Finite Group Databasebeta

← #34112008

Subgroups — #34112008

Order 20096.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757978C211yesmaximal, characteristic
2Q8×C22⋊D157C211yesmaximal, characteristic
3#23757622C211yesmaximal, characteristic, complemented
4#23757608C211yesmaximal, characteristic, complemented
5#23757607C211yesmaximal, characteristic, complemented
6#23757566C211yesmaximal, characteristic, complemented
7#23757514C211yesmaximal, characteristic, complemented
8#23757513C211yesmaximal, characteristic, complemented
9#23757482C211yesmaximal, characteristic, complemented
10#23757430C211yesmaximal, characteristic, complemented
11#23757429C211yesmaximal, characteristic, complemented
12#23757365C211yesmaximal, characteristic, complemented
13C8×C22⋊D157C211yesmaximal, characteristic
14#103×D157C211yesmaximal, characteristic, complemented
15C2×#103×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2211yescharacteristic
17C2×Dic628C2211yescharacteristic
18C2×#14578944C2211yescharacteristic
19C2×#14578944C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097769—21no—
23#19097769C2211yescharacteristic
24#19097761—21no—
25#19097751C2211yescharacteristic
26#19097750—21no—
27#19097750—21no—
28#19097750C2211yescharacteristic
29#19097750C2211yescharacteristic
30#19097749—21no—
31#19097747—21no—
32#19097747C2211yescharacteristic
33Q8×Dic157C2211yescharacteristic
34#19097736C2211yescharacteristic
35C4×C22⋊D157—21no—
36C4×C22⋊D157—21no—
37C4×C22⋊D157C2211yescharacteristic
38C4⋊C4×D157C2211yescharacteristic
39#19097705C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097704C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097694C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C22×Q8×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54C2×C4⋊C4×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59C2×Dic314—21no—
60C2×Dic314—21no—
61C2×Dic314C2311yescharacteristic
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—41no—
65C2×C22⋊D157D411yescharacteristic
66C2×C4×D157—21no—
67C2×C4×D157—21no—
68C2×C4×D157C2311yescharacteristic
69Dic628—41no—
70#14578944—22no—
71#14578944—41no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C8×D157—41no—
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933D411yescharacteristic
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90C22⋊Dic157D411yescharacteristic
91#14578931—21no—
92#14578931—21no—
93#14578931D411yescharacteristic
94C4×Dic157—21no—
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97C2×Q8×C157—21no—
98C2×Q8×C157—22no—
99C2×Q8×C157—41no—
100C2×Q8×C157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105C2×C8×C157—41no—
106C2×C8×C157C2311yescharacteristic
107C2×C8×C157C2311yescharacteristic
108C4⋊C4×C157—21no—
109C4⋊C4×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22×D157C2×D411yescharacteristic
112Dic314—21no—
113Dic314—21no—
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117Dic314—81no—
118C22⋊D157Q1614yescomplemented
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C4×D157—21no—
127C4×D157—21no—
128C4×D157—42no—
129C4×D157—81no—
130#11929434—21no—
131#11929434—21no—
132C23×C157C2×D411yescharacteristic
133Q8×C157—21no—
134Q8×C157—21no—
135Q8×C157—21no—
136Q8×C157—42no—
137Q8×C157#4112yes—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—42no—
145C2×C4×C157—81no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—21no—
149C8×C157—21no—
150C22×C157—21no—
151C22×C157—21no—
152C22×C157C2×Q1612yes—
153C22×C157C22×D411yescharacteristic
154C2×D157—21no—
155C2×D157—21no—
156C2×D157C2×Q1612yes—
157C4×C157—21no—
158C4×C157—22no—
159C4×C157—41no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157C2×#4111yescharacteristic
163C4×C157H(5,𝔽2)11yescharacteristic
164Dic157—21no—
165Dic157—21no—
166Dic157—41no—
167Dic157—41no—
168Dic157—41no—
169Dic157—41no—
170Dic157C2×Q1612yes—
171C2×C157—22no—
172C2×C157#54611yescharacteristic
173C2×C157C22×Q1611yescharacteristic
174C2×C157#57711yescharacteristic
175D157—22no—
176C157#329911yescharacteristic, complemented, Sylow
177#3299—1571nomaximal, Sylow
178#554—1571no—
179D4×Q8—1571no—
180#487—1571no—
181#484—1571no—
182#484—1571no—
183#477—1571no—
184#467—1571no—
185#467—1571no—
186#462—1571no—
187#451—1571no—
188#451—1571no—
189C4×Q16—1571no—
190C8×D4—1571no—
191C2×#103—1571no—
192C2×#103D15711yescharacteristic, complemented
193C22×Q8—1571no—
194C22×Q8C2×D15711yescharacteristic
195C2×Q16—1571no—
196C2×Q16—1571no—
197C2×Q16—1571no—
198C22×C8—1571no—
199C22×C8C2×D15711yescharacteristic
200C4⋊Q8—3141no—
201C4⋊Q8—1571no—
202C4⋊Q8—1571no—
203#125—3141no—
204C22⋊Q8—3141no—
205C22⋊Q8—3141no—
206C22⋊Q8—3141no—
207C22⋊Q8—3141no—
208C22⋊Q8—1571no—
209C22⋊Q8—1571no—
210C22⋊Q8—1571no—
211C22⋊Q8—1571no—
212C4×Q8—1571no—
213C4×Q8—1571no—
214C4×D4—3141no—
215C4×D4—3141no—
216C4×D4—1571no—
217C2×C4⋊C4—1571no—
218C2×C4⋊C4C2×D15711yescharacteristic
219#107—1571no—
220#107—1571no—
221#107—1571no—
222C4⋊C8—1571no—
223#103—1571no—
224#103—1571no—
225#103—1571no—
226#103—1571no—
227#103—1571no—
228#103—1571no—
229#103C2×D15711yescharacteristic
230#103C2×D15711yescharacteristic
231#103C2×D15711yescharacteristic
232#103C2×D15711yescharacteristic
233C22⋊C8—1571no—
234C22⋊C8—1571no—
235C4×C8—1571no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—3141no—
240C2×Q8—3141no—
241C2×Q8—3142no—
242C2×Q8—6281no—
243C2×Q8—1571no—
244C2×Q8—1571no—
245C2×Q8C22×D15711yescharacteristic
246C2×Q8C22×D15711yescharacteristic
247C2×D4—1571no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3141no—
251C22×C4—3141no—
252C22×C4—1571no—
253C22×C4C22×D15711yescharacteristic
254Q16—3142no—
255Q16—6281no—
256Q16—6281no—
257C2×C8—41no—
258C2×C8—6281no—
259C2×C8—1571no—
260C2×C8—1571no—
261C2×C8C22×D15711yescharacteristic
262C2×C8C22×D15711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—3141no—
273C4⋊C4—3141no—
274C4⋊C4—1571no—
275C4⋊C4—1571no—
276C4⋊C4—1571no—
277C4⋊C4C22×D15711yescharacteristic
278C4⋊C4C22×D15711yescharacteristic
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—1571no—
284C22⋊C4—1571no—
285C42—3141no—
286C42—3141no—
287C42—1571no—
288C23—1571no—
289C23D4×D15711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—3141no—
295Q8—3141no—
296Q8—3141no—
297Q8—3141no—
298Q8—6282no—
299Q8—12561no—
300Q8#1457895412yes—
301D4—1574no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—6282no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—1571no—
320C2×C4—12561no—
321C2×C4D4×D15711yescharacteristic
322C2×C4C23×D15711yescharacteristic
323C8—21no—
324C8—21no—
325C8—3141no—
326C8—3141no—
327C22—21no—
328C22—21no—
329C22—3141no—
330C22—3141no—
331C22—1572no—
332C22Q16×D15712yes—
333C22C2×D4×D15711yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3141no—
340C4—3141no—
341C4—6281no—
342C4—6281no—
343C4—6281no—
344C4—6281no—
345C4—1572no—
346C4C2×#1457895411yescharacteristic
347C4#1909781611yescharacteristic
348C2—22no—
349C2—3142no—
350C2#2375791311yescharacteristic
351C2C2×Q16×D15711yescharacteristic
352C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#103×C157C2
socleC22×C157C22×D4
prime radicalC2×C22⋊D157D4
hypercenterC2×#103D157
hypocenterC157#3299
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #23757622C2semidirect
4. #23757608C2semidirect
5. #23757607C2semidirect
6. #23757566C2semidirect
7. #23757514C2semidirect
8. #23757513C2semidirect
9. #23757482C2semidirect
10. #23757430C2semidirect
11. #23757429C2semidirect
12. #23757365C2semidirect
14. #103×D157C2semidirect
15. C2×#103×C157C2semidirect
118. C22⋊D157Q16semidirect
176. C157#3299semidirect
192. C2×#103D157semidirect

Maximal subgroups

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

subgroupindex
#237579782
Q8×C22⋊D1572
#237576222
#237576082
#237576072
#237575662
#237575142
#237575132
#237574822
#237574302
#237574292
#237573652
C8×C22⋊D1572
#103×D1572
C2×#103×C1572
#3299157