Finite Group Databasebeta

← #34055453

Subgroups — #34055453

Order 19072.

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

normal subgroups only

This group has 472 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
1#23750135C211yesmaximal, characteristic, complemented
2C2×#18011882C211yesmaximal, characteristic, complemented
3#23749961C211yesmaximal, characteristic, complemented
4#23749947C212yesmaximal, complemented
5#23749941C211yesmaximal, characteristic, complemented
6#23749858C212yesmaximal, complemented
7#23749853C212yesmaximal, complemented
8#23749776C211yesmaximal, characteristic, complemented
9#23749767C212yesmaximal, complemented
10C22⋊C8×D149C211yesmaximal, characteristic, complemented
11#447×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2211yescharacteristic, complemented
13C22×C22⋊D149—21no—
14C22×C4×D149C2211yescharacteristic
15C2×#14558724C2212yescomplemented
16C2×C8×D149C2212yescomplemented
17#18011893—21no—
18#18011891C2211yescharacteristic, complemented
19#18011890C2211yescharacteristic
20#18011882—21no—
21#18011882—21no—
22#18011882C2212yes—
23#18011882C2212yes—
24#18011882C2211yescharacteristic, complemented
25#18011879—21no—
26D4×Dic149—21no—
27#18011871—21no—
28#18011869C2211yescharacteristic, complemented
29C2×C4⋊Dic149C2211yescharacteristic
30C2×C22⋊Dic149—21no—
31#18011845C2212yescomplemented
32#18011842C2212yescomplemented
33#18011836C2212yescomplemented
34#18011832C2212yescomplemented
35#18011832C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic
37#18011823C2212yescomplemented
38C22×D4×C149C2211yescharacteristic
39C2×D8×C149C2212yescomplemented
40#121×C149C2211yescharacteristic
41#102×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic
43C23×D149D411yescharacteristic
44#14558735—42no—
45C2×Dic298C2311yescharacteristic
46C2×C22⋊D149—22no—
47C2×C22⋊D149—22no—
48C2×C22⋊D149—21no—
49C2×C22⋊D149—21no—
50C2×C22⋊D149—44no—
51C22×Dic149—21no—
52C22×Dic149—21no—
53C22×Dic149D411yescharacteristic
54C2×C4×D149—42no—
55C2×C4×D149D411yescharacteristic
56C2×C4×D149D411yescharacteristic
57C2×C4×D149C2311yescharacteristic
58C2×C4×D149C2311yescharacteristic
59#14558724—42no—
60C2×#11924746C2312yes—
61C8×D149—42no—
62C4⋊Dic149—21no—
63C4⋊Dic149—21no—
64C4⋊Dic149C2311yescharacteristic
65C4⋊Dic149C2312yes—
66#14558715—21no—
67C22⋊Dic149—22no—
68C22⋊Dic149—22no—
69C22⋊Dic149—21no—
70C22⋊Dic149—21no—
71#14558713—21no—
72#14558713—21no—
73C4×Dic149—21no—
74C24×C149—21no—
75C2×D4×C149—21no—
76C2×D4×C149—21no—
77C2×D4×C149—42no—
78C2×D4×C149C2312yes—
79C2×D4×C149C2311yescharacteristic
80C22×C4×C149C2311yescharacteristic
81D8×C149—42no—
82C2×C8×C149C2312yes—
83C4⋊C4×C149C2311yescharacteristic
84C22⋊C4×C149—21no—
85C22×D149—22no—
86C22×D149—22no—
87C22×D149—41no—
88C22×D149C2×D411yescharacteristic
89C22×D149C2×D411yescharacteristic
90Dic298—22no—
91C22⋊D149—44no—
92C22⋊D149—42no—
93C22⋊D149—42no—
94C22⋊D149—44no—
95C2×Dic149—22no—
96C2×Dic149—21no—
97C2×Dic149—21no—
98C2×Dic149—41no—
99C2×Dic149—42no—
100C2×Dic149—42no—
101C2×Dic149C2×D411yescharacteristic
102C2×Dic149C2×D411yescharacteristic
103C4×D149—22no—
104C4×D149—22no—
105C4×D149—41no—
106C4×D149—41no—
107#11924746—22no—
108C23×C149—21no—
109C23×C149—22no—
110C23×C149—42no—
111C23×C149—42no—
112C23×C149—42no—
113C23×C149C2×D411yescharacteristic
114D4×C149—22no—
115D4×C149—22no—
116D4×C149—22no—
117D4×C149—42no—
118D4×C149—42no—
119C2×C4×C149—22no—
120C2×C4×C149—21no—
121C2×C4×C149C2×D411yescharacteristic
122C2×C4×C149C2411yescharacteristic
123C8×C149—22no—
124C22×C149—21no—
125C22×C149—21no—
126C22×C149—42no—
127C22×C149—41no—
128C22×C149—41no—
129C22×C149—42no—
130C22×C149—42no—
131C22×C149—41no—
132C22×C149—84no—
133C22×C149#13512yes—
134C22×C149C22×D411yescharacteristic
135C2×D149—21no—
136C2×D149—21no—
137C2×D149—21no—
138C2×D149—21no—
139C2×D149—21no—
140C2×D149—44no—
141C2×D149AGL(1,ℤ/8)12yes—
142C4×C149—21no—
143C4×C149—41no—
144C4×C149H(5,𝔽2)12yes—
145Dic149—21no—
146Dic149—21no—
147Dic149—41no—
148Dic149—42no—
149Dic149—41no—
150C2×C149—22no—
151C2×C149—81no—
152C2×C149—82no—
153C2×C149#53411yescharacteristic
154C2×C149C2×#13511yescharacteristic
155C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
156D149—22no—
157D149—41no—
158C149#318811yescharacteristic, complemented, Sylow
159#3188—1491nomaximal, Sylow
160#525—1491no—
161C2×#121—1491no—
162#484—1491no—
163#482—1492no—
164#481—1491no—
165#466—1492no—
166#465—1492no—
167#450—1491no—
168#449—1492no—
169#447D14911yescharacteristic, complemented
170C2×C22⋊C8—1491no—
171C2×#41—1491no—
172C22×D4—2981no—
173C22×D4C2×D14911yescharacteristic
174C23×C4—1491no—
175C2×SD16—1492no—
176C2×D8C2×D14912yescomplemented
177C22×C8—1492no—
178#123—2981no—
179C22⋊Q8—1491no—
180C22⋊Q8—1491no—
181#121—2981no—
182#121—2981no—
183#121—1492no—
184#121—1492no—
185#121—1491no—
186#121C2×D14911yescharacteristic
187#120—2981no—
188C4×D4—2981no—
189C4×D4—2981no—
190#117—1491no—
191C2×C4⋊C4—1491no—
192C2×C22⋊C4—2981no—
193#107—1492no—
194#106—1492no—
195#103—1492no—
196#102—1492no—
197#102—1492no—
198#102C2×D14912yescomplemented
199C22⋊C8—1491no—
200C22⋊C8—1492no—
201C22⋊C8C2×D14911yescharacteristic
202C24—21no—
203C24—1491no—
204#41—5962no—
205C2×Q8—1491no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—2981no—
210C2×D4—2981no—
211C2×D4—2982no—
212C2×D4—2982no—
213C2×D4—5964no—
214C2×D4C22×D14911yescharacteristic
215C2×D4C22×D14912yes—
216C22×C4—2981no—
217C22×C4—2981no—
218C22×C4—5962no—
219C22×C4—1491no—
220C22×C4—1491no—
221C22×C4—1491no—
222C22×C4—1491no—
223C22×C4—1491no—
224C22×C4C22×D14911yescharacteristic
225SD16—5962no—
226D8—42no—
227C2×C8—5962no—
228C2×C8—1492no—
229C2×C8C22×D14912yes—
230C4⋊C4—2981no—
231C4⋊C4—2981no—
232C4⋊C4—2981no—
233C4⋊C4—1491no—
234C4⋊C4—1492no—
235C4⋊C4C22×D14911yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—2981no—
238C22⋊C4—2981no—
239C22⋊C4—2981no—
240C22⋊C4—2982no—
241C22⋊C4—2982no—
242C22⋊C4—2981no—
243C42—2981no—
244C23—21no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—2982no—
250C23—2982no—
251C23—5961no—
252C23—1491no—
253C23—1491no—
254C23D4×D14911yescharacteristic
255Q8—2982no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—5964no—
262D4—5962no—
263D4—5962no—
264D4—5964no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—2982no—
268C2×C4—2982no—
269C2×C4—2981no—
270C2×C4—2981no—
271C2×C4—2982no—
272C2×C4—5961no—
273C2×C4—5961no—
274C2×C4—5961no—
275C2×C4—5962no—
276C2×C4—5962no—
277C2×C4—1491no—
278C2×C4—1491no—
279C2×C4D4×D14911yescharacteristic
280C2×C4C23×D14911yescharacteristic
281C8—22no—
282C8—2982no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—84no—
292C22—2981no—
293C22—2981no—
294C22—2981no—
295C22—2981no—
296C22—2981no—
297C22—5964no—
298C22—1492no—
299C22#1801193012yes—
300C22C2×D4×D14911yescharacteristic
301C4—21no—
302C4—41no—
303C4—2981no—
304C4—2981no—
305C4—5962no—
306C4—5961no—
307C4—5961no—
308C4#1801195612yes—
309C2—22no—
310C2—82no—
311C2—81no—
312C2—2982no—
313C2—5961no—
314C2#2375018511yescharacteristic
315C2C2×#1801193011yescharacteristic
316C2AGL(1,ℤ/8)×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#447×C149C2
socleC22×C149C22×D4
hypercenter#447D149
hypocenterC149#3188
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23750135C2semidirect
2. C2×#18011882C2semidirect
3. #23749961C2semidirect
4. #23749947C2semidirect
5. #23749941C2semidirect
6. #23749858C2semidirect
7. #23749853C2semidirect
8. #23749776C2semidirect
9. #23749767C2semidirect
10. C22⋊C8×D149C2semidirect
11. #447×C149C2semidirect
12. C2×#14558735C22semidirect
15. C2×#14558724C22semidirect
16. C2×C8×D149C22semidirect
18. #18011891C22semidirect
24. #18011882C22semidirect
28. #18011869C22semidirect
31. #18011845C22semidirect
32. #18011842C22semidirect
33. #18011836C22semidirect
34. #18011832C22semidirect
35. #18011832C22semidirect
37. #18011823C22semidirect
39. C2×D8×C149C22semidirect
41. #102×C149C22semidirect
158. C149#3188semidirect
169. #447D149semidirect
176. C2×D8C2×D149semidirect
198. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237501352
C2×#180118822
#237499612
#237499472
#237499412
#237498582
#237498532
#237497762
#237497672
C22⋊C8×D1492
#447×C1492
#3188149