Finite Group Databasebeta

← #34053966

Subgroups — #34053966

Order 19072.

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

normal subgroups only

This group has 752 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
2C2×#18011942C211yesmaximal, characteristic, complemented
3C22×#14558723C211yesmaximal, characteristic, complemented
4#23749785C214yesmaximal, complemented
5#23749775C214yesmaximal, complemented
6#23749629C211yesmaximal, characteristic, complemented
7#23749619C211yesmaximal, characteristic, complemented
8#23749592C211yesmaximal, characteristic, complemented
9#536×C149C211yesmaximal, characteristic, complemented
10#18011957—24no—
11#18011957—24no—
12C2×#14558737—21no—
13C2×#14558736C2212yes—
14C2×#14558736C2211yescharacteristic
15C22×D596C2212yes—
16C22×D596C2211yescharacteristic
17C2×D4×D149—22no—
18C2×D4×D149—21no—
19#18011942—24no—
20C2×#14558726C2212yescomplemented
21C2×#14558723—24no—
22C2×#14558723C2212yescomplemented
23C2×#14558723C2214yescomplemented
24C2×#14558720C2211yescharacteristic, complemented
25C22×#11924746C2211yescharacteristic, complemented
26#18011834C2212yescomplemented
27#18011831C2212yescomplemented
28#18011831C2214yescomplemented
29C22⋊#11924746C2214yescomplemented
30C2×#41×C149C2211yescharacteristic
31C22×Q8×C149C2211yescharacteristic, complemented
32C4⋊Q8×C149—21no—
33#124×C149—21no—
34#123×C149—22no—
35C22⋊Q8×C149C2214yescomplemented
36#117×C149C2211yescharacteristic, complemented
37C23×D149—22no—
38C23×D149—21no—
39#14558737—28no—
40#14558736—24no—
41#14558736—24no—
42#14558736—22no—
43#14558736D414yes—
44C2×Dic298C2311yescharacteristic
45C2×D596—24no—
46C2×D596—24no—
47C2×D596—22no—
48C2×D596—21no—
49C2×D596—22no—
50C2×D596D414yes—
51C2×D596C2311yescharacteristic
52C2×D596C2312yes—
53C2×C22⋊D149—22no—
54C2×C22⋊D149—21no—
55D4×D149—216no—
56D4×D149—28no—
57C2×C4×D149—22no—
58C2×C4×D149—21no—
59#14558726—24no—
60#14558723—28no—
61#14558723—24no—
62#14558720—22no—
63C2×#11924746—22no—
64C2×#11924746C2312yes—
65C2×#11924746C2312yes—
66#41×C149—22no—
67#41×C149—22no—
68C2×Q8×C149—21no—
69C2×Q8×C149—22no—
70C2×Q8×C149—22no—
71C2×Q8×C149D411yescharacteristic
72C2×Q8×C149C2312yes—
73C2×D4×C149D412yes—
74C2×D4×C149D411yescharacteristic
75C22×C4×C149—21no—
76C22×C4×C149C2312yes—
77C22×C4×C149C2311yescharacteristic
78C4⋊C4×C149—24no—
79C4⋊C4×C149C2312yes—
80C22⋊C4×C149—21no—
81C22⋊C4×C149—24no—
82C42×C149—21no—
83C22×D149—22no—
84C22×D149—21no—
85C22×D149—28no—
86C22×D149—24no—
87C22×D149—82no—
88C22×D149—84no—
89Dic298—21no—
90Dic298C2×D412yes—
91D596—28no—
92D596—24no—
93D596—22no—
94D596—21no—
95D596C2×D412yes—
96D596C2×D414yes—
97C22⋊D149—28no—
98C22⋊D149—24no—
99C2×Dic149—21no—
100C4×D149—28no—
101C4×D149—24no—
102#11924746—22no—
103#11924746—22no—
104C23×C149C2×D412yes—
105C23×C149C2×D411yescharacteristic
106Q8×C149—22no—
107Q8×C149—22no—
108Q8×C149—22no—
109Q8×C149—42no—
110D4×C149—24no—
111D4×C149—22no—
112C2×C4×C149—22no—
113C2×C4×C149—22no—
114C2×C4×C149—22no—
115C2×C4×C149—22no—
116C2×C4×C149—42no—
117C2×C4×C149C2×D412yes—
118C2×C4×C149C2×D412yes—
119C2×C4×C149C2×D412yes—
120C2×C4×C149C2×D411yescharacteristic
121C2×C4×C149C2411yescharacteristic
122C22×C149—22no—
123C22×C149—22no—
124C22×C149—24no—
125C22×C149#12012yes—
126C22×C149C22×D411yescharacteristic
127C2×D149—24no—
128C2×D149—22no—
129C2×D149—44no—
130C2×D149—42no—
131C2×D149—84no—
132C2×D149—88no—
133C4×C149—22no—
134C4×C149—42no—
135C4×C149—42no—
136C4×C149#12012yes—
137C4×C149C22×D411yescharacteristic
138C4×C149C22×D411yescharacteristic
139Dic149—22no—
140C2×C149—22no—
141C2×C149—42no—
142C2×C149C2×#12011yescharacteristic
143C2×C149#57712yes—
144D149—44no—
145D149—42no—
146C149#300911yescharacteristic, complemented, Sylow
147#3009—1491nomaximal, Sylow
148C2×H(5,𝔽2)—1491no—
149C2×#137—1491no—
150C22×SD16—1491no—
151#536D14911yescharacteristic, complemented
152#452—1494no—
153#450—1494no—
154#418—1491no—
155#416—1491no—
156#408—1491no—
157H(5,𝔽2)—2984no—
158H(5,𝔽2)—2984no—
159C2×#41—2981no—
160C2×#41—1492no—
161C2×#41—1491no—
162C2×#41C2×D14911yescharacteristic
163C22×Q8C2×D14911yescharacteristic, complemented
164C22×D4—2982no—
165C22×D4—2981no—
166C22×D4—1492no—
167C22×D4—1491no—
168#137—2984no—
169C2×Q16—1492no—
170C2×SD16—2984no—
171C2×SD16—1494no—
172C2×SD16—1492no—
173C2×M16—1491no—
174C22×C8—1491no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D14914yescomplemented
179#117C2×D14911yescharacteristic, complemented
180#103—1492no—
181#102—1492no—
182#102—1494no—
183C22⋊C8—1494no—
184C24—2982no—
185C24—2981no—
186#41—22no—
187#41—22no—
188#41—2984no—
189#41—2984no—
190#41—2988no—
191#41—2982no—
192#41—1494no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1491no—
197C2×Q8C22⋊D14911yescharacteristic
198C2×Q8C22×D14912yes—
199C2×D4—2984no—
200C2×D4—2984no—
201C2×D4—29816no—
202C2×D4—2988no—
203C2×D4—2982no—
204C2×D4—2981no—
205C2×D4—2982no—
206C2×D4—2982no—
207C2×D4—2981no—
208C2×D4—1494no—
209C2×D4—1491no—
210C2×D4—1492no—
211C2×D4C22⋊D14912yes—
212C2×D4C22⋊D14911yescharacteristic
213C22×C4—21no—
214C22×C4—2982no—
215C22×C4—2981no—
216C22×C4C22×D14912yes—
217C22×C4C22×D14911yescharacteristic
218Q16—2984no—
219SD16—2988no—
220SD16—2984no—
221M16—2982no—
222C2×C8—2982no—
223C2×C8—1492no—
224C2×C8—1492no—
225C4⋊C4—24no—
226C4⋊C4C22×D14912yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—2988no—
231C23—2982no—
232C23—2981no—
233C23—2984no—
234C23—11922no—
235C23—11924no—
236C23C2×C22⋊D14912yes—
237C23C2×C22⋊D14911yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2981no—
243Q8—1492no—
244D4—24no—
245D4—22no—
246D4—2988no—
247D4—2988no—
248D4—2984no—
249D4—2982no—
250D4—2981no—
251D4—2984no—
252D4—1492no—
253D4—1494no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—2988no—
260C2×C4—2981no—
261C2×C4—2984no—
262C2×C4D4×D14912yes—
263C2×C4D4×D14912yes—
264C2×C4C2×C22⋊D14912yes—
265C2×C4C2×C22⋊D14911yescharacteristic
266C2×C4C23×D14911yescharacteristic
267C8—2982no—
268C8—2982no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—2984no—
273C22—2982no—
274C22—5964no—
275C22—5962no—
276C22—11924no—
277C22—11928no—
278C22#1801187912yes—
279C22C22×C22⋊D14911yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—2982no—
284C4#1801187912yes—
285C4C2×D4×D14911yescharacteristic
286C4C2×D4×D14911yescharacteristic
287C2—22no—
288C2—42no—
289C2—5964no—
290C2—5962no—
291C2C2×#1801187911yescharacteristic
292C2#2375043312yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#536×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011957C2
hypercenter#536D149
hypocenterC149#3009
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#18011942C2semidirect
3. C22×#14558723C2semidirect
4. #23749785C2semidirect
5. #23749775C2semidirect
6. #23749629C2semidirect
7. #23749619C2semidirect
8. #23749592C2semidirect
9. #536×C149C2semidirect
20. C2×#14558726C22semidirect
22. C2×#14558723C22semidirect
23. C2×#14558723C22semidirect
24. C2×#14558720C22semidirect
25. C22×#11924746C22semidirect
26. #18011834C22semidirect
27. #18011831C22semidirect
28. #18011831C22semidirect
29. C22⋊#11924746C22semidirect
31. C22×Q8×C149C22semidirect
35. C22⋊Q8×C149C22semidirect
36. #117×C149C22semidirect
146. C149#3009semidirect
151. #536D149semidirect
163. C22×Q8C2×D149semidirect
178. C22⋊Q8C2×D149semidirect
179. #117C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C2×#180119422
C22×#145587232
#237497852
#237497752
#237496292
#237496192
#237495922
#536×C1492
#3009149