Finite Group Databasebeta

← #34055524

Subgroups — #34055524

Order 19072.

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

normal subgroups only

This group has 414 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#23750138C211yesmaximal, characteristic
2C2×#18011887C211yesmaximal, characteristic, complemented
3#23749964C212yesmaximal, complemented
4#23749958C211yesmaximal, characteristic, complemented
5#23749944C211yesmaximal, characteristic, complemented
6#23749880C212yesmaximal, complemented
7#23749874C212yesmaximal, complemented
8#23749786C212yesmaximal, complemented
9#23749764C211yesmaximal, characteristic, complemented
10#23749587C211yesmaximal, characteristic, complemented
11#451×C149C211yesmaximal, characteristic, complemented
12C2×#14558737C2211yescharacteristic
13C22×C4×D149C2211yescharacteristic
14C2×#14558723C2212yescomplemented
15C2×#14558719C2212yes—
16#18011896—21no—
17#18011887—21no—
18#18011887—21no—
19#18011887C2212yes—
20#18011887C2212yes—
21#18011887C2211yescharacteristic
22#18011886C2211yescharacteristic
23#18011885C2211yescharacteristic
24#18011881—21no—
25#18011873—21no—
26#18011871—21no—
27#18011869C2211yescharacteristic
28C2×C4⋊Dic149C2211yescharacteristic
29C2×#14558715—21no—
30C2×#14558713—21no—
31#18011844C2212yescomplemented
32#18011843C2212yescomplemented
33#18011835C2212yescomplemented
34#18011835C2212yescomplemented
35#18011833C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic
37#18011823C2212yes—
38C22×Q8×C149C2211yescharacteristic
39C2×Q16×C149C2212yescomplemented
40C22⋊Q8×C149C2211yescharacteristic
41#103×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic
43C23×D149D411yescharacteristic
44#14558737—42no—
45C2×D596—21no—
46C2×D596—21no—
47C2×D596C2311yescharacteristic
48C2×C22⋊D149—21no—
49C22×Dic149—21no—
50C22×Dic149D411yescharacteristic
51C2×C4×D149—21no—
52C2×C4×D149—42no—
53C2×C4×D149D411yescharacteristic
54C2×C4×D149D411yescharacteristic
55C2×C4×D149C2311yescharacteristic
56C2×C4×D149C2311yescharacteristic
57#14558723—42no—
58#14558719—42no—
59C2×#11924746C2312yes—
60C4⋊Dic149—21no—
61C4⋊Dic149C2312yes—
62C4⋊Dic149C2311yescharacteristic
63#14558715—22no—
64#14558715—22no—
65#14558715—21no—
66#14558713—22no—
67#14558713—22no—
68#14558713—21no—
69#14558713—21no—
70#14558713—21no—
71#14558713—21no—
72C4×Dic149—21no—
73C2×Q8×C149—21no—
74C2×Q8×C149—42no—
75C2×Q8×C149C2312yes—
76C2×Q8×C149C2311yescharacteristic
77C22×C4×C149—21no—
78C22×C4×C149C2311yescharacteristic
79Q16×C149—42no—
80C2×C8×C149C2312yes—
81C4⋊C4×C149—21no—
82C4⋊C4×C149C2311yescharacteristic
83C22⋊C4×C149—21no—
84C22×D149—22no—
85C22×D149—21no—
86C22×D149—41no—
87C22×D149—41no—
88C22×D149C2×D411yescharacteristic
89C22×D149C2×D411yescharacteristic
90D596—22no—
91D596—42no—
92D596—42no—
93C22⋊D149—42no—
94C2×Dic149—22no—
95C2×Dic149—21no—
96C2×Dic149—41no—
97C2×Dic149—42no—
98C2×Dic149C2×D411yescharacteristic
99C2×Dic149C2×D411yescharacteristic
100C4×D149—22no—
101C4×D149—22no—
102C4×D149—41no—
103C4×D149—41no—
104C4×D149—42no—
105#11924746—22no—
106C23×C149C2×D411yescharacteristic
107Q8×C149—22no—
108Q8×C149—22no—
109Q8×C149—22no—
110Q8×C149—42no—
111C2×C4×C149—22no—
112C2×C4×C149—21no—
113C2×C4×C149—21no—
114C2×C4×C149—22no—
115C2×C4×C149—42no—
116C2×C4×C149C2×D411yescharacteristic
117C2×C4×C149C2411yescharacteristic
118C8×C149—22no—
119C22×C149—21no—
120C22×C149—21no—
121C22×C149#13712yes—
122C22×C149C22×D411yescharacteristic
123C2×D149—21no—
124C2×D149—21no—
125C2×D149—21no—
126C2×D149—21no—
127C2×D149—21no—
128C2×D149—21no—
129C2×D149—44no—
130C2×D149—41no—
131C2×D149—41no—
132C2×D149—41no—
133C4×C149—21no—
134C4×C149—42no—
135C4×C149—41no—
136C4×C149—41no—
137C4×C149H(5,𝔽2)12yes—
138Dic149—21no—
139Dic149—21no—
140Dic149—41no—
141Dic149—42no—
142C2×C149—22no—
143C2×C149#53411yescharacteristic
144C2×C149C2×#13711yescharacteristic
145C2×C149#57511yescharacteristic
146D149—41no—
147D149—41no—
148D149—81no—
149C149#319611yescharacteristic, complemented, Sylow
150#3196—1491nomaximal, Sylow
151#525—1491no—
152C2×C22⋊Q8—1491no—
153#485—1492no—
154#484—1491no—
155#481—1491no—
156#470—1492no—
157#468—1492no—
158#452—1492no—
159#451D14911yescharacteristic, complemented
160#448—1491no—
161#407—1491no—
162C2×#41—1491no—
163C22×Q8C2×D14911yescharacteristic
164C23×C4—1491no—
165C2×Q16C2×D14912yescomplemented
166C2×SD16—1492no—
167C2×M16—1492no—
168#123—2981no—
169C22⋊Q8—2981no—
170C22⋊Q8—2981no—
171C22⋊Q8—1492no—
172C22⋊Q8—1492no—
173C22⋊Q8—1491no—
174C22⋊Q8C2×D14911yescharacteristic
175#121—1491no—
176#121—1491no—
177#120—2981no—
178C4×D4—2981no—
179C4×D4—2981no—
180#117—1491no—
181C2×C4⋊C4—2981no—
182C2×C4⋊C4—1491no—
183C2×C22⋊C4—2981no—
184#107—1492no—
185#106—1492no—
186#103—1492no—
187#103—1492no—
188#103C2×D14912yescomplemented
189#102—1492no—
190C22⋊C8—1491no—
191C22⋊C8—1492no—
192C22⋊C8C2×D14911yescharacteristic
193C24—1491no—
194#41—5962no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8C22×D14911yescharacteristic
198C2×Q8C22×D14912yes—
199C2×D4—2981no—
200C2×D4—2981no—
201C2×D4—2981no—
202C2×D4—1491no—
203C22×C4—21no—
204C22×C4—2981no—
205C22×C4—2981no—
206C22×C4—5962no—
207C22×C4—1491no—
208C22×C4—1491no—
209C22×C4—1491no—
210C22×C4—1491no—
211C22×C4—1491no—
212C22×C4C22×D14911yescharacteristic
213Q16—42no—
214SD16—5962no—
215M16—5962no—
216C2×C8—1492no—
217C2×C8C22×D14912yes—
218C4⋊C4—21no—
219C4⋊C4—2982no—
220C4⋊C4—2982no—
221C4⋊C4—2981no—
222C4⋊C4—2981no—
223C4⋊C4—1492no—
224C4⋊C4—1491no—
225C4⋊C4C22×D14911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—2982no—
228C22⋊C4—2982no—
229C22⋊C4—2981no—
230C22⋊C4—2981no—
231C22⋊C4—2981no—
232C22⋊C4—2981no—
233C42—2981no—
234C23—2982no—
235C23—2981no—
236C23—5961no—
237C23—5961no—
238C23—1491no—
239C23—1491no—
240C23D4×D14911yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—42no—
245D4—2982no—
246D4—5962no—
247D4—5962no—
248D4—5962no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—2982no—
255C2×C4—2982no—
256C2×C4—2981no—
257C2×C4—2982no—
258C2×C4—5961no—
259C2×C4—5961no—
260C2×C4—5962no—
261C2×C4—5961no—
262C2×C4—5962no—
263C2×C4—1491no—
264C2×C4—1491no—
265C2×C4D4×D14911yescharacteristic
266C2×C4C23×D14911yescharacteristic
267C8—22no—
268C8—2982no—
269C22—21no—
270C22—21no—
271C22—2981no—
272C22—2981no—
273C22—2981no—
274C22—2981no—
275C22—2981no—
276C22—2981no—
277C22—5964no—
278C22—5961no—
279C22—5961no—
280C22—5961no—
281C22#1801194312yes—
282C22C2×D4×D14911yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—2981no—
288C4—2981no—
289C4—5961no—
290C4—5962no—
291C4#1801195612yes—
292C2—22no—
293C2—5961no—
294C2—5961no—
295C2—11921no—
296C2#2375018511yescharacteristic
297C2C2×#1801194311yescharacteristic
298C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#451×C149C2
socleC22×C149C22×D4
prime radical#23750138C2
hypercenter#451D149
hypocenterC149#3196
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#18011887C2semidirect
3. #23749964C2semidirect
4. #23749958C2semidirect
5. #23749944C2semidirect
6. #23749880C2semidirect
7. #23749874C2semidirect
8. #23749786C2semidirect
9. #23749764C2semidirect
10. #23749587C2semidirect
11. #451×C149C2semidirect
14. C2×#14558723C22semidirect
31. #18011844C22semidirect
32. #18011843C22semidirect
33. #18011835C22semidirect
34. #18011835C22semidirect
35. #18011833C22semidirect
39. C2×Q16×C149C22semidirect
41. #103×C149C22semidirect
149. C149#3196semidirect
159. #451D149semidirect
165. C2×Q16C2×D149semidirect
188. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237501382
C2×#180118872
#237499642
#237499582
#237499442
#237498802
#237498742
#237497862
#237497642
#237495872
#451×C1492
#3196149