Finite Group Databasebeta

← #34053980

Subgroups — #34053980

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
2C2×#18011936C211yesmaximal, characteristic, complemented
3C22×#14558725C211yesmaximal, characteristic, complemented
4#23750194C211yesmaximal, characteristic, complemented
5#23749771C214yesmaximal, complemented
6#23749765C214yesmaximal, complemented
7#23749633C211yesmaximal, characteristic, complemented
8#23749591C211yesmaximal, characteristic, complemented
9#416×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
19C22×D596C2211yescharacteristic
20#18011936—24no—
21C2×#14558725—24no—
22C2×#14558725C2214yes—
23C2×#14558723C2212yes—
24C2×#14558721C2212yes—
25C2×#14558720C2211yescharacteristic
26#18011908—21no—
27#18011899—21no—
28#18011886C2214yes—
29#18011881—22no—
30#18011834C2214yescomplemented
31#18011823C2214yescomplemented
32C2×#41×C149C2211yescharacteristic
33C22×C8×C149C2211yescharacteristic
34#117×C149C2211yescharacteristic
35#103×C149C2212yescomplemented
36#102×C149C2212yescomplemented
37C23×D149—21no—
38#14558736—28no—
39#14558736—22no—
40#14558736—22no—
41#14558735—216no—
42#14558735—24no—
43#14558735—24no—
44C2×Dic298—28no—
45C2×Dic298—22no—
46C2×Dic298—21no—
47C2×Dic298—22no—
48C2×Dic298D411yescharacteristic
49C2×Dic298D414yescomplemented
50C2×Dic298C2312yes—
51Q8×D149—24no—
52Q8×D149—24no—
53C2×D596—24no—
54C2×D596—22no—
55C2×D596—22no—
56C2×D596—21no—
57C2×D596D411yescharacteristic
58C2×D596C2312yes—
59C2×C22⋊D149—22no—
60C2×C22⋊D149D412yes—
61C22×Dic149—22no—
62C22×Dic149—21no—
63C2×C4×D149—22no—
64C2×C4×D149C2312yes—
65#14558725—28no—
66#14558723—24no—
67#14558721—24no—
68#14558720—22no—
69C2×#11924746C2312yes—
70#14558713—24no—
71#41×C149—22no—
72#41×C149D414yescomplemented
73C2×Q8×C149C2311yescharacteristic
74C2×D4×C149—21no—
75C2×D4×C149C2311yescharacteristic
76C22×C4×C149—21no—
77C22×C4×C149C2311yescharacteristic
78C2×C8×C149—22no—
79C2×C8×C149C2312yes—
80C4⋊C4×C149C2312yes—
81C22⋊C4×C149—21no—
82C42×C149—21no—
83C22×D149—22no—
84C22×D149—42no—
85C22×D149—82no—
86C22×D149C2×D412yes—
87Dic298—28no—
88Dic298—22no—
89Dic298—24no—
90Dic298—22no—
91Dic298C2×D414yes—
92D596—22no—
93D596—22no—
94D596—22no—
95D596—42no—
96D596—84no—
97C22⋊D149—28no—
98C22⋊D149—24no—
99C2×Dic149—28no—
100C2×Dic149—24no—
101C2×Dic149—22no—
102C2×Dic149C2×D412yes—
103C4×D149—28no—
104C4×D149—22no—
105C4×D149—22no—
106#11924746—22no—
107C23×C149—21no—
108C23×C149C2×D411yescharacteristic
109Q8×C149—21no—
110Q8×C149C2×D412yes—
111D4×C149—24no—
112D4×C149—21no—
113D4×C149C2×D412yes—
114C2×C4×C149—21no—
115C2×C4×C149—24no—
116C2×C4×C149—22no—
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#12012yes—
126C22×C149C22×D411yescharacteristic
127C2×D149—22no—
128C2×D149—24no—
129C2×D149—42no—
130C2×D149—44no—
131C2×D149—84no—
132C4×C149—22no—
133C4×C149—42no—
134C4×C149#12012yes—
135C4×C149C22×D411yescharacteristic
136C4×C149C22×D411yescharacteristic
137Dic149—24no—
138Dic149—22no—
139C2×C149—22no—
140C2×C149—42no—
141C2×C149C2×#12011yescharacteristic
142C2×C149#57712yes—
143D149—42no—
144D149—82no—
145C149#301011yescharacteristic, complemented, Sylow
146#3010—1491nomaximal, Sylow
147C2×2−1+4—1491no—
148C2×AGL(1,ℤ/8)—1491no—
149C22×SD16—1491no—
150#535—1491no—
151#449—1494no—
152#448—1494no—
153#419—1491no—
154#416D14911yescharacteristic, complemented
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×D4—1491no—
167AGL(1,ℤ/8)—2984no—
168C2×SD16—2984no—
169C2×SD16—1492no—
170C2×SD16—1494no—
171C2×D8—1492no—
172C2×M16—1491no—
173C22×C8C2×D14911yescharacteristic
174#127—2981no—
175#124—2981no—
176#121—1494no—
177#120—2982no—
178#117C2×D14911yescharacteristic
179#103—1494no—
180#103C2×D14912yescomplemented
181#102C2×D14912yescomplemented
182C22⋊C8—1494no—
183C24—2981no—
184#41—22no—
185#41—2988no—
186#41—29816no—
187#41—2984no—
188#41—2984no—
189#41—2982no—
190#41—2982no—
191#41D59614yescomplemented
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—2984no—
204C2×D4—2982no—
205C2×D4—2982no—
206C2×D4—2981no—
207C2×D4—2982no—
208C2×D4—1492no—
209C2×D4—1491no—
210C2×D4—1492no—
211C2×D4C22×D14911yescharacteristic
212C22×C4—21no—
213C22×C4—2982no—
214C22×C4—2981no—
215C22×C4—2982no—
216C22×C4—1492no—
217C22×C4C22×D14911yescharacteristic
218SD16—2988no—
219SD16—2984no—
220D8—2984no—
221M16—2982no—
222C2×C8—22no—
223C2×C8—1492no—
224C2×C8C22×D14912yes—
225C4⋊C4C22×D14912yes—
226C22⋊C4—21no—
227C22⋊C4—2984no—
228C42—21no—
229C23—21no—
230C23—2982no—
231C23—5962no—
232C23—1492no—
233C23—11922no—
234C23D4×D14911yescharacteristic
235Q8—21no—
236Q8—2988no—
237Q8—2982no—
238Q8—2984no—
239Q8—2982no—
240Q8—1494no—
241Q8C2×D59612yes—
242D4—24no—
243D4—21no—
244D4—2988no—
245D4—2984no—
246D4—2982no—
247D4—2982no—
248D4—2982no—
249D4—5962no—
250D4—11924no—
251D4C2×D59612yes—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—2988no—
256C2×C4—2988no—
257C2×C4—2982no—
258C2×C4—2984no—
259C2×C4—2982no—
260C2×C4—2982no—
261C2×C4—1492no—
262C2×C4D4×D14912yes—
263C2×C4D4×D14911yescharacteristic
264C2×C4C2×D59612yes—
265C2×C4C23×D14911yescharacteristic
266C8—22no—
267C8—2982no—
268C22—22no—
269C22—22no—
270C22—42no—
271C22—2984no—
272C22—2982no—
273C22—5964no—
274C22—5962no—
275C22—11924no—
276C22#1801188112yes—
277C22C2×D4×D14911yescharacteristic
278C4—22no—
279C4—42no—
280C4—2984no—
281C4—2982no—
282C4#1801188112yes—
283C4C2×D4×D14911yescharacteristic
284C4C22×D59611yescharacteristic
285C2—22no—
286C2—42no—
287C2—5962no—
288C2—11922no—
289C2C2×#1801188111yescharacteristic
290C2#2375043212yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#416×C149C2
socleC22×C149C22×D4
hypercenter#416D149
hypocenterC149#3010
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#18011958C2semidirect
2. C2×#18011936C2semidirect
3. C22×#14558725C2semidirect
4. #23750194C2semidirect
5. #23749771C2semidirect
6. #23749765C2semidirect
7. #23749633C2semidirect
8. #23749591C2semidirect
9. #416×C149C2semidirect
15. C2×#14558735C22semidirect
18. C2×Q8×D149C22semidirect
30. #18011834C22semidirect
31. #18011823C22semidirect
35. #103×C149C22semidirect
36. #102×C149C22semidirect
49. C2×Dic298D4semidirect
72. #41×C149D4semidirect
145. C149#3010semidirect
154. #416D149semidirect
180. #103C2×D149semidirect
181. #102C2×D149semidirect
191. #41D596semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
C2×#180119362
C22×#145587252
#237501942
#237497712
#237497652
#237496332
#237495912
#416×C1492
#3010149