Finite Group Databasebeta

← #34111860

Subgroups — #34111860

Order 20096.

494 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×#19097792C211yesmaximal, characteristic, complemented
2#23757942C211yesmaximal, characteristic, complemented
3#23757922C211yesmaximal, characteristic, complemented
4#23757644C211yesmaximal, characteristic, complemented
5#23757502C211yesmaximal, characteristic, complemented
6#23757499C211yesmaximal, characteristic, complemented
7#23757481C211yesmaximal, characteristic, complemented
8#23757466C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757434C211yesmaximal, characteristic, complemented
11#23757418C211yesmaximal, characteristic, complemented
12#23757415C211yesmaximal, characteristic, complemented
13SD16×Dic157C211yesmaximal, characteristic, complemented
14C8×C22⋊D157C211yesmaximal, characteristic, complemented
15C2×#135×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18#19097792—24no—
19C2×#14578944C2211yescharacteristic
20C2×#14578943C2211yescharacteristic
21C2×#14578942C2211yescharacteristic
22C2×#14578941C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24C2×C8×D157C2211yescharacteristic
25#19097769C2211yescharacteristic
26#19097767—21no—
27#19097767C2211yescharacteristic
28#19097754—21no—
29#19097752—21no—
30#19097750C2211yescharacteristic
31#19097747C2211yescharacteristic
32#19097745—21no—
33#19097745C2211yescharacteristic
34#19097743—21no—
35#19097742—21no—
36#19097742C2211yescharacteristic
37Q8×Dic157C2211yescharacteristic
38D4×Dic157C2211yescharacteristic
39C4×C22⋊D157—21no—
40C4×C22⋊D157C2211yescharacteristic
41C2×#14578933—21no—
42#19097703C2211yescharacteristic
43#19097700C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic
50C22⋊#11929434C2211yescharacteristic
51#19097683C2211yescharacteristic
52C8×Dic157C2211yescharacteristic
53C2×#41×C157C2211yescharacteristic
54C2×#41×C157C2211yescharacteristic
55#135×C157—24no—
56C2×Q16×C157C2211yescharacteristic
57C2×SD16×C157C2211yescharacteristic
58C2×SD16×C157C2211yescharacteristic
59C2×D8×C157C2211yescharacteristic
60C22×C8×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63#14578953—22no—
64#14578953—41no—
65C2×Dic314C2311yescharacteristic
66C2×D628—21no—
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157D411yescharacteristic
74C22×Dic157—21no—
75C22×Dic157—21no—
76C2×C4×D157—21no—
77C2×C4×D157C2311yescharacteristic
78#14578944—22no—
79#14578943—22no—
80#14578942—41no—
81#14578941—41no—
82#14578939—22no—
83C2×#11929434C2311yescharacteristic
84C2×#11929434C2311yescharacteristic
85C8×D157—22no—
86C4⋊Dic157—21no—
87C4⋊Dic157C2311yescharacteristic
88C4⋊Dic157C2311yescharacteristic
89#14578933—21no—
90#14578933—21no—
91#14578933—21no—
92#14578933—21no—
93#14578933D411yescharacteristic
94C22⋊Dic157—21no—
95C22⋊Dic157—21no—
96C22⋊Dic157—21no—
97C22⋊Dic157D411yescharacteristic
98#14578931—21no—
99#14578931—21no—
100#14578931D411yescharacteristic
101C4×Dic157—21no—
102C4×Dic157C2311yescharacteristic
103#41×C157—22no—
104#41×C157—22no—
105#41×C157—41no—
106#41×C157—41no—
107C2×Q8×C157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157—21no—
110C2×D4×C157—21no—
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157—21no—
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116Q16×C157—22no—
117SD16×C157—22no—
118SD16×C157D414yescomplemented
119D8×C157—22no—
120C2×C8×C157—22no—
121C2×C8×C157C2311yescharacteristic
122C2×C8×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157C2×D411yescharacteristic
125Dic314—21no—
126Dic314—21no—
127D628—21no—
128D628—21no—
129D628—42no—
130C22⋊D157—22no—
131C22⋊D157—42no—
132C22⋊D157—42no—
133C22⋊D157—81no—
134C22⋊D157—81no—
135C22⋊D157—81no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—42no—
140C2×Dic157—42no—
141C2×Dic157C2×D411yescharacteristic
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C4×D157—21no—
145C4×D157—21no—
146C4×D157—42no—
147#11929434—21no—
148#11929434—21no—
149C23×C157—21no—
150C23×C157—21no—
151C23×C157C2×D411yescharacteristic
152Q8×C157—21no—
153Q8×C157—21no—
154Q8×C157—21no—
155Q8×C157C2×D412yes—
156D4×C157—21no—
157D4×C157—21no—
158D4×C157—21no—
159D4×C157—42no—
160D4×C157—42no—
161D4×C157C2×D412yes—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—42no—
167C2×C4×C157—42no—
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157—21no—
173C22×C157—22no—
174C22×C157—21no—
175C22×C157—21no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157C22×D411yescharacteristic
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—21no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C4×C157—21no—
189C4×C157—22no—
190C4×C157—41no—
191C4×C157C22×D411yescharacteristic
192C4×C157H(5,𝔽2)11yescharacteristic
193Dic157—21no—
194Dic157—21no—
195Dic157—41no—
196Dic157—41no—
197Dic157—41no—
198Dic157#13512yescomplemented
199C2×C157—41no—
200C2×C157—42no—
201C2×C157—81no—
202C2×C157D4211yescharacteristic
203C2×C157C2×#13511yescharacteristic
204C2×C157#57711yescharacteristic
205D157—41no—
206D157—81no—
207C157#328411yescharacteristic, complemented, Sylow
208#3284—1571nomaximal, Sylow
209C2×#135—1571no—
210C2×#135D15711yescharacteristic, complemented
211#550—1571no—
212#547—1571no—
213#492—1571no—
214#465—1571no—
215#465—1571no—
216#462—1571no—
217#459—1571no—
218#452—1571no—
219#452—1571no—
220#449—1571no—
221#449—1571no—
222C4×SD16—1571no—
223C8×D4—1571no—
224C2×#41—1571no—
225C2×#41—1571no—
226C2×#41C2×D15711yescharacteristic
227C2×#41C2×D15711yescharacteristic
228#135—24no—
229#135—3144no—
230C2×Q16—1571no—
231C2×Q16C2×D15711yescharacteristic
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16C2×D15711yescharacteristic
236C2×SD16C2×D15711yescharacteristic
237C2×D8—1571no—
238C2×D8C2×D15711yescharacteristic
239C22×C8—1571no—
240C22×C8C2×D15711yescharacteristic
241C4⋊Q8—1571no—
242#127—3141no—
243#127—1571no—
244#123—3141no—
245#123—3141no—
246C22⋊Q8—1571no—
247C22⋊Q8—1571no—
248#121—3141no—
249#121—3141no—
250#121—3141no—
251#121—1571no—
252#121—1571no—
253C4×Q8—1571no—
254C4×D4—3141no—
255C4×D4—1571no—
256C4×D4—1571no—
257C2×C4⋊C4—3141no—
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—
268C4×C8—1571no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—3142no—
274#41—3142no—
275#41—6281no—
276#41—6281no—
277C2×Q8—1571no—
278C2×Q8C22×D15711yescharacteristic
279C2×Q8C22×D15711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—3141no—
288C2×D4—1571no—
289C2×D4—1571no—
290C2×D4C22×D15711yescharacteristic
291C2×D4C22×D15711yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—3141no—
297C22×C4—1571no—
298C22×C4C22×D15711yescharacteristic
299Q16—22no—
300Q16—3142no—
301SD16—22no—
302SD16—3142no—
303SD16—6281no—
304SD16—6281no—
305SD16C22⋊D15714yescomplemented
306D8—22no—
307D8—3142no—
308C2×C8—22no—
309C2×C8—3142no—
310C2×C8—1571no—
311C2×C8—1571no—
312C2×C8C22×D15711yescharacteristic
313C2×C8C22×D15711yescharacteristic
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—3141no—
318C4⋊C4—3141no—
319C4⋊C4—1571no—
320C4⋊C4—1571no—
321C4⋊C4—1571no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—1571no—
328C22⋊C4—1571no—
329C42—3141no—
330C42—1571no—
331C23—21no—
332C23—21no—
333C23—3141no—
334C23—1571no—
335C23D4×D15711yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3141no—
340Q8—3141no—
341Q8C2×C22⋊D15712yes—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—42no—
347D4—3142no—
348D4—3141no—
349D4—3141no—
350D4—6282no—
351D4—6282no—
352D4—6282no—
353D4—12561no—
354D4—12561no—
355D4—12561no—
356D4C2×C22⋊D15712yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4—1571no—
374C2×C4D4×D15711yescharacteristic
375C2×C4C23×D15711yescharacteristic
376C8—21no—
377C8—3141no—
378C8—3141no—
379C8C2×C22⋊D15712yes—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3141no—
390C22—3141no—
391C22—3141no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22C2×D4×D15711yescharacteristic
396C4—21no—
397C4—22no—
398C4—41no—
399C4—3141no—
400C4—3141no—
401C4—6281no—
402C4—6281no—
403C4—6281no—
404C4—1572no—
405C4C22×C22⋊D15711yescharacteristic
406C4#1909781611yescharacteristic
407C2—41no—
408C2—42no—
409C2—81no—
410C2—6281no—
411C2—12561no—
412C2D4×C22⋊D15711yescharacteristic
413C2#135×D15711yescharacteristic
414C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#135×C157C2
socleC22×C157C22×D4
hypercenterC2×#135D157
hypocenterC157#3284
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#19097792C2semidirect
2. #23757942C2semidirect
3. #23757922C2semidirect
4. #23757644C2semidirect
5. #23757502C2semidirect
6. #23757499C2semidirect
7. #23757481C2semidirect
8. #23757466C2semidirect
9. #23757435C2semidirect
10. #23757434C2semidirect
11. #23757418C2semidirect
12. #23757415C2semidirect
13. SD16×Dic157C2semidirect
14. C8×C22⋊D157C2semidirect
15. C2×#135×C157C2semidirect
118. SD16×C157D4semidirect
198. Dic157#135semidirect
207. C157#3284semidirect
210. C2×#135D157semidirect
305. SD16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977922
#237579422
#237579222
#237576442
#237575022
#237574992
#237574812
#237574662
#237574352
#237574342
#237574182
#237574152
SD16×Dic1572
C8×C22⋊D1572
C2×#135×C1572
#3284157