Finite Group Databasebeta

← #34112120

Subgroups — #34112120

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757930C211yesmaximal, characteristic, complemented
2#23757923C211yesmaximal, characteristic, complemented
3#23757631C211yesmaximal, characteristic, complemented
4#23757609C211yesmaximal, characteristic, complemented
5#23757585C211yesmaximal, characteristic, complemented
6#23757566C211yesmaximal, characteristic, complemented
7#23757517C211yesmaximal, characteristic, complemented
8#23757506C211yesmaximal, characteristic, complemented
9#23757486C211yesmaximal, characteristic, complemented
10#23757410C211yesmaximal, characteristic, complemented
11#23757374C211yesmaximal, characteristic, complemented
12#23757347C211yesmaximal, characteristic, complemented
13#23757274C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×Q8×D157C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578938C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097758—21no—
23#19097758C2211yescharacteristic
24#19097753—21no—
25#19097752—21no—
26#19097750—21no—
27#19097750C2211yescharacteristic
28#19097748—21no—
29#19097742—21no—
30#19097742C2211yescharacteristic
31#19097742C2211yescharacteristic
32Q8×Dic157C2211yescharacteristic
33D4×Dic157C2211yescharacteristic
34C4×C22⋊D157—21no—
35C4×C22⋊D157C2211yescharacteristic
36C2×C4⋊Dic157C2211yescharacteristic
37C4⋊C4×D157C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097705C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097681C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×D8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578953—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—41no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157D411yescharacteristic
68C22×Dic157—21no—
69C22×Dic157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157C2311yescharacteristic
73#14578942—41no—
74#14578938—41no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C8×D157—41no—
78C4⋊Dic157—21no—
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90C22⋊Dic157—21no—
91C22⋊Dic157—21no—
92C22⋊Dic157—21no—
93C22⋊Dic157D411yescharacteristic
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97#41×C157—22no—
98#41×C157—41no—
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157—21no—
102C2×D4×C157C2311yescharacteristic
103C2×D4×C157C2311yescharacteristic
104C22×C4×C157—21no—
105C22×C4×C157C2311yescharacteristic
106SD16×C157—22no—
107D8×C157—41no—
108C2×C8×C157C2311yescharacteristic
109C2×C8×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—42no—
116Dic314—42no—
117C22⋊D157—22no—
118C22⋊D157—42no—
119C22⋊D157—42no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—42no—
125C2×Dic157—42no—
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—42no—
132C4×D157—42no—
133#11929434—21no—
134#11929434—21no—
135C23×C157—21no—
136C23×C157—21no—
137C23×C157C2×D411yescharacteristic
138Q8×C157—21no—
139Q8×C157#4112yescomplemented
140D4×C157—21no—
141D4×C157—21no—
142D4×C157—21no—
143D4×C157—21no—
144D4×C157—42no—
145D4×C157—42no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—42no—
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—21no—
167C2×D157AGL(1,ℤ/8)12yes—
168C4×C157—21no—
169C4×C157—22no—
170C4×C157—41no—
171C4×C157C2×#4111yescharacteristic
172C4×C157H(5,𝔽2)11yescharacteristic
173Dic157—21no—
174Dic157—21no—
175Dic157—21no—
176Dic157—41no—
177Dic157—41no—
178Dic157—41no—
179Dic157—41no—
180C2×C157—41no—
181C2×C157—81no—
182C2×C157—81no—
183C2×C157#54611yescharacteristic
184C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C157#57811yescharacteristic
186D157—22no—
187C157#330811yescharacteristic, complemented, Sylow
188#3308—1571nomaximal, Sylow
189#548—1571no—
190#547—1571no—
191#489—1571no—
192#484—1571no—
193#480—1571no—
194#477—1571no—
195#468—1571no—
196#466—1571no—
197#463—1571no—
198#449D15711yescharacteristic, complemented
199#448—1571no—
200#440—1571no—
201#435—1571no—
202#417—1571no—
203C2×#103—1571no—
204C2×#41—3141no—
205C2×#41C2×D15711yescharacteristic
206C22×Q8—1571no—
207C2×SD16—1571no—
208C2×SD16C2×D15711yescharacteristic
209C2×D8C2×D15711yescharacteristic
210C2×M16—1571no—
211C22×C8—1571no—
212C4⋊Q8—1571no—
213#124—3141no—
214#124—1571no—
215#123—3141no—
216#123—3141no—
217C22⋊Q8—3141no—
218C22⋊Q8—3141no—
219C22⋊Q8—1571no—
220#121—3141no—
221#121—1571no—
222#121—1571no—
223#121C2×D15711yescharacteristic
224C4×Q8—1571no—
225C4×D4—3141no—
226C4×D4—1571no—
227C4×D4—1571no—
228C2×C4⋊C4—1571no—
229C2×C4⋊C4—1571no—
230#107—1571no—
231#107—1571no—
232#107—1571no—
233C4⋊C8—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#103—1571no—
238#103—1571no—
239#103C2×D15711yescharacteristic
240#102—1571no—
241#102—1571no—
242#102—1571no—
243#102C2×D15711yescharacteristic
244C22⋊C8—1571no—
245C22⋊C8C2×D15711yescharacteristic
246#97—1571no—
247#41—22no—
248#41—41no—
249#41—6284no—
250C2×Q8—3141no—
251C2×Q8—3141no—
252C2×Q8—3142no—
253C2×Q8—6281no—
254C2×Q8—1571no—
255C2×Q8C22×D15711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—1571no—
261C2×D4C22×D15711yescharacteristic
262C2×D4C22×D15711yescharacteristic
263C22×C4—21no—
264C22×C4—3141no—
265C22×C4—3141no—
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—1571no—
269C22×C4C22×D15711yescharacteristic
270SD16—22no—
271SD16—6281no—
272D8—41no—
273M16—6281no—
274C2×C8—6281no—
275C2×C8—1571no—
276C2×C8—1571no—
277C2×C8C22×D15711yescharacteristic
278C2×C8C22×D15711yescharacteristic
279C4⋊C4—3141no—
280C4⋊C4—3141no—
281C4⋊C4—3141no—
282C4⋊C4—3141no—
283C4⋊C4—3141no—
284C4⋊C4—1571no—
285C4⋊C4—1571no—
286C4⋊C4—1571no—
287C4⋊C4—1571no—
288C4⋊C4C22×D15711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—1571no—
297C22⋊C4—1571no—
298C42—3141no—
299C42—1571no—
300C23—21no—
301C23—21no—
302C23—1571no—
303C23D4×D15711yescharacteristic
304Q8—21no—
305Q8—3141no—
306Q8—3141no—
307Q8—6282no—
308Q8—6282no—
309Q8#1457895312yescomplemented
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—42no—
316D4—3142no—
317D4—6282no—
318D4—6282no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—3141no—
327C2×C4—3141no—
328C2×C4—3141no—
329C2×C4—3141no—
330C2×C4—6282no—
331C2×C4—6282no—
332C2×C4—6282no—
333C2×C4—6282no—
334C2×C4—1571no—
335C2×C4—1571no—
336C2×C4—1571no—
337C2×C4D4×D15711yescharacteristic
338C2×C4C23×D15711yescharacteristic
339C8—21no—
340C8—21no—
341C8—3141no—
342C8—3141no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—3141no—
353C22—3141no—
354C22—1572no—
355C22C2×D4×D15711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—3141no—
360C4—3141no—
361C4—3141no—
362C4—6281no—
363C4—6281no—
364C4—6281no—
365C4—6281no—
366C4C2×#1457895311yescharacteristic
367C4#1909781611yescharacteristic
368C2—41no—
369C2—81no—
370C2—81no—
371C2—3142no—
372C2#2375791011yescharacteristic
373C2AGL(1,ℤ/8)×D15711yescharacteristic
374C2#2375808711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3308
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. #23757930C2semidirect
2. #23757923C2semidirect
3. #23757631C2semidirect
4. #23757609C2semidirect
5. #23757585C2semidirect
6. #23757566C2semidirect
7. #23757517C2semidirect
8. #23757506C2semidirect
9. #23757486C2semidirect
10. #23757410C2semidirect
11. #23757374C2semidirect
12. #23757347C2semidirect
13. #23757274C2semidirect
14. #103×D157C2semidirect
15. #449×C157C2semidirect
139. Q8×C157#41semidirect
187. C157#3308semidirect
198. #449D157semidirect
309. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237579302
#237579232
#237576312
#237576092
#237575852
#237575662
#237575172
#237575062
#237574862
#237574102
#237573742
#237573472
#237572742
#103×D1572
#449×C1572
#3308157