Finite Group Databasebeta

← #34058676

Subgroups — #34058676

Order 19072.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750351C211yesmaximal, characteristic, complemented
2#23750345C213yesmaximal, complemented
3#23750330C213yesmaximal, complemented
4#23750325C213yesmaximal, complemented
5#23750310C213yesmaximal, complemented
6Q8×D596C211yesmaximal, characteristic, complemented
7#23750282C211yesmaximal, characteristic, complemented
8#23750282C211yesmaximal, characteristic, complemented
9#23750240C213yesmaximal, complemented
10#23750237C213yesmaximal, complemented
11#23750157C213yesmaximal, complemented
12#23750153C213yesmaximal, complemented
13#23750101C211yesmaximal, characteristic, complemented
14C4×Q8×D149C211yesmaximal, characteristic, complemented
15#520×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18#18011911C2213yes—
19#18011910C2213yes—
20#18011906C2213yes—
21#18011904C2213yes—
22#18011904C2216yes—
23#18011904C2216yes—
24#18011903C2213yes—
25#18011902C2213yes—
26#18011901C2213yes—
27#18011901C2216yes—
28#18011898C2213yes—
29#18011898C2213yes—
30#18011897C2213yes—
31#18011896C2211yescharacteristic
32#18011894C2216yes—
33#18011893C2213yes—
34#18011891C2213yes—
35#18011891C2213yes—
36#18011890C2213yes—
37#18011890C2213yes—
38#18011890C2216yes—
39#18011889C2213yes—
40#18011884C2213yes—
41#18011882C2213yes—
42#18011882C2213yes—
43Q8×Dic149C2211yescharacteristic
44Q8×Dic149C2211yescharacteristic
45C4×Dic298C2213yes—
46C4×Dic298C2213yes—
47C4×Dic298C2213yes—
48#18011876C2213yes—
49#18011876C2213yes—
50C4×D596C2213yes—
51C4×C22⋊D149C2213yes—
52C4×C22⋊D149C2213yes—
53C4×C22⋊D149C2216yes—
54#18011870C2216yes—
55#18011869C2213yes—
56#18011868C2213yes—
57C4⋊C4×D149C2213yes—
58C42×D149C2213yes—
59C2×#41×C149C2211yescharacteristic
60C4×Q8×C149C2211yescharacteristic
61C4×Q8×C149C2211yescharacteristic
62C4×D4×C149C2213yes—
63C4×D4×C149C2213yes—
64#117×C149C2213yes—
65C2×C4⋊C4×C149C2213yes—
66C2×Dic298C2313yes—
67C2×Dic298C2313yes—
68C2×Dic298C2313yes—
69Q8×D149—24no—
70Q8×D149—24no—
71C2×D596C2311yescharacteristic
72C2×C22⋊D149C2313yes—
73C2×C22⋊D149C2313yes—
74C2×C4×D149C2313yes—
75C2×C4×D149C2313yes—
76C2×C4×D149C2311yescharacteristic
77C2×C4×D149C2313yes—
78C4⋊Dic149C2313yes—
79C4⋊Dic149C2316yes—
80C4⋊Dic149C2313yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2313yes—
83#14558715C2311yescharacteristic
84#14558715C2313yes—
85#14558715C2313yes—
86#14558715C2316yes—
87#14558715C2316yes—
88#14558715C2313yes—
89#14558715C2313yes—
90#14558715C2313yes—
91C22⋊Dic149C2316yes—
92C22⋊Dic149C2316yes—
93C22⋊Dic149C2313yes—
94C22⋊Dic149C2313yes—
95#14558713C2311yescharacteristic
96#14558713C2313yes—
97#14558713C2313yes—
98#14558713C2311yescharacteristic
99#14558713C2311yescharacteristic
100#14558713C2313yes—
101C4×Dic149C2313yes—
102C4×Dic149C2313yes—
103C4×Dic149C2316yes—
104C4×Dic149C2313yes—
105C4×Dic149C2313yes—
106#41×C149—24no—
107C2×Q8×C149C2311yescharacteristic
108C2×D4×C149C2313yes—
109C22×C4×C149C2313yes—
110C22×C4×C149C2316yes—
111C4⋊C4×C149C2313yes—
112C4⋊C4×C149C2313yes—
113C4⋊C4×C149C2313yes—
114C4⋊C4×C149C2311yescharacteristic
115C22⋊C4×C149C2313yes—
116C22⋊C4×C149C2313yes—
117C42×C149C2313yes—
118C42×C149C2313yes—
119C22×D149C2411yescharacteristic
120C22×D149C2411yescharacteristic
121Dic298—26no—
122Dic298—26no—
123Dic298—43no—
124D596—22no—
125C22⋊D149—26no—
126C22⋊D149—43no—
127C2×Dic149C2413yes—
128C2×Dic149C2413yes—
129C2×Dic149C2413yes—
130C2×Dic149C2411yescharacteristic
131C2×Dic149C2413yes—
132C2×Dic149C2411yescharacteristic
133C4×D149—22no—
134C4×D149—26no—
135C4×D149—26no—
136C4×D149—43no—
137C23×C149C2413yes—
138Q8×C149#4114yes—
139D4×C149—26no—
140C2×C4×C149—26no—
141C2×C4×C149—46no—
142C2×C4×C149C2411yescharacteristic
143C2×C4×C149C2413yes—
144C2×C4×C149C2411yescharacteristic
145C2×C4×C149C2413yes—
146C2×C4×C149C2411yescharacteristic
147C2×C4×C149C2413yes—
148C22×C149—23no—
149C22×C149—23no—
150C22×C149—23no—
151C22×C149C2511yescharacteristic
152C2×D149—21no—
153C2×D149—21no—
154C2×D149—21no—
155C2×D149—21no—
156C2×D149—21no—
157C2×D1492−1+412yes—
158C4×C149—21no—
159C4×C149—21no—
160C4×C149—21no—
161C4×C149—23no—
162C4×C149—23no—
163C4×C149C2×#4116yes—
164Dic149—23no—
165Dic149—23no—
166Dic149—21no—
167Dic149—23no—
168Dic149—23no—
169Dic149—21no—
170C2×C149—43no—
171C2×C149C22×#4111yescharacteristic
172C2×C149C2×2−1+411yescharacteristic
173C2×C149#58511yescharacteristic
174D149—22no—
175D149—41no—
176C149#351611yescharacteristic, complemented, Sylow
177#3516—1491nomaximal, Sylow
178#557—1491no—
179#556—1493no—
180#554—1493no—
181#553—1493no—
182#552—1493no—
183D4×Q8—1491no—
184#548—1491no—
185#548—1491no—
186#542—1493no—
187#541—1493no—
188#529—1493no—
189#529—1493no—
190#520D14911yescharacteristic, complemented
191#519—1491no—
192C2×C4×Q8—1491no—
193C2×#41C2×D14911yescharacteristic
194C22×Q8—1491no—
195C22×Q8—1491no—
196C4⋊Q8—1493no—
197C4⋊Q8—1493no—
198#126—1493no—
199#126—1496no—
200#126—1493no—
201#126—1496no—
202#125—1493no—
203#125—1493no—
204#125—1496no—
205#125—1493no—
206#124—1493no—
207#124—1493no—
208#124—1493no—
209#123—1496no—
210#123—1491no—
211#123—1493no—
212C22⋊Q8—1493no—
213C22⋊Q8—1493no—
214C22⋊Q8—1496no—
215C22⋊Q8—1493no—
216C22⋊Q8—1493no—
217C22⋊Q8—1493no—
218#121—1493no—
219#121—1493no—
220#121—1493no—
221C4×Q8—1491no—
222C4×Q8—1491no—
223C4×Q8—1493no—
224C4×Q8—1493no—
225C4×Q8—1493no—
226C4×Q8—1493no—
227C4×Q8—1493no—
228C4×Q8C2×D14911yescharacteristic
229C4×Q8C2×D14911yescharacteristic
230C4×D4—1493no—
231C4×D4—1493no—
232C4×D4—1496no—
233C4×D4—1493no—
234C4×D4C2×D14913yes—
235C4×D4C2×D14913yes—
236#117—1496no—
237#117—1493no—
238#117—1493no—
239#117C2×D14913yes—
240C2×C4⋊C4—1493no—
241C2×C4⋊C4C2×D14913yes—
242C2×C42—1493no—
243#41—24no—
244C2×Q8—2984no—
245C2×Q8—2984no—
246C2×Q8—1493no—
247C2×Q8—1493no—
248C2×Q8—1493no—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—1493no—
251C2×D4—1493no—
252C2×D4—1491no—
253C2×D4C22×D14913yes—
254C22×C4—1491no—
255C22×C4—1493no—
256C22×C4—1493no—
257C22×C4—1493no—
258C22×C4C22×D14913yes—
259C22×C4C22×D14916yes—
260C4⋊C4—1493no—
261C4⋊C4—1496no—
262C4⋊C4—1493no—
263C4⋊C4—1493no—
264C4⋊C4—1493no—
265C4⋊C4—1493no—
266C4⋊C4—1496no—
267C4⋊C4—1493no—
268C4⋊C4—1493no—
269C4⋊C4—1496no—
270C4⋊C4—1493no—
271C4⋊C4—1491no—
272C4⋊C4—1493no—
273C4⋊C4C22×D14913yes—
274C4⋊C4C22×D14913yes—
275C4⋊C4C22×D14913yes—
276C4⋊C4C22×D14911yescharacteristic
277C22⋊C4—1493no—
278C22⋊C4—1496no—
279C22⋊C4—1496no—
280C22⋊C4—1493no—
281C22⋊C4—1493no—
282C22⋊C4—1491no—
283C22⋊C4—1491no—
284C22⋊C4—1493no—
285C22⋊C4—1491no—
286C22⋊C4—1493no—
287C22⋊C4C22×D14913yes—
288C22⋊C4C22×D14913yes—
289C42—1493no—
290C42—1493no—
291C42—1493no—
292C42—1496no—
293C42—1493no—
294C42C22×D14913yes—
295C42C22×D14913yes—
296C23—1491no—
297C23—1491no—
298C23C23×D14913yes—
299Q8—2986no—
300Q8—2986no—
301Q8—5963no—
302Q8#1455873614yes—
303D4—26no—
304D4—2982no—
305D4—2986no—
306D4—5963no—
307C2×C4—26no—
308C2×C4—46no—
309C2×C4—2982no—
310C2×C4—2986no—
311C2×C4—2986no—
312C2×C4—5963no—
313C2×C4—1491no—
314C2×C4—1491no—
315C2×C4—1493no—
316C2×C4—1493no—
317C2×C4—1493no—
318C2×C4—1493no—
319C2×C4C23×D14911yescharacteristic
320C2×C4C23×D14913yes—
321C2×C4C23×D14911yescharacteristic
322C2×C4C23×D14913yes—
323C2×C4C23×D14911yescharacteristic
324C2×C4C23×D14913yes—
325C22—23no—
326C22—23no—
327C22—23no—
328C22—2981no—
329C22—2981no—
330C22—2981no—
331C22—2981no—
332C22—2981no—
333C22—1492no—
334C22C24×D14911yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—2983no—
341C4—2983no—
342C4—2981no—
343C4—2983no—
344C4—2983no—
345C4—2981no—
346C4C2×#1455873616yes—
347C2—43no—
348C2—2982no—
349C2—5961no—
350C2C22×#1455873611yescharacteristic
351C22−1+4×D14911yescharacteristic
352C2#2375046011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#520×C149C2
hypercenter#520D149
hypocenterC149#3516
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23750351C2semidirect
2. #23750345C2semidirect
3. #23750330C2semidirect
4. #23750325C2semidirect
5. #23750310C2semidirect
6. Q8×D596C2semidirect
7. #23750282C2semidirect
8. #23750282C2semidirect
9. #23750240C2semidirect
10. #23750237C2semidirect
11. #23750157C2semidirect
12. #23750153C2semidirect
13. #23750101C2semidirect
14. C4×Q8×D149C2semidirect
15. #520×C149C2semidirect
176. C149#3516semidirect
190. #520D149semidirect

Maximal subgroups

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

subgroupindex
#237503512
#237503452
#237503302
#237503252
#237503102
Q8×D5962
#237502822
#237502402
#237502372
#237501572
#237501532
#237501012
C4×Q8×D1492
#520×C1492
#3516149