Finite Group Databasebeta

← #34058691

Subgroups — #34058691

Order 19072.

1122 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750351C211yesmaximal, characteristic, complemented
2#23750325C213yesmaximal, complemented
3#23750323C213yesmaximal, complemented
4#23750297C211yesmaximal, characteristic, complemented
5#23750296C213yesmaximal, complemented
6#23750294C211yesmaximal, characteristic, complemented
7#23750294C211yesmaximal, characteristic, complemented
8#23750276C213yesmaximal, complemented
9#23750230C213yesmaximal, complemented
10#23750226C213yesmaximal, complemented
11#23750172C213yesmaximal, complemented
12#23750153C213yesmaximal, complemented
13#23750107C211yesmaximal, characteristic, complemented
14C4×#14558737C211yesmaximal, characteristic, complemented
15#520×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558737C2211yescharacteristic
18C2×#14558736C2216yes—
19#18011910C2213yes—
20#18011908C2213yes—
21#18011907C2213yes—
22#18011907C2213yes—
23#18011906C2213yes—
24#18011904C2213yes—
25#18011903C2213yes—
26#18011902C2213yes—
27#18011901C2211yescharacteristic
28#18011897C2213yes—
29#18011896C2213yes—
30#18011896C2216yes—
31#18011894C2216yes—
32#18011893C2213yes—
33#18011889C2213yes—
34#18011886C2213yes—
35#18011886C2213yes—
36#18011885C2216yes—
37#18011885C2213yes—
38#18011885C2213yes—
39#18011884C2213yes—
40#18011884C2216yes—
41#18011884C2216yes—
42#18011882C2213yes—
43#18011882C2213yes—
44Q8×Dic149C2211yescharacteristic
45Q8×Dic149C2211yescharacteristic
46C4×Dic298C2213yes—
47C4×D596C2213yes—
48C4×D596C2213yes—
49C4×D596C2213yes—
50#18011873C2213yes—
51#18011873C2213yes—
52C4×C22⋊D149C2213yes—
53C4×C22⋊D149C2213yes—
54C4×C22⋊D149C2216yes—
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—
66#14558737—24no—
67#14558737—24no—
68#14558736—412no—
69C2×Dic298C2313yes—
70C2×D596C2313yes—
71C2×D596C2313yes—
72C2×D596C2313yes—
73C2×D596C2316yes—
74C2×C22⋊D149C2313yes—
75C2×C22⋊D149C2313yes—
76C2×C22⋊D149C2316yes—
77C2×C22⋊D149C2316yes—
78C2×C4×D149C2313yes—
79C2×C4×D149C2313yes—
80C2×C4×D149C2313yes—
81C2×C4×D149C2316yes—
82C2×C4×D149C2313yes—
83C4⋊Dic149C2311yescharacteristic
84C4⋊Dic149C2313yes—
85C4⋊Dic149C2313yes—
86C4⋊Dic149C2313yes—
87#14558715C2313yes—
88#14558715C2313yes—
89#14558715C2311yescharacteristic
90#14558715C2316yes—
91#14558715C2311yescharacteristic
92#14558715C2311yescharacteristic
93#14558715C2311yescharacteristic
94C22⋊Dic149C2313yes—
95C22⋊Dic149C2313yes—
96#14558713C2313yes—
97#14558713C2313yes—
98#14558713C2313yes—
99#14558713C2313yes—
100#14558713C2316yes—
101#14558713C2313yes—
102#14558713C2313yes—
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C4×Dic149C2313yes—
106C4×Dic149C2311yescharacteristic
107#41×C149—24no—
108C2×Q8×C149C2311yescharacteristic
109C2×D4×C149C2313yes—
110C22×C4×C149C2313yes—
111C22×C4×C149C2316yes—
112C4⋊C4×C149C2313yes—
113C4⋊C4×C149C2313yes—
114C4⋊C4×C149C2313yes—
115C4⋊C4×C149C2311yescharacteristic
116C22⋊C4×C149C2313yes—
117C22⋊C4×C149C2313yes—
118C42×C149C2313yes—
119C42×C149C2313yes—
120C22×D149C2413yes—
121C22×D149C2413yes—
122Dic298—26no—
123D596—26no—
124D596—26no—
125D596—43no—
126D596—46no—
127C22⋊D149—26no—
128C22⋊D149—46no—
129C22⋊D149—43no—
130C22⋊D149—46no—
131C2×Dic149C2411yescharacteristic
132C2×Dic149C2411yescharacteristic
133C2×Dic149C2413yes—
134C2×Dic149C2413yes—
135C2×Dic149C2411yescharacteristic
136C2×Dic149C2411yescharacteristic
137C4×D149—26no—
138C4×D149—26no—
139C4×D149—26no—
140C4×D149—43no—
141C4×D149—46no—
142C23×C149C2413yes—
143Q8×C149#4114yescomplemented
144D4×C149—26no—
145C2×C4×C149—26no—
146C2×C4×C149—46no—
147C2×C4×C149C2411yescharacteristic
148C2×C4×C149C2413yes—
149C2×C4×C149C2411yescharacteristic
150C2×C4×C149C2413yes—
151C2×C4×C149C2411yescharacteristic
152C2×C4×C149C2413yes—
153C22×C149—23no—
154C22×C149—23no—
155C22×C149—23no—
156C22×C149C2511yescharacteristic
157C2×D149—23no—
158C2×D149—23no—
159C2×D149—23no—
160C2×D149—23no—
161C2×D149—23no—
162C2×D149—23no—
163C4×C149—21no—
164C4×C149—23no—
165C4×C149—23no—
166C4×C149—21no—
167C4×C149—21no—
168C4×C149C2×#4116yes—
169Dic149—21no—
170Dic149—21no—
171Dic149—23no—
172Dic149—23no—
173Dic149—21no—
174Dic149H(5,𝔽2)12yes—
175C2×C149—43no—
176C2×C149C22×#4111yescharacteristic
177C2×C149C2×H(5,𝔽2)11yescharacteristic
178C2×C149#58511yescharacteristic
179D149—43no—
180D149—43no—
181C149#351711yescharacteristic, complemented, Sylow
182#3517—1491nomaximal, Sylow
183#557—1491no—
184#553—1493no—
185#553—1493no—
186#550—1491no—
187#550—1491no—
188#550—1491no—
189#550—1493no—
190#547—1493no—
191#540—1493no—
192#540—1493no—
193#532—1493no—
194#529—1493no—
195#520—1491no—
196#520D14911yescharacteristic, complemented
197C4×#41—1491no—
198C2×#41—1491no—
199C2×#41—1491no—
200C2×#41—1496no—
201C2×#41C2×D14911yescharacteristic
202C4⋊Q8—1493no—
203#127—1493no—
204#127—1493no—
205#127—1493no—
206#126—1493no—
207#126—1493no—
208#125—1491no—
209#125—1493no—
210#125—1493no—
211#124—1493no—
212#123—1496no—
213#123—1493no—
214#123—1496no—
215#123—1493no—
216C22⋊Q8—1493no—
217#121—1493no—
218#121—1493no—
219#121—1493no—
220#121—1496no—
221#121—1493no—
222#121—1493no—
223#121—1493no—
224#121—1493no—
225#121—1496no—
226#121—1496no—
227C4×Q8—1491no—
228C4×Q8—1491no—
229C4×Q8—1493no—
230C4×Q8C2×D14911yescharacteristic
231C4×Q8C2×D14911yescharacteristic
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4—1496no—
235C4×D4—1493no—
236C4×D4—1493no—
237C4×D4—1493no—
238C4×D4—1493no—
239C4×D4—1493no—
240C4×D4C2×D14913yes—
241C4×D4C2×D14913yes—
242#117—1493no—
243#117—1493no—
244#117C2×D14913yes—
245C2×C4⋊C4—1493no—
246C2×C4⋊C4C2×D14913yes—
247C2×C42—1493no—
248#41—24no—
249#41—2984no—
250#41—2984no—
251#41—59612no—
252C2×Q8—1493no—
253C2×Q8C22×D14911yescharacteristic
254C2×D4—1493no—
255C2×D4—1496no—
256C2×D4—1493no—
257C2×D4—1496no—
258C2×D4—1493no—
259C2×D4—1493no—
260C2×D4—1493no—
261C2×D4—1496no—
262C2×D4C22×D14913yes—
263C22×C4—1493no—
264C22×C4—1493no—
265C22×C4—1493no—
266C22×C4—1493no—
267C22×C4—1496no—
268C22×C4C22×D14913yes—
269C22×C4C22×D14916yes—
270C4⋊C4—1493no—
271C4⋊C4—1493no—
272C4⋊C4—1491no—
273C4⋊C4—1493no—
274C4⋊C4—1491no—
275C4⋊C4—1491no—
276C4⋊C4—1491no—
277C4⋊C4—1496no—
278C4⋊C4—1491no—
279C4⋊C4—1493no—
280C4⋊C4—1493no—
281C4⋊C4C22×D14913yes—
282C4⋊C4C22×D14913yes—
283C4⋊C4C22×D14913yes—
284C4⋊C4C22×D14911yescharacteristic
285C22⋊C4—1493no—
286C22⋊C4—1493no—
287C22⋊C4—1493no—
288C22⋊C4—1496no—
289C22⋊C4—1493no—
290C22⋊C4—1493no—
291C22⋊C4—1493no—
292C22⋊C4—1493no—
293C22⋊C4—1493no—
294C22⋊C4C22×D14913yes—
295C22⋊C4C22×D14913yes—
296C42—1491no—
297C42—1493no—
298C42—1493no—
299C42—1493no—
300C42C22×D14913yes—
301C42C22×D14913yes—
302C23—1493no—
303C23—1493no—
304C23C23×D14913yes—
305Q8—2986no—
306Q8#1455873614yescomplemented
307D4—26no—
308D4—2986no—
309D4—2986no—
310D4—2986no—
311D4—5963no—
312D4—5966no—
313D4—5966no—
314D4—5963no—
315D4—5966no—
316C2×C4—26no—
317C2×C4—46no—
318C2×C4—2986no—
319C2×C4—2986no—
320C2×C4—2986no—
321C2×C4—5963no—
322C2×C4—5966no—
323C2×C4—1491no—
324C2×C4—1491no—
325C2×C4—1493no—
326C2×C4—1493no—
327C2×C4—1491no—
328C2×C4—1491no—
329C2×C4C23×D14911yescharacteristic
330C2×C4C23×D14913yes—
331C2×C4C23×D14911yescharacteristic
332C2×C4C23×D14913yes—
333C2×C4C23×D14911yescharacteristic
334C2×C4C23×D14913yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—2983no—
339C22—2983no—
340C22—2983no—
341C22—2983no—
342C22—2983no—
343C22—2983no—
344C22C24×D14911yescharacteristic
345C4—21no—
346C4—23no—
347C4—23no—
348C4—21no—
349C4—21no—
350C4—2981no—
351C4—2981no—
352C4—2983no—
353C4—2983no—
354C4—2981no—
355C4—1492no—
356C4C2×#1455873616yes—
357C2—43no—
358C2—5963no—
359C2—5963no—
360C2C22×#1455873611yescharacteristic
361C2H(5,𝔽2)×D14911yescharacteristic
362C2#2375046211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#520×C149C2
hypercenter#520D149
hypocenterC149#3517
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23750351C2semidirect
2. #23750325C2semidirect
3. #23750323C2semidirect
4. #23750297C2semidirect
5. #23750296C2semidirect
6. #23750294C2semidirect
7. #23750294C2semidirect
8. #23750276C2semidirect
9. #23750230C2semidirect
10. #23750226C2semidirect
11. #23750172C2semidirect
12. #23750153C2semidirect
13. #23750107C2semidirect
14. C4×#14558737C2semidirect
15. #520×C149C2semidirect
143. Q8×C149#41semidirect
181. C149#3517semidirect
196. #520D149semidirect
306. Q8#14558736semidirect

Maximal subgroups

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

subgroupindex
#237503512
#237503252
#237503232
#237502972
#237502962
#237502942
#237502762
#237502302
#237502262
#237501722
#237501532
#237501072
C4×#145587372
#520×C1492
#3517149