Finite Group Databasebeta

← #34053967

Subgroups — #34053967

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
2C22×#14558723C211yesmaximal, characteristic, complemented
3#23750199C211yesmaximal, characteristic, complemented
4#23749786C214yesmaximal, complemented
5#23749774C214yesmaximal, complemented
6#23749631C211yesmaximal, characteristic, complemented
7#23749621C211yesmaximal, characteristic, complemented
8#23749593C211yesmaximal, characteristic, complemented
9C2×#137×C149C211yesmaximal, characteristic, complemented
10#18011957—24no—
11#18011957—24no—
12C2×#14558737C2212yes—
13C2×#14558736—21no—
14C2×#14558736C2211yescharacteristic
15C22×D596—21no—
16C22×D596C2211yescharacteristic
17C2×D4×D149—22no—
18C2×D4×D149C2212yes—
19C2×#14558723—24no—
20C2×#14558723C2214yescomplemented
21C22×#11924746C2211yescharacteristic, complemented
22#18011909—21no—
23#18011900—21no—
24#18011894—22no—
25#18011887C2214yescomplemented
26#18011870C2211yescharacteristic, complemented
27#18011835C2212yescomplemented
28#18011833C2214yescomplemented
29#18011832C2212yescomplemented
30#18011823C2214yescomplemented
31C2×#41×C149C2211yescharacteristic
32C22×Q8×C149C2211yescharacteristic, complemented
33#137×C149—24no—
34C2×Q16×C149C2212yescomplemented
35C2×SD16×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×D596D414yes—
51C2×D596C2312yes—
52C2×C22⋊D149—22no—
53C2×C22⋊D149D412yes—
54D4×D149—216no—
55D4×D149—24no—
56D4×D149—24no—
57C2×C4×D149—22no—
58C2×C4×D149C2312yes—
59#14558723—28no—
60C2×#11924746—22no—
61C2×#11924746C2312yes—
62C4⋊Dic149C2312yes—
63#14558715—24no—
64C22⋊Dic149—21no—
65#14558713—24no—
66C4×Dic149—21no—
67#41×C149—22no—
68#41×C149D414yes—
69C2×Q8×C149—21no—
70C2×Q8×C149—22no—
71C2×Q8×C149—22no—
72C2×Q8×C149C2312yes—
73C2×Q8×C149C2311yescharacteristic
74C2×D4×C149—21no—
75C2×D4×C149C2311yescharacteristic
76C22×C4×C149—21no—
77C22×C4×C149—21no—
78C22×C4×C149C2311yescharacteristic
79Q16×C149—24no—
80SD16×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—
97C2×Dic149—22no—
98C2×Dic149C2×D412yes—
99C4×D149—28no—
100C4×D149—22no—
101C4×D149—22no—
102#11924746—22no—
103C23×C149—21no—
104C23×C149C2×D411yescharacteristic
105Q8×C149—21no—
106Q8×C149—22no—
107Q8×C149—22no—
108Q8×C149—42no—
109Q8×C149C2×D412yes—
110D4×C149—24no—
111D4×C149—21no—
112D4×C149C2×D412yes—
113C2×C4×C149—21no—
114C2×C4×C149—24no—
115C2×C4×C149—22no—
116C2×C4×C149—42no—
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—24no—
130C2×D149—44no—
131C2×D149—84no—
132C2×D149—88no—
133C4×C149—22no—
134C4×C149—42no—
135C4×C149#12012yes—
136C4×C149C22×D411yescharacteristic
137C4×C149C22×D411yescharacteristic
138Dic149—22no—
139Dic149—42no—
140C2×C149—22no—
141C2×C149—42no—
142C2×C149C2×#12011yescharacteristic
143C2×C149#57712yes—
144D149—42no—
145D149—44no—
146C149#300911yescharacteristic, complemented, Sylow
147#3009—1491nomaximal, Sylow
148C2×H(5,𝔽2)—1491no—
149C2×#137D14911yescharacteristic, complemented
150C22×SD16—1491no—
151#536—1491no—
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—2981no—
165C22×D4—2982no—
166C22×D4—1492no—
167C22×D4—1491no—
168#137—24no—
169C2×Q16C2×D14912yescomplemented
170C2×SD16—2984no—
171C2×SD16—1494no—
172C2×SD16C2×D14912yescomplemented
173C2×M16C2×D14911yescharacteristic, complemented
174C22×C8—1491no—
175C4⋊Q8—2981no—
176#124—2981no—
177#123—2982no—
178C22⋊Q8—1494no—
179#117—1491no—
180#103—1492no—
181#102—1492no—
182#102—1494no—
183C22⋊C8—1494no—
184C24—2982no—
185C24—2981no—
186#41—22no—
187#41—2988no—
188#41—2984no—
189#41—2984no—
190#41—2982no—
191#41—2982no—
192#41C22⋊D14914yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1491no—
197C2×Q8C22×D14912yes—
198C2×Q8C22×D14911yescharacteristic
199C2×D4—21no—
200C2×D4—2988no—
201C2×D4—2982no—
202C2×D4—29816no—
203C2×D4—2984no—
204C2×D4—2984no—
205C2×D4—2982no—
206C2×D4—2981no—
207C2×D4—2982no—
208C2×D4—1492no—
209C2×D4—1491no—
210C2×D4—1494no—
211C2×D4—1492no—
212C2×D4C22×D14911yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2982no—
216C22×C4—1492no—
217C22×C4C22×D14911yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2988no—
221M16—22no—
222C2×C8—2982no—
223C2×C8—1492no—
224C2×C8C22×D14912yes—
225C4⋊C4—2984no—
226C4⋊C4—1492no—
227C22⋊C4—2984no—
228C22⋊C4—2981no—
229C42—2981no—
230C23—21no—
231C23—2988no—
232C23—2982no—
233C23—2984no—
234C23—1492no—
235C23—11922no—
236C23—11924no—
237C23D4×D14911yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2982no—
243Q8C2×C22⋊D14912yes—
244D4—24no—
245D4—21no—
246D4—2988no—
247D4—2988no—
248D4—2984no—
249D4—2982no—
250D4—2982no—
251D4—2984no—
252D4—1494no—
253D4C2×C22⋊D14912yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—2988no—
259C2×C4—2982no—
260C2×C4—2982no—
261C2×C4—2982no—
262C2×C4—1492no—
263C2×C4D4×D14912yes—
264C2×C4D4×D14911yescharacteristic
265C2×C4C2×C22⋊D14912yes—
266C2×C4C23×D14911yescharacteristic
267C8—22no—
268C8—2982no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—2984no—
273C22—2982no—
274C22—2984no—
275C22—5964no—
276C22—11924no—
277C22—11928no—
278C22#1801187912yes—
279C22C2×D4×D14911yescharacteristic
280C4—22no—
281C4—42no—
282C4—2982no—
283C4—5962no—
284C4#1801187912yes—
285C4C2×D4×D14911yescharacteristic
286C4C22×C22⋊D14911yescharacteristic
287C2—22no—
288C2—42no—
289C2—5962no—
290C2—5964no—
291C2C2×#1801187911yescharacteristic
292C2#2375043412yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×#137×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011957C2
hypercenterC2×#137D149
hypocenterC149#3009
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14558723C2semidirect
3. #23750199C2semidirect
4. #23749786C2semidirect
5. #23749774C2semidirect
6. #23749631C2semidirect
7. #23749621C2semidirect
8. #23749593C2semidirect
9. C2×#137×C149C2semidirect
20. C2×#14558723C22semidirect
21. C22×#11924746C22semidirect
25. #18011887C22semidirect
26. #18011870C22semidirect
27. #18011835C22semidirect
28. #18011833C22semidirect
29. #18011832C22semidirect
30. #18011823C22semidirect
32. C22×Q8×C149C22semidirect
34. C2×Q16×C149C22semidirect
35. C2×SD16×C149C22semidirect
36. C2×M16×C149C22semidirect
146. C149#3009semidirect
149. C2×#137D149semidirect
163. C22×Q8C2×D149semidirect
169. C2×Q16C2×D149semidirect
172. C2×SD16C2×D149semidirect
173. C2×M16C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C22×#145587232
#237501992
#237497862
#237497742
#237496312
#237496212
#237495932
C2×#137×C1492
#3009149