Finite Group Databasebeta

← #34109868

Subgroups — #34109868

Order 20096.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758069C211yesmaximal, characteristic
2C2×#19097800C211yesmaximal, characteristic, complemented
3#137×D157C211yesmaximal, characteristic, complemented
4#23758038C211yesmaximal, characteristic, complemented
5#23757549C211yesmaximal, characteristic, complemented
6#23757548C211yesmaximal, characteristic, complemented
7#23757547C211yesmaximal, characteristic, complemented
8#23757546C211yesmaximal, characteristic, complemented
9#23757535C211yesmaximal, characteristic, complemented
10#23757534C211yesmaximal, characteristic, complemented
11#23757533C211yesmaximal, characteristic, complemented
12#23757320C211yesmaximal, characteristic, complemented
13#23757254C211yesmaximal, characteristic, complemented
14#101×D157C211yesmaximal, characteristic, complemented
15#471×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17#41×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097804C2211yescharacteristic
20#19097804C2211yescharacteristic
21#19097803—21no—
22#19097802C2211yescharacteristic
23#19097801C2211yescharacteristic
24#19097801C2211yescharacteristic
25#19097800—21no—
26#19097800—21no—
27#19097800—21no—
28#19097800—21no—
29#19097800—21no—
30#19097800—21no—
31#19097797C2211yescharacteristic
32#19097795—21no—
33#19097792—21no—
34#19097790—21no—
35C2×Dic628C2211yescharacteristic, complemented
36C2×#14578944C2211yescharacteristic, complemented
37Q16×D157—21no—
38C2×#14578942C2211yescharacteristic, complemented
39SD16×D157—21no—
40#19097779—21no—
41#19097779C2211yescharacteristic
42#19097778C2211yescharacteristic
43#19097777—21no—
44#19097777C2211yescharacteristic
45C2×#14578937—21no—
46C2×#14578937C2211yescharacteristic
47M16×D157—21no—
48M16×D157C2211yescharacteristic
49M16×D157C2211yescharacteristic
50M16×D157C2211yescharacteristic
51#19097707C2211yescharacteristic, complemented
52#19097706C2211yescharacteristic, complemented
53#19097706C2211yescharacteristic, complemented
54#19097690C2211yescharacteristic, complemented
55#19097689C2211yescharacteristic, complemented
56#19097689C2211yescharacteristic, complemented
57#19097689C2211yescharacteristic, complemented
58#19097689C2211yescharacteristic, complemented
59#19097688C2211yescharacteristic, complemented
60#137×C157C2211yescharacteristic
61AGL(1,ℤ/8)×C157C2211yescharacteristic
62C2×SD16×C157C2211yescharacteristic, complemented
63#131×C157C2211yescharacteristic
64#108×C157C2211yescharacteristic, complemented
65#101×C157C2211yescharacteristic, complemented
66#100×C157C2211yescharacteristic, complemented
67#14578955D411yescharacteristic
68#14578954—21no—
69#14578954C2311yescharacteristic
70#14578953—21no—
71#14578953—21no—
72#14578953—21no—
73#14578953—21no—
74#14578953—21no—
75#14578953D411yescharacteristic
76C2×Dic314—21no—
77C2×Dic314C2311yescharacteristic
78C2×Dic314C2311yescharacteristic
79Q8×D157—21no—
80Q8×D157—21no—
81Q8×D157—21no—
82Q8×D157—21no—
83Q8×D157D411yescharacteristic
84C2×C22⋊D157—21no—
85D4×D157—21no—
86D4×D157D411yescharacteristic
87C22×Dic157—21no—
88C2×C4×D157—21no—
89C2×C4×D157—21no—
90C2×C4×D157C2311yescharacteristic
91Dic628—21no—
92Dic628—21no—
93Dic628—21no—
94Dic628—21no—
95#14578944—21no—
96#14578944—21no—
97#14578944—21no—
98#14578944—21no—
99#14578944—21no—
100#14578943—21no—
101#14578943—21no—
102#14578942—21no—
103#14578942—21no—
104#14578942—21no—
105#14578942—21no—
106#14578942—21no—
107#14578941—21no—
108#14578939—21no—
109#14578938—21no—
110#14578938C2311yescharacteristic
111#14578938C2311yescharacteristic
112#14578938C2311yescharacteristic
113#14578937—21no—
114#14578937—21no—
115#14578937—21no—
116#14578937—21no—
117#14578937—21no—
118#14578937D411yescharacteristic
119#14578937D411yescharacteristic
120#14578937D411yescharacteristic
121#14578937D411yescharacteristic
122C2×#11929434—21no—
123C2×#11929434C2311yescharacteristic
124C8×D157—21no—
125C8×D157—21no—
126C8×D157—21no—
127C8×D157—21no—
128C8×D157—21no—
129#41×C157C2311yescharacteristic
130C2×Q8×C157C2311yescharacteristic
131C2×D4×C157C2311yescharacteristic
132Q16×C157—21no—
133SD16×C157—21no—
134SD16×C157—21no—
135SD16×C157—21no—
136SD16×C157—21no—
137D8×C157—21no—
138M16×C157—21no—
139M16×C157C2311yescharacteristic
140M16×C157C2311yescharacteristic
141M16×C157C2311yescharacteristic
142C2×C8×C157—21no—
143C2×C8×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157#4111yescharacteristic
146Dic314—21no—
147Dic314—21no—
148Dic314—21no—
149Dic314—21no—
150Dic314—21no—
151Dic314—41no—
152Dic314—41no—
153Dic314C2×D411yescharacteristic
154D628—21no—
155D628C2×D411yescharacteristic
156C22⋊D157—21no—
157C22⋊D157—21no—
158C22⋊D157—21no—
159C22⋊D157—41no—
160C22⋊D157—41no—
161C2×Dic157—21no—
162C2×Dic157—21no—
163C2×Dic157—21no—
164C2×Dic157—41no—
165C2×Dic157—41no—
166C2×Dic157#4111yescharacteristic
167C4×D157—21no—
168C4×D157—21no—
169C4×D157—21no—
170C4×D157—41no—
171C4×D157—41no—
172C4×D157C2×D411yescharacteristic
173C4×D157C2×D411yescharacteristic
174C4×D157C2×D411yescharacteristic
175C4×D157C2×D411yescharacteristic
176#11929434—21no—
177#11929434—21no—
178#11929434—21no—
179#11929434C2×D411yescharacteristic
180#11929434C2×D411yescharacteristic
181C23×C157—21no—
182Q8×C157—21no—
183Q8×C157—21no—
184Q8×C157C2×D411yescharacteristic
185D4×C157—21no—
186D4×C157—21no—
187D4×C157—21no—
188D4×C157C2×D411yescharacteristic
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157C2411yescharacteristic
192C8×C157—21no—
193C8×C157—21no—
194C8×C157—21no—
195C8×C157C2×D411yescharacteristic
196C8×C157C2×D411yescharacteristic
197C22×C157—21no—
198C22×C157—41no—
199C22×C157—81no—
200C22×C157C2×#4111yescharacteristic
201C2×D157—21no—
202C2×D157—41no—
203C2×D157—81no—
204C2×D157#12111yescharacteristic
205C2×D157#12111yescharacteristic
206C4×C157—21no—
207C4×C157—41no—
208C4×C157C22×D411yescharacteristic
209C4×C157C22×D411yescharacteristic
210Dic157—21no—
211Dic157—41no—
212Dic157—41no—
213Dic157#12111yescharacteristic
214Dic157#12111yescharacteristic
215C2×C157—21no—
216C2×C157—41no—
217C2×C157—81no—
218C2×C157C2×#12111yescharacteristic
219D157—21no—
220D157—21no—
221D157—41no—
222C157#306911yescharacteristic, complemented, Sylow
223#3069—1571nomaximal, Sylow
224#575—1571no—
225C2×#137—1571no—
226C2×#137—1571no—
227#568—1571no—
228#473—1571no—
229#473—1571no—
230#473—1571no—
231#473—1571no—
232#471—1571no—
233#471—1571no—
234#471—1571no—
235#471D15711yescharacteristic, complemented
236#430—1571no—
237#413—1571no—
238C2×#101—1571no—
239C2×#41—1571no—
240C2×#41—1571no—
241C22×Q8—1571no—
242#137—3141no—
243#137—3141no—
244#137—3141no—
245#137—3141no—
246#137—3141no—
247#137—3141no—
248#137—3141no—
249#137—1571no—
250#137—1571no—
251#137—1571no—
252#137—1571no—
253#137—1571no—
254#137C2×D15711yescharacteristic
255AGL(1,ℤ/8)—3141no—
256AGL(1,ℤ/8)—1571no—
257AGL(1,ℤ/8)C2×D15711yescharacteristic
258#135—3141no—
259#135—3141no—
260C2×Q16—3141no—
261C2×Q16—1571no—
262C2×Q16—1571no—
263C2×SD16—3141no—
264C2×SD16—1571no—
265C2×SD16C2×D15711yescharacteristic, complemented
266#131—3141no—
267#131—3141no—
268#131—1571no—
269#131—1571no—
270#131—1571no—
271#131C2×D15711yescharacteristic
272C2×M16—3141no—
273C2×M16—3141no—
274C2×M16—1571no—
275C2×M16—1571no—
276C2×M16—1571no—
277C2×M16—1571no—
278#108—1571no—
279#108—1571no—
280#108—1571no—
281#108C2×D15711yescharacteristic, complemented
282#101—1571no—
283#101—1571no—
284#101—1571no—
285#101—1571no—
286#101—1571no—
287#101C2×D15711yescharacteristic, complemented
288#100—1571no—
289#100C2×D15711yescharacteristic, complemented
290#41—3141no—
291#41—3141no—
292#41—3141no—
293#41—3141no—
294#41—3141no—
295#41—3141no—
296#41—1571no—
297#41—1571no—
298#41—1571no—
299#41C22×D15711yescharacteristic
300C2×Q8—3141no—
301C2×Q8—3141no—
302C2×Q8—3141no—
303C2×Q8—3141no—
304C2×Q8—3141no—
305C2×Q8—1571no—
306C2×Q8—1571no—
307C2×Q8—1571no—
308C2×Q8C22×D15711yescharacteristic
309C2×D4—3141no—
310C2×D4—3141no—
311C2×D4—1571no—
312C2×D4C22×D15711yescharacteristic
313C22×C4—3141no—
314C22×C4—3141no—
315C22×C4—3141no—
316C22×C4—1571no—
317Q16—21no—
318Q16—3141no—
319Q16—3141no—
320Q16—3141no—
321Q16—3141no—
322Q16—3141no—
323Q16—3141no—
324Q16—3141no—
325Q16—3141no—
326Q16—3141no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—21no—
331SD16—3141no—
332SD16—3141no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339D8—21no—
340D8—3141no—
341M16—21no—
342M16—3141no—
343M16—3141no—
344M16—3141no—
345M16—3141no—
346M16—3141no—
347M16—3141no—
348M16—1571no—
349M16—1571no—
350M16—1571no—
351M16—1571no—
352M16—1571no—
353M16—1571no—
354M16—1571no—
355M16C22×D15711yescharacteristic
356M16C22×D15711yescharacteristic
357M16C22×D15711yescharacteristic
358C2×C8—21no—
359C2×C8—3141no—
360C2×C8—3141no—
361C2×C8—3141no—
362C2×C8—3141no—
363C2×C8—3141no—
364C2×C8—3141no—
365C2×C8—1571no—
366C2×C8C22×D15711yescharacteristic
367C23—21no—
368C23—3141no—
369C23—1571no—
370Q8—21no—
371Q8—21no—
372Q8—3141no—
373Q8—3141no—
374Q8—3141no—
375Q8—3141no—
376Q8—3141no—
377Q8—6281no—
378Q8—6281no—
379Q8—1571no—
380Q8D4×D15711yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—3141no—
385D4—3141no—
386D4—3141no—
387D4—3141no—
388D4—6281no—
389D4—6281no—
390D4—1571no—
391D4D4×D15711yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—3141no—
395C2×C4—3141no—
396C2×C4—3141no—
397C2×C4—3141no—
398C2×C4—3141no—
399C2×C4—3141no—
400C2×C4—6281no—
401C2×C4—6281no—
402C2×C4—6281no—
403C2×C4—6281no—
404C2×C4—1571no—
405C2×C4—1571no—
406C2×C4—1571no—
407C2×C4—1571no—
408C2×C4—1571no—
409C2×C4C23×D15711yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—3141no—
414C8—3141no—
415C8—3141no—
416C8—1571no—
417C8—1571no—
418C8D4×D15711yescharacteristic
419C8D4×D15711yescharacteristic
420C22—21no—
421C22—41no—
422C22—81no—
423C22—3141no—
424C22—6281no—
425C22—1571no—
426C22—1571no—
427C22—12561no—
428C22#41×D15711yescharacteristic
429C4—21no—
430C4—41no—
431C4—3141no—
432C4—6281no—
433C4—6281no—
434C4—1571no—
435C4—1571no—
436C4C2×D4×D15711yescharacteristic
437C4C2×D4×D15711yescharacteristic
438C2—21no—
439C2—41no—
440C2—81no—
441C2—3141no—
442C2—3141no—
443C2—6281no—
444C2#121×D15711yescharacteristic
subgroupG/H
centerC2#121×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#471×C157C2
socleC2×C157C2×#121
prime radical#23758069C2
hypercenter#471D157
hypocenterC157#3069
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
2. C2×#19097800C2semidirect
3. #137×D157C2semidirect
4. #23758038C2semidirect
5. #23757549C2semidirect
6. #23757548C2semidirect
7. #23757547C2semidirect
8. #23757546C2semidirect
9. #23757535C2semidirect
10. #23757534C2semidirect
11. #23757533C2semidirect
12. #23757320C2semidirect
13. #23757254C2semidirect
14. #101×D157C2semidirect
15. #471×C157C2semidirect
35. C2×Dic628C22semidirect
36. C2×#14578944C22semidirect
38. C2×#14578942C22semidirect
51. #19097707C22semidirect
52. #19097706C22semidirect
53. #19097706C22semidirect
54. #19097690C22semidirect
55. #19097689C22semidirect
56. #19097689C22semidirect
57. #19097689C22semidirect
58. #19097689C22semidirect
59. #19097688C22semidirect
62. C2×SD16×C157C22semidirect
64. #108×C157C22semidirect
65. #101×C157C22semidirect
66. #100×C157C22semidirect
222. C157#3069semidirect
235. #471D157semidirect
265. C2×SD16C2×D157semidirect
281. #108C2×D157semidirect
287. #101C2×D157semidirect
289. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580692
C2×#190978002
#137×D1572
#237580382
#237575492
#237575482
#237575472
#237575462
#237575352
#237575342
#237575332
#237573202
#237572542
#101×D1572
#471×C1572
#3069157