Finite Group Databasebeta

← #34111717

Subgroups — #34111717

Order 20096.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097801C211yesmaximal, characteristic, complemented
2C2×#19097791C211yesmaximal, characteristic, complemented
3#23757932C211yesmaximal, characteristic, complemented
4#23757660C211yesmaximal, characteristic, complemented
5#23757524C211yesmaximal, characteristic, complemented
6#23757500C211yesmaximal, characteristic, complemented
7#23757495C211yesmaximal, characteristic, complemented
8#23757470C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757418C211yesmaximal, characteristic, complemented
11#23757416C211yesmaximal, characteristic, complemented
12#23757411C211yesmaximal, characteristic, complemented
13#23757378C211yesmaximal, characteristic, complemented
14#23757345C211yesmaximal, characteristic, complemented
15#550×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18#19097801—24no—
19#19097791—24no—
20C2×#14578944C2211yescharacteristic
21C2×#14578944C2211yescharacteristic
22C2×#14578942C2211yescharacteristic
23C2×#14578942C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578938C2211yescharacteristic
30C22×#11929434C2211yescharacteristic
31#19097759C2211yescharacteristic
32#19097757—21no—
33#19097750C2211yescharacteristic
34#19097749—21no—
35#19097745C2211yescharacteristic
36#19097743—21no—
37C4×Dic314C2211yescharacteristic
38#19097731—21no—
39#19097702C2211yescharacteristic
40C4⋊#11929434C2211yescharacteristic
41#19097695C2211yescharacteristic
42#19097694C2211yescharacteristic
43#19097694C2211yescharacteristic
44#19097692C2211yescharacteristic
45#19097692C2211yescharacteristic
46#19097691C2211yescharacteristic
47C22⋊#11929434C2211yescharacteristic
48C22⋊#11929434C2211yescharacteristic
49#19097682C2211yescharacteristic
50C2×#41×C157C2211yescharacteristic
51C2×#41×C157C2211yescharacteristic
52#127×C157—21no—
53#127×C157C2211yescharacteristic
54#121×C157—21no—
55#121×C157—21no—
56#121×C157C2211yescharacteristic
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#14578944D414yescomplemented
75#14578942—22no—
76#14578942—22no—
77#14578942—22no—
78#14578942—41no—
79#14578941—22no—
80#14578939—22no—
81#14578939—41no—
82#14578938—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—22no—
98#41×C157—41no—
99#41×C157—41no—
100C2×Q8×C157C2311yescharacteristic
101C2×D4×C157—21no—
102C2×D4×C157—21no—
103C2×D4×C157—21no—
104C2×D4×C157—21no—
105C2×D4×C157—21no—
106C2×D4×C157—21no—
107C2×D4×C157D411yescharacteristic
108C2×D4×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110C22×C4×C157—21no—
111C22×C4×C157—21no—
112C22×C4×C157C2311yescharacteristic
113C22×C4×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157D411yescharacteristic
116C4⋊C4×C157C2311yescharacteristic
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×C157—21no—
144C23×C157C2×D411yescharacteristic
145C23×C157C2×D411yescharacteristic
146Q8×C157—21no—
147Q8×C157C2×D412yes—
148D4×C157—21no—
149D4×C157—21no—
150D4×C157—21no—
151D4×C157—21no—
152D4×C157—41no—
153D4×C157—42no—
154D4×C157—42no—
155D4×C157—42no—
156D4×C157—81no—
157D4×C157—81no—
158D4×C157—81no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—42no—
166C2×C4×C157—42no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2×D411yescharacteristic
170C2×C4×C157C2×D411yescharacteristic
171C2×C4×C157C2411yescharacteristic
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—21no—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157—21no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157C22×D411yescharacteristic
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C4×C157—21no—
189C4×C157—21no—
190C4×C157—22no—
191C4×C157—21no—
192C4×C157—21no—
193C4×C157—41no—
194C4×C157C22×D411yescharacteristic
195C4×C157H(5,𝔽2)11yescharacteristic
196Dic157—22no—
197Dic157—41no—
198Dic157—41no—
199C2×C157—41no—
200C2×C157—41no—
201C2×C157—81no—
202C2×C157—81no—
203C2×C157D4211yescharacteristic
204C2×C157#57511yescharacteristic
205C2×C157#57711yescharacteristic
206D157—81no—
207C157#327111yescharacteristic, complemented, Sylow
208#3271—1571nomaximal, Sylow
209C2×#137—1571no—
210C2×#135—1571no—
211#550D15711yescharacteristic, complemented
212#548—1571no—
213#496—1571no—
214#469—1571no—
215#465—1571no—
216#464—1571no—
217#460—1571no—
218#452—1571no—
219#449—1571no—
220#449—1571no—
221#448—1571no—
222#441—1571no—
223#435—1571no—
224C2×#41—1571no—
225C2×#41C2×D15711yescharacteristic
226C2×#41C2×D15711yescharacteristic
227C22×Q8—1571no—
228#137—3144no—
229#135—3144no—
230C2×Q16—1571no—
231C2×Q16—1571no—
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×D8—1571no—
238C2×D8—1571no—
239C2×M16—1571no—
240C22×C8—1571no—
241#127—21no—
242#127C2×D15711yescharacteristic
243#124—3141no—
244#124—1571no—
245C22⋊Q8—3141no—
246C22⋊Q8—1571no—
247#121—21no—
248#121—21no—
249#121—3141no—
250#121—1571no—
251#121C2×D15711yescharacteristic
252#121C2×D15711yescharacteristic
253C4×Q8—1571no—
254C4×Q8C2×D15711yescharacteristic
255C4×D4—21no—
256C4×D4—3141no—
257C4×D4C2×D15711yescharacteristic
258#106—1571no—
259C4⋊C8—1571no—
260#103—1571no—
261#103—1571no—
262#103—1571no—
263#102—1571no—
264#102—1571no—
265#102—1571no—
266C22⋊C8—1571no—
267C22⋊C8—1571no—
268#97—1571no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—3142no—
274#41—6281no—
275C2×Q8—3142no—
276C2×Q8—3141no—
277C2×Q8—6281no—
278C2×Q8—1571no—
279C2×Q8—1571no—
280C2×Q8C22×D15711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—1571no—
290C2×D4C22⋊D15711yescharacteristic
291C2×D4C22×D15711yescharacteristic
292C2×D4C22×D15711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—3141no—
296C22×C4—3141no—
297C22×C4C22×D15711yescharacteristic
298C22×C4C22×D15711yescharacteristic
299Q16—3142no—
300Q16—1574no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—3142no—
305SD16—6281no—
306D8—3142no—
307D8—6281no—
308M16—3142no—
309C2×C8—3142no—
310C2×C8—1571no—
311C2×C8—1571no—
312C2×C8—1571no—
313C2×C8—1571no—
314C4⋊C4—21no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—1571no—
318C4⋊C4C22⋊D15711yescharacteristic
319C4⋊C4C22×D15711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4C22⋊D15711yescharacteristic
327C22⋊C4C22⋊D15711yescharacteristic
328C42—21no—
329C42—3141no—
330C42C22×D15711yescharacteristic
331C23—21no—
332C23—21no—
333C23—3141no—
334C23C2×C22⋊D15711yescharacteristic
335C23C2×C22⋊D15711yescharacteristic
336Q8—21no—
337Q8—3141no—
338Q8—3141no—
339Q8—3141no—
340Q8—6282no—
341Q8—1572no—
342Q8D4×D15712yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—3141no—
355D4—3141no—
356D4—6282no—
357D4—6282no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—3141no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4C2×C22⋊D15711yescharacteristic
372C2×C4C2×C22⋊D15711yescharacteristic
373C2×C4C2×C22⋊D15711yescharacteristic
374C2×C4C2×C22⋊D15711yescharacteristic
375C2×C4C23×D15711yescharacteristic
376C8—3141no—
377C8—3141no—
378C8—3141no—
379C8—1572no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22C22×C22⋊D15711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—21no—
401C4—41no—
402C4—3142no—
403C4—6281no—
404C4—6281no—
405C4C2×D4×D15711yescharacteristic
406C4#1909781611yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—81no—
411C2—12561no—
412C2D4×C22⋊D15711yescharacteristic
413C2#2375807111yescharacteristic
414C2#2375808411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#550×C157C2
socleC22×C157C22×D4
hypercenter#550D157
hypocenterC157#3271
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#19097801C2semidirect
2. C2×#19097791C2semidirect
3. #23757932C2semidirect
4. #23757660C2semidirect
5. #23757524C2semidirect
6. #23757500C2semidirect
7. #23757495C2semidirect
8. #23757470C2semidirect
9. #23757435C2semidirect
10. #23757418C2semidirect
11. #23757416C2semidirect
12. #23757411C2semidirect
13. #23757378C2semidirect
14. #23757345C2semidirect
15. #550×C157C2semidirect
74. #14578944D4semidirect
207. C157#3271semidirect
211. #550D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C2×#190977912
#237579322
#237576602
#237575242
#237575002
#237574952
#237574702
#237574352
#237574182
#237574162
#237574112
#237573782
#237573452
#550×C1572
#3271157