Finite Group Databasebeta

← #34055710

Subgroups — #34055710

Order 19072.

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

normal subgroups only

This group has 490 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#127×D149C211yesmaximal, characteristic, complemented
2#23750160C211yesmaximal, characteristic, complemented
3#23750014C211yesmaximal, characteristic, complemented
4#23750012C211yesmaximal, characteristic, complemented
5#23749991C211yesmaximal, characteristic, complemented
6#23749824C212yesmaximal, complemented
7#23749819C212yesmaximal, complemented
8#23749777C212yesmaximal, complemented
9#23749760C212yesmaximal, complemented
10#23749680C211yesmaximal, characteristic, complemented
11#490×C149C211yesmaximal, characteristic, complemented
12C2×D4×D149—21no—
13C2×D4×D149—21no—
14C2×D4×D149C2212yes—
15C2×#14558725C2212yescomplemented
16C2×#14558724C2212yescomplemented
17C2×D1192C2212yescomplemented
18C2×#14558721C2212yescomplemented
19#18011908C2211yescharacteristic
20#18011907—21no—
21#18011907—21no—
22#18011907C2212yes—
23#18011906—21no—
24#18011899C2211yescharacteristic, complemented
25#18011896—21no—
26#18011891C2211yescharacteristic
27#18011886C2211yescharacteristic
28#18011876C2211yescharacteristic, complemented
29#18011873—21no—
30#18011873C2211yescharacteristic, complemented
31#18011869—21no—
32C42×D149C2211yescharacteristic
33#18011840C2212yescomplemented
34#18011831C2212yescomplemented
35#18011831C2212yescomplemented
36#18011823C2212yes—
37#18011822C2211yescharacteristic, complemented
38#127×C149C2211yescharacteristic
39#125×C149C2211yescharacteristic, complemented
40#102×C149C2212yescomplemented
41#102×C149C2212yescomplemented
42#97×C149C2211yescharacteristic, complemented
43C23×D149—22no—
44C2×Dic298C2311yescharacteristic
45C2×D596—21no—
46C2×D596—21no—
47C2×D596—21no—
48C2×D596C2312yes—
49C2×D596C2311yescharacteristic
50C2×C22⋊D149—22no—
51C2×C22⋊D149—22no—
52C2×C22⋊D149—22no—
53D4×D149—42no—
54D4×D149—42no—
55D4×D149—44no—
56D4×D149—44no—
57C2×C4×D149—21no—
58C2×C4×D149D411yescharacteristic
59C2×C4×D149D411yescharacteristic
60C2×C4×D149C2311yescharacteristic
61#14558725—42no—
62#14558724—42no—
63D1192—42no—
64#14558721—42no—
65C2×#11924746C2312yes—
66C4⋊Dic149—21no—
67#14558715—21no—
68#14558713—21no—
69#14558713—21no—
70#14558713—21no—
71#14558713—21no—
72#14558713—21no—
73C4×Dic149—21no—
74C4×Dic149D411yescharacteristic
75C4×Dic149D411yescharacteristic
76C4×Dic149C2311yescharacteristic
77C2×D4×C149—21no—
78C2×D4×C149—21no—
79C2×D4×C149C2312yes—
80C2×C8×C149C2312yes—
81C4⋊C4×C149—21no—
82C4⋊C4×C149—21no—
83C4⋊C4×C149C2311yescharacteristic
84C4⋊C4×C149C2311yescharacteristic
85C42×C149C2311yescharacteristic
86C22×D149—21no—
87C22×D149—22no—
88C22×D149—44no—
89C22×D149—44no—
90C22×D149—44no—
91C22×D149C2×D411yescharacteristic
92Dic298—22no—
93D596—22no—
94D596—22no—
95D596—22no—
96D596—42no—
97D596—42no—
98D596—42no—
99C22⋊D149—44no—
100C22⋊D149—44no—
101C22⋊D149—44no—
102C2×Dic149—21no—
103C2×Dic149C2×D411yescharacteristic
104C2×Dic149C2×D411yescharacteristic
105C2×Dic149C2×D411yescharacteristic
106C4×D149—22no—
107C4×D149—22no—
108C4×D149—22no—
109C4×D149—42no—
110#11924746—22no—
111C23×C149—22no—
112D4×C149—22no—
113D4×C149—22no—
114D4×C149—42no—
115D4×C149—42no—
116C2×C4×C149—21no—
117C2×C4×C149—21no—
118C2×C4×C149C2×D411yescharacteristic
119C2×C4×C149C2×D411yescharacteristic
120C2×C4×C149C2411yescharacteristic
121C8×C149—22no—
122C22×C149—42no—
123C22×C149—42no—
124C22×C149—42no—
125C22×C149C22×D411yescharacteristic
126C2×D149—21no—
127C2×D149—21no—
128C2×D149—42no—
129C2×D149—41no—
130C2×D149—41no—
131C2×D149—42no—
132C2×D149—42no—
133C2×D149—41no—
134C2×D149—88no—
135C2×D149AGL(1,ℤ/8)12yes—
136C4×C149—21no—
137C4×C149—21no—
138C4×C149—41no—
139C4×C149—41no—
140C4×C149H(5,𝔽2)12yes—
141Dic149—21no—
142Dic149—21no—
143Dic149—41no—
144Dic149AGL(1,ℤ/8)12yes—
145C2×C149—82no—
146C2×C149#53511yescharacteristic
147C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
149D149—22no—
150D149—81no—
151D149—82no—
152C149#321711yescharacteristic, complemented, Sylow
153#3217—1491nomaximal, Sylow
154C2×#127—1491no—
155#529—1491no—
156#496—1491no—
157#496—1491no—
158#490—1491no—
159#490D14911yescharacteristic, complemented
160#460—1492no—
161#459—1492no—
162#450—1492no—
163#447—1492no—
164#432—1491no—
165C22×D4—2981no—
166C22×D4—2981no—
167C22×D4—1492no—
168C2×SD16—1492no—
169C2×SD16—1492no—
170C2×D8—1492no—
171C2×D8—1492no—
172#127—2981no—
173#127—2981no—
174#127—1492no—
175#127—1491no—
176#127C2×D14911yescharacteristic
177#126—2981no—
178#125C2×D14911yescharacteristic, complemented
179#124—1491no—
180#123—2981no—
181C22⋊Q8—1491no—
182#121—1491no—
183C4×Q8—1491no—
184C4×D4—2981no—
185C4×D4—1491no—
186#117—2981no—
187C2×C42—1491no—
188C4⋊C8—1492no—
189#102—1492no—
190#102—1492no—
191#102C2×D14912yescomplemented
192#102C2×D14912yescomplemented
193C22⋊C8—1492no—
194#97—1491no—
195#97C2×D14911yescharacteristic, complemented
196C24—2982no—
197C2×Q8—1491no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—2981no—
201C2×D4—2982no—
202C2×D4—2982no—
203C2×D4—2982no—
204C2×D4—2981no—
205C2×D4—2981no—
206C2×D4—5962no—
207C2×D4—5962no—
208C2×D4—5964no—
209C2×D4—5964no—
210C2×D4—1492no—
211C2×D4—1491no—
212C2×D4C22×D14912yes—
213C22×C4—2981no—
214C22×C4—1491no—
215C22×C4—1491no—
216C22×C4—1491no—
217SD16—5962no—
218SD16—5962no—
219D8—5962no—
220D8—5962no—
221C2×C8—1492no—
222C2×C8C22×D14912yes—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—2981no—
226C4⋊C4—2981no—
227C4⋊C4C22×D14911yescharacteristic
228C4⋊C4C22×D14911yescharacteristic
229C22⋊C4—2981no—
230C22⋊C4—2981no—
231C22⋊C4—2981no—
232C22⋊C4—2981no—
233C22⋊C4—2981no—
234C42—2981no—
235C42—1491no—
236C42—1491no—
237C42—1491no—
238C42C22×D14911yescharacteristic
239C23—22no—
240C23—2981no—
241C23—2982no—
242C23—5964no—
243C23—5964no—
244C23—5964no—
245C23—1491no—
246Q8—2982no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—42no—
251D4—2982no—
252D4—2982no—
253D4—2982no—
254D4—5962no—
255D4—5962no—
256D4—5962no—
257D4—5964no—
258D4—5964no—
259D4—5964no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—2982no—
263C2×C4—2982no—
264C2×C4—2982no—
265C2×C4—2981no—
266C2×C4—5962no—
267C2×C4—1491no—
268C2×C4—1491no—
269C2×C4—1491no—
270C2×C4D4×D14911yescharacteristic
271C2×C4D4×D14911yescharacteristic
272C2×C4C23×D14911yescharacteristic
273C8—22no—
274C8—2982no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—2981no—
279C22—2981no—
280C22—5961no—
281C22—5962no—
282C22—5962no—
283C22—5961no—
284C22—5962no—
285C22—5961no—
286C22—1492no—
287C22—11928no—
288C22C2×D4×D14911yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—2981no—
294C4—2981no—
295C4—5961no—
296C4—1492no—
297C4#1801195712yes—
298C2—82no—
299C2—2982no—
300C2—11922no—
301C2—11921no—
302C2#2375019311yescharacteristic
303C2AGL(1,ℤ/8)×D14911yescharacteristic
304C2AGL(1,ℤ/8)×D14911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#490×C149C2
socleC22×C149C22×D4
hypercenter#490D149
hypocenterC149#3217
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #127×D149C2semidirect
2. #23750160C2semidirect
3. #23750014C2semidirect
4. #23750012C2semidirect
5. #23749991C2semidirect
6. #23749824C2semidirect
7. #23749819C2semidirect
8. #23749777C2semidirect
9. #23749760C2semidirect
10. #23749680C2semidirect
11. #490×C149C2semidirect
15. C2×#14558725C22semidirect
16. C2×#14558724C22semidirect
17. C2×D1192C22semidirect
18. C2×#14558721C22semidirect
24. #18011899C22semidirect
28. #18011876C22semidirect
30. #18011873C22semidirect
33. #18011840C22semidirect
34. #18011831C22semidirect
35. #18011831C22semidirect
37. #18011822C22semidirect
39. #125×C149C22semidirect
40. #102×C149C22semidirect
41. #102×C149C22semidirect
42. #97×C149C22semidirect
152. C149#3217semidirect
159. #490D149semidirect
178. #125C2×D149semidirect
191. #102C2×D149semidirect
192. #102C2×D149semidirect
195. #97C2×D149semidirect

Maximal subgroups

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

subgroupindex
#127×D1492
#237501602
#237500142
#237500122
#237499912
#237498242
#237498192
#237497772
#237497602
#237496802
#490×C1492
#3217149