Finite Group Databasebeta

← #34055659

Subgroups — #34055659

Order 19072.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#23750161C211yesmaximal, characteristic, complemented
2C2×#18011898C211yesmaximal, characteristic, complemented
3#23750006C212yesmaximal, complemented
4#23749978C212yesmaximal, complemented
5#23749845C212yesmaximal, complemented
6#23749839C212yesmaximal, complemented
7#23749780C211yesmaximal, characteristic, complemented
8#23749766C211yesmaximal, characteristic, complemented
9#23749759C211yesmaximal, characteristic
10#23749675C211yesmaximal, characteristic, complemented
11#450×C149C211yesmaximal, characteristic, complemented
12C22×Dic298C2211yescharacteristic
13C2×#14558726C2212yescomplemented
14C2×#14558725C2212yescomplemented
15C2×#14558721C2212yes—
16#18011906—21no—
17#18011902C2211yescharacteristic, complemented
18#18011900C2211yescharacteristic, complemented
19#18011898—21no—
20#18011898—21no—
21#18011898C2212yes—
22#18011898C2212yes—
23#18011895—21no—
24#18011885C2211yescharacteristic
25Q8×Dic149C2211yescharacteristic, complemented
26#18011873C2211yescharacteristic, complemented
27#18011871—21no—
28#18011869—21no—
29C2×C22⋊Dic149—21no—
30C2×C22⋊Dic149—21no—
31C2×C4×Dic149C2211yescharacteristic
32#18011840C2212yescomplemented
33#18011836C2212yescomplemented
34#18011834C2212yescomplemented
35#18011832C2212yes—
36C22⋊#11924746C2211yescharacteristic
37C8×Dic149C2212yescomplemented
38C22×D4×C149C2211yescharacteristic
39C2×SD16×C149C2212yes—
40C22⋊Q8×C149C2211yescharacteristic
41#102×C149C2212yes—
42C22⋊C8×C149C2211yescharacteristic
43C2×Dic298—21no—
44C2×Dic298—42no—
45C2×Dic298C2312yes—
46C2×D596C2311yescharacteristic
47C2×C22⋊D149—21no—
48C22×Dic149—21no—
49C22×Dic149D411yescharacteristic
50C22×Dic149D411yescharacteristic
51C2×C4×D149—21no—
52#14558726—42no—
53#14558725—42no—
54#14558721—42no—
55C2×#11924746C2312yes—
56C4⋊Dic149—21no—
57C4⋊Dic149—21no—
58C4⋊Dic149C2311yescharacteristic
59#14558715—21no—
60C22⋊Dic149—22no—
61C22⋊Dic149—22no—
62C22⋊Dic149—22no—
63C22⋊Dic149—22no—
64#14558713—21no—
65#14558713—21no—
66#14558713—21no—
67#14558713—21no—
68C4×Dic149—21no—
69C4×Dic149D411yescharacteristic
70C4×Dic149D411yescharacteristic
71C4×Dic149C2311yescharacteristic
72C4×Dic149C2311yescharacteristic
73C24×C149—21no—
74C2×Q8×C149C2311yescharacteristic
75C2×D4×C149—21no—
76C2×D4×C149—42no—
77C2×D4×C149C2312yes—
78C22×C4×C149C2311yescharacteristic
79SD16×C149—42no—
80C2×C8×C149C2312yes—
81C4⋊C4×C149—21no—
82C4⋊C4×C149C2311yescharacteristic
83C22⋊C4×C149—21no—
84C22×D149—21no—
85Dic298—22no—
86Dic298—22no—
87Dic298—42no—
88D596—22no—
89C22⋊D149—42no—
90C2×Dic149—22no—
91C2×Dic149—22no—
92C2×Dic149—21no—
93C2×Dic149—41no—
94C2×Dic149—42no—
95C2×Dic149C2×D411yescharacteristic
96C2×Dic149C2×D411yescharacteristic
97C2×Dic149C2×D411yescharacteristic
98C2×Dic149C2×D411yescharacteristic
99C4×D149—42no—
100#11924746—22no—
101C23×C149—22no—
102C23×C149—42no—
103C23×C149—42no—
104C23×C149—42no—
105C23×C149C2×D411yescharacteristic
106Q8×C149—22no—
107D4×C149—22no—
108D4×C149—22no—
109D4×C149—42no—
110C2×C4×C149—22no—
111C2×C4×C149—21no—
112C2×C4×C149—21no—
113C2×C4×C149C2×D411yescharacteristic
114C2×C4×C149C2411yescharacteristic
115C8×C149—22no—
116C22×C149—21no—
117C22×C149—21no—
118C22×C149—42no—
119C22×C149—42no—
120C22×C149—42no—
121C22×C149—84no—
122C22×C149#13512yes—
123C22×C149C22×D411yescharacteristic
124C2×D149—41no—
125C2×D149—41no—
126C2×D149—41no—
127C4×C149—21no—
128C4×C149—41no—
129C4×C149—41no—
130C4×C149H(5,𝔽2)12yes—
131Dic149—21no—
132Dic149—21no—
133Dic149—21no—
134Dic149—21no—
135Dic149—42no—
136Dic149—41no—
137C2×C149—22no—
138C2×C149—82no—
139C2×C149#53511yescharacteristic
140C2×C149C2×#13511yescharacteristic
141C2×C149#57511yescharacteristic
142D149—81no—
143C149#321111yescharacteristic, complemented, Sylow
144#3211—1491nomaximal, Sylow
145#529—1491no—
146C2×#124—1491no—
147#495—1492no—
148#488—1492no—
149#464—1492no—
150#463—1492no—
151#451—1491no—
152#450D14911yescharacteristic, complemented
153#448—1491no—
154#447—1491no—
155#431—1491no—
156C22×Q8—1491no—
157C22×D4C2×D14911yescharacteristic
158C2×Q16—1492no—
159C2×SD16—1492no—
160C2×SD16C2×D14912yes—
161C2×D8—1492no—
162#126—2981no—
163#125—1491no—
164#124—2981no—
165#124—2981no—
166#124—1491no—
167#124—1492no—
168#124—1492no—
169#123—2981no—
170C22⋊Q8C2×D14911yescharacteristic
171#121—1491no—
172C4×Q8—1491no—
173C4×D4—2981no—
174C4×D4—1491no—
175#117—2981no—
176C2×C22⋊C4—2981no—
177C2×C22⋊C4—2981no—
178C2×C42—1491no—
179C4⋊C8—1492no—
180#103—1492no—
181#103—1492no—
182#102—1492no—
183#102C2×D14912yes—
184C22⋊C8—1491no—
185C22⋊C8C2×D14911yescharacteristic
186C4×C8—1492no—
187C24—21no—
188C2×Q8—2981no—
189C2×Q8—5962no—
190C2×Q8—1492no—
191C2×Q8C22×D14911yescharacteristic
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—2981no—
195C2×D4—1491no—
196C2×D4C22×D14912yes—
197C22×C4—2981no—
198C22×C4—2981no—
199C22×C4—1491no—
200C22×C4—1491no—
201C22×C4C22×D14911yescharacteristic
202Q16—5962no—
203SD16—42no—
204SD16—5962no—
205D8—5962no—
206C2×C8—1492no—
207C2×C8C22×D14912yes—
208C4⋊C4—21no—
209C4⋊C4—2981no—
210C4⋊C4—2981no—
211C4⋊C4—2981no—
212C4⋊C4—1491no—
213C4⋊C4C22×D14911yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—2981no—
216C22⋊C4—2981no—
217C22⋊C4—2981no—
218C22⋊C4—2982no—
219C22⋊C4—2982no—
220C22⋊C4—2982no—
221C22⋊C4—2982no—
222C22⋊C4—2981no—
223C42—2981no—
224C42—1491no—
225C42—1491no—
226C42—1491no—
227C42—1491no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—2981no—
233C23D4×D14911yescharacteristic
234Q8—22no—
235Q8—2982no—
236Q8—2982no—
237Q8—5962no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—2982no—
242D4—5962no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—2982no—
247C2×C4—2981no—
248C2×C4—2982no—
249C2×C4—5962no—
250C2×C4—5961no—
251C2×C4—5962no—
252C2×C4—1491no—
253C2×C4—1491no—
254C2×C4—1491no—
255C2×C4—1491no—
256C2×C4D4×D14911yescharacteristic
257C2×C4C23×D14911yescharacteristic
258C8—22no—
259C8—2982no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—5961no—
267C22—5961no—
268C22—5961no—
269C22#1801193212yes—
270C22C2×D4×D14911yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—2981no—
275C4—2981no—
276C4—2981no—
277C4—2981no—
278C4—5961no—
279C4—5962no—
280C4#1801195612yes—
281C2—22no—
282C2—82no—
283C2—11921no—
284C2#2375019011yescharacteristic
285C2C2×#1801193211yescharacteristic
286C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#450×C149C2
socleC22×C149C22×D4
prime radical#23749759C2
hypercenter#450D149
hypocenterC149#3211
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23750161C2semidirect
2. C2×#18011898C2semidirect
3. #23750006C2semidirect
4. #23749978C2semidirect
5. #23749845C2semidirect
6. #23749839C2semidirect
7. #23749780C2semidirect
8. #23749766C2semidirect
10. #23749675C2semidirect
11. #450×C149C2semidirect
13. C2×#14558726C22semidirect
14. C2×#14558725C22semidirect
17. #18011902C22semidirect
18. #18011900C22semidirect
25. Q8×Dic149C22semidirect
26. #18011873C22semidirect
32. #18011840C22semidirect
33. #18011836C22semidirect
34. #18011834C22semidirect
37. C8×Dic149C22semidirect
143. C149#3211semidirect
152. #450D149semidirect

Maximal subgroups

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

subgroupindex
#237501612
C2×#180118982
#237500062
#237499782
#237498452
#237498392
#237497802
#237497662
#237497592
#237496752
#450×C1492
#3211149