Finite Group Databasebeta

← #34113782

Subgroups — #34113782

Order 20096.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097818C211yesmaximal, characteristic, complemented
2C22×Q8×D157C211yesmaximal, characteristic, complemented
3Q8×D628C211yesmaximal, characteristic, complemented
4D4×Dic314C213yesmaximal, complemented
5Q8×C22⋊D157C211yesmaximal, characteristic, complemented
6#23757936C213yesmaximal, complemented
7#23757931C213yesmaximal, complemented
8#23757930C211yesmaximal, characteristic, complemented
9#23757929C213yesmaximal, complemented
10#23757846C213yesmaximal, complemented
11#23757843C213yesmaximal, complemented
12#23757783C213yesmaximal, complemented
13#23757782C213yesmaximal, complemented
14#23757749C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16#19097818—28no—
17C2×#14578954C2213yes—
18C2×#14578953C2213yes—
19C2×#14578953C2213yescomplemented
20C22×Dic314C2213yes—
21C22×Dic314C2213yescomplemented
22C2×Q8×D157—28no—
23C2×Q8×D157C2213yes—
24C2×Q8×D157C2211yescharacteristic
25C2×Q8×D157C2213yes—
26C2×Q8×D157C2211yescharacteristic
27C2×Q8×D157C2211yescharacteristic, complemented
28C22×C4×D157C2213yes—
29#19097771C2213yescomplemented
30#19097770C2216yescomplemented
31#19097769C2213yescomplemented
32#19097758C2213yescomplemented
33#19097757C2216yescomplemented
34#19097756C2213yes—
35#19097753C2213yes—
36#19097753C2216yes—
37#19097751C2213yes—
38#19097751C2213yescomplemented
39#19097750C2213yes—
40#19097750C2213yescomplemented
41#19097749C2213yes—
42#19097748C2213yes—
43#19097748C2216yescomplemented
44#19097748C2216yescomplemented
45#19097747C2213yescomplemented
46#19097744C2213yes—
47#19097743C2213yes—
48#19097742C2213yes—
49#19097741C2211yescharacteristic
50#19097739C2213yes—
51Q8×Dic157C2211yescharacteristic, complemented
52C4×Dic314C2213yescomplemented
53#19097736C2213yescomplemented
54C4×D628C2213yes—
55D4×Dic157C2216yes—
56C4×C22⋊D157C2213yes—
57C4×C22⋊D157C2213yes—
58#19097731C2216yes—
59#19097729C2213yes—
60#19097728C2213yes—
61C2×#41×C157C2211yescharacteristic, complemented
62C22×Q8×C157C2211yescharacteristic
63#124×C157C2213yescomplemented
64C22⋊Q8×C157C2213yescomplemented
65#121×C157C2213yes—
66C4×Q8×C157C2211yescharacteristic, complemented
67C4×D4×C157C2213yes—
68C23×D157C2311yescharacteristic
69#14578954—212no—
70#14578953—212no—
71#14578953—212no—
72C2×Dic314—212no—
73C2×Dic314—212no—
74C2×Dic314C2313yes—
75C2×Dic314C2316yes—
76C2×Dic314C2313yes—
77C2×Dic314C2313yes—
78C2×Dic314C2313yes—
79Q8×D157—212no—
80Q8×D157—212no—
81Q8×D157—24no—
82Q8×D157—24no—
83Q8×D157D418yescomplemented
84C2×D628C2311yescharacteristic
85C2×D628C2311yescharacteristic
86C2×C22⋊D157C2313yes—
87C2×C22⋊D157C2311yescharacteristic
88C2×C22⋊D157C2313yes—
89C2×C22⋊D157C2313yes—
90C22×Dic157C2316yes—
91C22×Dic157C2313yes—
92C2×C4×D157—212no—
93C2×C4×D157C2313yes—
94C2×C4×D157C2313yes—
95C2×C4×D157C2313yes—
96C2×C4×D157C2313yes—
97C2×C4×D157C2313yes—
98C4⋊Dic157C2313yes—
99C4⋊Dic157C2316yes—
100C4⋊Dic157C2313yes—
101#14578933C2313yes—
102#14578933C2313yes—
103#14578933C2316yes—
104#14578933C2313yes—
105#14578933C2313yes—
106C22⋊Dic157C2313yes—
107C22⋊Dic157C2313yes—
108C22⋊Dic157C2316yes—
109#14578931C2311yescharacteristic
110#14578931C2313yes—
111#14578931C2313yes—
112#14578931C2311yescharacteristic
113#14578931C2313yes—
114#14578931C2313yes—
115C4×Dic157C2313yes—
116C4×Dic157C2316yes—
117#41×C157—24no—
118C2×Q8×C157—24no—
119C2×Q8×C157C2313yes—
120C2×Q8×C157C2311yescharacteristic
121C2×D4×C157C2313yes—
122C2×D4×C157C2313yes—
123C22×C4×C157C2313yes—
124C22×C4×C157C2313yes—
125C4⋊C4×C157C2313yes—
126C4⋊C4×C157C2313yes—
127C22⋊C4×C157C2311yescharacteristic
128C22⋊C4×C157C2313yes—
129C22⋊C4×C157C2316yes—
130C42×C157C2313yes—
131C22×D157—22no—
132C22×D157—22no—
133C22×D157—22no—
134C22×D157C2411yescharacteristic
135C22×D157C2411yescharacteristic
136C22×D157C2411yescharacteristic
137Dic314—26no—
138Dic314—26no—
139Dic314—212no—
140Dic314—26no—
141Dic314C2×D4112yes—
142D628—22no—
143D628—22no—
144C22⋊D157—22no—
145C22⋊D157—26no—
146C22⋊D157—26no—
147C22⋊D157—26no—
148C2×Dic157—26no—
149C2×Dic157—212no—
150C2×Dic157C2413yes—
151C2×Dic157C2413yes—
152C2×Dic157C2411yescharacteristic
153C2×Dic157C2413yes—
154C2×Dic157C2413yes—
155C4×D157—26no—
156C4×D157—26no—
157C4×D157—26no—
158C4×D157—26no—
159C4×D157C2×D4112yes—
160C23×C157C2413yes—
161C23×C157C2411yescharacteristic
162Q8×C157—26no—
163Q8×C157C2×D414yes—
164D4×C157—26no—
165D4×C157—26no—
166C2×C4×C157—26no—
167C2×C4×C157—26no—
168C2×C4×C157C2413yes—
169C2×C4×C157C2411yescharacteristic
170C2×C4×C157C2411yescharacteristic
171C2×C4×C157C2413yes—
172C2×C4×C157C2413yes—
173C22×C157—21no—
174C22×C157—21no—
175C22×C157—23no—
176C22×C157—23no—
177C22×C157—23no—
178C22×C1572−1+412yescomplemented
179C22×C157C2511yescharacteristic
180C2×D157—28no—
181C2×D157—21no—
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—21no—
185C2×D157—21no—
186C2×D157—21no—
187C2×D157—21no—
188C2×D157C22×D412yes—
189C2×D1572−1+412yescomplemented
190C4×C157—21no—
191C4×C157—23no—
192C4×C157—23no—
193C4×C157—21no—
194C4×C157C22×D416yes—
195Dic157—23no—
196Dic157—23no—
197Dic157—23no—
198Dic157—21no—
199Dic157C22×D416yes—
200C2×C157—22no—
201C2×C157—43no—
202C2×C157C23×D411yescharacteristic
203C2×C157C2×2−1+411yescharacteristic
204C2×C157C2×2−1+411yescharacteristic
205D157—22no—
206D157—22no—
207D157—41no—
208C157#348611yescharacteristic, complemented, Sylow
209#3486—1571nomaximal, Sylow
210C2×2−1+4—1571no—
211C23×Q8—1571no—
212D4×Q8—1571no—
213D4×Q8—1573no—
214D4×Q8—1571no—
215D4×Q8—1573no—
216#548—1571no—
217#548—1573no—
218#548—1573no—
219#548D15711yescharacteristic, complemented
220#536—1573no—
221#536—1573no—
222#525—1573no—
223#525—1573no—
224#519—1571no—
2252−1+4—3148no—
226C2×#41—1573no—
227C2×#41—1573no—
228C2×#41—1573no—
229C2×#41C2×D15711yescharacteristic, complemented
230C22×Q8—3148no—
231C22×Q8—1573no—
232C22×Q8—1573no—
233C22×Q8—1571no—
234C22×Q8—1571no—
235C22×Q8—1571no—
236C22×Q8—1573no—
237C22×Q8—1573no—
238C22×Q8C2×D15711yescharacteristic
239C23×C4—1573no—
240C4⋊Q8—1573no—
241C4⋊Q8—1576no—
242C4⋊Q8—1573no—
243#124—1573no—
244#124—1576no—
245#124C2×D15713yescomplemented
246#123—1573no—
247#123—1576no—
248#123—1573no—
249C22⋊Q8—1573no—
250C22⋊Q8—1573no—
251C22⋊Q8—1573no—
252C22⋊Q8—1573no—
253C22⋊Q8—1573no—
254C22⋊Q8—1573no—
255C22⋊Q8—1573no—
256C22⋊Q8—1576no—
257C22⋊Q8—1576no—
258C22⋊Q8C2×D15713yescomplemented
259#121—1573no—
260#121—1573no—
261#121—1573no—
262#121C2×D15713yes—
263#120—1573no—
264#120—1571no—
265C4×Q8—1571no—
266C4×Q8—1573no—
267C4×Q8—1573no—
268C4×Q8C2×D15711yescharacteristic, complemented
269C4×D4—1576no—
270C4×D4—1573no—
271C4×D4—1573no—
272C4×D4—1576no—
273C4×D4—1573no—
274C4×D4C2×D15713yes—
275#117—1573no—
276#117—1573no—
277C24—1571no—
278#41—24no—
279#41—31412no—
280#41—31412no—
281#41—31412no—
282C2×Q8—24no—
283C2×Q8—31412no—
284C2×Q8—31412no—
285C2×Q8—31412no—
286C2×Q8—3144no—
287C2×Q8—3144no—
288C2×Q8—31412no—
289C2×Q8—1578no—
290C2×Q8—1573no—
291C2×Q8—1576no—
292C2×Q8—1573no—
293C2×Q8—1573no—
294C2×Q8—1573no—
295C2×Q8C22×D15713yes—
296C2×Q8C22×D15711yescharacteristic
297C2×D4—1573no—
298C2×D4—1571no—
299C2×D4—1573no—
300C2×D4—1573no—
301C2×D4—1571no—
302C2×D4—1571no—
303C2×D4C22×D15713yes—
304C2×D4C22×D15713yes—
305C22×C4—31412no—
306C22×C4—1573no—
307C22×C4—1576no—
308C22×C4—1573no—
309C22×C4—1573no—
310C22×C4—1573no—
311C22×C4—1573no—
312C22×C4—1573no—
313C22×C4C22×D15713yes—
314C22×C4C22×D15713yes—
315C4⋊C4—1573no—
316C4⋊C4—1573no—
317C4⋊C4—1573no—
318C4⋊C4—1573no—
319C4⋊C4—1573no—
320C4⋊C4—1576no—
321C4⋊C4—1576no—
322C4⋊C4—1573no—
323C4⋊C4C22×D15713yes—
324C4⋊C4C22×D15713yes—
325C22⋊C4—1573no—
326C22⋊C4—1576no—
327C22⋊C4—1573no—
328C22⋊C4—1573no—
329C22⋊C4—1571no—
330C22⋊C4—1571no—
331C22⋊C4—1573no—
332C22⋊C4—1573no—
333C22⋊C4—1573no—
334C22⋊C4C22×D15716yes—
335C22⋊C4C22×D15711yescharacteristic
336C22⋊C4C22×D15713yes—
337C42—1573no—
338C42—1576no—
339C42C22×D15713yes—
340C23—3142no—
341C23—3142no—
342C23—3142no—
343C23—1571no—
344C23—1571no—
345C23—1571no—
346C23C23×D15713yes—
347C23C23×D15711yescharacteristic
348Q8—26no—
349Q8—31412no—
350Q8—3146no—
351Q8—3146no—
352Q8—3146no—
353Q8—15712no—
354Q8D4×D15714yes—
355D4—26no—
356D4—26no—
357D4—3142no—
358D4—3142no—
359D4—3146no—
360D4—3146no—
361D4—3142no—
362D4—3146no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—3146no—
366C2×C4—3146no—
367C2×C4—3146no—
368C2×C4—31412no—
369C2×C4—3146no—
370C2×C4—3146no—
371C2×C4—1573no—
372C2×C4—1571no—
373C2×C4—1573no—
374C2×C4—1573no—
375C2×C4—1573no—
376C2×C4—15712no—
377C2×C4C23×D15713yes—
378C2×C4C23×D15711yescharacteristic
379C2×C4C23×D15713yes—
380C2×C4C23×D15711yescharacteristic
381C2×C4C23×D15713yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—3148no—
388C22—3141no—
389C22—3141no—
390C22—3141no—
391C22—3141no—
392C22—3141no—
393C22—3141no—
394C22—3141no—
395C22—1572no—
396C22—1572no—
397C22#1909781812yescomplemented
398C22C24×D15711yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—3143no—
404C4—3143no—
405C4—3143no—
406C4—3141no—
407C4—1576no—
408C4C2×D4×D15716yes—
409C2—22no—
410C2—43no—
411C2—3142no—
412C2—3142no—
413C2—6281no—
414C2C22×D4×D15711yescharacteristic
415C22−1+4×D15711yescharacteristic
416C2C2×#1909781811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#548×C157C2
hypercenter#548D157
hypocenterC157#3486
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19097818C2semidirect
2. C22×Q8×D157C2semidirect
3. Q8×D628C2semidirect
4. D4×Dic314C2semidirect
5. Q8×C22⋊D157C2semidirect
6. #23757936C2semidirect
7. #23757931C2semidirect
8. #23757930C2semidirect
9. #23757929C2semidirect
10. #23757846C2semidirect
11. #23757843C2semidirect
12. #23757783C2semidirect
13. #23757782C2semidirect
14. #23757749C2semidirect
15. #548×C157C2semidirect
19. C2×#14578953C22semidirect
21. C22×Dic314C22semidirect
27. C2×Q8×D157C22semidirect
29. #19097771C22semidirect
30. #19097770C22semidirect
31. #19097769C22semidirect
32. #19097758C22semidirect
33. #19097757C22semidirect
38. #19097751C22semidirect
40. #19097750C22semidirect
43. #19097748C22semidirect
44. #19097748C22semidirect
45. #19097747C22semidirect
51. Q8×Dic157C22semidirect
52. C4×Dic314C22semidirect
53. #19097736C22semidirect
61. C2×#41×C157C22semidirect
63. #124×C157C22semidirect
64. C22⋊Q8×C157C22semidirect
66. C4×Q8×C157C22semidirect
83. Q8×D157D4semidirect
178. C22×C1572−1+4semidirect
189. C2×D1572−1+4semidirect
208. C157#3486semidirect
219. #548D157semidirect
229. C2×#41C2×D157semidirect
245. #124C2×D157semidirect
258. C22⋊Q8C2×D157semidirect
268. C4×Q8C2×D157semidirect
397. C22#19097818semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
C22×Q8×D1572
Q8×D6282
D4×Dic3142
Q8×C22⋊D1572
#237579362
#237579312
#237579302
#237579292
#237578462
#237578432
#237577832
#237577822
#237577492
#548×C1572
#3486157