Finite Group Databasebeta

← #34047321

Subgroups — #34047321

Order 19072.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011871C211yesmaximal, characteristic
2#23749518C211yesmaximal, characteristic, complemented
3#23749509C211yesmaximal, characteristic, complemented
4#23749507C211yesmaximal, characteristic, complemented
5#23749501C211yesmaximal, characteristic
6#23749501C211yesmaximal, characteristic
7#23749499C211yesmaximal, characteristic, complemented
8#23749496C211yesmaximal, characteristic, complemented
9#23749492C211yesmaximal, characteristic, complemented
10#23749490C211yesmaximal, characteristic, complemented
11#23749486C211yesmaximal, characteristic, complemented
12#23749483C211yesmaximal, characteristic, complemented
13C4×#14558713C211yesmaximal, characteristic, complemented
14#95×D149C211yesmaximal, characteristic, complemented
15#388×C149C211yesmaximal, characteristic, complemented
16C22×C22⋊D149C2211yescharacteristic
17C23×Dic149C2211yescharacteristic
18C22×C4×D149C2211yescharacteristic
19C22×C4×D149C2211yescharacteristic
20C22×C4×D149C2211yescharacteristic
21#18011871C418yescomplemented
22C2×C4⋊Dic149C2211yescharacteristic
23C2×#14558715C2211yescharacteristic
24C2×#14558715C2211yescharacteristic
25C2×#14558715C2211yescharacteristic
26C2×C22⋊Dic149C2211yescharacteristic
27C2×C22⋊Dic149C2211yescharacteristic
28C2×#14558713C2211yescharacteristic
29C2×#14558713C2211yescharacteristic
30C2×#14558713C2211yescharacteristic
31C2×C4×Dic149C2211yescharacteristic
32C2×C4×Dic149C2211yescharacteristic
33#18011818C2211yescharacteristic
34#18011818C2211yescharacteristic
35#18011818C2211yescharacteristic
36#18011818C2211yescharacteristic
37#18011818C2211yescharacteristic
38#18011818C2211yescharacteristic
39#18011818C2211yescharacteristic
40#18011818C2211yescharacteristic
41#18011818C2211yescharacteristic
42#18011818C2211yescharacteristic
43#18011818C2211yescharacteristic
44#18011818C2211yescharacteristic
45C2×C4⋊C4×C149C2211yescharacteristic
46C2×C22⋊C4×C149C2211yescharacteristic
47C2×C22⋊C4×C149C2211yescharacteristic
48C2×C22⋊C4×C149C2211yescharacteristic
49C2×C42×C149C2211yescharacteristic
50#95×C149C2211yescharacteristic
51#95×C149C2211yescharacteristic
52C23×D149C2311yescharacteristic
53C2×C22⋊D149—22no—
54C2×C22⋊D149—22no—
55C2×C22⋊D149C2×C414yes—
56C22×Dic149—41no—
57C22×Dic149—41no—
58C22×Dic149C2×C414yescomplemented
59C22×Dic149C2311yescharacteristic
60C22×Dic149C2311yescharacteristic
61C22×Dic149C2311yescharacteristic
62C22×Dic149C2311yescharacteristic
63C22×Dic149C2311yescharacteristic
64C22×Dic149C2311yescharacteristic
65C22×Dic149C2311yescharacteristic
66C2×C4×D149—22no—
67C2×C4×D149—22no—
68C2×C4×D149—41no—
69C2×C4×D149—41no—
70C2×C4×D149—41no—
71C2×C4×D149—41no—
72C2×C4×D149—41no—
73C2×C4×D149—41no—
74C2×C4×D149C2×C414yes—
75C4⋊Dic149—41no—
76#14558715—41no—
77#14558715—41no—
78#14558715C2×C414yescomplemented
79C22⋊Dic149—41no—
80C22⋊Dic149—41no—
81#14558713—22no—
82#14558713—22no—
83#14558713C2×C414yes—
84C4×Dic149—41no—
85C4×Dic149C2×C414yescomplemented
86C24×C149C2311yescharacteristic
87C22×C4×C149C2311yescharacteristic
88C22×C4×C149C2311yescharacteristic
89C22×C4×C149C2311yescharacteristic
90C22×C4×C149C2311yescharacteristic
91C22×C4×C149C2311yescharacteristic
92C22×C4×C149C2311yescharacteristic
93C4⋊C4×C149—41no—
94C22⋊C4×C149—41no—
95C22⋊C4×C149—41no—
96C22⋊C4×C149C2×C414yescomplemented
97C42×C149—41no—
98C22×D149—21no—
99C22×D149—21no—
100C22×D149—21no—
101C22×D149—21no—
102C22×D149C22×C412yes—
103C22×D149#4112yes—
104C22×D149#4112yes—
105C22⋊D149—28no—
106C2×Dic149—21no—
107C2×Dic149—21no—
108C2×Dic149—21no—
109C2×Dic149—21no—
110C2×Dic149—21no—
111C2×Dic149—21no—
112C2×Dic149—21no—
113C2×Dic149—21no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149—21no—
117C2×Dic149—21no—
118C2×Dic149—21no—
119C2×Dic149—21no—
120C2×Dic149—21no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149—21no—
124C2×Dic149—44no—
125C2×Dic149C22×C412yes—
126C2×Dic149C22×C412yes—
127C2×Dic149C22×C412yes—
128C4×D149—44no—
129C4×D149—44no—
130C4×D149—44no—
131C23×C149—21no—
132C23×C149—21no—
133C23×C149—21no—
134C23×C149—21no—
135C23×C149—21no—
136C23×C149—21no—
137C23×C149C22×C412yes—
138C23×C149C2411yescharacteristic
139C2×C4×C149—21no—
140C2×C4×C149—21no—
141C2×C4×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149—21no—
148C2×C4×C149—21no—
149C2×C4×C149—21no—
150C2×C4×C149—21no—
151C2×C4×C149—21no—
152C2×C4×C149—21no—
153C2×C4×C149—21no—
154C2×C4×C149—21no—
155C2×C4×C149C22×C412yes—
156C2×C4×C149C22×C412yes—
157C22×C149—22no—
158C22×C149—22no—
159C22×C149—22no—
160C22×C149—41no—
161C22×C149—41no—
162C22×C149—41no—
163C22×C149—41no—
164C22×C149C23×C411yescharacteristic
165C22×C149C2×#4111yescharacteristic
166C22×C149C2×#4111yescharacteristic
167C22×C149H(5,𝔽2)11yescharacteristic
168C22×C149H(5,𝔽2)11yescharacteristic
169C22×C149H(5,𝔽2)11yescharacteristic
170C22×C1492−1+411yescharacteristic
171C2×D149—22no—
172C2×D149—22no—
173C2×D149—22no—
174C2×D149—22no—
175C2×D149—22no—
176C2×D149—22no—
177C2×D149#11714yes—
178C4×C149—22no—
179C4×C149—22no—
180C4×C149—41no—
181C4×C149—41no—
182C4×C149—41no—
183C4×C149—41no—
184Dic149—22no—
185Dic149—22no—
186Dic149—22no—
187Dic149—41no—
188Dic149—41no—
189Dic149—41no—
190Dic149—41no—
191C2×C149—42no—
192C2×C149C2×#11711yescharacteristic
193C2×C149#51811yescharacteristic
194C2×C149#52011yescharacteristic
195C2×C149#53811yescharacteristic
196C2×C149#53811yescharacteristic
197C2×C149#53911yescharacteristic
198C2×C149#53911yescharacteristic
199D149—24no—
200C149#231811yescharacteristic, complemented, Sylow
201#2318—1491nomaximal, Sylow
202C2×C4×D4—1491no—
203#388—1491no—
204#388D14911yescharacteristic, complemented
205#387—1491no—
206#387—1491no—
207#386—1491no—
208#386—1491no—
209#385—1491no—
210#385—1491no—
211#383—1491no—
212#382—1491no—
213#381—1491no—
214#380—1491no—
215C4×C22⋊C4—1491no—
216C2×#95—1491no—
217C22×D4—1491no—
218C23×C4—1491no—
219C23×C4—1491no—
220C23×C4—1491no—
221C23×C4—1491no—
222C4×D4—1498no—
223C2×C4⋊C4—1491no—
224C2×C4⋊C4—1491no—
225C2×C4⋊C4—1491no—
226C2×C4⋊C4—1491no—
227C2×C4⋊C4C2×D14911yescharacteristic
228C2×C22⋊C4—1491no—
229C2×C22⋊C4—1491no—
230C2×C22⋊C4—1491no—
231C2×C22⋊C4—1491no—
232C2×C22⋊C4—1491no—
233C2×C22⋊C4C2×D14911yescharacteristic
234C2×C22⋊C4C2×D14911yescharacteristic
235C2×C22⋊C4C2×D14911yescharacteristic
236C2×C42—1491no—
237C2×C42—1491no—
238C2×C42C2×D14911yescharacteristic
239#95—1491no—
240#95—1491no—
241#95—1491no—
242#95—1491no—
243#95—1491no—
244#95—1491no—
245#95—1491no—
246#95—1491no—
247#95—1491no—
248#95—1491no—
249#95—1491no—
250#95—1491no—
251#95C2×D14911yescharacteristic
252#95C2×D14911yescharacteristic
253C24—1491no—
254C24C22×D14911yescharacteristic
255C2×D4—2982no—
256C2×D4—2982no—
257C2×D4—1494no—
258C22×C4—2982no—
259C22×C4—2982no—
260C22×C4—5961no—
261C22×C4—5961no—
262C22×C4—5961no—
263C22×C4—5961no—
264C22×C4—5961no—
265C22×C4—5961no—
266C22×C4—5961no—
267C22×C4—5961no—
268C22×C4—1491no—
269C22×C4—1491no—
270C22×C4—1491no—
271C22×C4—1491no—
272C22×C4—1491no—
273C22×C4—1491no—
274C22×C4—1491no—
275C22×C4—1494no—
276C22×C4—1494no—
277C22×C4C22×D14911yescharacteristic
278C22×C4C22×D14911yescharacteristic
279C22×C4C22×D14911yescharacteristic
280C22×C4C22×D14911yescharacteristic
281C22×C4C22×D14911yescharacteristic
282C22×C4C22×D14911yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—5961no—
285C4⋊C4—5961no—
286C4⋊C4—5961no—
287C4⋊C4—1494no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—2982no—
291C22⋊C4—2982no—
292C22⋊C4—5961no—
293C22⋊C4—5961no—
294C22⋊C4—1494no—
295C22⋊C4C4×D14914yescomplemented
296C42—41no—
297C42—5961no—
298C42—1494no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—2981no—
306C23—2981no—
307C23—2981no—
308C23—2981no—
309C23—1492no—
310C23—1492no—
311C23—1492no—
312C23C2×C4×D14912yes—
313C23C23×D14911yescharacteristic
314D4—2988no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—2981no—
332C2×C4—2981no—
333C2×C4—2981no—
334C2×C4—2981no—
335C2×C4—2981no—
336C2×C4—2981no—
337C2×C4—2981no—
338C2×C4—2981no—
339C2×C4—2981no—
340C2×C4—2981no—
341C2×C4—2981no—
342C2×C4—2981no—
343C2×C4—2981no—
344C2×C4—2981no—
345C2×C4—2981no—
346C2×C4—2981no—
347C2×C4—2981no—
348C2×C4—2981no—
349C2×C4—5964no—
350C2×C4—5964no—
351C2×C4—5964no—
352C2×C4—5964no—
353C2×C4—1492no—
354C2×C4—1492no—
355C2×C4—1492no—
356C2×C4C2×C4×D14912yes—
357C2×C4C2×C4×D14912yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—2982no—
366C22—2982no—
367C22—2982no—
368C22—2982no—
369C22—2982no—
370C22—2982no—
371C22—1494no—
372C22C22×C4×D14911yescharacteristic
373C22#41×D14911yescharacteristic
374C22#41×D14911yescharacteristic
375C22#1801195611yescharacteristic
376C22#1801195611yescharacteristic
377C22#1801195711yescharacteristic
378C22#1801195811yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—2982no—
386C4—2982no—
387C4—2982no—
388C4—5961no—
389C4—5961no—
390C4—5961no—
391C4—5961no—
392C2—42no—
393C2—2984no—
394C2#117×D14911yescharacteristic
395C2#2375009711yescharacteristic
396C2#2375010411yescharacteristic
397C2#2375020711yescharacteristic
398C2#2375020911yescharacteristic
399C2#2375021611yescharacteristic
400C2#2375021811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#388×C149C2
socle, cosocleC23×C149C24
prime radicalC22×C22⋊D149C22
hypercenter#388D149
hypocenterC149#2318
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #23749518C2semidirect
3. #23749509C2semidirect
4. #23749507C2semidirect
7. #23749499C2semidirect
8. #23749496C2semidirect
9. #23749492C2semidirect
10. #23749490C2semidirect
11. #23749486C2semidirect
12. #23749483C2semidirect
13. C4×#14558713C2semidirect
14. #95×D149C2semidirect
15. #388×C149C2semidirect
21. #18011871C4semidirect
58. C22×Dic149C2×C4semidirect
78. #14558715C2×C4semidirect
85. C4×Dic149C2×C4semidirect
96. C22⋊C4×C149C2×C4semidirect
200. C149#2318semidirect
204. #388D149semidirect
295. C22⋊C4C4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118712
#237495182
#237495092
#237495072
#237495012
#237494992
#237494962
#237494922
#237494902
#237494862
#237494832
C4×#145587132
#95×D1492
#388×C1492
#2318149