Finite Group Databasebeta

← #34055789

Subgroups — #34055789

Order 19072.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750248C211yesmaximal, characteristic, complemented
2#23750192C211yesmaximal, characteristic, complemented
3#23750011C211yesmaximal, characteristic, complemented
4#23750006C211yesmaximal, characteristic, complemented
5#23749983C211yesmaximal, characteristic, complemented
6#23749970C211yesmaximal, characteristic, complemented
7#23749845C211yesmaximal, characteristic, complemented
8#23749839C211yesmaximal, characteristic, complemented
9#23749823C211yesmaximal, characteristic, complemented
10#23749819C211yesmaximal, characteristic, complemented
11#23749785C211yesmaximal, characteristic, complemented
12#23749772C211yesmaximal, characteristic, complemented
13#23749759C211yesmaximal, characteristic, complemented
14#23749683C211yesmaximal, characteristic, complemented
15#450×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic
17C2×#14558726C2211yescharacteristic, complemented
18C2×#14558725C2211yescharacteristic, complemented
19C2×#14558723C2211yescharacteristic, complemented
20C2×D1192C2211yescharacteristic, complemented
21C2×#14558721C2211yescharacteristic
22C2×#14558721C2211yescharacteristic
23#18011910C2211yescharacteristic, complemented
24#18011907C2211yescharacteristic
25#18011907C2211yescharacteristic
26#18011906—21no—
27#18011900C2211yescharacteristic, complemented
28#18011898C2211yescharacteristic
29#18011898C2211yescharacteristic
30#18011897—21no—
31#18011893—21no—
32#18011891—21no—
33#18011885C2211yescharacteristic
34#18011882—21no—
35#18011882—21no—
36#18011880—21no—
37#18011880—21no—
38Q8×Dic149C2211yescharacteristic, complemented
39#18011873C2211yescharacteristic, complemented
40#18011870C2211yescharacteristic
41#18011840C2211yescharacteristic, complemented
42#18011840C2211yescharacteristic, complemented
43#18011836C2211yescharacteristic, complemented
44#18011834C2211yescharacteristic, complemented
45#18011833C2211yescharacteristic, complemented
46#18011832C2211yescharacteristic
47#18011832C2211yescharacteristic
48#18011831C2211yescharacteristic, complemented
49C22⋊#11924746C2211yescharacteristic
50#18011821C2211yescharacteristic, complemented
51C8×Dic149C2211yescharacteristic, complemented
52C22×D4×C149C2211yescharacteristic
53C2×SD16×C149C2211yescharacteristic
54C2×SD16×C149C2211yescharacteristic
55C22⋊Q8×C149C2211yescharacteristic
56#102×C149C2211yescharacteristic
57#102×C149C2211yescharacteristic
58C22⋊C8×C149C2211yescharacteristic
59#14558736—41no—
60#14558736—41no—
61C2×Dic298—21no—
62C2×Dic298C2311yescharacteristic
63C2×D596C2311yescharacteristic
64C2×D596C2311yescharacteristic
65C2×C22⋊D149—21no—
66C2×C22⋊D149—21no—
67C2×C22⋊D149—21no—
68C2×C22⋊D149—21no—
69C2×C22⋊D149—21no—
70C2×C22⋊D149—21no—
71C2×C4×D149—21no—
72C2×C4×D149—21no—
73#14558726—41no—
74#14558725—41no—
75#14558723—41no—
76D1192—41no—
77#14558721—41no—
78#14558721—41no—
79C2×#11924746C2311yescharacteristic
80C2×#11924746C2311yescharacteristic
81C4⋊Dic149—21no—
82C4⋊Dic149D411yescharacteristic
83C4⋊Dic149D411yescharacteristic
84C4⋊Dic149C2311yescharacteristic
85#14558715—21no—
86C22⋊Dic149—21no—
87C22⋊Dic149—21no—
88C22⋊Dic149—21no—
89C22⋊Dic149—21no—
90C22⋊Dic149D411yescharacteristic
91C22⋊Dic149D411yescharacteristic
92#14558713—21no—
93#14558713—21no—
94#14558713—21no—
95#14558713—21no—
96C4×Dic149—21no—
97C4×Dic149C2311yescharacteristic
98C4×Dic149C2311yescharacteristic
99C24×C149—21no—
100C2×Q8×C149C2311yescharacteristic
101C2×D4×C149—21no—
102C2×D4×C149—41no—
103C2×D4×C149—41no—
104C2×D4×C149C2311yescharacteristic
105C2×D4×C149C2311yescharacteristic
106C22×C4×C149C2311yescharacteristic
107SD16×C149—41no—
108SD16×C149—41no—
109C2×C8×C149C2311yescharacteristic
110C2×C8×C149C2311yescharacteristic
111C4⋊C4×C149—21no—
112C4⋊C4×C149C2311yescharacteristic
113C22⋊C4×C149—21no—
114C22×D149—21no—
115C22×D149—21no—
116Dic298—21no—
117Dic298—21no—
118Dic298—81no—
119D596—21no—
120D596—21no—
121D596—21no—
122D596—21no—
123C22⋊D149—42no—
124C22⋊D149—81no—
125C22⋊D149—81no—
126C22⋊D149—81no—
127C22⋊D149—81no—
128C22⋊D149—81no—
129C2×Dic149—21no—
130C2×Dic149—21no—
131C2×Dic149C2×D411yescharacteristic
132C2×Dic149C2×D411yescharacteristic
133C2×Dic149C2×D411yescharacteristic
134C2×Dic149C2×D411yescharacteristic
135C4×D149—42no—
136C4×D149—81no—
137#11924746—21no—
138#11924746—21no—
139C23×C149—21no—
140C23×C149—21no—
141C23×C149—42no—
142C23×C149—81no—
143C23×C149—81no—
144C23×C149C2×D411yescharacteristic
145Q8×C149—21no—
146Q8×C149—21no—
147D4×C149—21no—
148D4×C149—21no—
149D4×C149—21no—
150D4×C149—21no—
151D4×C149—42no—
152C2×C4×C149—21no—
153C2×C4×C149—21no—
154C2×C4×C149—21no—
155C2×C4×C149—21no—
156C2×C4×C149C2×D411yescharacteristic
157C2×C4×C149C2411yescharacteristic
158C8×C149—21no—
159C8×C149—21no—
160C22×C149—21no—
161C22×C149—21no—
162C22×C149—21no—
163C22×C149—41no—
164C22×C149—41no—
165C22×C149—41no—
166C22×C149—41no—
167C22×C149—41no—
168C22×C149—41no—
169C22×C149—84no—
170C22×C149C22×D411yescharacteristic
171C2×D149—41no—
172C2×D149—41no—
173C2×D149—41no—
174C2×D149—41no—
175C2×D149—41no—
176C2×D149—41no—
177C4×C149—21no—
178C4×C149—41no—
179C4×C149—41no—
180C4×C149H(5,𝔽2)11yescharacteristic
181C4×C149H(5,𝔽2)11yescharacteristic
182Dic149—21no—
183Dic149—21no—
184Dic149—21no—
185Dic149—21no—
186Dic149—41no—
187Dic149—41no—
188C2×C149—41no—
189C2×C149—81no—
190C2×C149—81no—
191C2×C149#53511yescharacteristic
192C2×C149#57611yescharacteristic
193C2×C149#57711yescharacteristic
194D149—81no—
195D149—81no—
196C149#322411yescharacteristic, complemented, Sylow
197#3224—1491nomaximal, Sylow
198#542—1491no—
199#535—1491no—
200#496—1491no—
201#495—1491no—
202#489—1491no—
203#486—1491no—
204#464—1491no—
205#463—1491no—
206#460—1491no—
207#459—1491no—
208#452—1491no—
209#450D14911yescharacteristic, complemented
210#449—1491no—
211#447—1491no—
212#433—1491no—
213C2×#41—1491no—
214C22×D4C2×D14911yescharacteristic
215C2×Q16—1491no—
216C2×SD16—1491no—
217C2×SD16—1491no—
218C2×SD16C2×D14911yescharacteristic
219C2×SD16C2×D14911yescharacteristic
220C2×D8—1491no—
221C2×D8—1491no—
222C2×D8—1491no—
223C4⋊Q8—1491no—
224#127—1491no—
225#127—1491no—
226#126—2981no—
227#124—2981no—
228#124—1491no—
229#124—1491no—
230#124—1491no—
231#123—2981no—
232C22⋊Q8—2981no—
233C22⋊Q8C2×D14911yescharacteristic
234#121—2981no—
235#121—2981no—
236#121—1491no—
237#120—2981no—
238#120—2981no—
239C4×Q8—1491no—
240C4×D4—1491no—
241#117—1491no—
242C4⋊C8—1491no—
243C4⋊C8—1491no—
244#103—1491no—
245#103—1491no—
246#102—1491no—
247#102—1491no—
248#102—1491no—
249#102—1491no—
250#102C2×D14911yescharacteristic
251#102C2×D14911yescharacteristic
252C22⋊C8—1491no—
253C22⋊C8C2×D14911yescharacteristic
254#97—1491no—
255C4×C8—1491no—
256C24—21no—
257#41—5961no—
258#41—5961no—
259C2×Q8—2981no—
260C2×Q8—1491no—
261C2×Q8C22×D14911yescharacteristic
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—2981no—
266C2×D4—2981no—
267C2×D4—2981no—
268C2×D4—2981no—
269C2×D4—2981no—
270C2×D4—2981no—
271C2×D4—1491no—
272C2×D4—1491no—
273C2×D4C22×D14911yescharacteristic
274C2×D4C22×D14911yescharacteristic
275C22×C4—2981no—
276C22×C4—2981no—
277C22×C4C22×D14911yescharacteristic
278Q16—5961no—
279SD16—41no—
280SD16—41no—
281SD16—5961no—
282SD16—5961no—
283D8—5961no—
284D8—5961no—
285D8—5961no—
286C2×C8—1491no—
287C2×C8—1491no—
288C2×C8C22×D14911yescharacteristic
289C2×C8C22×D14911yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2981no—
292C4⋊C4—2981no—
293C4⋊C4—1491no—
294C4⋊C4—1491no—
295C4⋊C4—1491no—
296C4⋊C4C22×D14911yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—2981no—
299C22⋊C4—2981no—
300C22⋊C4—2981no—
301C22⋊C4—2981no—
302C22⋊C4—2981no—
303C22⋊C4—2981no—
304C22⋊C4—2981no—
305C22⋊C4—2981no—
306C22⋊C4—1491no—
307C22⋊C4—1491no—
308C42—2981no—
309C42—1491no—
310C42—1491no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—2981no—
317C23—2981no—
318C23D4×D14911yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—2981no—
322Q8—2981no—
323Q8—11921no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—2981no—
330D4—2981no—
331D4—2981no—
332D4—2981no—
333D4—5962no—
334D4—11921no—
335D4—11921no—
336D4—11921no—
337D4—11921no—
338D4—11921no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—2981no—
344C2×C4—2981no—
345C2×C4—5962no—
346C2×C4—1491no—
347C2×C4—1491no—
348C2×C4—1491no—
349C2×C4—1491no—
350C2×C4—11921no—
351C2×C4D4×D14911yescharacteristic
352C2×C4C23×D14911yescharacteristic
353C8—21no—
354C8—21no—
355C8—2981no—
356C8—2981no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—5961no—
368C22—5961no—
369C22—5961no—
370C22—5961no—
371C22—5961no—
372C22—5961no—
373C22C2×D4×D14911yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—2981no—
378C4—2981no—
379C4—2981no—
380C4—2981no—
381C4—5961no—
382C4—5961no—
383C4#1801195611yescharacteristic
384C4#1801195611yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—11921no—
389C2—11921no—
390C2#2375019011yescharacteristic
391C2#2375042911yescharacteristic
392C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#450×C149C2
socleC22×C149C22×D4
hypercenter#450D149
hypocenterC149#3224
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23750248C2semidirect
2. #23750192C2semidirect
3. #23750011C2semidirect
4. #23750006C2semidirect
5. #23749983C2semidirect
6. #23749970C2semidirect
7. #23749845C2semidirect
8. #23749839C2semidirect
9. #23749823C2semidirect
10. #23749819C2semidirect
11. #23749785C2semidirect
12. #23749772C2semidirect
13. #23749759C2semidirect
14. #23749683C2semidirect
15. #450×C149C2semidirect
17. C2×#14558726C22semidirect
18. C2×#14558725C22semidirect
19. C2×#14558723C22semidirect
20. C2×D1192C22semidirect
23. #18011910C22semidirect
27. #18011900C22semidirect
38. Q8×Dic149C22semidirect
39. #18011873C22semidirect
41. #18011840C22semidirect
42. #18011840C22semidirect
43. #18011836C22semidirect
44. #18011834C22semidirect
45. #18011833C22semidirect
48. #18011831C22semidirect
50. #18011821C22semidirect
51. C8×Dic149C22semidirect
196. C149#3224semidirect
209. #450D149semidirect

Maximal subgroups

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

subgroupindex
#237502482
#237501922
#237500112
#237500062
#237499832
#237499702
#237498452
#237498392
#237498232
#237498192
#237497852
#237497722
#237497592
#237496832
#450×C1492
#3224149