Finite Group Databasebeta

← #34058789

Subgroups — #34058789

Order 19072.

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#23750361C213yesmaximal, complemented
2#23750351C211yesmaximal, characteristic, complemented
3#23750339C213yesmaximal, complemented
4#23750327C213yesmaximal, complemented
5#23750315C213yesmaximal, complemented
6#23750282C211yesmaximal, characteristic, complemented
7#23750282C211yesmaximal, characteristic, complemented
8#23750242C213yesmaximal, complemented
9#23750229C213yesmaximal, complemented
10#23750156C213yesmaximal, complemented
11#23750155C213yesmaximal, complemented
12#23750108C211yesmaximal, characteristic, complemented
13#23750101C211yesmaximal, characteristic, complemented
14#41×Dic149C211yesmaximal, characteristic, complemented
15#550×C149C211yesmaximal, characteristic, complemented
16C2×Q8×D149C2211yescharacteristic
17#18011910C2213yes—
18#18011906C2211yescharacteristic
19#18011904C2213yes—
20#18011904C2216yes—
21#18011903C2213yes—
22#18011902C2213yes—
23#18011901C2213yes—
24#18011898C2213yes—
25#18011898C2213yes—
26#18011898C2216yes—
27#18011897C2213yes—
28#18011895C2213yes—
29#18011892C2213yes—
30#18011892C2216yes—
31#18011892C2216yes—
32#18011890C2213yes—
33#18011890C2213yes—
34#18011888C2213yes—
35#18011883C2213yes—
36#18011883C2216yes—
37#18011882C2213yes—
38#18011882C2213yes—
39Q8×Dic149C2211yescharacteristic
40Q8×Dic149C2211yescharacteristic
41Q8×Dic149C2211yescharacteristic
42C4×Dic298C2213yes—
43C4×Dic298C2213yes—
44#18011876C2213yes—
45D4×Dic149C2213yes—
46D4×Dic149C2216yes—
47D4×Dic149C2213yes—
48D4×Dic149C2213yes—
49D4×Dic149C2213yes—
50C4×C22⋊D149C2213yes—
51C4×C22⋊D149C2213yes—
52#18011870C2213yes—
53#18011870C2213yes—
54#18011869C2213yes—
55#18011868C2213yes—
56#18011867C2216yes—
57C2×C4⋊Dic149C2213yes—
58C2×C4×Dic149C2213yes—
59C2×#41×C149C2211yescharacteristic
60C2×#41×C149C2211yescharacteristic
61#127×C149C2213yes—
62#121×C149C2213yes—
63#121×C149C2213yes—
64C4×Q8×C149C2211yescharacteristic
65C4×D4×C149C2213yes—
66C2×Dic298C2313yes—
67C2×Dic298C2313yes—
68Q8×D149—24no—
69C2×C22⋊D149C2313yes—
70C2×C22⋊D149C2313yes—
71C22×Dic149C2313yes—
72C22×Dic149C2313yes—
73C22×Dic149C2316yes—
74C2×C4×D149C2313yes—
75C4⋊Dic149C2313yes—
76C4⋊Dic149C2313yes—
77C4⋊Dic149C2311yescharacteristic
78C4⋊Dic149C2313yes—
79C4⋊Dic149C2313yes—
80C4⋊Dic149C2313yes—
81#14558715C2313yes—
82#14558715C2316yes—
83#14558715C2313yes—
84#14558715C2313yes—
85#14558715C2311yescharacteristic
86#14558715C2313yes—
87#14558715C2313yes—
88C22⋊Dic149C2313yes—
89C22⋊Dic149C2316yes—
90C22⋊Dic149C2313yes—
91C22⋊Dic149C2316yes—
92C22⋊Dic149C2316yes—
93C22⋊Dic149C2313yes—
94C22⋊Dic149C2313yes—
95#14558713C2311yescharacteristic
96#14558713C2311yescharacteristic
97#14558713C2311yescharacteristic
98#14558713C2313yes—
99C4×Dic149C2311yescharacteristic
100C4×Dic149C2316yes—
101C4×Dic149C2313yes—
102C4×Dic149C2313yes—
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105#41×C149—24no—
106#41×C149—24no—
107C2×Q8×C149C2311yescharacteristic
108C2×D4×C149C2313yes—
109C2×D4×C149C2313yes—
110C2×D4×C149C2313yes—
111C2×D4×C149C2316yes—
112C22×C4×C149C2313yes—
113C22×C4×C149C2313yes—
114C4⋊C4×C149C2311yescharacteristic
115C4⋊C4×C149C2313yes—
116C22⋊C4×C149C2313yes—
117C22⋊C4×C149C2313yes—
118C42×C149C2313yes—
119C22×D149C2411yescharacteristic
120Dic298—26no—
121Dic298—43no—
122C22⋊D149—43no—
123C22⋊D149—43no—
124C2×Dic149—43no—
125C2×Dic149—46no—
126C2×Dic149—43no—
127C2×Dic149C2413yes—
128C2×Dic149C2411yescharacteristic
129C2×Dic149C2411yescharacteristic
130C2×Dic149C2413yes—
131C2×Dic149C2413yes—
132C2×Dic149C2413yes—
133C2×Dic149C2411yescharacteristic
134C4×D149—26no—
135C23×C149C2413yes—
136C23×C149C2413yes—
137Q8×C149#4114yes—
138D4×C149—26no—
139D4×C149—26no—
140D4×C149—43no—
141D4×C149—46no—
142C2×C4×C149—26no—
143C2×C4×C149—26no—
144C2×C4×C149C2413yes—
145C2×C4×C149C2411yescharacteristic
146C2×C4×C149C2411yescharacteristic
147C2×C4×C149C2411yescharacteristic
148C2×C4×C149C2413yes—
149C22×C149—23no—
150C22×C149—23no—
151C22×C149—23no—
152C22×C149—23no—
153C22×C149—23no—
154C22×C149—23no—
155C22×C149C2511yescharacteristic
156C2×D149—21no—
157C2×D149—21no—
158C2×D149—21no—
159C4×C149—21no—
160C4×C149—21no—
161C4×C149—21no—
162C4×C149—23no—
163C4×C149C2×#4116yes—
164Dic149—23no—
165Dic149—21no—
166Dic149—21no—
167Dic149—23no—
168Dic149—23no—
169Dic149—23no—
170Dic149—21no—
171C2×C149—43no—
172C2×C149—43no—
173C2×C149C22×#4111yescharacteristic
174C2×C149#58511yescharacteristic
175C2×C149#58511yescharacteristic
176D149—41no—
177C149#352411yescharacteristic, complemented, Sylow
178#3524—1491nomaximal, Sylow
179#559—1493no—
180#557—1491no—
181#555—1493no—
182#553—1493no—
183#552—1493no—
184#550D14911yescharacteristic, complemented
185#548—1491no—
186#548—1491no—
187#542—1493no—
188#540—1493no—
189#529—1493no—
190#529—1493no—
191#520—1491no—
192#519—1491no—
193C4×#41—1491no—
194C2×#41C2×D14911yescharacteristic
195C2×#41C2×D14911yescharacteristic
196C22×Q8—1491no—
197C4⋊Q8—1493no—
198#127C2×D14913yes—
199#126—1491no—
200#126—1493no—
201#126—1496no—
202#125—1493no—
203#125—1493no—
204#125—1493no—
205#124—1493no—
206#124—1493no—
207#124—1496no—
208#124—1493no—
209#123—1496no—
210#123—1493no—
211#123—1496no—
212#123—1493no—
213C22⋊Q8—1493no—
214C22⋊Q8—1493no—
215C22⋊Q8—1493no—
216#121—1493no—
217#121—1493no—
218#121—1493no—
219#121—1496no—
220#121C2×D14913yes—
221#121C2×D14913yes—
222C4×Q8—1491no—
223C4×Q8—1491no—
224C4×Q8—1491no—
225C4×Q8—1493no—
226C4×Q8—1493no—
227C4×Q8—1493no—
228C4×Q8C2×D14911yescharacteristic
229C4×D4—1493no—
230C4×D4—1493no—
231C4×D4—1493no—
232C4×D4—1493no—
233C4×D4—1496no—
234C4×D4—1493no—
235C4×D4—1493no—
236C4×D4C2×D14913yes—
237#117—1493no—
238#117—1493no—
239#117—1493no—
240#117—1496no—
241#117—1493no—
242C2×C4⋊C4—1493no—
243C2×C42—1493no—
244#41—24no—
245#41—24no—
246C2×Q8—2984no—
247C2×Q8—1493no—
248C2×Q8—1493no—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—1493no—
251C2×D4—1493no—
252C2×D4C22×D14913yes—
253C2×D4C22×D14913yes—
254C2×D4C22×D14916yes—
255C2×D4C22×D14913yes—
256C22×C4—1493no—
257C22×C4—1493no—
258C22×C4—1493no—
259C22×C4—1496no—
260C22×C4C22×D14913yes—
261C22×C4C22×D14913yes—
262C4⋊C4—1493no—
263C4⋊C4—1493no—
264C4⋊C4—1493no—
265C4⋊C4—1491no—
266C4⋊C4—1493no—
267C4⋊C4—1493no—
268C4⋊C4—1493no—
269C4⋊C4—1493no—
270C4⋊C4—1496no—
271C4⋊C4—1493no—
272C4⋊C4—1493no—
273C4⋊C4—1491no—
274C4⋊C4—1493no—
275C4⋊C4C22×D14913yes—
276C4⋊C4C22×D14911yescharacteristic
277C22⋊C4—1493no—
278C22⋊C4—1496no—
279C22⋊C4—1493no—
280C22⋊C4—1496no—
281C22⋊C4—1493no—
282C22⋊C4—1496no—
283C22⋊C4—1493no—
284C22⋊C4—1491no—
285C22⋊C4—1493no—
286C22⋊C4—1491no—
287C22⋊C4—1491no—
288C22⋊C4C22×D14913yes—
289C22⋊C4C22×D14913yes—
290C42—1491no—
291C42—1493no—
292C42—1493no—
293C42—1493no—
294C42—1496no—
295C42—1493no—
296C42C22×D14913yes—
297C23—1491no—
298C23C23×D14913yes—
299C23C23×D14913yes—
300Q8—2986no—
301Q8—5963no—
302Q8#1455873514yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
307D4—5963no—
308D4—5963no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—2986no—
312C2×C4—5963no—
313C2×C4—5963no—
314C2×C4—5966no—
315C2×C4—1491no—
316C2×C4—1491no—
317C2×C4—1493no—
318C2×C4—1491no—
319C2×C4—1493no—
320C2×C4—1493no—
321C2×C4—1493no—
322C2×C4C23×D14911yescharacteristic
323C2×C4C23×D14913yes—
324C2×C4C23×D14911yescharacteristic
325C2×C4C23×D14911yescharacteristic
326C2×C4C23×D14913yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—2981no—
334C22—2981no—
335C22—2981no—
336C22C24×D14911yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—2983no—
342C4—2981no—
343C4—2983no—
344C4—2981no—
345C4—2981no—
346C4—2983no—
347C4—2983no—
348C4C2×#1455873516yes—
349C2—43no—
350C2—43no—
351C2—5961no—
352C2C22×#1455873511yescharacteristic
353C2#2375046011yescharacteristic
354C2#2375046111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#550×C149C2
hypercenter#550D149
hypocenterC149#3524
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #23750361C2semidirect
2. #23750351C2semidirect
3. #23750339C2semidirect
4. #23750327C2semidirect
5. #23750315C2semidirect
6. #23750282C2semidirect
7. #23750282C2semidirect
8. #23750242C2semidirect
9. #23750229C2semidirect
10. #23750156C2semidirect
11. #23750155C2semidirect
12. #23750108C2semidirect
13. #23750101C2semidirect
14. #41×Dic149C2semidirect
15. #550×C149C2semidirect
177. C149#3524semidirect
184. #550D149semidirect

Maximal subgroups

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

subgroupindex
#237503612
#237503512
#237503392
#237503272
#237503152
#237502822
#237502422
#237502292
#237501562
#237501552
#237501082
#237501012
#41×Dic1492
#550×C1492
#3524149