Finite Group Databasebeta

← #34054483

Subgroups — #34054483

Order 19072.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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#23750180C211yesmaximal, characteristic, complemented
2C4×D4×D149C211yesmaximal, characteristic
3#23749928C212yesmaximal, complemented
4#23749927C212yesmaximal, complemented
5#23749916C211yesmaximal, characteristic, complemented
6#23749915C211yesmaximal, characteristic, complemented
7#23749902C211yesmaximal, characteristic, complemented
8#23749666C212yesmaximal, complemented
9#23749647C211yesmaximal, characteristic, complemented
10#23749628C212yesmaximal
11#474×C149C211yesmaximal, characteristic, complemented
12C2×D4×D149C2211yescharacteristic
13C22×C4×D149—21no—
14C2×#14558719C2212yes—
15#18011911C2211yescharacteristic, complemented
16#18011902C2212yescomplemented
17#18011891—21no—
18#18011887—21no—
19Q8×Dic149—21no—
20#18011876—21no—
21C4×D596C2211yescharacteristic
22D4×Dic149C2211yescharacteristic
23#18011873C2211yescharacteristic
24C4×C22⋊D149—21no—
25#18011871—21no—
26#18011869C2212yes—
27C4⋊C4×D149C2211yescharacteristic
28C22⋊C4×D149—21no—
29C42×D149C2211yescharacteristic
30#18011844C2212yescomplemented
31#18011843C2212yescomplemented
32#18011842C2212yescomplemented
33C4⋊#11924746C2211yescharacteristic, complemented
34#18011840C2212yescomplemented
35#18011833C2212yes—
36#18011832C2212yes—
37#18011831C2212yes—
38C4⋊Q8×C149C2211yescharacteristic, complemented
39C4×D4×C149C2211yescharacteristic
40#107×C149C2212yescomplemented
41C4⋊C8×C149C2211yescharacteristic, complemented
42#102×C149C2212yes—
43C23×D149—21no—
44C2×Dic298—21no—
45C2×D596C2311yescharacteristic
46C2×C22⋊D149—21no—
47D4×D149—22no—
48D4×D149Q814yescomplemented
49C22×Dic149—21no—
50C2×C4×D149—21no—
51C2×C4×D149—21no—
52C2×C4×D149—21no—
53C2×C4×D149—44no—
54C2×C4×D149D411yescharacteristic
55C2×C4×D149D411yescharacteristic
56C2×C4×D149C2311yescharacteristic
57#14558719—24no—
58C2×#11924746C2312yes—
59C4⋊Dic149—22no—
60C4⋊Dic149—21no—
61C4⋊Dic149C2312yes—
62C4⋊Dic149C2311yescharacteristic
63#14558715—21no—
64#14558715—22no—
65C22⋊Dic149—21no—
66#14558713—22no—
67#14558713—21no—
68#14558713—21no—
69C4×Dic149—22no—
70C4×Dic149D411yescharacteristic
71C4×Dic149D411yescharacteristic
72C4×Dic149C2311yescharacteristic
73C2×Q8×C149—21no—
74C2×D4×C149C2311yescharacteristic
75C22×C4×C149—21no—
76C2×C8×C149C2312yes—
77C4⋊C4×C149—21no—
78C4⋊C4×C149C2312yes—
79C4⋊C4×C149C2311yescharacteristic
80C22⋊C4×C149—21no—
81C42×C149C2311yescharacteristic
82C22×D149—21no—
83C22×D149—24no—
84C22×D149—42no—
85C22×D149—42no—
86C22×D149C2×D411yescharacteristic
87Dic298—42no—
88D596—21no—
89D596C2×Q812yes—
90C22⋊D149—24no—
91C2×Dic149—21no—
92C2×Dic149—22no—
93C2×Dic149—42no—
94C2×Dic149C2×D411yescharacteristic
95C2×Dic149C2×D411yescharacteristic
96C2×Dic149C2×D411yescharacteristic
97C4×D149—22no—
98C4×D149—41no—
99C4×D149—42no—
100C4×D149—42no—
101C4×D149—42no—
102C4×D149C2×Q812yes—
103C4×D149#4112yes—
104#11924746—22no—
105C23×C149—21no—
106Q8×C149—42no—
107D4×C149—21no—
108D4×C149C2×Q812yes—
109C2×C4×C149—21no—
110C2×C4×C149—22no—
111C2×C4×C149—42no—
112C2×C4×C149C2×D411yescharacteristic
113C2×C4×C149C2×D411yescharacteristic
114C2×C4×C149C2411yescharacteristic
115C8×C149—22no—
116C22×C149—22no—
117C22×C149—41no—
118C22×C149—41no—
119C22×C149C22×D411yescharacteristic
120C2×D149—21no—
121C2×D149—21no—
122C2×D149—22no—
123C2×D149—48no—
124C2×D149—41no—
125C2×D149—41no—
126C2×D149C22⋊Q812yes—
127C4×C149—21no—
128C4×C149—41no—
129C4×C149—42no—
130C4×C149AGL(1,ℤ/8)12yes—
131C4×C149C22×Q811yescharacteristic
132C4×C149C2×#4111yescharacteristic
133Dic149—21no—
134Dic149—21no—
135Dic149—41no—
136Dic149—42no—
137Dic149C22⋊Q812yes—
138C2×C149—42no—
139C2×C149C2×C22⋊Q811yescharacteristic
140C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C149#57511yescharacteristic
142D149—22no—
143D149—42no—
144C149#307711yescharacteristic, complemented, Sylow
145#3077—1491nomaximal, Sylow
146#533—1491no—
147C2×C4×D4—1491no—
148#478—1492no—
149#478—1492no—
150#476—1491no—
151#476—1491no—
152#474—1491no—
153#474D14911yescharacteristic, complemented
154#428—1492no—
155#423—1491no—
156#418—1492no—
157C22×D4—1491no—
158C23×C4—2981no—
159C2×M16—1492no—
160C4⋊Q8—1491no—
161C4⋊Q8C2×D14911yescharacteristic, complemented
162#125—1492no—
163C22⋊Q8—2981no—
164C22⋊Q8—2981no—
165C4×Q8—2981no—
166C4×Q8—2981no—
167C4×D4—2981no—
168C4×D4—2981no—
169C4×D4—1491no—
170C4×D4—1491no—
171C4×D4—1491no—
172C4×D4C2×D14911yescharacteristic
173#117—1492no—
174C2×C4⋊C4—1491no—
175C2×C22⋊C4—2981no—
176C2×C42—1491no—
177#107—1492no—
178#107C2×D14912yescomplemented
179#106—1492no—
180#106—1492no—
181C4⋊C8—1492no—
182C4⋊C8—1491no—
183C4⋊C8C2×D14911yescharacteristic, complemented
184#102—1492no—
185#102—1492no—
186#102—1492no—
187#102C2×D14912yes—
188C24—2981no—
189C2×Q8—21no—
190C2×Q8—2981no—
191C2×D4—2982no—
192C2×D4—2981no—
193C2×D4—1494no—
194C2×D4—1491no—
195C2×D4C22×D14911yescharacteristic
196C22×C4—21no—
197C22×C4—2981no—
198C22×C4—2981no—
199C22×C4—2981no—
200C22×C4—2981no—
201C22×C4—5964no—
202C22×C4—1491no—
203C22×C4—1491no—
204C22×C4—1491no—
205M16—2984no—
206C2×C8—1492no—
207C2×C8C22×D14912yes—
208C4⋊C4—21no—
209C4⋊C4—2981no—
210C4⋊C4—2982no—
211C4⋊C4—2982no—
212C4⋊C4—2981no—
213C4⋊C4—1492no—
214C4⋊C4—1491no—
215C4⋊C4C22×D14912yes—
216C4⋊C4C22×D14911yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—2982no—
219C22⋊C4—2981no—
220C22⋊C4—2981no—
221C22⋊C4—2981no—
222C42—2982no—
223C42—1491no—
224C42—1491no—
225C42—1491no—
226C42C22×D14911yescharacteristic
227C23—21no—
228C23—2984no—
229C23—2981no—
230C23—5962no—
231C23—5962no—
232C23—1491no—
233Q8—42no—
234Q8—5962no—
235D4—21no—
236D4—2984no—
237D4—2981no—
238D4—1492no—
239D4Q8×D14912yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—2982no—
244C2×C4—2982no—
245C2×C4—2981no—
246C2×C4—5961no—
247C2×C4—5962no—
248C2×C4—5962no—
249C2×C4—5962no—
250C2×C4—5962no—
251C2×C4—1491no—
252C2×C4—1491no—
253C2×C4—1491no—
254C2×C4—1492no—
255C2×C4—1492no—
256C2×C4D4×D14911yescharacteristic
257C2×C4D4×D14911yescharacteristic
258C2×C4C23×D14911yescharacteristic
259C8—22no—
260C8—2982no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—2981no—
265C22—2981no—
266C22—2982no—
267C22—5968no—
268C22—5961no—
269C22—5961no—
270C22—1492no—
271C22C2×D4×D14911yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—2981no—
276C4—2981no—
277C4—5961no—
278C4—5962no—
279C4—1492no—
280C4#1801193512yes—
281C4C2×Q8×D14911yescharacteristic
282C4#41×D14911yescharacteristic
283C2—42no—
284C2—2982no—
285C2—5962no—
286C2C22⋊Q8×D14911yescharacteristic
287C2C2×#1801193511yescharacteristic
288C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#474×C149C2
socleC22×C149C22×D4
prime radicalC2×D4×D149C22
hypercenter#474D149
hypocenterC149#3077
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23750180C2semidirect
3. #23749928C2semidirect
4. #23749927C2semidirect
5. #23749916C2semidirect
6. #23749915C2semidirect
7. #23749902C2semidirect
8. #23749666C2semidirect
9. #23749647C2semidirect
11. #474×C149C2semidirect
15. #18011911C22semidirect
16. #18011902C22semidirect
30. #18011844C22semidirect
31. #18011843C22semidirect
32. #18011842C22semidirect
33. C4⋊#11924746C22semidirect
34. #18011840C22semidirect
38. C4⋊Q8×C149C22semidirect
40. #107×C149C22semidirect
41. C4⋊C8×C149C22semidirect
48. D4×D149Q8semidirect
144. C149#3077semidirect
153. #474D149semidirect
161. C4⋊Q8C2×D149semidirect
178. #107C2×D149semidirect
183. C4⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
C4×D4×D1492
#237499282
#237499272
#237499162
#237499152
#237499022
#237496662
#237496472
#237496282
#474×C1492
#3077149