Finite Group Databasebeta

← #34056271

Subgroups — #34056271

Order 19072.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011940C211yesmaximal, characteristic, complemented
2C2×#18011932C211yesmaximal, characteristic, complemented
3#23750295C211yesmaximal, characteristic, complemented
4#23750287C211yesmaximal, characteristic, complemented
5#23750012C211yesmaximal, characteristic, complemented
6#23749877C211yesmaximal, characteristic, complemented
7#23749853C211yesmaximal, characteristic, complemented
8#23749848C211yesmaximal, characteristic, complemented
9#23749788C211yesmaximal, characteristic, complemented
10#23749771C211yesmaximal, characteristic, complemented
11#23749769C211yesmaximal, characteristic, complemented
12#23749762C211yesmaximal, characteristic, complemented
13#23749730C211yesmaximal, characteristic, complemented
14#23749695C211yesmaximal, characteristic, complemented
15#460×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558735C2211yescharacteristic
18C2×#14558735C2211yescharacteristic
19C2×Q8×D149C2211yescharacteristic
20#18011940—24no—
21#18011932—24no—
22C2×Dic596C2211yescharacteristic
23C2×#14558726C2211yescharacteristic
24C2×#14558725C2211yescharacteristic
25C2×#14558724C2211yescharacteristic
26C2×#14558724C2211yescharacteristic
27C2×#14558721C2211yescharacteristic
28C2×#14558721C2211yescharacteristic
29C2×#14558719C2211yescharacteristic
30C2×C8×D149C2211yescharacteristic
31#18011907—21no—
32#18011899—21no—
33#18011899C2211yescharacteristic
34#18011891—21no—
35#18011891C2211yescharacteristic
36#18011886—21no—
37#18011886C2211yescharacteristic
38#18011883—21no—
39#18011883—21no—
40#18011882C2211yescharacteristic
41#18011882C2211yescharacteristic
42C4×Dic298C2211yescharacteristic
43C4×D596—21no—
44C4×D596C2211yescharacteristic
45D4×Dic149—21no—
46#18011842C2211yescharacteristic
47#18011836C2211yescharacteristic
48#18011835C2211yescharacteristic
49#18011834C2211yescharacteristic
50#18011833C2211yescharacteristic
51#18011823C2211yescharacteristic
52#18011823C2211yescharacteristic
53#18011822C2211yescharacteristic
54C2×SD16×C149C2211yescharacteristic
55C2×SD16×C149C2211yescharacteristic
56#127×C149C2211yescharacteristic
57C4×Q8×C149C2211yescharacteristic
58C4⋊C8×C149C2211yescharacteristic
59#102×C149C2211yescharacteristic
60#102×C149C2211yescharacteristic
61#14558737—22no—
62#14558737—41no—
63#14558735—22no—
64#14558735—22no—
65#14558735—41no—
66#14558735—41no—
67C2×Dic298—21no—
68C2×Dic298C2311yescharacteristic
69C2×Dic298C2311yescharacteristic
70Q8×D149—22no—
71Q8×D149—41no—
72C2×D596—21no—
73C2×D596—21no—
74C2×D596D411yescharacteristic
75C2×D596C2311yescharacteristic
76C2×C22⋊D149—21no—
77C2×C22⋊D149—21no—
78C2×C22⋊D149—21no—
79C2×C22⋊D149—21no—
80C22×Dic149—21no—
81C22×Dic149—21no—
82C2×C4×D149—21no—
83C2×C4×D149—21no—
84C2×C4×D149C2311yescharacteristic
85C2×C4×D149C2311yescharacteristic
86Dic596—22no—
87#14558726D414yescomplemented
88#14558725—22no—
89#14558724—22no—
90#14558724—41no—
91#14558721—22no—
92#14558721—41no—
93#14558719—22no—
94C2×#11924746C2311yescharacteristic
95C2×#11924746C2311yescharacteristic
96C8×D149—22no—
97C4⋊Dic149—21no—
98C4⋊Dic149D411yescharacteristic
99C4⋊Dic149C2311yescharacteristic
100#14558715—21no—
101C22⋊Dic149—21no—
102C22⋊Dic149—21no—
103#14558713—21no—
104#14558713—21no—
105#14558713—21no—
106#14558713—21no—
107#14558713D411yescharacteristic
108#14558713D411yescharacteristic
109C4×Dic149—21no—
110C2×Q8×C149C2311yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149—21no—
113C2×D4×C149C2311yescharacteristic
114C2×D4×C149C2311yescharacteristic
115SD16×C149—22no—
116SD16×C149—22no—
117C2×C8×C149C2311yescharacteristic
118C2×C8×C149C2311yescharacteristic
119C4⋊C4×C149—21no—
120C4⋊C4×C149C2311yescharacteristic
121C42×C149—21no—
122C42×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149C2×D411yescharacteristic
125C22×D149C2×D411yescharacteristic
126Dic298—21no—
127Dic298—21no—
128Dic298—21no—
129Dic298—42no—
130Dic298C2×D412yes—
131D596—21no—
132D596—21no—
133D596—41no—
134D596—42no—
135D596—42no—
136C22⋊D149—42no—
137C22⋊D149—42no—
138C22⋊D149—81no—
139C22⋊D149—81no—
140C2×Dic149—21no—
141C2×Dic149—21no—
142C2×Dic149—21no—
143C2×Dic149—42no—
144C2×Dic149—42no—
145C2×Dic149C2×D411yescharacteristic
146C2×Dic149C2×D411yescharacteristic
147C4×D149—21no—
148C4×D149—21no—
149C4×D149—21no—
150C4×D149—21no—
151C4×D149—42no—
152C4×D149—42no—
153#11924746—21no—
154#11924746C2×D412yes—
155C23×C149—21no—
156C23×C149—21no—
157Q8×C149—21no—
158Q8×C149C2×D412yes—
159D4×C149—21no—
160D4×C149—21no—
161D4×C149—21no—
162D4×C149—21no—
163D4×C149—42no—
164D4×C149—81no—
165C2×C4×C149—21no—
166C2×C4×C149—21no—
167C2×C4×C149C2×D411yescharacteristic
168C2×C4×C149C2×D411yescharacteristic
169C2×C4×C149C2411yescharacteristic
170C8×C149—21no—
171C8×C149—21no—
172C22×C149—41no—
173C22×C149—41no—
174C22×C149—41no—
175C22×C149—41no—
176C22×C149—41no—
177C22×C149—41no—
178C22×C149C22×D411yescharacteristic
179C2×D149—21no—
180C2×D149—21no—
181C2×D149—21no—
182C2×D149—21no—
183C2×D149—21no—
184C2×D149—21no—
185C2×D149—41no—
186C2×D149—41no—
187C2×D149—41no—
188C4×C149—22no—
189C4×C149—21no—
190C4×C149—21no—
191C4×C149—41no—
192C4×C149C22×D411yescharacteristic
193C4×C149H(5,𝔽2)11yescharacteristic
194Dic149—21no—
195Dic149—21no—
196Dic149—22no—
197Dic149—41no—
198Dic149—41no—
199C2×C149—81no—
200C2×C149—81no—
201C2×C149D4211yescharacteristic
202C2×C149#57511yescharacteristic
203C2×C149#57711yescharacteristic
204D149—41no—
205D149—41no—
206D149—81no—
207C149#327111yescharacteristic, complemented, Sylow
208#3271—1491nomaximal, Sylow
209C2×#137—1491no—
210C2×#135—1491no—
211#550—1491no—
212#548—1491no—
213#496—1491no—
214#469—1491no—
215#465—1491no—
216#464—1491no—
217#460D14911yescharacteristic, complemented
218#452—1491no—
219#449—1491no—
220#449—1491no—
221#448—1491no—
222#441—1491no—
223#435—1491no—
224C2×#41—1491no—
225C2×#41—1491no—
226C2×#41—1491no—
227C22×Q8—1491no—
228#137—2984no—
229#135—2984no—
230C2×Q16—1491no—
231C2×Q16—1491no—
232C2×SD16—1491no—
233C2×SD16—1491no—
234C2×SD16—1491no—
235C2×SD16C2×D14911yescharacteristic
236C2×SD16C2×D14911yescharacteristic
237C2×D8—1491no—
238C2×D8—1491no—
239C2×M16—1491no—
240C22×C8—1491no—
241#127—2981no—
242#127C2×D14911yescharacteristic
243#124—2981no—
244#124—1491no—
245C22⋊Q8—2981no—
246C22⋊Q8—1491no—
247#121—2981no—
248#121—2981no—
249#121—2981no—
250#121—1491no—
251#121—1491no—
252#121—1491no—
253C4×Q8—1491no—
254C4×Q8C2×D14911yescharacteristic
255C4×D4—2981no—
256C4×D4—2981no—
257C4×D4—1491no—
258#106—1491no—
259C4⋊C8C2×D14911yescharacteristic
260#103—1491no—
261#103—1491no—
262#103—1491no—
263#102—1491no—
264#102C2×D14911yescharacteristic
265#102C2×D14911yescharacteristic
266C22⋊C8—1491no—
267C22⋊C8—1491no—
268#97—1491no—
269#41—2982no—
270#41—2982no—
271#41—2982no—
272#41—5961no—
273#41—5961no—
274#41—5961no—
275C2×Q8—2981no—
276C2×Q8—2982no—
277C2×Q8—5961no—
278C2×Q8—1491no—
279C2×Q8—1491no—
280C2×Q8C22×D14911yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—2981no—
284C2×D4—2981no—
285C2×D4—2981no—
286C2×D4—2981no—
287C2×D4—2981no—
288C2×D4—2981no—
289C2×D4—1491no—
290C2×D4—1491no—
291C2×D4C22×D14911yescharacteristic
292C2×D4C22×D14911yescharacteristic
293C22×C4—2981no—
294C22×C4—2981no—
295C22×C4—2981no—
296C22×C4—2981no—
297C22×C4—1491no—
298C22×C4—1491no—
299Q16—2982no—
300Q16—1494no—
301SD16—22no—
302SD16—22no—
303SD16—2982no—
304SD16—2982no—
305SD16—5961no—
306D8—2982no—
307D8—5961no—
308M16—2982no—
309C2×C8—2982no—
310C2×C8—1491no—
311C2×C8—1491no—
312C2×C8C22×D14911yescharacteristic
313C2×C8C22×D14911yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—2981no—
316C4⋊C4—2981no—
317C4⋊C4—1491no—
318C4⋊C4—1491no—
319C4⋊C4C22×D14911yescharacteristic
320C22⋊C4—2981no—
321C22⋊C4—2981no—
322C22⋊C4—2981no—
323C22⋊C4—2981no—
324C22⋊C4—2981no—
325C22⋊C4—2981no—
326C22⋊C4—1491no—
327C22⋊C4—1491no—
328C42—21no—
329C42—2981no—
330C42C22×D14911yescharacteristic
331C23—21no—
332C23—21no—
333C23—2981no—
334C23—1491no—
335C23—1491no—
336Q8—21no—
337Q8—2981no—
338Q8—2981no—
339Q8—2981no—
340Q8—5962no—
341Q8—1492no—
342Q8D4×D14912yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—2981no—
350D4—2981no—
351D4—5961no—
352D4—5962no—
353D4—5962no—
354D4—5962no—
355D4—5962no—
356D4—11921no—
357D4—11921no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—2981no—
361C2×C4—2981no—
362C2×C4—2981no—
363C2×C4—2981no—
364C2×C4—2981no—
365C2×C4—2981no—
366C2×C4—2981no—
367C2×C4—5962no—
368C2×C4—5962no—
369C2×C4—5962no—
370C2×C4—5962no—
371C2×C4—1491no—
372C2×C4—1491no—
373C2×C4D4×D14911yescharacteristic
374C2×C4D4×D14911yescharacteristic
375C2×C4C23×D14911yescharacteristic
376C8—21no—
377C8—21no—
378C8—2981no—
379C8—1492no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—2981no—
387C22—2981no—
388C22—2981no—
389C22—2981no—
390C22—2981no—
391C22—2981no—
392C22—5961no—
393C22—5961no—
394C22—5961no—
395C22C2×D4×D14911yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—41no—
400C4—2981no—
401C4—2981no—
402C4—2982no—
403C4—5961no—
404C4—5961no—
405C4C2×D4×D14911yescharacteristic
406C4#1801195711yescharacteristic
407C2—81no—
408C2—81no—
409C2—5961no—
410C2—5961no—
411C2—11921no—
412C2#2375025911yescharacteristic
413C2#2375042111yescharacteristic
414C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#460×C149C2
socleC22×C149C22×D4
hypercenter#460D149
hypocenterC149#3271
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#18011940C2semidirect
2. C2×#18011932C2semidirect
3. #23750295C2semidirect
4. #23750287C2semidirect
5. #23750012C2semidirect
6. #23749877C2semidirect
7. #23749853C2semidirect
8. #23749848C2semidirect
9. #23749788C2semidirect
10. #23749771C2semidirect
11. #23749769C2semidirect
12. #23749762C2semidirect
13. #23749730C2semidirect
14. #23749695C2semidirect
15. #460×C149C2semidirect
87. #14558726D4semidirect
207. C149#3271semidirect
217. #460D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119402
C2×#180119322
#237502952
#237502872
#237500122
#237498772
#237498532
#237498482
#237497882
#237497712
#237497692
#237497622
#237497302
#237496952
#460×C1492
#3271149