Finite Group Databasebeta

← #34058281

Subgroups — #34058281

Order 19072.

1112 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#23750357C213yesmaximal, complemented
2#23750351C211yesmaximal, characteristic, complemented
3#23750346C213yesmaximal, complemented
4#23750327C213yesmaximal, complemented
5#23750321C213yesmaximal, complemented
6#23750294C211yesmaximal, characteristic, complemented
7#23750286C211yesmaximal, characteristic, complemented
8#23750170C213yesmaximal, complemented
9#23750161C213yesmaximal, complemented
10#23750154C213yesmaximal, complemented
11#41×Dic149C211yesmaximal, characteristic, complemented
12C4×#14558737C211yesmaximal, characteristic, complemented
13C4×#14558735C213yesmaximal, complemented
14C2×Q8×Dic149C211yesmaximal, characteristic, complemented
15#548×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558735C2213yes—
18#18011910C2213yes—
19#18011907C2213yes—
20#18011906C2216yes—
21#18011904C2216yes—
22#18011903C2213yes—
23#18011902C2213yes—
24#18011900C2213yes—
25#18011897C2216yes—
26#18011895C2216yes—
27#18011893C2216yes—
28#18011887C2213yes—
29#18011885C2213yes—
30#18011885C2213yes—
31#18011883C2216yes—
32#18011882C2213yes—
33Q8×Dic149C2213yes—
34Q8×Dic149C2211yescharacteristic
35Q8×Dic149C2211yescharacteristic
36Q8×Dic149C2213yes—
37Q8×Dic149C2211yescharacteristic
38C4×Dic298C2213yes—
39#18011876C2213yes—
40C4×D596C2213yes—
41D4×Dic149C2213yes—
42D4×Dic149C2213yes—
43D4×Dic149C2213yes—
44#18011873C2213yes—
45C4×C22⋊D149C2213yes—
46C4×C22⋊D149C2213yes—
47#18011871C2216yes—
48#18011871C2213yes—
49#18011871C2213yes—
50#18011870C2213yes—
51#18011869C2213yes—
52#18011869C2213yes—
53#18011867C2213yes—
54#18011867C2211yescharacteristic
55C2×C4⋊Dic149C2213yes—
56C2×C4×Dic149C2213yes—
57C2×C4×Dic149C2213yes—
58C42×D149C2213yes—
59C2×#41×C149C2211yescharacteristic
60C22×Q8×C149C2211yescharacteristic
61#124×C149C2213yes—
62C22⋊Q8×C149C2213yes—
63#121×C149C2213yes—
64C4×Q8×C149C2211yescharacteristic
65C4×D4×C149C2213yes—
66#14558737—24no—
67#14558735—212no—
68C2×Dic298C2313yes—
69C2×D596C2313yes—
70C2×C22⋊D149C2313yes—
71C2×C22⋊D149C2313yes—
72C2×C22⋊D149C2316yes—
73C22×Dic149C2313yes—
74C22×Dic149C2313yes—
75C22×Dic149C2313yes—
76C22×Dic149C2311yescharacteristic
77C2×C4×D149C2313yes—
78C2×C4×D149C2313yes—
79C2×C4×D149C2313yes—
80C4⋊Dic149C2316yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2313yes—
83C4⋊Dic149C2313yes—
84C4⋊Dic149C2313yes—
85#14558715C2313yes—
86#14558715C2316yes—
87#14558715C2311yescharacteristic
88#14558715C2313yes—
89#14558715C2311yescharacteristic
90C22⋊Dic149C2313yes—
91C22⋊Dic149C2316yes—
92C22⋊Dic149C2311yescharacteristic
93#14558713C2313yes—
94#14558713C2316yes—
95#14558713C2316yes—
96#14558713C2313yes—
97C4×Dic149C2313yes—
98C4×Dic149C2313yes—
99C4×Dic149C2313yes—
100C4×Dic149C2311yescharacteristic
101C4×Dic149C2313yes—
102C4×Dic149C2311yescharacteristic
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C4×Dic149C2313yes—
106#41×C149—24no—
107C2×Q8×C149—24no—
108C2×Q8×C149C2313yes—
109C2×Q8×C149C2311yescharacteristic
110C2×D4×C149C2313yes—
111C2×D4×C149C2313yes—
112C22×C4×C149C2313yes—
113C22×C4×C149C2313yes—
114C4⋊C4×C149C2313yes—
115C4⋊C4×C149C2313yes—
116C22⋊C4×C149C2316yes—
117C22⋊C4×C149C2311yescharacteristic
118C22⋊C4×C149C2313yes—
119C42×C149C2313yes—
120C22×D149C2413yes—
121Dic298—26no—
122D596—26no—
123C22⋊D149—26no—
124C22⋊D149—26no—
125C22⋊D149—46no—
126C2×Dic149—26no—
127C2×Dic149—43no—
128C2×Dic149—43no—
129C2×Dic149#4114yescomplemented
130C2×Dic149C2411yescharacteristic
131C2×Dic149C2413yes—
132C2×Dic149C2411yescharacteristic
133C2×Dic149C2413yes—
134C2×Dic149C2413yes—
135C2×Dic149C2411yescharacteristic
136C2×Dic149C2411yescharacteristic
137C4×D149—26no—
138C4×D149—26no—
139C4×D149—43no—
140C23×C149C2413yes—
141C23×C149C2411yescharacteristic
142Q8×C149—26no—
143Q8×C149#4114yescomplemented
144D4×C149—26no—
145D4×C149—26no—
146C2×C4×C149—26no—
147C2×C4×C149—26no—
148C2×C4×C149C2411yescharacteristic
149C2×C4×C149C2413yes—
150C2×C4×C149C2413yes—
151C2×C4×C149C2411yescharacteristic
152C2×C4×C149C2413yes—
153C22×C149—21no—
154C22×C149—21no—
155C22×C149—23no—
156C22×C149—23no—
157C22×C149—23no—
158C22×C149C2×#4112yes—
159C22×C149C2511yescharacteristic
160C2×D149—23no—
161C2×D149—23no—
162C2×D149—23no—
163C4×C149—21no—
164C4×C149—23no—
165C4×C149—23no—
166C4×C149—21no—
167C4×C149C2×#4116yes—
168Dic149—21no—
169Dic149—23no—
170Dic149—23no—
171Dic149—21no—
172Dic149—23no—
173Dic149C2×#4112yes—
174Dic149C2×#4112yes—
175C2×C149—22no—
176C2×C149—43no—
177C2×C149C22×#4111yescharacteristic
178C2×C149C22×#4111yescharacteristic
179C2×C149#58511yescharacteristic
180D149—43no—
181C149#348011yescharacteristic, complemented, Sylow
182#3480—1491nomaximal, Sylow
183#559—1493no—
184#557—1491no—
185#556—1493no—
186#553—1493no—
187#553—1493no—
188#550—1491no—
189#548—1491no—
190#548D14911yescharacteristic, complemented
191#532—1493no—
192#529—1493no—
193#529—1493no—
194C4×#41—1491no—
195C4×#41—1491no—
196C4×#41—1493no—
197C2×C4×Q8—1491no—
198C2×#41—1491no—
199C2×#41—1493no—
200C2×#41C2×D14911yescharacteristic
201C22×Q8C2×D14911yescharacteristic
202C4⋊Q8—1493no—
203#127—1493no—
204#126—1496no—
205#126—1496no—
206#125—1493no—
207#125—1493no—
208#124—1493no—
209#124—1496no—
210#124C2×D14913yes—
211#123—1496no—
212#123—1496no—
213C22⋊Q8—1493no—
214C22⋊Q8C2×D14913yes—
215#121—1493no—
216#121—1496no—
217#121—1493no—
218#121—1493no—
219#121C2×D14913yes—
220C4×Q8—1491no—
221C4×Q8—1493no—
222C4×Q8—1491no—
223C4×Q8—1493no—
224C4×Q8—1491no—
225C4×Q8—1493no—
226C4×Q8—1493no—
227C4×Q8C2×D14911yescharacteristic
228C4×D4—1493no—
229C4×D4—1493no—
230C4×D4—1493no—
231C4×D4—1493no—
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4—1493no—
235C4×D4—1493no—
236C4×D4—1496no—
237C4×D4—1493no—
238C4×D4C2×D14913yes—
239#117—1493no—
240#117—1493no—
241#117—1493no—
242#117—1491no—
243#117—1493no—
244C2×C4⋊C4—1493no—
245C2×C42—1493no—
246C2×C42—1493no—
247C2×C42—1493no—
248#41—24no—
249#41—2984no—
250#41—29812no—
251C2×Q8—24no—
252C2×Q8—1493no—
253C2×Q8C22×D14913yes—
254C2×Q8C22×D14911yescharacteristic
255C2×D4—1493no—
256C2×D4—1493no—
257C2×D4—1496no—
258C2×D4—1493no—
259C2×D4C22×D14913yes—
260C2×D4C22×D14913yes—
261C22×C4—1493no—
262C22×C4—1493no—
263C22×C4—1493no—
264C22×C4—1493no—
265C22×C4—1493no—
266C22×C4—1493no—
267C22×C4—1491no—
268C22×C4C22×D14913yes—
269C22×C4C22×D14913yes—
270C4⋊C4—1493no—
271C4⋊C4—1493no—
272C4⋊C4—1496no—
273C4⋊C4—1491no—
274C4⋊C4—1491no—
275C4⋊C4—1493no—
276C4⋊C4—1493no—
277C4⋊C4—1493no—
278C4⋊C4—1493no—
279C4⋊C4—1496no—
280C4⋊C4C22×D14913yes—
281C4⋊C4C22×D14913yes—
282C22⋊C4—1491no—
283C22⋊C4—1493no—
284C22⋊C4—1496no—
285C22⋊C4—1493no—
286C22⋊C4—1496no—
287C22⋊C4—1493no—
288C22⋊C4—1496no—
289C22⋊C4C22×D14911yescharacteristic
290C22⋊C4C22×D14913yes—
291C22⋊C4C22×D14916yes—
292C42—1491no—
293C42—1491no—
294C42—1493no—
295C42—1493no—
296C42—1493no—
297C42—1493no—
298C42—1493no—
299C42—1493no—
300C42—1493no—
301C42C22×D14913yes—
302C23—1493no—
303C23C23×D14913yes—
304C23C23×D14911yescharacteristic
305Q8—26no—
306Q8—2986no—
307Q8#1455873514yescomplemented
308D4—26no—
309D4—26no—
310D4—2986no—
311D4—2986no—
312D4—2986no—
313D4—5966no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2986no—
317C2×C4—2986no—
318C2×C4—2986no—
319C2×C4—5963no—
320C2×C4—5963no—
321C2×C4—5963no—
322C2×C4—1491no—
323C2×C4—1493no—
324C2×C4—1491no—
325C2×C4—1493no—
326C2×C4—1493no—
327C2×C4—1494no—
328C2×C4—1491no—
329C2×C4—1491no—
330C2×C4C23×D14911yescharacteristic
331C2×C4C23×D14913yes—
332C2×C4C23×D14911yescharacteristic
333C2×C4C23×D14913yes—
334C2×C4C23×D14913yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—2983no—
341C22—2983no—
342C22—2983no—
343C22#41×D14912yes—
344C22C24×D14911yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—2983no—
350C4—2981no—
351C4—2983no—
352C4—2981no—
353C4—2983no—
354C4—1492no—
355C4—1492no—
356C4C2×#1455873516yes—
357C2—22no—
358C2—43no—
359C2—5963no—
360C2C2×#41×D14911yescharacteristic
361C2C22×#1455873511yescharacteristic
362C2#2375046211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#548×C149C2
hypercenter#548D149
hypocenterC149#3480
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23750357C2semidirect
2. #23750351C2semidirect
3. #23750346C2semidirect
4. #23750327C2semidirect
5. #23750321C2semidirect
6. #23750294C2semidirect
7. #23750286C2semidirect
8. #23750170C2semidirect
9. #23750161C2semidirect
10. #23750154C2semidirect
11. #41×Dic149C2semidirect
12. C4×#14558737C2semidirect
13. C4×#14558735C2semidirect
14. C2×Q8×Dic149C2semidirect
15. #548×C149C2semidirect
129. C2×Dic149#41semidirect
143. Q8×C149#41semidirect
181. C149#3480semidirect
190. #548D149semidirect
307. Q8#14558735semidirect

Maximal subgroups

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

subgroupindex
#237503572
#237503512
#237503462
#237503272
#237503212
#237502942
#237502862
#237501702
#237501612
#237501542
#41×Dic1492
C4×#145587372
C4×#145587352
C2×Q8×Dic1492
#548×C1492
#3480149