Finite Group Databasebeta

← #34055472

Subgroups — #34055472

Order 19072.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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#23750136C211yesmaximal, characteristic
2#23749963C212yesmaximal, complemented
3#23749955C211yesmaximal, characteristic, complemented
4#23749938C211yesmaximal, characteristic, complemented
5#23749864C212yesmaximal, complemented
6#23749856C212yesmaximal, complemented
7#23749785C212yesmaximal, complemented
8#23749781C211yesmaximal, characteristic, complemented
9#23749764C211yesmaximal, characteristic, complemented
10C2×C22⋊#11924746C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C2×#14558726C2212yescomplemented
14C2×#14558723C2212yescomplemented
15C22×#11924746C2212yes—
16#18011894—21no—
17#18011890C2211yescharacteristic
18#18011890C2211yescharacteristic
19#18011885C2211yescharacteristic
20#18011885C2211yescharacteristic
21#18011880—21no—
22C4×C22⋊D149—21no—
23C4×C22⋊D149—21no—
24#18011870C2211yescharacteristic
25#18011844C2212yescomplemented
26#18011842C2212yescomplemented
27#18011835C2212yescomplemented
28#18011835C2212yescomplemented
29#18011834C2212yescomplemented
30#18011831C2212yescomplemented
31C22⋊#11924746C2211yescharacteristic
32C22⋊#11924746C2211yescharacteristic
33C22⋊#11924746C2212yes—
34C22×Q8×C149C2211yescharacteristic
35C23×C4×C149C2211yescharacteristic
36C22⋊Q8×C149—21no—
37C22⋊Q8×C149—21no—
38C22⋊Q8×C149C2212yes—
39C22⋊Q8×C149C2212yes—
40C2×C4⋊C4×C149—21no—
41C2×C4⋊C4×C149C2211yescharacteristic
42C2×C22⋊C4×C149—21no—
43#14558736—42no—
44C2×Dic298C2311yescharacteristic
45C2×D596C2311yescharacteristic
46C2×C22⋊D149—21no—
47C2×C22⋊D149—21no—
48C2×C22⋊D149—21no—
49C2×C4×D149—21no—
50#14558726—42no—
51#14558723—42no—
52C2×#11924746—42no—
53C2×#11924746C2312yes—
54C2×#11924746C2312yes—
55C4⋊Dic149C2311yescharacteristic
56C4⋊Dic149C2311yescharacteristic
57#14558715—21no—
58#14558715—21no—
59C22⋊Dic149—21no—
60C22⋊Dic149—21no—
61C22⋊Dic149—21no—
62#14558713—21no—
63#14558713—21no—
64C4×Dic149—21no—
65C24×C149D411yescharacteristic
66C2×Q8×C149—21no—
67C2×Q8×C149—42no—
68C2×Q8×C149C2312yes—
69C22×C4×C149—21no—
70C22×C4×C149—21no—
71C22×C4×C149—42no—
72C22×C4×C149D411yescharacteristic
73C22×C4×C149D411yescharacteristic
74C22×C4×C149D411yescharacteristic
75C22×C4×C149C2311yescharacteristic
76C22×C4×C149C2311yescharacteristic
77C22×C4×C149C2311yescharacteristic
78C4⋊C4×C149—22no—
79C4⋊C4×C149—22no—
80C4⋊C4×C149—21no—
81C4⋊C4×C149C2312yes—
82C22⋊C4×C149—22no—
83C22⋊C4×C149—22no—
84C22×D149—21no—
85Dic298—22no—
86D596—22no—
87C22⋊D149—42no—
88C22⋊D149—42no—
89C22⋊D149—42no—
90C2×Dic149—21no—
91C2×Dic149—21no—
92C2×Dic149—21no—
93C4×D149—42no—
94#11924746—22no—
95#11924746—22no—
96C23×C149—22no—
97C23×C149—22no—
98C23×C149—41no—
99C23×C149C2×D411yescharacteristic
100C23×C149C2×D411yescharacteristic
101C23×C149C2×D411yescharacteristic
102Q8×C149—22no—
103Q8×C149—22no—
104Q8×C149—42no—
105C2×C4×C149—22no—
106C2×C4×C149—22no—
107C2×C4×C149—22no—
108C2×C4×C149—22no—
109C2×C4×C149—22no—
110C2×C4×C149—41no—
111C2×C4×C149—41no—
112C2×C4×C149—41no—
113C2×C4×C149—42no—
114C2×C4×C149—42no—
115C2×C4×C149C2×D411yescharacteristic
116C2×C4×C149C2×D411yescharacteristic
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2411yescharacteristic
119C22×C149—21no—
120C22×C149—21no—
121C22×C149—21no—
122C22×C149—21no—
123C22×C149—21no—
124C22×C149—21no—
125C22×C149—21no—
126C22×C149—44no—
127C22×C149#13512yes—
128C22×C149#13712yes—
129C22×C149C22×D411yescharacteristic
130C2×D149—41no—
131C2×D149—41no—
132C2×D149—41no—
133C4×C149—21no—
134C4×C149—21no—
135C4×C149—21no—
136C4×C149—42no—
137C4×C149—42no—
138C4×C149H(5,𝔽2)12yes—
139Dic149—41no—
140Dic149—41no—
141Dic149—41no—
142C2×C149—22no—
143C2×C149—22no—
144C2×C149—41no—
145C2×C149#53411yescharacteristic
146C2×C149C2×#13511yescharacteristic
147C2×C149C2×#13711yescharacteristic
148D149—81no—
149C149#319011yescharacteristic, complemented, Sylow
150#3190—1491nomaximal, Sylow
151#525—1491no—
152C2×C22⋊Q8D14911yescharacteristic, complemented
153#485—1492no—
154#483—1491no—
155#480—1491no—
156#467—1492no—
157#465—1492no—
158#452—1492no—
159#451—1491no—
160#448—1491no—
161C2×C22⋊C8—1491no—
162C2×#41—1491no—
163C22×Q8C2×D14911yescharacteristic
164C23×C4C2×D14911yescharacteristic
165C2×Q16—1492no—
166C2×SD16—1492no—
167C22×C8—1492no—
168#123—2981no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—1491no—
172C22⋊Q8—1491no—
173C22⋊Q8C2×D14912yes—
174C22⋊Q8C2×D14912yes—
175#121—1491no—
176#121—1491no—
177#120—2981no—
178C4×D4—2981no—
179C4×D4—2981no—
180#117—1491no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×D14911yescharacteristic
183C2×C22⋊C4—21no—
184#107—1492no—
185#106—1492no—
186#103—1492no—
187#103—1492no—
188#103—1492no—
189#102—1492no—
190C22⋊C8—1491no—
191C22⋊C8—1491no—
192C22⋊C8—1492no—
193C24C22⋊D14911yescharacteristic
194#41—5962no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—1491no—
198C2×Q8C22×D14912yes—
199C2×D4—2981no—
200C2×D4—2981no—
201C2×D4—2981no—
202C2×D4—1491no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—2981no—
207C22×C4C22⋊D14911yescharacteristic
208C22×C4C22⋊D14911yescharacteristic
209C22×C4C22⋊D14911yescharacteristic
210C22×C4C22×D14911yescharacteristic
211C22×C4C22×D14911yescharacteristic
212C22×C4C22×D14911yescharacteristic
213Q16—5962no—
214SD16—5962no—
215C2×C8—5962no—
216C2×C8—1492no—
217C2×C8—1492no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—2981no—
222C4⋊C4—2981no—
223C4⋊C4—1491no—
224C4⋊C4—1491no—
225C4⋊C4C22×D14912yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—2981no—
229C22⋊C4—2981no—
230C22⋊C4—2981no—
231C22⋊C4—2981no—
232C22⋊C4—2981no—
233C42—2981no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—2981no—
238C23C2×C22⋊D14911yescharacteristic
239C23C2×C22⋊D14911yescharacteristic
240C23C2×C22⋊D14911yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—2982no—
245D4—2982no—
246D4—5962no—
247D4—5962no—
248D4—5962no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—2981no—
260C2×C4—2981no—
261C2×C4—2981no—
262C2×C4—5962no—
263C2×C4C2×C22⋊D14911yescharacteristic
264C2×C4C2×C22⋊D14911yescharacteristic
265C2×C4C2×C22⋊D14911yescharacteristic
266C2×C4C23×D14911yescharacteristic
267C8—2982no—
268C8—2982no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—5961no—
278C22—5961no—
279C22—5961no—
280C22#1801193112yes—
281C22#1801194212yes—
282C22C22×C22⋊D14911yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—5961no—
289C4—5961no—
290C4—5961no—
291C4#1801195612yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—11921no—
296C2#2375018411yescharacteristic
297C2C2×#1801193111yescharacteristic
298C2C2×#1801194211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×C22⋊Q8×C149C2
socleC22×C149C22×D4
prime radical#23750136C2
hypercenterC2×C22⋊Q8D149
hypocenterC149#3190
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #23749963C2semidirect
3. #23749955C2semidirect
4. #23749938C2semidirect
5. #23749864C2semidirect
6. #23749856C2semidirect
7. #23749785C2semidirect
8. #23749781C2semidirect
9. #23749764C2semidirect
10. C2×C22⋊#11924746C2semidirect
11. C2×C22⋊Q8×C149C2semidirect
13. C2×#14558726C22semidirect
14. C2×#14558723C22semidirect
25. #18011844C22semidirect
26. #18011842C22semidirect
27. #18011835C22semidirect
28. #18011835C22semidirect
29. #18011834C22semidirect
30. #18011831C22semidirect
149. C149#3190semidirect
152. C2×C22⋊Q8D149semidirect

Maximal subgroups

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

subgroupindex
#237501362
#237499632
#237499552
#237499382
#237498642
#237498562
#237497852
#237497812
#237497642
C2×C22⋊#119247462
C2×C22⋊Q8×C1492
#3190149