Finite Group Databasebeta

← #34055446

Subgroups — #34055446

Order 19072.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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×#18011887C211yesmaximal, characteristic, complemented
2#121×D149C211yesmaximal, characteristic
3#23749954C212yesmaximal, complemented
4#23749944C211yesmaximal, characteristic, complemented
5#23749941C211yesmaximal, characteristic, complemented
6#23749857C212yesmaximal, complemented
7#23749851C212yesmaximal, complemented
8#23749774C212yesmaximal, complemented
9#23749764C211yesmaximal, characteristic, complemented
10C22⋊C8×D149C211yesmaximal, characteristic, complemented
11#448×C149C211yesmaximal, characteristic, complemented
12C2×D4×D149—21no—
13C2×D4×D149C2211yescharacteristic
14C22×C4×D149C2211yescharacteristic
15C2×#14558723C2212yescomplemented
16C2×C8×D149C2212yescomplemented
17#18011887—21no—
18#18011887—21no—
19#18011887C2212yes—
20#18011887C2212yescomplemented
21#18011886C2211yescharacteristic
22#18011885C2211yescharacteristic
23#18011884—21no—
24#18011883—21no—
25#18011882C2211yescharacteristic
26C2×C4⋊Dic149C2211yescharacteristic, complemented
27C2×#14558715—21no—
28C4⋊C4×D149C2211yescharacteristic
29C2×#14558713—21no—
30C22⋊C4×D149—21no—
31#18011843C2212yescomplemented
32#18011842C2212yescomplemented
33#18011835C2212yescomplemented
34#18011833C2212yescomplemented
35#18011832C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic, complemented
37#18011823C2212yes—
38C22×Q8×C149C2211yescharacteristic, complemented
39C2×SD16×C149C2212yescomplemented
40#121×C149C2211yescharacteristic
41#103×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic, complemented
43C23×D149—21no—
44C23×D149D411yescharacteristic
45C2×D596—21no—
46C2×D596C2311yescharacteristic
47C2×C22⋊D149—21no—
48C2×C22⋊D149—21no—
49C2×C22⋊D149—21no—
50D4×D149—42no—
51D4×D149—44no—
52C22×Dic149—21no—
53C22×Dic149D411yescharacteristic
54C2×C4×D149—21no—
55C2×C4×D149—42no—
56C2×C4×D149D411yescharacteristic
57C2×C4×D149D411yescharacteristic
58C2×C4×D149C2311yescharacteristic
59C2×C4×D149C2311yescharacteristic
60#14558723—42no—
61C2×#11924746C2312yes—
62C8×D149—42no—
63C4⋊Dic149—21no—
64C4⋊Dic149C2312yes—
65C4⋊Dic149C2311yescharacteristic
66#14558715—21no—
67#14558715—22no—
68#14558715—22no—
69C22⋊Dic149—21no—
70#14558713—22no—
71#14558713—22no—
72#14558713—21no—
73#14558713—21no—
74C2×Q8×C149—21no—
75C2×Q8×C149—42no—
76C2×Q8×C149C2312yes—
77C2×D4×C149—21no—
78C2×D4×C149C2311yescharacteristic
79C22×C4×C149—21no—
80C22×C4×C149C2311yescharacteristic
81SD16×C149—42no—
82C2×C8×C149C2312yes—
83C4⋊C4×C149C2311yescharacteristic
84C22⋊C4×C149—21no—
85C22×D149—21no—
86C22×D149—41no—
87C22×D149—41no—
88C22×D149—42no—
89C22×D149—42no—
90C22×D149—42no—
91C22×D149SD1614yescomplemented
92C22×D149C2×D411yescharacteristic
93C22×D149C2×D411yescharacteristic
94D596—22no—
95D596—42no—
96C22⋊D149—42no—
97C22⋊D149—42no—
98C22⋊D149—42no—
99C2×Dic149—21no—
100C2×Dic149—22no—
101C2×Dic149—41no—
102C2×Dic149—42no—
103C2×Dic149C2×D411yescharacteristic
104C2×Dic149C2×D411yescharacteristic
105C4×D149—22no—
106C4×D149—22no—
107C4×D149—41no—
108C4×D149—41no—
109C4×D149—42no—
110#11924746—22no—
111C23×C149—21no—
112C23×C149C2×D411yescharacteristic
113Q8×C149—22no—
114Q8×C149—22no—
115Q8×C149—42no—
116D4×C149—22no—
117D4×C149—42no—
118C2×C4×C149—22no—
119C2×C4×C149—21no—
120C2×C4×C149—22no—
121C2×C4×C149—42no—
122C2×C4×C149C2×D411yescharacteristic
123C2×C4×C149C2411yescharacteristic
124C8×C149—22no—
125C22×C149—21no—
126C22×C149—21no—
127C22×C149—41no—
128C22×C149—41no—
129C22×C149—41no—
130C22×C149C2×SD1612yes—
131C22×C149C22×D411yescharacteristic
132C2×D149—21no—
133C2×D149—21no—
134C2×D149—21no—
135C2×D149—21no—
136C2×D149—44no—
137C2×D149—41no—
138C2×D149—41no—
139C2×D149—41no—
140C2×D149—84no—
141C2×D149C2×SD1612yes—
142C2×D149C2×SD1612yes—
143C4×C149—21no—
144C4×C149—42no—
145C4×C149—41no—
146C4×C149H(5,𝔽2)12yes—
147Dic149—21no—
148Dic149—21no—
149Dic149—41no—
150Dic149—42no—
151C2×C149—22no—
152C2×C149—81no—
153C2×C149#53411yescharacteristic
154C2×C149C22×SD1611yescharacteristic
155C2×C149#57511yescharacteristic
156D149—22no—
157D149—22no—
158D149—81no—
159C149#318711yescharacteristic, complemented, Sylow
160#3187—1491nomaximal, Sylow
161C2×C22⋊Q8—1491no—
162C2×#121—1491no—
163#483—1492no—
164#481—1491no—
165#481—1491no—
166#465—1492no—
167#465—1492no—
168#450—1492no—
169#448—1491no—
170#448D14911yescharacteristic, complemented
171C2×C22⋊C8—1491no—
172C22×Q8C2×D14911yescharacteristic, complemented
173C22×D4—2981no—
174C22×D4—1491no—
175C23×C4—1491no—
176C2×SD16—1492no—
177C2×SD16C2×D14912yescomplemented
178C22×C8—1492no—
179C22⋊Q8—2981no—
180C22⋊Q8—2981no—
181C22⋊Q8—1492no—
182C22⋊Q8—1492no—
183#121—2981no—
184#121—2981no—
185#121—1491no—
186#121—1491no—
187#121—1491no—
188#121C2×D14911yescharacteristic
189C2×C4⋊C4—2981no—
190C2×C4⋊C4—1491no—
191C2×C4⋊C4—1491no—
192C2×C22⋊C4—2981no—
193C2×C22⋊C4—2981no—
194#106—1492no—
195#106—1492no—
196#103—1492no—
197#103C2×D14912yescomplemented
198#102—1492no—
199#102—1492no—
200C22⋊C8—1491no—
201C22⋊C8—1492no—
202C22⋊C8C2×D14911yescharacteristic, complemented
203C24—2981no—
204C24—1491no—
205C2×Q8—21no—
206C2×Q8—42no—
207C2×Q8C22×D14912yes—
208C2×D4—21no—
209C2×D4—2981no—
210C2×D4—2981no—
211C2×D4—2981no—
212C2×D4—2981no—
213C2×D4—5962no—
214C2×D4—5964no—
215C2×D4—1491no—
216C2×D4C22×D14911yescharacteristic
217C22×C4—21no—
218C22×C4—2981no—
219C22×C4—2981no—
220C22×C4—5962no—
221C22×C4—1491no—
222C22×C4—1491no—
223C22×C4—1491no—
224C22×C4—1491no—
225C22×C4—1491no—
226C22×C4C22×D14911yescharacteristic
227SD16—42no—
228SD16—5962no—
229C2×C8—5962no—
230C2×C8—1492no—
231C2×C8C22×D14912yes—
232C4⋊C4—2982no—
233C4⋊C4—2982no—
234C4⋊C4—2981no—
235C4⋊C4—2981no—
236C4⋊C4—1492no—
237C4⋊C4—1491no—
238C4⋊C4C22×D14911yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—2982no—
241C22⋊C4—2982no—
242C22⋊C4—2981no—
243C22⋊C4—2981no—
244C22⋊C4—2981no—
245C23—21no—
246C23—2981no—
247C23—5961no—
248C23—5961no—
249C23—5962no—
250C23—5962no—
251C23—5962no—
252C23—1491no—
253C23—1491no—
254C23—1494no—
255C23D4×D14911yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—42no—
259D4—22no—
260D4—42no—
261D4—2982no—
262D4—5962no—
263D4—5962no—
264D4—5962no—
265D4—5962no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—2982no—
271C2×C4—2982no—
272C2×C4—2982no—
273C2×C4—2981no—
274C2×C4—5961no—
275C2×C4—5961no—
276C2×C4—5962no—
277C2×C4—5961no—
278C2×C4—5962no—
279C2×C4—1491no—
280C2×C4—1491no—
281C2×C4D4×D14911yescharacteristic
282C2×C4C23×D14911yescharacteristic
283C8—22no—
284C8—2982no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—2981no—
291C22—2981no—
292C22—2981no—
293C22—2981no—
294C22—5964no—
295C22—5961no—
296C22—5961no—
297C22—5961no—
298C22—1492no—
299C22—1492no—
300C22—11924no—
301C22SD16×D14912yes—
302C22C2×D4×D14911yescharacteristic
303C4—21no—
304C4—42no—
305C4—41no—
306C4—2981no—
307C4—2981no—
308C4—5962no—
309C4—5961no—
310C4#1801195612yes—
311C2—22no—
312C2—81no—
313C2—2982no—
314C2—2982no—
315C2—11921no—
316C2#2375018511yescharacteristic
317C2C2×SD16×D14911yescharacteristic
318C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#448×C149C2
socleC22×C149C22×D4
prime radical#121×D149C2
hypercenter#448D149
hypocenterC149#3187
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#18011887C2semidirect
3. #23749954C2semidirect
4. #23749944C2semidirect
5. #23749941C2semidirect
6. #23749857C2semidirect
7. #23749851C2semidirect
8. #23749774C2semidirect
9. #23749764C2semidirect
10. C22⋊C8×D149C2semidirect
11. #448×C149C2semidirect
15. C2×#14558723C22semidirect
16. C2×C8×D149C22semidirect
20. #18011887C22semidirect
26. C2×C4⋊Dic149C22semidirect
31. #18011843C22semidirect
32. #18011842C22semidirect
33. #18011835C22semidirect
34. #18011833C22semidirect
35. #18011832C22semidirect
36. C22⋊#11924746C22semidirect
38. C22×Q8×C149C22semidirect
39. C2×SD16×C149C22semidirect
41. #103×C149C22semidirect
42. C22⋊C8×C149C22semidirect
91. C22×D149SD16semidirect
159. C149#3187semidirect
170. #448D149semidirect
172. C22×Q8C2×D149semidirect
177. C2×SD16C2×D149semidirect
197. #103C2×D149semidirect
202. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118872
#121×D1492
#237499542
#237499442
#237499412
#237498572
#237498512
#237497742
#237497642
C22⋊C8×D1492
#448×C1492
#3187149