Finite Group Databasebeta

← #34112838

Subgroups — #34112838

Order 20096.

396 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#23757982C211yesmaximal, characteristic, complemented
2#23757943C211yesmaximal, characteristic, complemented
3#23757637C211yesmaximal, characteristic, complemented
4#23757636C211yesmaximal, characteristic, complemented
5#23757618C211yesmaximal, characteristic, complemented
6#23757574C211yesmaximal, characteristic, complemented
7#23757571C211yesmaximal, characteristic, complemented
8#23757561C211yesmaximal, characteristic, complemented
9#23757466C211yesmaximal, characteristic, complemented
10#23757466C211yesmaximal, characteristic, complemented
11#23757402C211yesmaximal, characteristic, complemented
12#23757385C211yesmaximal, characteristic, complemented
13#23757383C211yesmaximal, characteristic, complemented
14D8×Dic157C211yesmaximal, characteristic, complemented
15#459×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×#14578939C2211yescharacteristic
18#19097770C2211yescharacteristic
19#19097767C2211yescharacteristic
20#19097767C2211yescharacteristic
21#19097763C2211yescharacteristic
22#19097762C2211yescharacteristic
23#19097750—21no—
24#19097748—21no—
25#19097748—21no—
26#19097743—21no—
27#19097743—21no—
28#19097742—21no—
29C4×Dic314C2211yescharacteristic
30C4×D628C2211yescharacteristic
31D4×Dic157C2211yescharacteristic
32D4×Dic157C2211yescharacteristic
33D4×Dic157C2211yescharacteristic
34D4×Dic157C2211yescharacteristic
35C2×#14578933—21no—
36#19097705C2211yescharacteristic
37#19097703C2211yescharacteristic
38#19097702C2211yescharacteristic
39#19097700C2211yescharacteristic
40#19097700C2211yescharacteristic
41#19097693C2211yescharacteristic
42#19097693C2211yescharacteristic
43#19097692C2211yescharacteristic
44#19097692C2211yescharacteristic
45#19097692C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097682C2211yescharacteristic
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—44no—
60C2×Dic314—21no—
61C2×Dic314D411yescharacteristic
62C2×D628C2311yescharacteristic
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C22×Dic157—21no—
68C22×Dic157—21no—
69C22×Dic157—21no—
70C22×Dic157—21no—
71C2×C4×D157—21no—
72#14578939—22no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C4⋊Dic157—21no—
76C4⋊Dic157D411yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933D411yescharacteristic
85#14578933D411yescharacteristic
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90#14578931—21no—
91C4×Dic157C2311yescharacteristic
92C4×Dic157C2311yescharacteristic
93C2×D4×C157—21no—
94C2×D4×C157—21no—
95C2×D4×C157C2311yescharacteristic
96C2×D4×C157C2311yescharacteristic
97C2×D4×C157C2311yescharacteristic
98C22×C4×C157—21no—
99D8×C157—22no—
100D8×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C22⋊C4×C157—21no—
105C42×C157C2311yescharacteristic
106C22×D157—21no—
107Dic314—22no—
108Dic314—81no—
109D628—21no—
110D628—21no—
111C22⋊D157—42no—
112C22⋊D157—42no—
113C22⋊D157—81no—
114C22⋊D157—81no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—42no—
119C2×Dic157—42no—
120C2×Dic157—42no—
121C2×Dic157—81no—
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C4×D157—42no—
127#11929434—21no—
128#11929434—21no—
129C23×C157—21no—
130C23×C157—21no—
131C23×C157—21no—
132D4×C157—21no—
133D4×C157—21no—
134D4×C157—21no—
135D4×C157—21no—
136D4×C157—21no—
137D4×C157—42no—
138D4×C157—81no—
139D4×C157#4112yescomplemented
140C2×C4×C157—21no—
141C2×C4×C157—81no—
142C2×C4×C157C2×D411yescharacteristic
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2411yescharacteristic
145C8×C157—21no—
146C8×C157—21no—
147C22×C157—22no—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157C22×D411yescharacteristic
157C2×D157—41no—
158C2×D157—41no—
159C2×D157—41no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—41no—
163C4×C157C2×#4111yescharacteristic
164C4×C1572−1+411yescharacteristic
165Dic157—21no—
166Dic157—21no—
167Dic157—21no—
168Dic157—41no—
169Dic157—41no—
170Dic157—41no—
171Dic157AGL(1,ℤ/8)12yes—
172C2×C157—42no—
173C2×C157—81no—
174C2×C157—81no—
175C2×C157#54811yescharacteristic
176C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C157#57611yescharacteristic
178D157—81no—
179C157#336611yescharacteristic, complemented, Sylow
180#3366—1571nomaximal, Sylow
181#554—1571no—
182#550—1571no—
183#490—1571no—
184#490—1571no—
185#486—1571no—
186#478—1571no—
187#478—1571no—
188#476—1571no—
189#459—1571no—
190#459—1571no—
191#459D15711yescharacteristic, complemented
192#446—1571no—
193#442—1571no—
194#442—1571no—
195C4×D8—1571no—
196C2×#41—3141no—
197C2×D8—1571no—
198C2×D8C2×D15711yescharacteristic
199C2×D8C2×D15711yescharacteristic
200C4⋊Q8—1571no—
201#127—1571no—
202#127—1571no—
203#127C2×D15711yescharacteristic
204#125—1571no—
205#125—1571no—
206C22⋊Q8—3141no—
207C22⋊Q8—3141no—
208C22⋊Q8—3141no—
209#121—3141no—
210#121—3141no—
211#121—3141no—
212C4×Q8—1571no—
213C4×D4—1571no—
214C4×D4—1571no—
215C4×D4—1571no—
216C4×D4—1571no—
217C4×D4—1571no—
218C4×D4C2×D15711yescharacteristic
219C2×C4⋊C4—3141no—
220#107—1571no—
221#106—1571no—
222#106—1571no—
223C4⋊C8—1571no—
224C4⋊C8—1571no—
225C4⋊C8C2×D15711yescharacteristic
226#102—1571no—
227#102—1571no—
228#102—1571no—
229#102—1571no—
230#102—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102C2×D15711yescharacteristic
235#102C2×D15711yescharacteristic
236#97—1571no—
237#97—1571no—
238C4×C8—1571no—
239#41—6284no—
240C2×Q8—3141no—
241C2×Q8—1571no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3141no—
245C2×D4—3141no—
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4C22×D15711yescharacteristic
250C2×D4C22×D15711yescharacteristic
251C2×D4C22×D15711yescharacteristic
252C22×C4—21no—
253C22×C4—3141no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—3141no—
257C22×C4—3141no—
258D8—22no—
259D8—22no—
260D8—3142no—
261C2×C8—1571no—
262C2×C8—1571no—
263C2×C8C22×D15711yescharacteristic
264C2×C8C22×D15711yescharacteristic
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—1571no—
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—
283C42—1571no—
284C42—1571no—
285C42C22×D15711yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—3141no—
290Q8—3142no—
291Q8—12561no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—81no—
299D4—3141no—
300D4—3141no—
301D4—6282no—
302D4—6282no—
303D4—12561no—
304D4—12561no—
305D4#1457895312yescomplemented
306C2×C4—21no—
307C2×C4—81no—
308C2×C4—3141no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—6282no—
312C2×C4—6282no—
313C2×C4—6282no—
314C2×C4—6282no—
315C2×C4—1571no—
316C2×C4—1571no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—12561no—
320C2×C4D4×D15711yescharacteristic
321C2×C4D4×D15711yescharacteristic
322C2×C4C23×D15711yescharacteristic
323C8—21no—
324C8—21no—
325C8—3141no—
326C8—3141no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—6281no—
337C22—6281no—
338C22—6281no—
339C22C2×D4×D15711yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—3141no—
344C4—3141no—
345C4—3141no—
346C4—6281no—
347C4—6281no—
348C4—6281no—
349C4—1572no—
350C4C2×#1457895311yescharacteristic
351C4#1909781811yescharacteristic
352C2—42no—
353C2—81no—
354C2—81no—
355C2—12561no—
356C2#2375792911yescharacteristic
357C2AGL(1,ℤ/8)×D15711yescharacteristic
358C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3366
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. #23757982C2semidirect
2. #23757943C2semidirect
3. #23757637C2semidirect
4. #23757636C2semidirect
5. #23757618C2semidirect
6. #23757574C2semidirect
7. #23757571C2semidirect
8. #23757561C2semidirect
9. #23757466C2semidirect
10. #23757466C2semidirect
11. #23757402C2semidirect
12. #23757385C2semidirect
13. #23757383C2semidirect
14. D8×Dic157C2semidirect
15. #459×C157C2semidirect
139. D4×C157#41semidirect
179. C157#3366semidirect
191. #459D157semidirect
305. D4#14578953semidirect

Maximal subgroups

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

subgroupindex
#237579822
#237579432
#237576372
#237576362
#237576182
#237575742
#237575712
#237575612
#237574662
#237574022
#237573852
#237573832
D8×Dic1572
#459×C1572
#3366157