Finite Group Databasebeta

← #34110256

Subgroups — #34110256

Order 20096.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757820C211yesmaximal, characteristic
2C4×Q8×D157C211yesmaximal, characteristic, complemented
3#23757588C211yesmaximal, characteristic, complemented
4#23757585C211yesmaximal, characteristic, complemented
5#23757567C211yesmaximal, characteristic, complemented
6#23757566C211yesmaximal, characteristic, complemented
7#23757472C211yesmaximal, characteristic, complemented
8#23757471C211yesmaximal, characteristic, complemented
9#23757413C211yesmaximal, characteristic, complemented
10#23757410C211yesmaximal, characteristic, complemented
11#23757375C211yesmaximal, characteristic, complemented
12#23757373C211yesmaximal, characteristic, complemented
13#23757371C211yesmaximal, characteristic, complemented
14#23757328C211yesmaximal, characteristic, complemented
15#440×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×Q8×D157C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097767C2211yescharacteristic
23#19097757—21no—
24#19097746C2211yescharacteristic
25#19097744—21no—
26#19097742C2211yescharacteristic
27Q8×Dic157C2211yescharacteristic
28C4×Dic314—21no—
29C4×Dic314C2211yescharacteristic
30#19097736—21no—
31#19097736C2211yescharacteristic
32C4×D628C2211yescharacteristic
33C4×C22⋊D157—21no—
34C4⋊C4×D157—21no—
35C4⋊C4×D157C2211yescharacteristic
36C42×D157—21no—
37C42×D157C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097694C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097682C2211yescharacteristic
52C2×SD16×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59#14578954—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—22no—
65C2×D628—21no—
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157D411yescharacteristic
74C2×C4×D157D411yescharacteristic
75C2×C4×D157C2311yescharacteristic
76#14578943—22no—
77#14578942—22no—
78#14578941—22no—
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C4⋊Dic157—21no—
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88C22⋊Dic157—21no—
89#14578931—21no—
90#14578931—21no—
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157D411yescharacteristic
95C4×Dic157D411yescharacteristic
96C4×Dic157C2311yescharacteristic
97C2×Q8×C157C2311yescharacteristic
98C2×D4×C157C2311yescharacteristic
99C22×C4×C157—21no—
100SD16×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157—21no—
104C4⋊C4×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C42×C157—21no—
108C42×C157C2311yescharacteristic
109C22×D157—21no—
110C22×D157C2×D411yescharacteristic
111Dic314—24no—
112Dic314—21no—
113Dic314—42no—
114Dic314#4112yescomplemented
115D628—21no—
116D628—21no—
117D628—42no—
118C22⋊D157—42no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—24no—
128C4×D157—22no—
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—22no—
132C4×D157—42no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136C23×C157—21no—
137Q8×C157—21no—
138Q8×C157#4112yescomplemented
139D4×C157—21no—
140D4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—42no—
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—21no—
149C8×C157—21no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157C22×D411yescharacteristic
154C2×D157—21no—
155C2×D157—21no—
156C2×D157—41no—
157C2×D157—41no—
158C2×D157—41no—
159C2×D157AGL(1,ℤ/8)12yes—
160C4×C157—22no—
161C4×C157—21no—
162C4×C157—21no—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157C2×#4111yescharacteristic
166C4×C157C2×#4111yescharacteristic
167Dic157—22no—
168Dic157—21no—
169Dic157—21no—
170Dic157—41no—
171Dic157—41no—
172Dic157#13712yes—
173C2×C157—81no—
174C2×C157#52511yescharacteristic
175C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C157C2×#13711yescharacteristic
177D157—22no—
178D157—81no—
179C157#311211yescharacteristic, complemented, Sylow
180#3112—1571nomaximal, Sylow
181#532—1571no—
182C2×C4×Q8—1571no—
183#480—1571no—
184#480—1571no—
185#477—1571no—
186#477—1571no—
187#460—1571no—
188#460—1571no—
189#448—1571no—
190#448—1571no—
191#440—1571no—
192#440—1571no—
193#440—1571no—
194#440D15711yescharacteristic, complemented
195#432—1571no—
196C2×#41—3141no—
197C22×Q8—1571no—
198C2×SD16—1571no—
199C2×SD16—1571no—
200C2×SD16—1571no—
201C2×SD16C2×D15711yescharacteristic
202C4⋊Q8—1571no—
203#127—1571no—
204#124—3141no—
205#121—3141no—
206#121—1571no—
207#121—1571no—
208C4×Q8—3141no—
209C4×Q8—3141no—
210C4×Q8—1571no—
211C4×Q8—1571no—
212C4×Q8—1571no—
213C4×Q8C2×D15711yescharacteristic
214C4×D4—3141no—
215C4×D4—1571no—
216C4×D4C2×D15711yescharacteristic
217C2×C4⋊C4—3141no—
218C2×C4⋊C4—1571no—
219C2×C42—3141no—
220C2×C42—1571no—
221#107—1571no—
222#107—1571no—
223#107—1571no—
224#107C2×D15711yescharacteristic
225C4⋊C8—1571no—
226C4⋊C8—1571no—
227#103—1571no—
228#103—1571no—
229#103—1571no—
230#103C2×D15711yescharacteristic
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102C2×D15711yescharacteristic
235C22⋊C8—1571no—
236C22⋊C8—1571no—
237#97—1571no—
238#97C2×D15711yescharacteristic
239#41—6284no—
240C2×Q8—3141no—
241C2×Q8—3141no—
242C2×Q8—3142no—
243C2×Q8—3142no—
244C2×Q8—1571no—
245C2×Q8C22×D15711yescharacteristic
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—3141no—
249C2×D4—3141no—
250C2×D4—1571no—
251C2×D4C22×D15711yescharacteristic
252C22×C4—21no—
253C22×C4—3141no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—1571no—
257C22×C4—1571no—
258C22×C4—1571no—
259SD16—22no—
260SD16—3142no—
261SD16—3142no—
262SD16—3142no—
263C2×C8—1571no—
264C2×C8—1571no—
265C2×C8C22×D15711yescharacteristic
266C2×C8C22×D15711yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—3141no—
273C4⋊C4—1571no—
274C4⋊C4—1571no—
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C42—21no—
282C42—3141no—
283C42—3141no—
284C42—3141no—
285C42—1571no—
286C42—1571no—
287C42—1571no—
288C42C22×D15711yescharacteristic
289C23—21no—
290C23—3141no—
291C23—1571no—
292Q8—21no—
293Q8—3144no—
294Q8—3141no—
295Q8—6282no—
296Q8—1572no—
297Q8#1457895412yescomplemented
298D4—21no—
299D4—21no—
300D4—3141no—
301D4—3141no—
302D4—6282no—
303D4—6282no—
304D4—6282no—
305D4—6282no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—3144no—
311C2×C4—3142no—
312C2×C4—3142no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—3141no—
318C2×C4—6282no—
319C2×C4—6282no—
320C2×C4—1571no—
321C2×C4—1571no—
322C2×C4—1571no—
323C2×C4D4×D15711yescharacteristic
324C2×C4D4×D15711yescharacteristic
325C2×C4C23×D15711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3141no—
329C8—3141no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—3141no—
334C22—3141no—
335C22—6281no—
336C22—6281no—
337C22—6281no—
338C22—1572no—
339C22C2×D4×D15711yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—3142no—
346C4—3141no—
347C4—3141no—
348C4—6281no—
349C4—6281no—
350C4—1572no—
351C4#41×D15711yescharacteristic
352C4C2×#1457895411yescharacteristic
353C2—81no—
354C2—3142no—
355C2—12561no—
356C2#2375778211yescharacteristic
357C2AGL(1,ℤ/8)×D15711yescharacteristic
358C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#440×C157C2
socleC22×C157C22×D4
prime radical#23757820C2
hypercenter#440D157
hypocenterC157#3112
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D157C2semidirect
3. #23757588C2semidirect
4. #23757585C2semidirect
5. #23757567C2semidirect
6. #23757566C2semidirect
7. #23757472C2semidirect
8. #23757471C2semidirect
9. #23757413C2semidirect
10. #23757410C2semidirect
11. #23757375C2semidirect
12. #23757373C2semidirect
13. #23757371C2semidirect
14. #23757328C2semidirect
15. #440×C157C2semidirect
114. Dic314#41semidirect
138. Q8×C157#41semidirect
179. C157#3112semidirect
194. #440D157semidirect
297. Q8#14578954semidirect

Maximal subgroups

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

subgroupindex
#237578202
C4×Q8×D1572
#237575882
#237575852
#237575672
#237575662
#237574722
#237574712
#237574132
#237574102
#237573752
#237573732
#237573712
#237573282
#440×C1572
#3112157