Finite Group Databasebeta

← #34053976

Subgroups — #34053976

Order 19072.

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

normal subgroups only

This group has 732 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×#18011958C211yesmaximal, characteristic, complemented
2C22×#14558724C211yesmaximal, characteristic, complemented
3#23750192C211yesmaximal, characteristic, complemented
4#23749767C214yesmaximal, complemented
5#23749762C214yesmaximal, complemented
6#23749635C211yesmaximal, characteristic, complemented
7#23749621C211yesmaximal, characteristic, complemented
8#23749593C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
10#18011958—24no—
11#18011958—24no—
12C2×#14558736—21no—
13C2×#14558736C2211yescharacteristic
14C2×#14558735—22no—
15C2×#14558735C2212yescomplemented
16C22×Dic298—21no—
17C22×Dic298C2211yescharacteristic
18C2×Q8×D149C2212yescomplemented
19C2×#14558724—24no—
20C2×#14558724C2214yes—
21C22×#11924746C2211yescharacteristic
22#18011907—21no—
23#18011898—21no—
24#18011882C2214yes—
25#18011880—22no—
26#18011870C2211yescharacteristic
27#18011836C2214yescomplemented
28#18011835C2212yescomplemented
29#18011832C2212yescomplemented
30#18011823C2214yescomplemented
31C2×#41×C149C2211yescharacteristic
32C22×D4×C149C2211yescharacteristic
33AGL(1,ℤ/8)×C149—24no—
34C2×SD16×C149C2212yes—
35C2×D8×C149C2212yes—
36C2×M16×C149C2211yescharacteristic
37#14558736—28no—
38#14558736—22no—
39#14558736—22no—
40#14558735—216no—
41#14558735—24no—
42#14558735—24no—
43C2×Dic298—28no—
44C2×Dic298—22no—
45C2×Dic298—21no—
46C2×Dic298—22no—
47C2×Dic298D411yescharacteristic
48C2×Dic298D414yescomplemented
49C2×Dic298C2312yes—
50Q8×D149—24no—
51Q8×D149—24no—
52C2×D596D411yescharacteristic
53C2×C22⋊D149—22no—
54C2×C22⋊D149—24no—
55C2×C22⋊D149D412yes—
56C22×Dic149—22no—
57C22×Dic149—21no—
58C2×C4×D149—22no—
59C2×C4×D149C2312yes—
60#14558724—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×D149C2×D412yes—
84Dic298—28no—
85Dic298—22no—
86Dic298—24no—
87Dic298—22no—
88Dic298C2×D414yes—
89D596—22no—
90C22⋊D149—28no—
91C22⋊D149—24no—
92C22⋊D149—84no—
93C2×Dic149—28no—
94C2×Dic149—24no—
95C2×Dic149—22no—
96C2×Dic149—22no—
97C2×Dic149C2×D412yes—
98C4×D149—28no—
99C4×D149—22no—
100C4×D149—22no—
101#11924746—22no—
102C23×C149—21no—
103C23×C149—22no—
104C23×C149—42no—
105C23×C149—82no—
106C23×C149C2×D411yescharacteristic
107Q8×C149—21no—
108Q8×C149C2×D412yes—
109D4×C149—24no—
110D4×C149—21no—
111D4×C149—22no—
112D4×C149—22no—
113D4×C149—42no—
114D4×C149C2×D412yes—
115C2×C4×C149—21no—
116C2×C4×C149—24no—
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2×D412yes—
119C2×C4×C149C2×D412yes—
120C2×C4×C149C2411yescharacteristic
121C8×C149—22no—
122C22×C149—22no—
123C22×C149—22no—
124C22×C149—42no—
125C22×C149—42no—
126C22×C149—44no—
127C22×C149—84no—
128C22×C149#12012yes—
129C22×C149C22×D411yescharacteristic
130C2×D149—22no—
131C2×D149—24no—
132C4×C149—22no—
133C4×C149#12012yes—
134C4×C149C22×D411yescharacteristic
135C4×C149C22×D411yescharacteristic
136Dic149—24no—
137Dic149—22no—
138Dic149—42no—
139C2×C149—22no—
140C2×C149—42no—
141C2×C149—82no—
142C2×C149C2×#12011yescharacteristic
143C2×C149#57712yes—
144D149—42no—
145C149#301011yescharacteristic, complemented, Sylow
146#3010—1491nomaximal, Sylow
147C2×2−1+4—1491no—
148C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
149C22×SD16—1491no—
150#535—1491no—
151#449—1494no—
152#448—1494no—
153#419—1491no—
154#416—1491no—
155#408—1491no—
1562−1+4—2984no—
1572−1+4—2984no—
158C2×#41—2981no—
159C2×#41—2982no—
160C2×#41—1492no—
161C2×#41—1491no—
162C2×#41C2×D14911yescharacteristic
163C22×Q8—2981no—
164C22×Q8—1492no—
165C22×Q8—1491no—
166C22×D4C2×D14911yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16—2984no—
169C2×SD16—1494no—
170C2×SD16C2×D14912yes—
171C2×D8C2×D14912yes—
172C2×M16C2×D14911yescharacteristic
173C22×C8—1491no—
174#127—2981no—
175#124—2981no—
176#121—1494no—
177#120—2982no—
178#117—1491no—
179#103—1492no—
180#103—1494no—
181#102—1492no—
182C22⋊C8—1494no—
183C24—21no—
184#41—22no—
185#41—2988no—
186#41—29816no—
187#41—2984no—
188#41—2984no—
189#41—2982no—
190#41—2982no—
191#41C22⋊D14914yescomplemented
192C2×Q8—2988no—
193C2×Q8—2982no—
194C2×Q8—2984no—
195C2×Q8—2984no—
196C2×Q8—2981no—
197C2×Q8—2982no—
198C2×Q8—1494no—
199C2×Q8—1492no—
200C2×Q8—1491no—
201C2×Q8C22×D14911yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—2982no—
207C2×D4—2984no—
208C2×D4—1492no—
209C2×D4—1491no—
210C2×D4C22×D14912yes—
211C2×D4C22×D14911yescharacteristic
212C22×C4—21no—
213C22×C4—2982no—
214C22×C4—2981no—
215C22×C4—2982no—
216C22×C4—1492no—
217C22×C4C22×D14911yescharacteristic
218SD16—24no—
219SD16—2988no—
220D8—24no—
221M16—22no—
222C2×C8—2982no—
223C2×C8—1492no—
224C2×C8C22×D14912yes—
225C4⋊C4—1492no—
226C22⋊C4—2981no—
227C22⋊C4—2984no—
228C42—2981no—
229C23—21no—
230C23—22no—
231C23—42no—
232C23—82no—
233C23—1492no—
234C23D4×D14911yescharacteristic
235Q8—21no—
236Q8—2988no—
237Q8—2982no—
238Q8—2984no—
239Q8—2982no—
240Q8—1494no—
241Q8C2×C22⋊D14912yes—
242D4—24no—
243D4—21no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—2988no—
248D4—2984no—
249D4—2982no—
250D4—11924no—
251D4C2×C22⋊D14912yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—2988no—
255C2×C4—2988no—
256C2×C4—2982no—
257C2×C4—2984no—
258C2×C4—2982no—
259C2×C4—2982no—
260C2×C4—2982no—
261C2×C4—1492no—
262C2×C4D4×D14912yes—
263C2×C4D4×D14911yescharacteristic
264C2×C4C2×C22⋊D14912yes—
265C2×C4C23×D14911yescharacteristic
266C8—22no—
267C8—2982no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—2984no—
275C22—2982no—
276C22#1801187912yes—
277C22C2×D4×D14911yescharacteristic
278C4—22no—
279C4—2984no—
280C4—2982no—
281C4—5962no—
282C4#1801187912yes—
283C4C2×D4×D14911yescharacteristic
284C4C22×C22⋊D14911yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—5962no—
289C2C2×#1801187911yescharacteristic
290C2#2375043512yes—
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#3010
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×#18011958C2semidirect
2. C22×#14558724C2semidirect
3. #23750192C2semidirect
4. #23749767C2semidirect
5. #23749762C2semidirect
6. #23749635C2semidirect
7. #23749621C2semidirect
8. #23749593C2semidirect
9. C2×AGL(1,ℤ/8)×C149C2semidirect
15. C2×#14558735C22semidirect
18. C2×Q8×D149C22semidirect
27. #18011836C22semidirect
28. #18011835C22semidirect
29. #18011832C22semidirect
30. #18011823C22semidirect
48. C2×Dic298D4semidirect
69. #41×C149D4semidirect
145. C149#3010semidirect
148. C2×AGL(1,ℤ/8)D149semidirect
191. #41C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
C22×#145587242
#237501922
#237497672
#237497622
#237496352
#237496212
#237495932
C2×AGL(1,ℤ/8)×C1492
#3010149