Finite Group Databasebeta

← #34055494

Subgroups — #34055494

Order 19072.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D149C211yesmaximal, characteristic, complemented
2C2×#18011882C211yesmaximal, characteristic
3#23749961C211yesmaximal, characteristic, complemented
4#23749958C211yesmaximal, characteristic, complemented
5#23749937C212yesmaximal, complemented
6#23749870C212yesmaximal, complemented
7#23749868C212yesmaximal, complemented
8#23749776C211yesmaximal, characteristic, complemented
9#23749762C212yesmaximal, complemented
10#23749587C211yesmaximal, characteristic, complemented
11#450×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149C2211yescharacteristic, complemented
13C22×C22⋊D149—21no—
14C22×C4×D149C2211yescharacteristic
15C2×#14558724C2212yescomplemented
16C2×#14558719C2212yescomplemented
17#18011891C2211yescharacteristic, complemented
18#18011890C2211yescharacteristic
19#18011889—21no—
20#18011888—21no—
21#18011887C2211yescharacteristic, complemented
22#18011882—21no—
23#18011882—21no—
24#18011882C2212yes—
25#18011882C2212yes—
26C2×C4⋊Dic149C2211yescharacteristic
27C4⋊C4×D149—21no—
28C4⋊C4×D149C2211yescharacteristic, complemented
29C2×C22⋊Dic149—21no—
30C22⋊C4×D149—21no—
31#18011845C2212yescomplemented
32#18011844C2212yescomplemented
33#18011836C2212yescomplemented
34#18011835C2212yescomplemented
35#18011832C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic
37#18011823C2212yescomplemented
38C22×D4×C149C2211yescharacteristic
39C2×SD16×C149C2212yescomplemented
40C22⋊Q8×C149C2211yescharacteristic
41#102×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic
43C23×D149D411yescharacteristic
44C2×Dic298—21no—
45C2×Dic298C2311yescharacteristic
46Q8×D149—42no—
47C2×C22⋊D149—22no—
48C2×C22⋊D149—22no—
49C2×C22⋊D149—44no—
50C22×Dic149—21no—
51C22×Dic149D411yescharacteristic
52C2×C4×D149—21no—
53C2×C4×D149—21no—
54C2×C4×D149—42no—
55C2×C4×D149D411yescharacteristic
56C2×C4×D149D411yescharacteristic
57C2×C4×D149C2311yescharacteristic
58C2×C4×D149C2311yescharacteristic
59#14558724—42no—
60#14558719—42no—
61C2×#11924746C2312yes—
62C4⋊Dic149—21no—
63C4⋊Dic149—21no—
64C4⋊Dic149C2311yescharacteristic
65C4⋊Dic149C2312yes—
66#14558715—21no—
67#14558715—21no—
68#14558715—21no—
69C22⋊Dic149—22no—
70C22⋊Dic149—22no—
71C22⋊Dic149—21no—
72#14558713—21no—
73#14558713—21no—
74C24×C149—21no—
75C2×Q8×C149C2311yescharacteristic
76C2×D4×C149—21no—
77C2×D4×C149—42no—
78C2×D4×C149C2312yes—
79C22×C4×C149C2311yescharacteristic
80SD16×C149—42no—
81C2×C8×C149C2312yes—
82C4⋊C4×C149—21no—
83C4⋊C4×C149C2311yescharacteristic
84C22⋊C4×C149—21no—
85C22×D149—22no—
86C22×D149—22no—
87C22×D149—41no—
88C22×D149C2×D411yescharacteristic
89C22×D149C2×D411yescharacteristic
90Dic298—22no—
91Dic298—42no—
92C22⋊D149—44no—
93C22⋊D149—44no—
94C2×Dic149—21no—
95C2×Dic149—21no—
96C2×Dic149—22no—
97C2×Dic149—41no—
98C2×Dic149—42no—
99C2×Dic149C2×D411yescharacteristic
100C2×Dic149C2×D411yescharacteristic
101C4×D149—22no—
102C4×D149—22no—
103C4×D149—41no—
104C4×D149—41no—
105C4×D149—42no—
106C4×D149—42no—
107#11924746—22no—
108C23×C149—22no—
109C23×C149—42no—
110C23×C149—42no—
111C23×C149—42no—
112C23×C149C2×D411yescharacteristic
113Q8×C149—22no—
114D4×C149—22no—
115D4×C149—22no—
116D4×C149—42no—
117C2×C4×C149—22no—
118C2×C4×C149—21no—
119C2×C4×C149—21no—
120C2×C4×C149C2×D411yescharacteristic
121C2×C4×C149C2411yescharacteristic
122C8×C149—22no—
123C22×C149—21no—
124C22×C149—21no—
125C22×C149—42no—
126C22×C149—42no—
127C22×C149—42no—
128C22×C149—84no—
129C22×C149#13712yes—
130C22×C149C22×D411yescharacteristic
131C2×D149—21no—
132C2×D149—21no—
133C2×D149—21no—
134C2×D149—21no—
135C2×D149—21no—
136C2×D149—44no—
137C2×D149AGL(1,ℤ/8)12yes—
138C4×C149—21no—
139C4×C149—41no—
140C4×C149—41no—
141C4×C149H(5,𝔽2)12yes—
142Dic149—21no—
143Dic149—21no—
144Dic149—41no—
145Dic149—41no—
146Dic149—42no—
147C2×C149—22no—
148C2×C149—82no—
149C2×C149#53411yescharacteristic
150C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C149C2×#13711yescharacteristic
152D149—22no—
153D149—41no—
154C149#319311yescharacteristic, complemented, Sylow
155#3193—1491nomaximal, Sylow
156C2×C22⋊Q8—1491no—
157C2×#121—1491no—
158#484—1491no—
159#484—1491no—
160#480—1492no—
161#468—1492no—
162#468—1492no—
163#450—1491no—
164#450D14911yescharacteristic, complemented
165#448—1492no—
166#407—1491no—
167C22×Q8—1491no—
168C22×D4—2981no—
169C22×D4C2×D14911yescharacteristic
170C23×C4—1491no—
171C2×SD16—1492no—
172C2×SD16C2×D14912yescomplemented
173C2×M16—1492no—
174C22⋊Q8—2981no—
175C22⋊Q8—2981no—
176C22⋊Q8—1491no—
177C22⋊Q8—1491no—
178C22⋊Q8—1491no—
179C22⋊Q8C2×D14911yescharacteristic
180#121—2981no—
181#121—2981no—
182#121—1492no—
183#121—1492no—
184C2×C4⋊C4—2981no—
185C2×C4⋊C4—1491no—
186C2×C4⋊C4—1491no—
187C2×C22⋊C4—2981no—
188C2×C22⋊C4—2981no—
189#107—1492no—
190#107—1492no—
191#103—1492no—
192#103—1492no—
193#102—1492no—
194#102C2×D14912yescomplemented
195C22⋊C8—1491no—
196C22⋊C8—1492no—
197C22⋊C8C2×D14911yescharacteristic
198C24—21no—
199C24—1491no—
200C2×Q8—2981no—
201C2×Q8—5962no—
202C2×Q8—1491no—
203C2×Q8C22×D14911yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—2982no—
207C2×D4—2982no—
208C2×D4—5964no—
209C2×D4C22×D14912yes—
210C22×C4—2981no—
211C22×C4—2981no—
212C22×C4—2981no—
213C22×C4—5962no—
214C22×C4—1491no—
215C22×C4—1491no—
216C22×C4—1491no—
217C22×C4—1491no—
218C22×C4—1491no—
219C22×C4C22×D14911yescharacteristic
220SD16—42no—
221SD16—5962no—
222M16—5962no—
223C2×C8—1492no—
224C2×C8C22×D14912yes—
225C4⋊C4—21no—
226C4⋊C4—2981no—
227C4⋊C4—2981no—
228C4⋊C4—2981no—
229C4⋊C4—2981no—
230C4⋊C4—2981no—
231C4⋊C4—1491no—
232C4⋊C4—1492no—
233C4⋊C4C22×D14911yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—2981no—
236C22⋊C4—2981no—
237C22⋊C4—2981no—
238C22⋊C4—2982no—
239C22⋊C4—2982no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—2982no—
245C23—2982no—
246C23—5961no—
247C23—1491no—
248C23—1491no—
249C23D4×D14911yescharacteristic
250Q8—22no—
251Q8—2982no—
252Q8—5962no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—5964no—
257D4—5964no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—2982no—
262C2×C4—2982no—
263C2×C4—2981no—
264C2×C4—2981no—
265C2×C4—2982no—
266C2×C4—5961no—
267C2×C4—5961no—
268C2×C4—5962no—
269C2×C4—5962no—
270C2×C4—5961no—
271C2×C4—5962no—
272C2×C4—1491no—
273C2×C4—1491no—
274C2×C4D4×D14911yescharacteristic
275C2×C4C23×D14911yescharacteristic
276C8—22no—
277C8—2982no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—2981no—
285C22—2981no—
286C22—2981no—
287C22—2981no—
288C22—2981no—
289C22—5964no—
290C22—1492no—
291C22#1801194012yes—
292C22C2×D4×D14911yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—2981no—
297C4—2981no—
298C4—5962no—
299C4—5961no—
300C4—5961no—
301C4#1801195612yes—
302C2—22no—
303C2—82no—
304C2—2982no—
305C2—5961no—
306C2#2375018511yescharacteristic
307C2AGL(1,ℤ/8)×D14911yescharacteristic
308C2C2×#1801194011yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#450×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011882C2
hypercenter#450D149
hypocenterC149#3193
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. C22⋊Q8×D149C2semidirect
3. #23749961C2semidirect
4. #23749958C2semidirect
5. #23749937C2semidirect
6. #23749870C2semidirect
7. #23749868C2semidirect
8. #23749776C2semidirect
9. #23749762C2semidirect
10. #23749587C2semidirect
11. #450×C149C2semidirect
12. C2×Q8×D149C22semidirect
15. C2×#14558724C22semidirect
16. C2×#14558719C22semidirect
17. #18011891C22semidirect
21. #18011887C22semidirect
28. C4⋊C4×D149C22semidirect
31. #18011845C22semidirect
32. #18011844C22semidirect
33. #18011836C22semidirect
34. #18011835C22semidirect
35. #18011832C22semidirect
37. #18011823C22semidirect
39. C2×SD16×C149C22semidirect
41. #102×C149C22semidirect
154. C149#3193semidirect
164. #450D149semidirect
172. C2×SD16C2×D149semidirect
194. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1492
C2×#180118822
#237499612
#237499582
#237499372
#237498702
#237498682
#237497762
#237497622
#237495872
#450×C1492
#3193149