

Subgroups — #34110336

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#23757876C211yesmaximal, characteristic, complemented
2#23757756C211yesmaximal, characteristic, complemented
3#23757592C211yesmaximal, characteristic, complemented
4#23757584C211yesmaximal, characteristic, complemented
5#23757578C211yesmaximal, characteristic, complemented
6#23757575C211yesmaximal, characteristic, complemented
7#23757471C211yesmaximal, characteristic, complemented
8#23757466C211yesmaximal, characteristic, complemented
9#23757418C211yesmaximal, characteristic, complemented
10#23757385C211yesmaximal, characteristic, complemented
11#23757376C211yesmaximal, characteristic, complemented
12SD16×Dic157C211yesmaximal, characteristic, complemented
13#23757353C211yesmaximal, characteristic, complemented
14#23757331C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578941C2211yescharacteristic
17C2×#14578939C2211yescharacteristic
18#19097767C2211yescharacteristic
19#19097762C2211yescharacteristic
20#19097755—21no—
21#19097753—21no—
22#19097745C2211yescharacteristic
23#19097743—21no—
24#19097742—21no—
25Q8×Dic157C2211yescharacteristic
26Q8×Dic157C2211yescharacteristic
27D4×Dic157—21no—
28D4×Dic157—21no—
29D4×Dic157C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31D4×Dic157C2211yescharacteristic
32#19097733C2211yescharacteristic
33#19097730—21no—
34#19097730C2211yescharacteristic
35C2×C4⋊Dic157—21no—
36C2×C4⋊Dic157C2211yescharacteristic
37#19097704C2211yescharacteristic
38#19097704C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097703C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×D8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59C2×D628C2311yescharacteristic
60C2×C22⋊D157—21no—
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C22×Dic157—21no—
64C22×Dic157—21no—
65C22×Dic157—21no—
66C22×Dic157—21no—
67C2×C4×D157—21no—
68#14578941—22no—
69#14578939—22no—
70C2×#11929434C2311yescharacteristic
71C2×#11929434C2311yescharacteristic
72C4⋊Dic157—21no—
73C4⋊Dic157—21no—
74C4⋊Dic157—21no—
75C4⋊Dic157—21no—
76C4⋊Dic157D411yescharacteristic
77C4⋊Dic157D411yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81#14578933—21no—
82C22⋊Dic157—21no—
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85C22⋊Dic157D411yescharacteristic
86C22⋊Dic157D411yescharacteristic
87#14578931—21no—
88#14578931—21no—
89C4×Dic157—21no—
90C4×Dic157—21no—
91C4×Dic157C2311yescharacteristic
92C4×Dic157C2311yescharacteristic
93#41×C157—22no—
94#41×C157—22no—
95C2×Q8×C157C2311yescharacteristic
96C2×D4×C157—21no—
97C2×D4×C157—21no—
98C2×D4×C157C2311yescharacteristic
99C2×D4×C157C2311yescharacteristic
100C22×C4×C157—21no—
101C22×C4×C157C2311yescharacteristic
102SD16×C157—22no—
103D8×C157—22no—
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157C2311yescharacteristic
107C22⋊C4×C157—21no—
108C22×D157—21no—
109D628—21no—
110D628—21no—
111C22⋊D157—81no—
112C22⋊D157—81no—
113C22⋊D157—81no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—81no—
118C2×Dic157—81no—
119C2×Dic157—81no—
120C2×Dic157—81no—
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C4×D157—81no—
126#11929434—21no—
127#11929434—21no—
128C23×C157—21no—
129C23×C157—21no—
130C23×C157C2×D411yescharacteristic
131Q8×C157—21no—
132Q8×C157#4112yes—
133D4×C157—24no—
134D4×C157—21no—
135D4×C157—21no—
136D4×C157—21no—
137D4×C157—81no—
138D4×C157#4112yes—
139C2×C4×C157—24no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157—21no—
148C22×C157—21no—
149C22×C157—22no—
150C22×C157—21no—
151C22×C157—21no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157C22×D411yescharacteristic
158C2×D157—41no—
159C2×D157—41no—
160C2×D157—41no—
161C4×C157—21no—
162C4×C157—22no—
163C4×C157—41no—
164C4×C157C2×#4111yescharacteristic
165C4×C157C2×#4111yescharacteristic
166Dic157—21no—
167Dic157—21no—
168Dic157—21no—
169Dic157—21no—
170Dic157—41no—
171Dic157—41no—
172Dic157—41no—
173C2×C157—41no—
174C2×C157—42no—
175C2×C157—81no—
176C2×C157#52511yescharacteristic
177C2×C157#57611yescharacteristic
178C2×C157#57711yescharacteristic
179D157—81no—
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#459—1571no—
190#449—1571no—
191#449D15711yescharacteristic, complemented
192#442—1571no—
193#440—1571no—
194C4×SD16—1571no—
195C4×D8—1571no—
196#433—1571no—
197C2×#41C2×D15711yescharacteristic
198C2×SD16—1571no—
199C2×SD16C2×D15711yescharacteristic
200C2×D8—1571no—
201C2×D8C2×D15711yescharacteristic
202#127—1571no—
203#125—1571no—
204#123—3141no—
205#123—3141no—
206#121—3141no—
207#121—3141no—
208#121—1571no—
209#121C2×D15711yescharacteristic
210C4×Q8—1571no—
211C4×Q8—1571no—
212C4×D4—3141no—
213C4×D4—3141no—
214C4×D4—1571no—
215C4×D4—1571no—
216C4×D4—1571no—
217C4×D4—1571no—
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⋊C8—1571no—
228#103—1571no—
229#103C2×D15711yescharacteristic
230#102—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102—1571no—
235#102C2×D15711yescharacteristic
236C22⋊C8—1571no—
237C22⋊C8C2×D15711yescharacteristic
238#97—1571no—
239C4×C8—1571no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D15711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—3141no—
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—1571no—
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×C4C22×D15711yescharacteristic
258SD16—22no—
259SD16—3142no—
260D8—22no—
261D8—3142no—
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—
288C42—1571no—
289C23—21no—
290C23—21no—
291C23—3141no—
292C23D4×D15711yescharacteristic
293Q8—21no—
294Q8#1457895312yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—3141no—
301D4—3141no—
302D4—12561no—
303D4—12561no—
304D4—12561no—
305D4#1457895312yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—1571no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1571no—
318C2×C4—12561no—
319C2×C4—12561no—
320C2×C4—12561no—
321C2×C4—12561no—
322C2×C4—12561no—
323C2×C4D4×D15711yescharacteristic
324C2×C4C23×D15711yescharacteristic
325C8—21no—
326C8—21no—
327C8—3141no—
328C8—3141no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—6281no—
339C22—6281no—
340C22—6281no—
341C22C2×D4×D15711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—3141no—
346C4—3141no—
347C4—3141no—
348C4—3141no—
349C4—6281no—
350C4—6281no—
351C4—6281no—
352C4C2×#1457895311yescharacteristic
353C4C2×#1457895311yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—12561no—
358C2#2375777911yescharacteristic
359C2#2375807711yescharacteristic
360C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3119
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23757876C2semidirect
2. #23757756C2semidirect
3. #23757592C2semidirect
4. #23757584C2semidirect
5. #23757578C2semidirect
6. #23757575C2semidirect
7. #23757471C2semidirect
8. #23757466C2semidirect
9. #23757418C2semidirect
10. #23757385C2semidirect
11. #23757376C2semidirect
12. SD16×Dic157C2semidirect
13. #23757353C2semidirect
14. #23757331C2semidirect
15. #449×C157C2semidirect
180. C157#3119semidirect
191. #449D157semidirect

Maximal subgroups

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

subgroupindex
#237578762
#237577562
#237575922
#237575842
#237575782
#237575752
#237574712
#237574662
#237574182
#237573852
#237573762
SD16×Dic1572
#237573532
#237573312
#449×C1572
#3119157