Finite Group Databasebeta

← #34082763

Subgroups — #34082763

Order 19328.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012786C211yesmaximal, characteristic, complemented
2C2×#41×D151C211yesmaximal, characteristic, complemented
3#23752882C213yesmaximal, complemented
4#23752881C211yesmaximal, characteristic, complemented
5#23752880C213yesmaximal, complemented
6#23752874C211yesmaximal, characteristic, complemented
7#23752871C213yesmaximal, complemented
8#23752869C211yesmaximal, characteristic, complemented
9#23752868C213yesmaximal, complemented
10#23752778C213yesmaximal, complemented
11#23752775C213yesmaximal, complemented
12#23752722C213yesmaximal, complemented
13#23752721C213yesmaximal, complemented
14#23752688C211yesmaximal, characteristic, complemented
15#550×C151C211yesmaximal, characteristic, complemented
16#18012786—28no—
17C2×#14559005C2211yescharacteristic
18C2×#14559004C2213yes—
19C2×#14559004C2213yes—
20C2×#14559003C2213yes—
21C2×#14559003C2213yes—
22C2×#14559003C2213yescomplemented
23#41×D151—28no—
24C22×Dic302C2213yescomplemented
25C2×Q8×D151C2211yescharacteristic
26C2×Q8×D151C2211yescharacteristic, complemented
27C2×D4×D151C2213yes—
28C22×C4×D151C2213yes—
29#18012735C2213yescomplemented
30#18012735C2216yescomplemented
31#18012727C2213yescomplemented
32#18012726C2213yescomplemented
33#18012725C2216yescomplemented
34#18012724C2211yescharacteristic
35#18012721C2213yes—
36#18012719C2213yescomplemented
37#18012718C2213yescomplemented
38#18012717C2213yescomplemented
39#18012716C2213yescomplemented
40#18012714C2213yescomplemented
41#18012713C2213yescomplemented
42#18012712C2213yescomplemented
43#18012711C2213yescomplemented
44#18012711C2216yescomplemented
45#18012711C2216yescomplemented
46#18012710C2213yescomplemented
47#18012710C2213yescomplemented
48#18012709C2213yes—
49#18012707C2213yes—
50#18012707C2216yes—
51Q8×Dic151C2211yescharacteristic, complemented
52C4×Dic302C2213yescomplemented
53#18012704C2213yescomplemented
54C4×D604C2213yescomplemented
55D4×Dic151C2216yescomplemented
56C4×C22⋊D151C2213yescomplemented
57C4×C22⋊D151C2213yescomplemented
58(C22⋊D151)⋊C4C2216yescomplemented
59#18012697C2213yes—
60#18012696C2213yes—
61C2×#41×C151C2211yescharacteristic
62C2×#41×C151C2211yescharacteristic, complemented
63#127×C151C2213yescomplemented
64#121×C151C2213yescomplemented
65#121×C151C2213yescomplemented
66C4×Q8×C151C2211yescharacteristic, complemented
67C4×D4×C151C2213yescomplemented
68C23×D151C2313yes—
69#14559005—24no—
70#14559004—212no—
71#14559004—212no—
72#14559003—212no—
73#14559003—212no—
74#14559003—212no—
75C2×Dic302—212no—
76C2×Dic302C2313yes—
77C2×Dic302C2313yes—
78C2×Dic302C2313yes—
79Q8×D151—24no—
80Q8×D151D418yescomplemented
81C2×D604C2311yescharacteristic
82C2×D604C2313yes—
83C2×D604C2313yes—
84C2×C22⋊D151C2313yes—
85C2×C22⋊D151C2316yes—
86C2×C22⋊D151C2313yes—
87C2×C22⋊D151C2316yes—
88C2×C22⋊D151C2313yes—
89C2×C22⋊D151C2313yes—
90D4×D151—212no—
91C22×Dic151C2316yes—
92C22×Dic151C2313yes—
93C2×C4×D151—212no—
94C2×C4×D151C2313yes—
95C2×C4×D151C2313yes—
96C2×C4×D151C2313yes—
97C2×C4×D151C2311yescharacteristic
98C2×C4×D151C2313yes—
99C4⋊Dic151C2313yes—
100C4⋊Dic151C2313yes—
101Dic151⋊C4C2313yes—
102Dic151⋊C4C2313yes—
103Dic151⋊C4C2313yes—
104Dic151⋊C4C2311yescharacteristic
105C22⋊Dic151C2313yes—
106C22⋊Dic151C2313yes—
107C22⋊Dic151C2316yes—
108(C2×D151)⋊C4C2311yescharacteristic
109(C2×D151)⋊C4C2313yes—
110(C2×D151)⋊C4C2316yes—
111(C2×D151)⋊C4C2311yescharacteristic
112(C2×D151)⋊C4C2313yes—
113(C2×D151)⋊C4C2311yescharacteristic
114(C2×D151)⋊C4C2313yes—
115C4×Dic151C2313yes—
116C4×Dic151C2316yes—
117#41×C151—24no—
118#41×C151—24no—
119C2×Q8×C151C2311yescharacteristic
120C2×D4×C151C2313yes—
121C2×D4×C151C2313yes—
122C2×D4×C151C2316yes—
123C2×D4×C151C2313yes—
124C22×C4×C151C2313yes—
125C22×C4×C151C2313yes—
126C4⋊C4×C151C2313yes—
127C4⋊C4×C151C2311yescharacteristic
128C22⋊C4×C151C2313yes—
129C22⋊C4×C151C2313yes—
130C42×C151C2313yes—
131C22×D151—26no—
132C22×D151—43no—
133C22×D151—43no—
134C22×D151C2413yes—
135C22×D151C2411yescharacteristic
136C22×D151C2411yescharacteristic
137Dic302—26no—
138Dic302—26no—
139Dic302C2×D4112yes—
140D604—22no—
141D604—26no—
142D604—43no—
143C22⋊D151—26no—
144C22⋊D151—212no—
145C22⋊D151—26no—
146C22⋊D151—43no—
147C22⋊D151—46no—
148C22⋊D151—43no—
149C2×Dic151—26no—
150C2×Dic151—212no—
151C2×Dic151C2411yescharacteristic
152C2×Dic151C2413yes—
153C2×Dic151C2413yes—
154C2×Dic151C2411yescharacteristic
155C2×Dic151C2413yes—
156C4×D151—22no—
157C4×D151—26no—
158C4×D151—26no—
159C4×D151—26no—
160C4×D151C2×D4112yes—
161C23×C151C2413yes—
162C23×C151C2413yes—
163Q8×C151C2×D414yes—
164D4×C151—26no—
165D4×C151—26no—
166D4×C151—43no—
167D4×C151—46no—
168C2×C4×C151—26no—
169C2×C4×C151—26no—
170C2×C4×C151C2413yes—
171C2×C4×C151C2411yescharacteristic
172C2×C4×C151C2411yescharacteristic
173C2×C4×C151C2411yescharacteristic
174C2×C4×C151C2413yes—
175C22×C151—23no—
176C22×C151—23no—
177C22×C151—23no—
178C22×C151—23no—
179C22×C151—23no—
180C22×C151—23no—
181C22×C151C2511yescharacteristic
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—21no—
185C2×D151—23no—
186C2×D151—21no—
187C2×D151—23no—
188C2×D151—23no—
189C2×D151—21no—
190C2×D151—412no—
191C2×D151C22×D412yes—
192C4×C151—21no—
193C4×C151—21no—
194C4×C151—21no—
195C4×C151—23no—
196C4×C151C22×D416yes—
197Dic151—21no—
198Dic151—23no—
199Dic151—21no—
200Dic151—23no—
201Dic151C22×D416yes—
202C2×C151—43no—
203C2×C151—43no—
204C2×C151C23×D411yescharacteristic
205C2×C151#58511yescharacteristic
206C2×C151#58511yescharacteristic
207D151—22no—
208D151—41no—
209D151—43no—
210C151#348711yescharacteristic, complemented, Sylow
211#3487—1511nomaximal, Sylow
212C2×2−1+4—1511no—
213C22×#41—1511no—
214#550—1511no—
215#550—1513no—
216#550—1513no—
217#550D15111yescharacteristic, complemented
218#548—1511no—
219#548—1513no—
220#548—1511no—
221#548—1513no—
222#535—1513no—
223#535—1513no—
224#525—1513no—
225#525—1513no—
226#519—1511no—
2272−1+4—3028no—
228C2×#41—3028no—
229C2×#41—1511no—
230C2×#41—1513no—
231C2×#41—1513no—
232C2×#41—1513no—
233C2×#41—1513no—
234C2×#41—1513no—
235C2×#41C2×D15111yescharacteristic
236C2×#41C2×D15111yescharacteristic, complemented
237C22×Q8—1511no—
238C22×Q8—1511no—
239C22×Q8—1513no—
240C22×D4—1513no—
241C23×C4—1513no—
242#127—1516no—
243#127—1513no—
244#127C2×D15113yescomplemented
245#124—1513no—
246#124—1516no—
247#124—1513no—
248#123—1511no—
249#123—1513no—
250C22⋊Q8—1513no—
251C22⋊Q8—1513no—
252C22⋊Q8—1513no—
253C22⋊Q8—1513no—
254#121—1513no—
255#121—1513no—
256#121—1513no—
257#121—1516no—
258#121—1516no—
259#121—1513no—
260#121—1513no—
261#121—1513no—
262#121C2×D15113yescomplemented
263#121C2×D15113yescomplemented
264#120—1516no—
265#120—1513no—
266#120—1513no—
267C4×Q8—1511no—
268C4×Q8—1513no—
269C4×Q8—1513no—
270C4×Q8C2×D15111yescharacteristic, complemented
271C4×D4—1516no—
272C4×D4—1513no—
273C4×D4—1513no—
274C4×D4—1516no—
275C4×D4—1513no—
276C4×D4C2×D15113yescomplemented
277#117—1513no—
278#117—1513no—
279C24—1513no—
280#41—24no—
281#41—24no—
282#41—30212no—
283#41—30212no—
284#41—30212no—
285#41—30212no—
286#41—3024no—
287#41—30212no—
288C2×Q8—30212no—
289C2×Q8—3024no—
290C2×Q8—1518no—
291C2×Q8—1513no—
292C2×Q8—1513no—
293C2×Q8—1513no—
294C2×Q8C22×D15111yescharacteristic
295C2×D4—30212no—
296C2×D4—1513no—
297C2×D4—1516no—
298C2×D4—1513no—
299C2×D4—1516no—
300C2×D4—1513no—
301C2×D4—1513no—
302C2×D4—1511no—
303C2×D4—1513no—
304C2×D4—1513no—
305C2×D4C22×D15113yes—
306C2×D4C22×D15113yes—
307C2×D4C22×D15116yes—
308C2×D4C22×D15113yes—
309C22×C4—30212no—
310C22×C4—1516no—
311C22×C4—1513no—
312C22×C4—1513no—
313C22×C4—1513no—
314C22×C4—1513no—
315C22×C4—1511no—
316C22×C4—1513no—
317C22×C4C22×D15113yes—
318C22×C4C22×D15113yes—
319C4⋊C4—1513no—
320C4⋊C4—1513no—
321C4⋊C4—1513no—
322C4⋊C4—1511no—
323C4⋊C4—1513no—
324C4⋊C4—1513no—
325C4⋊C4C22×D15113yes—
326C4⋊C4C22×D15111yescharacteristic
327C22⋊C4—1513no—
328C22⋊C4—1513no—
329C22⋊C4—1516no—
330C22⋊C4—1513no—
331C22⋊C4—1511no—
332C22⋊C4—1511no—
333C22⋊C4—1511no—
334C22⋊C4—1513no—
335C22⋊C4—1516no—
336C22⋊C4—1513no—
337C22⋊C4C22×D15113yes—
338C22⋊C4C22×D15113yes—
339C42—1513no—
340C42—1516no—
341C42C22×D15113yes—
342C23—3026no—
343C23—6043no—
344C23—6043no—
345C23—1513no—
346C23—1511no—
347C23—1511no—
348C23C23×D15113yes—
349C23C23×D15113yes—
350Q8—3026no—
351Q8—3026no—
352Q8—15112no—
353Q8D4×D15114yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—3022no—
359D4—3026no—
360D4—30212no—
361D4—3026no—
362D4—3026no—
363D4—6043no—
364D4—6043no—
365D4—6046no—
366D4—6043no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—3022no—
370C2×C4—3026no—
371C2×C4—3026no—
372C2×C4—30212no—
373C2×C4—3026no—
374C2×C4—3026no—
375C2×C4—1511no—
376C2×C4—1513no—
377C2×C4—1513no—
378C2×C4—1511no—
379C2×C4—1513no—
380C2×C4—15112no—
381C2×C4C23×D15113yes—
382C2×C4C23×D15111yescharacteristic
383C2×C4C23×D15111yescharacteristic
384C2×C4C23×D15111yescharacteristic
385C2×C4C23×D15113yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—3021no—
393C22—3021no—
394C22—3021no—
395C22—3023no—
396C22—3021no—
397C22—3023no—
398C22—3021no—
399C22—3023no—
400C22—60412no—
401C22—1512no—
402C22C24×D15111yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—3021no—
408C4—3023no—
409C4—3021no—
410C4—3023no—
411C4—1516no—
412C4C2×D4×D15116yes—
413C2—43no—
414C2—43no—
415C2—3022no—
416C2—6043no—
417C2—6041no—
418C2C22×D4×D15111yescharacteristic
419C2#2375304711yescharacteristic
420C2#2375304811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
Fitting#550×C151C2
hypercenter#550D151
hypocenterC151#3487
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#18012786C2semidirect
2. C2×#41×D151C2semidirect
3. #23752882C2semidirect
4. #23752881C2semidirect
5. #23752880C2semidirect
6. #23752874C2semidirect
7. #23752871C2semidirect
8. #23752869C2semidirect
9. #23752868C2semidirect
10. #23752778C2semidirect
11. #23752775C2semidirect
12. #23752722C2semidirect
13. #23752721C2semidirect
14. #23752688C2semidirect
15. #550×C151C2semidirect
22. C2×#14559003C22semidirect
24. C22×Dic302C22semidirect
26. C2×Q8×D151C22semidirect
29. #18012735C22semidirect
30. #18012735C22semidirect
31. #18012727C22semidirect
32. #18012726C22semidirect
33. #18012725C22semidirect
36. #18012719C22semidirect
37. #18012718C22semidirect
38. #18012717C22semidirect
39. #18012716C22semidirect
40. #18012714C22semidirect
41. #18012713C22semidirect
42. #18012712C22semidirect
43. #18012711C22semidirect
44. #18012711C22semidirect
45. #18012711C22semidirect
46. #18012710C22semidirect
47. #18012710C22semidirect
51. Q8×Dic151C22semidirect
52. C4×Dic302C22semidirect
53. #18012704C22semidirect
54. C4×D604C22semidirect
55. D4×Dic151C22semidirect
56. C4×C22⋊D151C22semidirect
57. C4×C22⋊D151C22semidirect
58. (C22⋊D151)⋊C4C22semidirect
62. C2×#41×C151C22semidirect
63. #127×C151C22semidirect
64. #121×C151C22semidirect
65. #121×C151C22semidirect
66. C4×Q8×C151C22semidirect
67. C4×D4×C151C22semidirect
80. Q8×D151D4semidirect
210. C151#3487semidirect
217. #550D151semidirect
236. C2×#41C2×D151semidirect
244. #127C2×D151semidirect
262. #121C2×D151semidirect
263. #121C2×D151semidirect
270. C4×Q8C2×D151semidirect
276. C4×D4C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127862
C2×#41×D1512
#237528822
#237528812
#237528802
#237528742
#237528712
#237528692
#237528682
#237527782
#237527752
#237527222
#237527212
#237526882
#550×C1512
#3487151