Finite Group Databasebeta

← #34053958

Subgroups — #34053958

Order 19072.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#18011957C211yesmaximal, characteristic, complemented
2C22×#14558721C211yesmaximal, characteristic, complemented
3#23750192C211yesmaximal, characteristic, complemented
4#23749769C214yesmaximal, complemented
5#23749757C214yesmaximal, complemented
6#23749631C211yesmaximal, characteristic, complemented
7#23749621C211yesmaximal, characteristic, complemented
8#23749593C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
10#18011957—24no—
11#18011957—24no—
12C2×#14558737C2212yescomplemented
13C2×#14558736—21no—
14C2×#14558736C2211yescharacteristic
15C22×D596—21no—
16C22×D596C2211yescharacteristic
17C2×D4×D149—22no—
18C2×D4×D149C2212yescomplemented
19C2×#14558721—24no—
20C2×#14558721C2214yescomplemented
21C22×#11924746C2211yescharacteristic, complemented
22#18011907—21no—
23#18011898—21no—
24#18011882C2214yescomplemented
25#18011880—22no—
26#18011870C2211yescharacteristic, complemented
27#18011835C2212yescomplemented
28#18011833C2214yescomplemented
29#18011832C2212yescomplemented
30#18011823C2214yescomplemented
31C2×#41×C149C2211yescharacteristic
32C22×D4×C149C2211yescharacteristic, complemented
33AGL(1,ℤ/8)×C149—24no—
34C2×SD16×C149C2212yescomplemented
35C2×D8×C149C2212yescomplemented
36C2×M16×C149C2211yescharacteristic, complemented
37C23×D149—22no—
38C23×D149—21no—
39#14558737—24no—
40#14558737—24no—
41#14558736—28no—
42#14558736—22no—
43#14558736—22no—
44C2×Dic298D411yescharacteristic
45C2×D596—28no—
46C2×D596—22no—
47C2×D596—21no—
48C2×D596—22no—
49C2×D596D411yescharacteristic
50C2×D596D414yescomplemented
51C2×D596C2312yes—
52C2×C22⋊D149—24no—
53C2×C22⋊D149—22no—
54C2×C22⋊D149D412yes—
55D4×D149—216no—
56D4×D149—24no—
57D4×D149—24no—
58C2×C4×D149—22no—
59C2×C4×D149C2312yes—
60#14558721—28no—
61C2×#11924746—22no—
62C2×#11924746C2312yes—
63C4⋊Dic149C2312yes—
64C22⋊Dic149—24no—
65C22⋊Dic149—21no—
66C4×Dic149—21no—
67C24×C149—21no—
68#41×C149—22no—
69#41×C149D414yescomplemented
70C2×Q8×C149C2311yescharacteristic
71C2×D4×C149—21no—
72C2×D4×C149—22no—
73C2×D4×C149—22no—
74C2×D4×C149—21no—
75C2×D4×C149C2312yes—
76C2×D4×C149C2311yescharacteristic
77C22×C4×C149—21no—
78C22×C4×C149C2311yescharacteristic
79SD16×C149—24no—
80D8×C149—24no—
81M16×C149—22no—
82C2×C8×C149C2312yes—
83C22×D149—22no—
84C22×D149—28no—
85C22×D149—24no—
86C22×D149—84no—
87C22×D149—82no—
88C22×D149C2×D412yes—
89Dic298—22no—
90D596—28no—
91D596—22no—
92D596—24no—
93D596—22no—
94D596C2×D414yes—
95C22⋊D149—28no—
96C22⋊D149—24no—
97C22⋊D149—84no—
98C2×Dic149—22no—
99C2×Dic149C2×D412yes—
100C4×D149—28no—
101C4×D149—22no—
102C4×D149—22no—
103#11924746—22no—
104C23×C149—21no—
105C23×C149—22no—
106C23×C149—42no—
107C23×C149—82no—
108C23×C149C2×D411yescharacteristic
109Q8×C149—21no—
110Q8×C149C2×D412yes—
111D4×C149—24no—
112D4×C149—21no—
113D4×C149—22no—
114D4×C149—22no—
115D4×C149—42no—
116D4×C149C2×D412yes—
117C2×C4×C149—21no—
118C2×C4×C149—24no—
119C2×C4×C149C2×D411yescharacteristic
120C2×C4×C149C2×D412yes—
121C2×C4×C149C2×D412yes—
122C2×C4×C149C2411yescharacteristic
123C8×C149—22no—
124C22×C149—22no—
125C22×C149—22no—
126C22×C149—42no—
127C22×C149—42no—
128C22×C149—44no—
129C22×C149—84no—
130C22×C149#12012yes—
131C22×C149C22×D411yescharacteristic
132C2×D149—22no—
133C2×D149—24no—
134C2×D149—24no—
135C2×D149—44no—
136C2×D149—84no—
137C2×D149—88no—
138C4×C149—22no—
139C4×C149#12012yes—
140C4×C149C22×D411yescharacteristic
141C4×C149C22×D411yescharacteristic
142Dic149—22no—
143Dic149—42no—
144C2×C149—22no—
145C2×C149—42no—
146C2×C149—82no—
147C2×C149C2×#12011yescharacteristic
148C2×C149#57612yes—
149D149—42no—
150D149—44no—
151C149#300811yescharacteristic, complemented, Sylow
152#3008—1491nomaximal, Sylow
153C2×H(5,𝔽2)—1491no—
154C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
155C22×D8—1491no—
156#535—1491no—
157#449—1494no—
158#447—1494no—
159#418—1491no—
160#416—1491no—
161#408—1491no—
162H(5,𝔽2)—2984no—
163H(5,𝔽2)—2984no—
164C2×#41—2981no—
165C2×#41—1492no—
166C2×#41—1491no—
167C2×#41C2×D14911yescharacteristic
168C22×D4—2981no—
169C22×D4—2982no—
170C22×D4—1492no—
171C22×D4—1491no—
172C22×D4C2×D14911yescharacteristic, complemented
173AGL(1,ℤ/8)—24no—
174C2×SD16C2×D14912yescomplemented
175C2×D8—2984no—
176C2×D8—1494no—
177C2×D8C2×D14912yescomplemented
178C2×M16C2×D14911yescharacteristic, complemented
179C22×C8—1491no—
180#127—2981no—
181#124—2981no—
182#121—1494no—
183#120—2982no—
184#117—1491no—
185#103—1492no—
186#102—1492no—
187#102—1494no—
188C22⋊C8—1494no—
189C24—21no—
190C24—2982no—
191C24—2981no—
192#41—22no—
193#41—2988no—
194#41—2984no—
195#41—2984no—
196#41—2982no—
197#41—2982no—
198#41C22⋊D14914yescomplemented
199C2×Q8—1491no—
200C2×Q8C22×D14911yescharacteristic
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—2988no—
206C2×D4—2982no—
207C2×D4—29816no—
208C2×D4—2984no—
209C2×D4—2984no—
210C2×D4—2984no—
211C2×D4—2982no—
212C2×D4—2981no—
213C2×D4—2982no—
214C2×D4—1492no—
215C2×D4—1491no—
216C2×D4—1494no—
217C2×D4—1492no—
218C2×D4C22×D14912yes—
219C2×D4C22×D14911yescharacteristic
220C22×C4—21no—
221C22×C4—2982no—
222C22×C4—1492no—
223C22×C4C22×D14911yescharacteristic
224SD16—24no—
225D8—24no—
226D8—2988no—
227M16—22no—
228C2×C8—2982no—
229C2×C8—1492no—
230C2×C8C22×D14912yes—
231C4⋊C4—1492no—
232C22⋊C4—2981no—
233C22⋊C4—2984no—
234C42—2981no—
235C23—21no—
236C23—22no—
237C23—42no—
238C23—82no—
239C23—2988no—
240C23—2982no—
241C23—2984no—
242C23—1492no—
243C23—11922no—
244C23—11924no—
245C23D4×D14911yescharacteristic
246Q8—21no—
247Q8—2982no—
248Q8C2×C22⋊D14912yes—
249D4—24no—
250D4—21no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—2988no—
255D4—2988no—
256D4—2984no—
257D4—2982no—
258D4—2982no—
259D4—2984no—
260D4—1494no—
261D4—11924no—
262D4C2×C22⋊D14912yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—2988no—
266C2×C4—2982no—
267C2×C4—2982no—
268C2×C4—2982no—
269C2×C4—1492no—
270C2×C4D4×D14912yes—
271C2×C4D4×D14911yescharacteristic
272C2×C4C2×C22⋊D14912yes—
273C2×C4C23×D14911yescharacteristic
274C8—22no—
275C8—2982no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—44no—
280C22—42no—
281C22—84no—
282C22—2984no—
283C22—2982no—
284C22—2984no—
285C22—5964no—
286C22—11924no—
287C22—11928no—
288C22#1801187912yes—
289C22C2×D4×D14911yescharacteristic
290C4—22no—
291C4—2982no—
292C4—5962no—
293C4#1801187912yes—
294C4C2×D4×D14911yescharacteristic
295C4C22×C22⋊D14911yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—5962no—
300C2—5964no—
301C2C2×#1801187911yescharacteristic
302C2#2375042912yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×AGL(1,ℤ/8)×C149C2
socleC22×C149C22×D4
hypercenterC2×AGL(1,ℤ/8)D149
hypocenterC149#3008
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18011957C2semidirect
2. C22×#14558721C2semidirect
3. #23750192C2semidirect
4. #23749769C2semidirect
5. #23749757C2semidirect
6. #23749631C2semidirect
7. #23749621C2semidirect
8. #23749593C2semidirect
9. C2×AGL(1,ℤ/8)×C149C2semidirect
12. C2×#14558737C22semidirect
18. C2×D4×D149C22semidirect
20. C2×#14558721C22semidirect
21. C22×#11924746C22semidirect
24. #18011882C22semidirect
26. #18011870C22semidirect
27. #18011835C22semidirect
28. #18011833C22semidirect
29. #18011832C22semidirect
30. #18011823C22semidirect
32. C22×D4×C149C22semidirect
34. C2×SD16×C149C22semidirect
35. C2×D8×C149C22semidirect
36. C2×M16×C149C22semidirect
50. C2×D596D4semidirect
69. #41×C149D4semidirect
151. C149#3008semidirect
154. C2×AGL(1,ℤ/8)D149semidirect
172. C22×D4C2×D149semidirect
174. C2×SD16C2×D149semidirect
177. C2×D8C2×D149semidirect
178. C2×M16C2×D149semidirect
198. #41C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C22×#145587212
#237501922
#237497692
#237497572
#237496312
#237496212
#237495932
C2×AGL(1,ℤ/8)×C1492
#3008149