Finite Group Databasebeta

← #34053970

Subgroups — #34053970

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×#18011944C211yesmaximal, characteristic, complemented
3C22×#14558725C211yesmaximal, characteristic, complemented
4#23750196C211yesmaximal, characteristic, complemented
5#23749789C214yesmaximal, complemented
6#23749778C214yesmaximal, complemented
7#23749631C211yesmaximal, characteristic, complemented
8#23749622C211yesmaximal, characteristic, complemented
9#408×C149C211yesmaximal, characteristic, complemented
10#18011957—24no—
11#18011957—24no—
12C2×#14558737—21no—
13C2×#14558736C2212yes—
14C2×#14558736C2211yescharacteristic
15C22×Dic298C2211yescharacteristic, complemented
16C22×D596C2212yes—
17C22×D596C2211yescharacteristic
18C2×D4×D149—22no—
19C2×D4×D149—21no—
20#18011944—24no—
21C2×Dic596C2212yescomplemented
22C2×#14558725—24no—
23C2×#14558725C2212yescomplemented
24C2×#14558725C2214yescomplemented
25#18011909—21no—
26#18011898—21no—
27#18011892—22no—
28#18011890C2214yescomplemented
29#18011870C2211yescharacteristic, complemented
30#18011836C2212yescomplemented
31#18011833C2212yescomplemented
32#18011833C2214yescomplemented
33C2×#41×C149C2211yescharacteristic
34C2×M16×C149C2211yescharacteristic, complemented
35C22×C8×C149C2211yescharacteristic, complemented
36C22⋊C8×C149C2214yescomplemented
37C23×D149—22no—
38C23×D149—21no—
39#14558737—28no—
40#14558736—24no—
41#14558736—24no—
42#14558736—22no—
43#14558736D414yes—
44C2×Dic298—21no—
45C2×Dic298—22no—
46C2×Dic298—22no—
47C2×Dic298C2312yes—
48C2×Dic298C2311yescharacteristic
49C2×D596—24no—
50C2×D596—24no—
51C2×D596—22no—
52C2×D596—21no—
53C2×D596—22no—
54C2×D596D414yes—
55C2×D596C2311yescharacteristic
56C2×D596C2312yes—
57C2×C22⋊D149—22no—
58C2×C22⋊D149—21no—
59D4×D149—216no—
60D4×D149—28no—
61C22×Dic149—21no—
62C2×C4×D149—22no—
63C2×C4×D149—21no—
64Dic596—24no—
65#14558725—28no—
66#14558725—24no—
67C4⋊Dic149C2312yes—
68#14558715—24no—
69C22⋊Dic149—21no—
70C22⋊Dic149—24no—
71C4×Dic149—21no—
72#41×C149—22no—
73#41×C149—22no—
74C2×Q8×C149D411yescharacteristic
75C2×D4×C149D412yes—
76C2×D4×C149D411yescharacteristic
77C22×C4×C149C2312yes—
78C22×C4×C149C2311yescharacteristic
79M16×C149—22no—
80C2×C8×C149—22no—
81C2×C8×C149C2312yes—
82C2×C8×C149C2312yes—
83C22×D149—22no—
84C22×D149—21no—
85C22×D149—28no—
86C22×D149—24no—
87C22×D149—82no—
88C22×D149—84no—
89Dic298—21no—
90Dic298—22no—
91Dic298—22no—
92Dic298—42no—
93Dic298C2×D412yes—
94D596—28no—
95D596—24no—
96D596—22no—
97D596—21no—
98D596C2×D412yes—
99D596C2×D414yes—
100C22⋊D149—28no—
101C22⋊D149—24no—
102C2×Dic149—21no—
103C2×Dic149—22no—
104C2×Dic149—22no—
105C2×Dic149—42no—
106C4×D149—28no—
107C4×D149—24no—
108C23×C149C2×D412yes—
109C23×C149C2×D411yescharacteristic
110Q8×C149—22no—
111D4×C149—24no—
112D4×C149—22no—
113C2×C4×C149—22no—
114C2×C4×C149—22no—
115C2×C4×C149C2×D412yes—
116C2×C4×C149C2×D412yes—
117C2×C4×C149C2×D412yes—
118C2×C4×C149C2×D411yescharacteristic
119C2×C4×C149C2411yescharacteristic
120C8×C149—22no—
121C8×C149—22no—
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#12012yes—
135C4×C149C22×D411yescharacteristic
136C4×C149C22×D411yescharacteristic
137Dic149—22no—
138Dic149—42no—
139Dic149—42no—
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#536—1491no—
152#452—1494no—
153#450—1494no—
154#418—1491no—
155#416—1491no—
156#408D14911yescharacteristic, complemented
157H(5,𝔽2)—2984no—
158H(5,𝔽2)—2984no—
159C2×#41—2981no—
160C2×#41—1492no—
161C2×#41—1491no—
162C2×#41C2×D14911yescharacteristic
163C22×Q8—1491no—
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×M16C2×D14911yescharacteristic, complemented
174C22×C8C2×D14911yescharacteristic, complemented
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⋊C8C2×D14914yescomplemented
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—2981no—
194C2×Q8—2982no—
195C2×Q8—2982no—
196C2×Q8—1492no—
197C2×Q8—1491no—
198C2×Q8D59611yescharacteristic
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×D4D59612yes—
212C2×D4D59611yescharacteristic
213C22×C4—2982no—
214C22×C4—2981no—
215C22×C4—2981no—
216C22×C4C22×D14912yes—
217C22×C4C22×D14911yescharacteristic
218Q16—2984no—
219SD16—2988no—
220SD16—2984no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D14912yes—
224C2×C8C22×D14912yes—
225C4⋊C4—2984no—
226C4⋊C4—1492no—
227C22⋊C4—2981no—
228C22⋊C4—2984no—
229C42—2981no—
230C23—2988no—
231C23—2982no—
232C23—2981no—
233C23—2984no—
234C23—11922no—
235C23—11924no—
236C23C2×D59612yes—
237C23C2×D59611yescharacteristic
238Q8—22no—
239Q8—2981no—
240Q8—2982no—
241Q8—2982no—
242Q8—5962no—
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—2988no—
257C2×C4—2981no—
258C2×C4—2984no—
259C2×C4—2982no—
260C2×C4—2982no—
261C2×C4—5962no—
262C2×C4D4×D14912yes—
263C2×C4D4×D14912yes—
264C2×C4C2×D59612yes—
265C2×C4C2×D59611yescharacteristic
266C2×C4C23×D14911yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—2984no—
273C22—2982no—
274C22—5964no—
275C22—5962no—
276C22—11924no—
277C22—11928no—
278C22#1801188112yes—
279C22C22×D59611yescharacteristic
280C4—22no—
281C4—2982no—
282C4—5962no—
283C4—5962no—
284C4#1801188112yes—
285C4C2×D4×D14911yescharacteristic
286C4C2×D4×D14911yescharacteristic
287C2—22no—
288C2—42no—
289C2—5964no—
290C2—5962no—
291C2C2×#1801188111yescharacteristic
292C2#2375043712yes—
subgroupG/H
centerC22C22×D596
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#408×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011957C2
hypercenter#408D149
hypocenterC149#3009
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#18011944C2semidirect
3. C22×#14558725C2semidirect
4. #23750196C2semidirect
5. #23749789C2semidirect
6. #23749778C2semidirect
7. #23749631C2semidirect
8. #23749622C2semidirect
9. #408×C149C2semidirect
15. C22×Dic298C22semidirect
21. C2×Dic596C22semidirect
23. C2×#14558725C22semidirect
24. C2×#14558725C22semidirect
28. #18011890C22semidirect
29. #18011870C22semidirect
30. #18011836C22semidirect
31. #18011833C22semidirect
32. #18011833C22semidirect
34. C2×M16×C149C22semidirect
35. C22×C8×C149C22semidirect
36. C22⋊C8×C149C22semidirect
146. C149#3009semidirect
156. #408D149semidirect
173. C2×M16C2×D149semidirect
174. C22×C8C2×D149semidirect
183. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C2×#180119442
C22×#145587252
#237501962
#237497892
#237497782
#237496312
#237496222
#408×C1492
#3009149