Finite Group Databasebeta

← #34055165

Subgroups — #34055165

Order 19072.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18011944C211yesmaximal, characteristic, complemented
2C2×#18011931C211yesmaximal, characteristic, complemented
3C22×#14558724C211yesmaximal, characteristic, complemented
4#23750196C211yesmaximal, characteristic, complemented
5#23750192C211yesmaximal, characteristic, complemented
6#23750016C212yesmaximal, complemented
7#23750013C212yesmaximal, complemented
8#23750008C212yesmaximal, complemented
9#23749996C212yesmaximal, complemented
10#23749583C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C22×Dic298C2211yescharacteristic
14#18011944—24no—
15#18011931—24no—
16C2×Dic596C2212yescomplemented
17C2×#14558726C2211yescharacteristic, complemented
18C2×#14558725C2212yescomplemented
19C2×#14558724—24no—
20C2×#14558724C2212yes—
21C2×#14558724C2212yes—
22C2×#14558724C2211yescharacteristic, complemented
23C2×#14558723C2211yescharacteristic, complemented
24C2×#14558721C2211yescharacteristic, complemented
25C22×#11924746C2211yescharacteristic
26#18011909C2212yescomplemented
27#18011907C2212yes—
28#18011898C2212yes—
29#18011898C2212yescomplemented
30#18011892—22no—
31#18011890—22no—
32#18011882—22no—
33#18011880—22no—
34#18011870C2211yescharacteristic
35#18011821C2212yescomplemented
36C8×Dic149C2212yescomplemented
37C2×#41×C149C2211yescharacteristic
38C22×D4×C149C2211yescharacteristic
39AGL(1,ℤ/8)×C149—24no—
40C2×SD16×C149C2212yes—
41C2×D8×C149C2212yes—
42C2×M16×C149C2211yescharacteristic
43#14558736—22no—
44#14558736—22no—
45C2×Dic298—21no—
46C2×Dic298—22no—
47C2×Dic298—22no—
48C2×Dic298C2312yes—
49C2×Dic298C2311yescharacteristic
50C2×D596C2311yescharacteristic
51C2×C22⋊D149—24no—
52C2×C22⋊D149—21no—
53C22×Dic149—21no—
54C2×C4×D149—21no—
55Dic596—24no—
56#14558726—22no—
57#14558725—24no—
58#14558724—24no—
59#14558724—24no—
60#14558724—22no—
61#14558723—22no—
62#14558721—22no—
63C2×#11924746D414yescomplemented
64C2×#11924746C2311yescharacteristic
65C2×#11924746C2311yescharacteristic
66C4⋊Dic149D412yes—
67#14558715—24no—
68C22⋊Dic149—24no—
69C22⋊Dic149—24no—
70C22⋊Dic149D412yes—
71C4×Dic149C2312yes—
72C24×C149—21no—
73#41×C149—22no—
74#41×C149—22no—
75C2×Q8×C149C2311yescharacteristic
76C2×D4×C149—21no—
77C2×D4×C149—22no—
78C2×D4×C149—22no—
79C2×D4×C149—21no—
80C2×D4×C149C2312yes—
81C2×D4×C149C2311yescharacteristic
82C22×C4×C149—21no—
83C22×C4×C149C2311yescharacteristic
84SD16×C149—24no—
85D8×C149—24no—
86M16×C149D414yescomplemented
87C2×C8×C149C2312yes—
88C22×D149—21no—
89Dic298—22no—
90Dic298—21no—
91Dic298—22no—
92Dic298—21no—
93Dic298—42no—
94D596—21no—
95D596—21no—
96C22⋊D149—42no—
97C22⋊D149—84no—
98C2×Dic149—21no—
99C2×Dic149—22no—
100C2×Dic149—42no—
101C2×Dic149C2×D414yes—
102C4×D149—42no—
103#11924746C2×D412yes—
104#11924746C2×D412yes—
105C23×C149—22no—
106C23×C149—21no—
107C23×C149—42no—
108C23×C149—82no—
109C23×C149C2×D411yescharacteristic
110Q8×C149—21no—
111Q8×C149—21no—
112D4×C149—21no—
113D4×C149—22no—
114D4×C149—21no—
115D4×C149—22no—
116D4×C149—42no—
117D4×C149—42no—
118C2×C4×C149—21no—
119C2×C4×C149—42no—
120C2×C4×C149C2×D411yescharacteristic
121C2×C4×C149C2×D412yes—
122C2×C4×C149C2×D412yes—
123C2×C4×C149C2411yescharacteristic
124C8×C149C2×D414yes—
125C22×C149—22no—
126C22×C149—41no—
127C22×C149—42no—
128C22×C149—42no—
129C22×C149—44no—
130C22×C149—84no—
131C22×C149#12712yes—
132C22×C149C22×D411yescharacteristic
133C2×D149—41no—
134C2×D149—42no—
135C4×C149—41no—
136C4×C149#12712yes—
137C4×C149C22×D411yescharacteristic
138C4×C149C22×D411yescharacteristic
139Dic149—24no—
140Dic149—41no—
141Dic149—42no—
142C2×C149—22no—
143C2×C149—81no—
144C2×C149—82no—
145C2×C149C2×#12711yescharacteristic
146C2×C149#57712yes—
147D149—81no—
148C149#315411yescharacteristic, complemented, Sylow
149#3154—1491nomaximal, Sylow
150C2×#137—1491no—
151C2×AGL(1,ℤ/8)D14911yescharacteristic, complemented
152C2×#135—1491no—
153C22×SD16—1491no—
154#536—1491no—
155#535—1491no—
156#497—1492no—
157#496—1492no—
158#495—1492no—
159#492—1492no—
160#405—1491no—
161C2×#41—1491no—
162C2×#41C2×D14911yescharacteristic
163C22×Q8—1491no—
164C22×D4C2×D14911yescharacteristic
165#137—2984no—
166AGL(1,ℤ/8)—24no—
167#135—2984no—
168C2×Q16—1491no—
169C2×Q16—1492no—
170C2×SD16—2984no—
171C2×SD16—1491no—
172C2×SD16—1492no—
173C2×SD16—1492no—
174C2×SD16—1491no—
175C2×SD16—1492no—
176C2×SD16C2×D14912yes—
177C2×D8—1491no—
178C2×D8C2×D14912yes—
179C2×M16C2×D14911yescharacteristic
180C22×C8—1491no—
181C4⋊Q8—1492no—
182#127—1492no—
183#124—1492no—
184#124—1492no—
185#123—2982no—
186C22⋊Q8—2982no—
187#121—2982no—
188#120—2982no—
189#117—1491no—
190#97—1492no—
191C4×C8—1492no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—2982no—
196#41—2982no—
197C2×Q8—2981no—
198C2×Q8—2982no—
199C2×Q8—2982no—
200C2×Q8—1491no—
201C2×Q8—1492no—
202C2×Q8C22×D14911yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—2981no—
208C2×D4—2984no—
209C2×D4—1491no—
210C2×D4C22×D14911yescharacteristic
211C2×D4C22×D14912yes—
212C22×C4—21no—
213C22×C4—2981no—
214C22×C4—2981no—
215C22×C4C22×D14911yescharacteristic
216Q16—2982no—
217Q16—2984no—
218SD16—24no—
219SD16—2984no—
220SD16—2984no—
221SD16—2982no—
222SD16—2982no—
223SD16—2984no—
224D8—24no—
225D8—2982no—
226M16C22⋊D14914yescomplemented
227C2×C8—1494no—
228C2×C8—1491no—
229C2×C8—1491no—
230C2×C8C22×D14912yes—
231C4⋊C4—2984no—
232C4⋊C4—1492no—
233C22⋊C4—2984no—
234C22⋊C4—2984no—
235C22⋊C4—1492no—
236C42—1492no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—2981no—
242C23D4×D14911yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—2982no—
246Q8—2981no—
247Q8—2982no—
248Q8—2981no—
249Q8—5962no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—2981no—
257D4—2981no—
258D4—5962no—
259D4—11924no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—2981no—
263C2×C4—2982no—
264C2×C4—5962no—
265C2×C4—5962no—
266C2×C4—1494no—
267C2×C4D4×D14911yescharacteristic
268C2×C4D4×D14912yes—
269C2×C4C2×C22⋊D14912yes—
270C2×C4C23×D14911yescharacteristic
271C8—1492no—
272C8—1492no—
273C8C2×C22⋊D14914yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—5961no—
281C22—5962no—
282C22#1801190712yes—
283C22C2×D4×D14911yescharacteristic
284C4—41no—
285C4—2984no—
286C4—5961no—
287C4—5962no—
288C4#1801190712yes—
289C4C2×D4×D14911yescharacteristic
290C4C22×C22⋊D14911yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—11921no—
295C2C2×#1801190711yescharacteristic
296C2#2375043512yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×AGL(1,ℤ/8)×C149C2
socleC22×C149C22×D4
hypercenterC2×AGL(1,ℤ/8)D149
hypocenterC149#3154
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18011944C2semidirect
2. C2×#18011931C2semidirect
3. C22×#14558724C2semidirect
4. #23750196C2semidirect
5. #23750192C2semidirect
6. #23750016C2semidirect
7. #23750013C2semidirect
8. #23750008C2semidirect
9. #23749996C2semidirect
10. #23749583C2semidirect
11. C2×AGL(1,ℤ/8)×C149C2semidirect
16. C2×Dic596C22semidirect
17. C2×#14558726C22semidirect
18. C2×#14558725C22semidirect
22. C2×#14558724C22semidirect
23. C2×#14558723C22semidirect
24. C2×#14558721C22semidirect
26. #18011909C22semidirect
29. #18011898C22semidirect
35. #18011821C22semidirect
36. C8×Dic149C22semidirect
63. C2×#11924746D4semidirect
86. M16×C149D4semidirect
148. C149#3154semidirect
151. C2×AGL(1,ℤ/8)D149semidirect
226. M16C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119442
C2×#180119312
C22×#145587242
#237501962
#237501922
#237500162
#237500132
#237500082
#237499962
#237495832
C2×AGL(1,ℤ/8)×C1492
#3154149