Finite Group Databasebeta

← #34083211

Subgroups — #34083211

Order 19328.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752945C213yesmaximal, complemented
2#23752941C211yesmaximal, characteristic, complemented
3#23752927C213yesmaximal, complemented
4#23752910C213yesmaximal, complemented
5#23752897C213yesmaximal, complemented
6#23752881C211yesmaximal, characteristic, complemented
7#23752874C211yesmaximal, characteristic, complemented
8#23752869C211yesmaximal, characteristic, complemented
9#23752833C213yesmaximal, complemented
10#23752813C213yesmaximal, complemented
11#23752744C213yesmaximal, complemented
12#23752740C213yesmaximal, complemented
13#23752688C211yesmaximal, characteristic, complemented
14C4×#14559005C211yesmaximal, characteristic, complemented
15#520×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×Q8×D151C2211yescharacteristic
18#18012739C2213yes—
19#18012735C2213yes—
20#18012734C2211yescharacteristic
21#18012734C2216yes—
22#18012732C2213yes—
23#18012730C2213yes—
24#18012730C2213yes—
25#18012729C2213yes—
26#18012727C2213yes—
27#18012726C2213yes—
28#18012725C2213yes—
29#18012725C2216yes—
30#18012724C2213yes—
31#18012722C2216yes—
32#18012721C2213yes—
33#18012721C2216yes—
34#18012719C2213yes—
35#18012718C2213yes—
36#18012717C2213yes—
37#18012714C2213yes—
38#18012713C2216yes—
39#18012713C2213yes—
40#18012712C2213yes—
41#18012710C2213yes—
42#18012710C2213yes—
43Q8×Dic151C2211yescharacteristic
44Q8×Dic151C2211yescharacteristic
45C4×Dic302C2213yes—
46C4×Dic302C2213yes—
47#18012704C2213yes—
48C4×D604C2213yes—
49C4×D604C2213yes—
50#18012701C2213yes—
51C4×C22⋊D151C2213yes—
52C4×C22⋊D151C2213yes—
53C4×C22⋊D151C2216yes—
54#18012698C2216yes—
55#18012697C2213yes—
56#18012697C2213yes—
57#18012696C2213yes—
58C42×D151C2213yes—
59C2×#41×C151C2211yescharacteristic
60C4×Q8×C151C2211yescharacteristic
61C4×Q8×C151C2211yescharacteristic
62C4×D4×C151C2213yes—
63C4×D4×C151C2213yes—
64#117×C151C2213yes—
65C2×C4⋊C4×C151C2213yes—
66#14559005—24no—
67C2×Dic302C2313yes—
68C2×Dic302C2313yes—
69Q8×D151—24no—
70C2×D604C2313yes—
71C2×D604C2313yes—
72C2×C22⋊D151C2313yes—
73C2×C22⋊D151C2316yes—
74C2×C22⋊D151C2313yes—
75C2×C4×D151C2313yes—
76C2×C4×D151C2313yes—
77C2×C4×D151C2313yes—
78C2×C4×D151C2313yes—
79C4⋊Dic151C2313yes—
80C4⋊Dic151C2313yes—
81C4⋊Dic151C2313yes—
82C4⋊Dic151C2316yes—
83C4⋊Dic151C2311yescharacteristic
84Dic151⋊C4C2313yes—
85Dic151⋊C4C2313yes—
86Dic151⋊C4C2311yescharacteristic
87Dic151⋊C4C2313yes—
88Dic151⋊C4C2313yes—
89Dic151⋊C4C2313yes—
90C22⋊Dic151C2313yes—
91C22⋊Dic151C2316yes—
92C22⋊Dic151C2313yes—
93(C2×D151)⋊C4C2311yescharacteristic
94(C2×D151)⋊C4C2313yes—
95(C2×D151)⋊C4C2316yes—
96(C2×D151)⋊C4C2311yescharacteristic
97(C2×D151)⋊C4C2313yes—
98(C2×D151)⋊C4C2316yes—
99(C2×D151)⋊C4C2311yescharacteristic
100(C2×D151)⋊C4C2313yes—
101C4×Dic151C2311yescharacteristic
102C4×Dic151C2313yes—
103C4×Dic151C2313yes—
104C4×Dic151C2313yes—
105C4×Dic151C2316yes—
106#41×C151—24no—
107C2×Q8×C151C2311yescharacteristic
108C2×D4×C151C2313yes—
109C22×C4×C151C2313yes—
110C22×C4×C151C2316yes—
111C4⋊C4×C151C2313yes—
112C4⋊C4×C151C2313yes—
113C4⋊C4×C151C2311yescharacteristic
114C4⋊C4×C151C2313yes—
115C22⋊C4×C151C2313yes—
116C22⋊C4×C151C2313yes—
117C42×C151C2313yes—
118C42×C151C2313yes—
119C22×D151C2411yescharacteristic
120C22×D151C2413yes—
121Dic302—26no—
122Dic302—43no—
123D604—26no—
124D604—43no—
125C22⋊D151—43no—
126C22⋊D151—43no—
127C22⋊D151—46no—
128C2×Dic151C2411yescharacteristic
129C2×Dic151C2411yescharacteristic
130C2×Dic151C2413yes—
131C2×Dic151C2411yescharacteristic
132C2×Dic151C2413yes—
133C2×Dic151C2413yes—
134C4×D151—26no—
135C4×D151—26no—
136C4×D151—43no—
137C4×D151—43no—
138C23×C151C2413yes—
139Q8×C151#4114yes—
140D4×C151—26no—
141C2×C4×C151—26no—
142C2×C4×C151—46no—
143C2×C4×C151C2413yes—
144C2×C4×C151C2411yescharacteristic
145C2×C4×C151C2411yescharacteristic
146C2×C4×C151C2411yescharacteristic
147C2×C4×C151C2413yes—
148C2×C4×C151C2413yes—
149C22×C151—23no—
150C22×C151—23no—
151C22×C151—23no—
152C22×C151C2511yescharacteristic
153C2×D151—21no—
154C2×D151—23no—
155C2×D151—23no—
156C2×D151—23no—
157C2×D151—21no—
158C2×D151—21no—
159C4×C151—21no—
160C4×C151—23no—
161C4×C151—23no—
162C4×C151—21no—
163C4×C151—21no—
164C4×C151C2×#4116yes—
165Dic151—21no—
166Dic151—23no—
167Dic151—21no—
168Dic151—23no—
169Dic151—21no—
170Dic151—23no—
171C2×C151—43no—
172C2×C151C22×#4111yescharacteristic
173C2×C151#58511yescharacteristic
174C2×C151#58511yescharacteristic
175D151—41no—
176D151—43no—
177C151#352411yescharacteristic, complemented, Sylow
178#3524—1511nomaximal, Sylow
179#559—1513no—
180#557—1511no—
181#555—1513no—
182#553—1513no—
183#552—1513no—
184#550—1511no—
185#548—1511no—
186#548—1511no—
187#542—1513no—
188#540—1513no—
189#529—1513no—
190#529—1513no—
191#520D15111yescharacteristic, complemented
192#519—1511no—
193C4×#41—1511no—
194C2×#41—1511no—
195C2×#41C2×D15111yescharacteristic
196C22×Q8—1511no—
197C4⋊Q8—1513no—
198#127—1513no—
199#126—1511no—
200#126—1516no—
201#126—1513no—
202#125—1513no—
203#125—1513no—
204#125—1513no—
205#124—1513no—
206#124—1513no—
207#124—1516no—
208#124—1513no—
209#123—1516no—
210#123—1513no—
211#123—1516no—
212#123—1513no—
213C22⋊Q8—1513no—
214C22⋊Q8—1513no—
215C22⋊Q8—1513no—
216#121—1513no—
217#121—1513no—
218#121—1516no—
219#121—1513no—
220#121—1513no—
221#121—1513no—
222C4×Q8—1511no—
223C4×Q8—1511no—
224C4×Q8—1513no—
225C4×Q8—1513no—
226C4×Q8—1513no—
227C4×Q8C2×D15111yescharacteristic
228C4×Q8C2×D15111yescharacteristic
229C4×D4—1513no—
230C4×D4—1513no—
231C4×D4—1516no—
232C4×D4—1513no—
233C4×D4—1513no—
234C4×D4—1513no—
235C4×D4C2×D15113yes—
236C4×D4C2×D15113yes—
237#117—1516no—
238#117—1513no—
239#117—1513no—
240#117—1513no—
241#117C2×D15113yes—
242C2×C4⋊C4C2×D15113yes—
243C2×C42—1513no—
244#41—24no—
245#41—3024no—
246C2×Q8—3024no—
247C2×Q8—1513no—
248C2×Q8—1513no—
249C2×Q8C22×D15111yescharacteristic
250C2×D4—1513no—
251C2×D4—1516no—
252C2×D4—1513no—
253C2×D4—1513no—
254C2×D4—1513no—
255C2×D4C22×D15113yes—
256C22×C4—1513no—
257C22×C4—1513no—
258C22×C4—1513no—
259C22×C4—1513no—
260C22×C4C22×D15113yes—
261C22×C4C22×D15116yes—
262C4⋊C4—1513no—
263C4⋊C4—1513no—
264C4⋊C4—1511no—
265C4⋊C4—1513no—
266C4⋊C4—1513no—
267C4⋊C4—1513no—
268C4⋊C4—1513no—
269C4⋊C4—1513no—
270C4⋊C4—1513no—
271C4⋊C4—1516no—
272C4⋊C4—1511no—
273C4⋊C4C22×D15113yes—
274C4⋊C4C22×D15113yes—
275C4⋊C4C22×D15113yes—
276C4⋊C4C22×D15111yescharacteristic
277C22⋊C4—1513no—
278C22⋊C4—1516no—
279C22⋊C4—1513no—
280C22⋊C4—1511no—
281C22⋊C4—1513no—
282C22⋊C4—1516no—
283C22⋊C4—1511no—
284C22⋊C4—1511no—
285C22⋊C4—1513no—
286C22⋊C4—1516no—
287C22⋊C4—1513no—
288C22⋊C4C22×D15113yes—
289C22⋊C4C22×D15113yes—
290C42—1511no—
291C42—1513no—
292C42—1513no—
293C42—1516no—
294C42—1513no—
295C42C22×D15113yes—
296C42C22×D15113yes—
297C23—1511no—
298C23—1513no—
299C23C23×D15113yes—
300Q8—3026no—
301Q8—6043no—
302Q8#1455900414yes—
303D4—26no—
304D4—3026no—
305D4—6043no—
306D4—6043no—
307D4—6043no—
308D4—6046no—
309C2×C4—26no—
310C2×C4—46no—
311C2×C4—3026no—
312C2×C4—3026no—
313C2×C4—6043no—
314C2×C4—6043no—
315C2×C4—1511no—
316C2×C4—1511no—
317C2×C4—1513no—
318C2×C4—1511no—
319C2×C4—1513no—
320C2×C4—1513no—
321C2×C4C23×D15113yes—
322C2×C4C23×D15111yescharacteristic
323C2×C4C23×D15111yescharacteristic
324C2×C4C23×D15113yes—
325C2×C4C23×D15111yescharacteristic
326C2×C4C23×D15113yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—3021no—
331C22—3023no—
332C22—3023no—
333C22—3021no—
334C22—3023no—
335C22—3021no—
336C22C24×D15111yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—21no—
342C4—3021no—
343C4—3023no—
344C4—3021no—
345C4—3021no—
346C4—3023no—
347C4—3023no—
348C4C2×#1455900416yes—
349C2—43no—
350C2—6041no—
351C2—6043no—
352C2C22×#1455900411yescharacteristic
353C2#2375304711yescharacteristic
354C2#2375304911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
Fitting#520×C151C2
hypercenter#520D151
hypocenterC151#3524
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23752945C2semidirect
2. #23752941C2semidirect
3. #23752927C2semidirect
4. #23752910C2semidirect
5. #23752897C2semidirect
6. #23752881C2semidirect
7. #23752874C2semidirect
8. #23752869C2semidirect
9. #23752833C2semidirect
10. #23752813C2semidirect
11. #23752744C2semidirect
12. #23752740C2semidirect
13. #23752688C2semidirect
14. C4×#14559005C2semidirect
15. #520×C151C2semidirect
177. C151#3524semidirect
191. #520D151semidirect

Maximal subgroups

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

subgroupindex
#237529452
#237529412
#237529272
#237529102
#237528972
#237528812
#237528742
#237528692
#237528332
#237528132
#237527442
#237527402
#237526882
C4×#145590052
#520×C1512
#3524151