Finite Group Databasebeta

← #34111873

Subgroups — #34111873

Order 20096.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097791C211yesmaximal, characteristic, complemented
2C22×#14578944C211yesmaximal, characteristic, complemented
3#23757932C211yesmaximal, characteristic, complemented
4#23757655C211yesmaximal, characteristic, complemented
5#23757512C211yesmaximal, characteristic, complemented
6#23757507C211yesmaximal, characteristic, complemented
7#23757495C211yesmaximal, characteristic, complemented
8#23757492C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757433C211yesmaximal, characteristic, complemented
11#23757412C211yesmaximal, characteristic, complemented
12#23757411C211yesmaximal, characteristic, complemented
13C4×#14578944C211yesmaximal, characteristic, complemented
14D4×#11929434C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18#19097791—24no—
19C2×#14578944—24no—
20C2×#14578944C2211yescharacteristic
21C2×#14578944C2211yescharacteristic
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578941C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C22×#11929434C2211yescharacteristic
30C22×#11929434C2211yescharacteristic
31#19097759C2211yescharacteristic
32#19097757—21no—
33#19097750C2211yescharacteristic
34#19097749—21no—
35#19097745C2211yescharacteristic
36#19097743—21no—
37C4×Dic314C2211yescharacteristic
38#19097731—21no—
39#19097704C2211yescharacteristic
40C4⋊#11929434C2211yescharacteristic
41#19097695C2211yescharacteristic
42#19097695C2211yescharacteristic
43#19097694C2211yescharacteristic
44#19097694C2211yescharacteristic
45#19097692C2211yescharacteristic
46#19097691C2211yescharacteristic
47C22⋊#11929434C2211yescharacteristic
48C22⋊#11929434C2211yescharacteristic
49C4×#11929434C2211yescharacteristic
50C2×#41×C157C2211yescharacteristic
51C22×Q8×C157C2211yescharacteristic
52#124×C157—21no—
53#124×C157C2211yescharacteristic
54C22⋊Q8×C157—21no—
55C22⋊Q8×C157C2211yescharacteristic
56#121×C157—21no—
57#121×C157C2211yescharacteristic
58C4×Q8×C157C2211yescharacteristic
59C4×D4×C157—21no—
60C4×D4×C157C2211yescharacteristic
61#14578954—22no—
62#14578954—41no—
63C2×Dic314—22no—
64C2×Dic314—21no—
65C2×Dic314—41no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73#14578944—22no—
74#14578944—22no—
75#14578944—22no—
76#14578944D414yescomplemented
77#14578942—22no—
78#14578942—41no—
79#14578941—22no—
80#14578941—41no—
81#14578939—22no—
82C2×#11929434—22no—
83C2×#11929434—22no—
84C2×#11929434C2311yescharacteristic
85C2×#11929434C2311yescharacteristic
86C2×#11929434C2311yescharacteristic
87C2×#11929434C2311yescharacteristic
88C4⋊Dic157C2311yescharacteristic
89#14578933—21no—
90#14578933—21no—
91C22⋊Dic157—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95C4×Dic157—21no—
96#41×C157—22no—
97#41×C157—41no—
98C2×Q8×C157—21no—
99C2×Q8×C157—22no—
100C2×Q8×C157—41no—
101C2×Q8×C157C2311yescharacteristic
102C2×Q8×C157C2311yescharacteristic
103C2×D4×C157—21no—
104C2×D4×C157—21no—
105C2×D4×C157D411yescharacteristic
106C2×D4×C157C2311yescharacteristic
107C22×C4×C157—21no—
108C22×C4×C157—21no—
109C22×C4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111C4⋊C4×C157—21no—
112C4⋊C4×C157—21no—
113C4⋊C4×C157D411yescharacteristic
114C4⋊C4×C157C2311yescharacteristic
115C22⋊C4×C157—21no—
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C22⋊C4×C157—21no—
119C22⋊C4×C157D411yescharacteristic
120C22⋊C4×C157D411yescharacteristic
121C42×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—42no—
128Dic314C2×D412yes—
129D628—21no—
130D628—21no—
131C22⋊D157—42no—
132C22⋊D157—42no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—42no—
137C4×D157—42no—
138#11929434—21no—
139#11929434—21no—
140#11929434—21no—
141#11929434C2×D412yes—
142C23×C157—21no—
143C23×C157C2×D411yescharacteristic
144C23×C157C2×D411yescharacteristic
145Q8×C157—21no—
146Q8×C157—21no—
147Q8×C157—21no—
148Q8×C157—42no—
149Q8×C157C2×D412yes—
150D4×C157—22no—
151D4×C157—21no—
152D4×C157—21no—
153D4×C157—42no—
154D4×C157—42no—
155C2×C4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—42no—
163C2×C4×C157—42no—
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2411yescharacteristic
169C22×C157—21no—
170C22×C157—21no—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157#13512yescomplemented
178C22×C157C22×D411yescharacteristic
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C4×C157—21no—
183C4×C157—21no—
184C4×C157—22no—
185C4×C157—21no—
186C4×C157—21no—
187C4×C157—41no—
188C4×C157—41no—
189C4×C157C22×D411yescharacteristic
190C4×C157H(5,𝔽2)11yescharacteristic
191Dic157—22no—
192Dic157—41no—
193Dic157—41no—
194C2×C157—22no—
195C2×C157—41no—
196C2×C157—81no—
197C2×C157D4211yescharacteristic
198C2×C157C2×#13511yescharacteristic
199C2×C157#57811yescharacteristic
200D157—81no—
201C157#328511yescharacteristic, complemented, Sylow
202#3285—1571nomaximal, Sylow
203C2×#135—1571no—
204C22×Q16—1571no—
205#548—1571no—
206#548D15711yescharacteristic, complemented
207#495—1571no—
208#467—1571no—
209#466—1571no—
210#464—1571no—
211#464—1571no—
212#452—1571no—
213#452—1571no—
214#448—1571no—
215#448—1571no—
216C4×Q16—1571no—
217C8×D4—1571no—
218C2×#41—1571no—
219C2×#41C2×D15711yescharacteristic
220C22×Q8—1571no—
221C22×Q8C2×D15711yescharacteristic
222#135—3144no—
223C2×Q16—3144no—
224C2×Q16—1571no—
225C2×Q16—1571no—
226C2×Q16—1571no—
227C2×Q16—1571no—
228C2×SD16—1571no—
229C2×SD16—1571no—
230C2×SD16—1571no—
231C2×SD16—1571no—
232C2×D8—1571no—
233C22×C8—1571no—
234C22×C8—1571no—
235#124—21no—
236#124—3141no—
237#124—1571no—
238#124C2×D15711yescharacteristic
239C22⋊Q8—21no—
240C22⋊Q8—3141no—
241C22⋊Q8—1571no—
242C22⋊Q8C2×D15711yescharacteristic
243#121—21no—
244#121—3141no—
245#121—1571no—
246#121C2×D15711yescharacteristic
247C4×Q8—1571no—
248C4×Q8C2×D15711yescharacteristic
249C4×D4—21no—
250C4×D4—3141no—
251C4×D4C2×D15711yescharacteristic
252#107—1571no—
253C4⋊C8—1571no—
254#103—1571no—
255#103—1571no—
256#103—1571no—
257#103—1571no—
258#102—1571no—
259#102—1571no—
260C22⋊C8—1571no—
261C22⋊C8—1571no—
262C4×C8—1571no—
263#41—22no—
264#41—41no—
265#41—3142no—
266#41—6281no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—3142no—
271C2×Q8—3141no—
272C2×Q8—6281no—
273C2×Q8—1571no—
274C2×Q8—1571no—
275C2×Q8C22×D15711yescharacteristic
276C2×Q8C22×D15711yescharacteristic
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—3141no—
280C2×D4—3141no—
281C2×D4—1571no—
282C2×D4C22⋊D15711yescharacteristic
283C2×D4C22×D15711yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—3141no—
287C22×C4—3141no—
288C22×C4C22×D15711yescharacteristic
289C22×C4C22×D15711yescharacteristic
290Q16—3142no—
291Q16—3142no—
292Q16—3142no—
293Q16—1574no—
294SD16—3142no—
295SD16—3142no—
296SD16—6281no—
297SD16—6281no—
298D8—3142no—
299C2×C8—3142no—
300C2×C8—3142no—
301C2×C8—1571no—
302C2×C8—1571no—
303C2×C8—1571no—
304C2×C8—1571no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3141no—
308C4⋊C4—3141no—
309C4⋊C4—1571no—
310C4⋊C4C22⋊D15711yescharacteristic
311C4⋊C4C22×D15711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4C22⋊D15711yescharacteristic
321C22⋊C4C22⋊D15711yescharacteristic
322C42—21no—
323C42—3141no—
324C42C22×D15711yescharacteristic
325C23—21no—
326C23—3141no—
327C23C2×C22⋊D15711yescharacteristic
328C23C2×C22⋊D15711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—3141no—
334Q8—3141no—
335Q8—3141no—
336Q8—6282no—
337Q8—1572no—
338Q8D4×D15712yes—
339D4—22no—
340D4—21no—
341D4—21no—
342D4—42no—
343D4—42no—
344D4—3141no—
345D4—3141no—
346D4—6282no—
347D4—6282no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—3141no—
360C2×C4—6282no—
361C2×C4—6282no—
362C2×C4C2×C22⋊D15711yescharacteristic
363C2×C4C2×C22⋊D15711yescharacteristic
364C2×C4C2×C22⋊D15711yescharacteristic
365C2×C4C2×C22⋊D15711yescharacteristic
366C2×C4C23×D15711yescharacteristic
367C8—3141no—
368C8—3141no—
369C8—3141no—
370C8—1572no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—6281no—
380C22—6281no—
381C22—6281no—
382C22#1909779112yescomplemented
383C22C22×C22⋊D15711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—3142no—
392C4—6281no—
393C4—6281no—
394C4C2×D4×D15711yescharacteristic
395C4#1909781611yescharacteristic
396C2—22no—
397C2—41no—
398C2—81no—
399C2—12561no—
400C2D4×C22⋊D15711yescharacteristic
401C2C2×#1909779111yescharacteristic
402C2#2375808811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#548×C157C2
socleC22×C157C22×D4
hypercenter#548D157
hypocenterC157#3285
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19097791C2semidirect
2. C22×#14578944C2semidirect
3. #23757932C2semidirect
4. #23757655C2semidirect
5. #23757512C2semidirect
6. #23757507C2semidirect
7. #23757495C2semidirect
8. #23757492C2semidirect
9. #23757435C2semidirect
10. #23757433C2semidirect
11. #23757412C2semidirect
12. #23757411C2semidirect
13. C4×#14578944C2semidirect
14. D4×#11929434C2semidirect
15. #548×C157C2semidirect
76. #14578944D4semidirect
177. C22×C157#135semidirect
201. C157#3285semidirect
206. #548D157semidirect
382. C22#19097791semidirect

Maximal subgroups

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

subgroupindex
C2×#190977912
C22×#145789442
#237579322
#237576552
#237575122
#237575072
#237574952
#237574922
#237574352
#237574332
#237574122
#237574112
C4×#145789442
D4×#119294342
#548×C1572
#3285157