Finite Group Databasebeta

← #34109687

Subgroups — #34109687

Order 20096.

472 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097800C211yesmaximal, characteristic, complemented
2C2×#19097790C211yesmaximal, characteristic
3#23757843C211yesmaximal, characteristic, complemented
4#23757752C211yesmaximal, characteristic, complemented
5#23757494C211yesmaximal, characteristic, complemented
6#23757492C211yesmaximal, characteristic, complemented
7#23757486C211yesmaximal, characteristic, complemented
8#23757484C211yesmaximal, characteristic, complemented
9#23757482C211yesmaximal, characteristic, complemented
10#23757478C211yesmaximal, characteristic, complemented
11#23757476C211yesmaximal, characteristic, complemented
12#23757298C211yesmaximal, characteristic, complemented
13#23757271C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097800—24no—
20#19097790—24no—
21C2×Dic628C2211yescharacteristic
22C2×Dic628C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578942C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097771C2211yescharacteristic
30#19097769C2211yescharacteristic
31#19097758C2211yescharacteristic
32#19097753—21no—
33#19097753—21no—
34#19097748—21no—
35#19097748—21no—
36C4×Dic314C2211yescharacteristic
37#19097736C2211yescharacteristic
38D4×Dic157C2211yescharacteristic
39C4×C22⋊D157—21no—
40#19097731—21no—
41#19097728C2211yescharacteristic
42#19097727—21no—
43C2×#14578933—21no—
44C4⋊C4×D157C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic
46#19097700C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097696C2211yescharacteristic
50#19097695C2211yescharacteristic
51#19097694C2211yescharacteristic
52#19097694C2211yescharacteristic
53#19097692C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C2×D8×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×D4×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#102×C157C2211yescharacteristic
61#14578953—22no—
62#14578953—22no—
63#14578953—22no—
64#14578953D414yes—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—22no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C22×Dic157—21no—
74C22×Dic157—21no—
75C22×Dic157—21no—
76C2×C4×D157—21no—
77C2×C4×D157—21no—
78C2×C4×D157C2311yescharacteristic
79Dic628—22no—
80Dic628—22no—
81#14578944—22no—
82#14578942—22no—
83#14578942—22no—
84#14578942—22no—
85#14578937—22no—
86C2×#11929434C2311yescharacteristic
87C2×#11929434C2311yescharacteristic
88C8×D157—22no—
89C4⋊Dic157—21no—
90C4⋊Dic157—21no—
91C4⋊Dic157C2311yescharacteristic
92#14578933—21no—
93#14578933—21no—
94#14578933—21no—
95#14578933—21no—
96#14578933—21no—
97#14578933D411yescharacteristic
98#14578933D411yescharacteristic
99C22⋊Dic157—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157—21no—
102#14578931D411yescharacteristic
103#14578931D411yescharacteristic
104C4×Dic157—21no—
105C4×Dic157—21no—
106C4×Dic157C2311yescharacteristic
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110C22×C4×C157—21no—
111SD16×C157—22no—
112D8×C157—22no—
113C2×C8×C157C2311yescharacteristic
114C2×C8×C157C2311yescharacteristic
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C22⋊C4×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157C2×D411yescharacteristic
121Dic314—21no—
122Dic314—21no—
123Dic314—21no—
124Dic314—21no—
125Dic314—21no—
126Dic314—42no—
127Dic314C2×D412yes—
128C22⋊D157—24no—
129C22⋊D157—42no—
130C2×Dic157—21no—
131C2×Dic157—24no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—42no—
136C2×Dic157—81no—
137C2×Dic157C2×D411yescharacteristic
138C2×Dic157C2×D411yescharacteristic
139C2×Dic157C2×D411yescharacteristic
140C4×D157—42no—
141C4×D157—81no—
142C4×D157C2×D412yes—
143C4×D157#4112yes—
144#11929434—21no—
145#11929434—21no—
146C23×C157—21no—
147C23×C157—21no—
148Q8×C157—21no—
149Q8×C157—21no—
150D4×C157—21no—
151D4×C157—21no—
152D4×C157—21no—
153D4×C157C2×D412yes—
154C2×C4×C157—21no—
155C2×C4×C157—21no—
156C2×C4×C157—81no—
157C2×C4×C157C2×D411yescharacteristic
158C2×C4×C157C2×D411yescharacteristic
159C2×C4×C157C2411yescharacteristic
160C8×C157—21no—
161C8×C157—21no—
162C22×C157—22no—
163C22×C157—41no—
164C22×C157—41no—
165C22×C157—41no—
166C22×C157—41no—
167C22×C157—41no—
168C22×C157C22×D411yescharacteristic
169C2×D157—21no—
170C2×D157—21no—
171C2×D157#12112yes—
172C4×C157—21no—
173C4×C157—21no—
174C4×C157—41no—
175C4×C157—41no—
176C4×C157C22×D411yescharacteristic
177C4×C157C2×#4111yescharacteristic
178Dic157—22no—
179Dic157—21no—
180Dic157—21no—
181Dic157—41no—
182Dic157—41no—
183Dic157—41no—
184Dic157#12112yes—
185C2×C157—42no—
186C2×C157—81no—
187C2×C157C2×#12111yescharacteristic
188C2×C157#57711yescharacteristic
189C2×C157#57811yescharacteristic
190D157—22no—
191C157#304611yescharacteristic, complemented, Sylow
192#3046—1571nomaximal, Sylow
193C2×#137—1571no—
194C2×#135—1571no—
195#536—1571no—
196#520—1571no—
197#464—1571no—
198#464—1571no—
199#463—1571no—
200#463D15711yescharacteristic, complemented
201#462—1571no—
202#462—1571no—
203#461—1571no—
204#461—1571no—
205#424—1571no—
206#417—1571no—
207C2×#103—1571no—
208C2×#41—1571no—
209C2×#41—1571no—
210C22×Q8—1571no—
211#137—3144no—
212#135—3144no—
213C2×Q16—1571no—
214C2×Q16—1571no—
215C2×Q16—1571no—
216C2×SD16—1571no—
217C2×SD16—1571no—
218C2×SD16—1571no—
219C2×SD16C2×D15711yescharacteristic
220C2×D8C2×D15711yescharacteristic
221C2×M16—1571no—
222C22×C8—1571no—
223C4⋊Q8—1571no—
224C4⋊Q8—1571no—
225#124—1571no—
226#124C2×D15711yescharacteristic
227#123—3141no—
228#123—3141no—
229C22⋊Q8—3141no—
230C22⋊Q8—3141no—
231C4×Q8—1571no—
232C4×Q8—1571no—
233C4×D4—3141no—
234C4×D4—3141no—
235C4×D4—1571no—
236C4×D4C2×D15711yescharacteristic
237#117—3141no—
238#117—1571no—
239C2×C4⋊C4—3141no—
240C2×C4⋊C4—1571no—
241C4⋊C8—1571no—
242C4⋊C8—1571no—
243C4⋊C8—1571no—
244C4⋊C8C2×D15711yescharacteristic
245#103—1571no—
246#103—1571no—
247#103—1571no—
248#103—1571no—
249#103—1571no—
250#103C2×D15711yescharacteristic
251#102—1571no—
252#102C2×D15711yescharacteristic
253#41—3142no—
254#41—3142no—
255#41—3142no—
256#41—1574no—
257C2×Q8—3141no—
258C2×Q8—3142no—
259C2×Q8—3142no—
260C2×Q8—1571no—
261C2×Q8—1571no—
262C2×Q8—1571no—
263C2×Q8C22×D15711yescharacteristic
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4C22×D15711yescharacteristic
267C2×D4C22×D15711yescharacteristic
268C22×C4—21no—
269C22×C4—3141no—
270C22×C4—3141no—
271C22×C4—3141no—
272C22×C4—3141no—
273C22×C4—3141no—
274C22×C4—1571no—
275Q16—3142no—
276Q16—3142no—
277Q16—3142no—
278SD16—22no—
279SD16—3142no—
280SD16—3142no—
281SD16—3142no—
282D8—22no—
283M16—3142no—
284C2×C8—3142no—
285C2×C8—1571no—
286C2×C8—1571no—
287C2×C8C22×D15711yescharacteristic
288C2×C8C22×D15711yescharacteristic
289C4⋊C4—3141no—
290C4⋊C4—3141no—
291C4⋊C4—3141no—
292C4⋊C4—3141no—
293C4⋊C4—3141no—
294C4⋊C4—3141no—
295C4⋊C4—3141no—
296C4⋊C4—1571no—
297C4⋊C4—1571no—
298C4⋊C4—1571no—
299C4⋊C4C22×D15711yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—3141no—
304C22⋊C4—3141no—
305C22⋊C4—3141no—
306C22⋊C4—1571no—
307C22⋊C4—1571no—
308C42—3141no—
309C42—3141no—
310C42—1571no—
311C42C22×D15711yescharacteristic
312C23—21no—
313C23—21no—
314C23—1571no—
315Q8—21no—
316Q8—21no—
317Q8—3141no—
318Q8—3141no—
319Q8—3141no—
320Q8—3141no—
321Q8—3141no—
322Q8—6282no—
323Q8—1572no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—3144no—
328D4—6282no—
329D4D4×D15712yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—3141no—
334C2×C4—3144no—
335C2×C4—3141no—
336C2×C4—3141no—
337C2×C4—3141no—
338C2×C4—6282no—
339C2×C4—6282no—
340C2×C4—1571no—
341C2×C4—1571no—
342C2×C4—1571no—
343C2×C4—1572no—
344C2×C4—1572no—
345C2×C4—12561no—
346C2×C4—12561no—
347C2×C4D4×D15711yescharacteristic
348C2×C4D4×D15711yescharacteristic
349C2×C4C23×D15711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3141no—
353C8—3141no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3141no—
361C22—3141no—
362C22—1572no—
363C22C2×D4×D15711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3141no—
369C4—3141no—
370C4—3142no—
371C4—6281no—
372C4—6281no—
373C4—6281no—
374C4—1572no—
375C4C2×D4×D15711yescharacteristic
376C4#41×D15711yescharacteristic
377C2—42no—
378C2—81no—
379C2—3142no—
380C2#121×D15711yescharacteristic
381C2#2375808311yescharacteristic
382C2#2375808711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097790C2
hypercenter#463D157
hypocenterC157#3046
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
3. #23757843C2semidirect
4. #23757752C2semidirect
5. #23757494C2semidirect
6. #23757492C2semidirect
7. #23757486C2semidirect
8. #23757484C2semidirect
9. #23757482C2semidirect
10. #23757478C2semidirect
11. #23757476C2semidirect
12. #23757298C2semidirect
13. #23757271C2semidirect
14. #103×D157C2semidirect
15. #463×C157C2semidirect
191. C157#3046semidirect
200. #463D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C2×#190977902
#237578432
#237577522
#237574942
#237574922
#237574862
#237574842
#237574822
#237574782
#237574762
#237572982
#237572712
#103×D1572
#463×C1572
#3046157