Finite Group Databasebeta

← #16829971

Subgroups — #16829971

Order 3712.

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×#11745270C211yesmaximal, characteristic, complemented
2C2×#11745312C211yesmaximal, characteristic, complemented
3C22×#90784C211yesmaximal, characteristic, complemented
4#14271737C211yesmaximal, characteristic, complemented
5#14271727C211yesmaximal, characteristic, complemented
6#14271556C212yesmaximal, complemented
7#14271549C212yesmaximal, complemented
8#14271548C212yesmaximal, complemented
9#14271534C212yesmaximal, complemented
10#14271121C211yesmaximal, characteristic, complemented
11C2×C29×#137C211yesmaximal, characteristic, complemented
12C2×#41×C29C2211yescharacteristic
13C22×Q8×C29C2211yescharacteristic
14C2×#90805C2211yescharacteristic
15C22×D116C2211yescharacteristic
16C29×#137—24no—
17C2×Q16×C29C2212yescomplemented
18C2×SD16×C29C2212yescomplemented
19C2×M16×C29C2211yescharacteristic
20#11745312—24no—
21#11745309C2212yes—
22#11745309C2212yescomplemented
23#11745308—22no—
24#11745306C2212yescomplemented
25C2×#90785C2211yescharacteristic, complemented
26C2×#90784—24no—
27C2×#90784C2212yes—
28C2×#90784C2212yes—
29C2×#90784C2211yescharacteristic, complemented
30#11745302C2212yes—
31#11745299—22no—
32C2×#90783C2211yescharacteristic, complemented
33C2×#90782C2211yescharacteristic, complemented
34#11745292—22no—
35#11745291—22no—
36#11745288C2211yescharacteristic
37C22×#6597C2211yescharacteristic
38#11745270—24no—
39C2×D232C2212yes—
40C2×#90773C2212yes—
41#11745163C2212yescomplemented
42C8×Dic29C2212yescomplemented
43C23×D29—21no—
44#41×C29—22no—
45#41×C29—22no—
46C2×Q8×C29—21no—
47C2×Q8×C29—22no—
48C2×Q8×C29—22no—
49C2×Q8×C29C2312yes—
50C2×Q8×C29C2311yescharacteristic
51C2×D4×C29—21no—
52C2×D4×C29C2311yescharacteristic
53C22×C4×C29—21no—
54C22×C4×C29—21no—
55C22×C4×C29C2311yescharacteristic
56C2×C22⋊D29—21no—
57C2×C22⋊D29—24no—
58#90805—22no—
59#90805—22no—
60C2×D116—22no—
61C2×D116—22no—
62C2×D116—21no—
63C2×D116C2311yescharacteristic
64C2×D116C2312yes—
65C2×C4×D29—21no—
66C2×Dic58C2311yescharacteristic
67Q16×C29—24no—
68SD16×C29—24no—
69M16×C29D414yescomplemented
70C2×C8×C29C2312yes—
71C22⋊Dic29D412yes—
72#90785—22no—
73#90784—24no—
74#90784—24no—
75#90784—22no—
76#90783—22no—
77#90782—22no—
78#90781—24no—
79#90781—24no—
80C4⋊Dic29D412yes—
81#90779—24no—
82C4×Dic29C2312yes—
83C2×#6597D414yescomplemented
84C2×#6597C2311yescharacteristic
85C2×#6597C2311yescharacteristic
86D232—24no—
87#90773—24no—
88C23×C29—21no—
89C23×C29C2×D411yescharacteristic
90C22×D29—21no—
91C22×D29—22no—
92C22×D29—42no—
93C22×D29—82no—
94Q8×C29—22no—
95Q8×C29—21no—
96Q8×C29—22no—
97Q8×C29—21no—
98Q8×C29—42no—
99D4×C29—21no—
100D4×C29—21no—
101D4×C29—42no—
102C2×C4×C29—22no—
103C2×C4×C29—21no—
104C2×C4×C29—42no—
105C2×C4×C29—42no—
106C2×C4×C29C2×D411yescharacteristic
107C2×C4×C29C2×D412yes—
108C2×C4×C29C2×D412yes—
109C2×C4×C29C2411yescharacteristic
110C22⋊D29—42no—
111C22⋊D29—84no—
112C2×Dic29—21no—
113C2×Dic29C2×D414yes—
114D116—22no—
115D116—21no—
116D116—21no—
117D116—22no—
118D116—42no—
119C4×D29—42no—
120Dic58—21no—
121Dic58—21no—
122C8×C29C2×D414yes—
123#6597C2×D412yes—
124#6597C2×D412yes—
125#3154—291nomaximal, Sylow
126C22×C29—22no—
127C22×C29—41no—
128C22×C29—42no—
129C22×C29#12712yes—
130C22×C29C22×D411yescharacteristic
131C2×D29—41no—
132C2×D29—42no—
133C2×D29—44no—
134C2×D29—42no—
135C2×D29—84no—
136C4×C29—41no—
137C4×C29—42no—
138C4×C29#12712yes—
139C4×C29C22×D411yescharacteristic
140C4×C29C22×D411yescharacteristic
141Dic29—24no—
142Dic29—41no—
143C2×#137D2911yescharacteristic, complemented
144C2×AGL(1,ℤ/8)—291no—
145C2×#135—291no—
146C22×SD16—291no—
147#536—291no—
148#535—291no—
149#497—292no—
150#496—292no—
151#495—292no—
152#492—292no—
153#405—291no—
154C2×C29—22no—
155C2×C29—81no—
156C2×C29C2×#12711yescharacteristic
157C2×C29#57712yes—
158D29—81no—
159D29—82no—
160C2×#41—291no—
161C2×#41C2×D2911yescharacteristic
162C22×Q8C2×D2911yescharacteristic
163C22×D4—291no—
164#137—24no—
165AGL(1,ℤ/8)—584no—
166#135—584no—
167C2×Q16—291no—
168C2×Q16C2×D2912yescomplemented
169C2×SD16—292no—
170C2×SD16—292no—
171C2×SD16—291no—
172C2×SD16—291no—
173C2×SD16—292no—
174C2×SD16—584no—
175C2×SD16C2×D2912yescomplemented
176C2×D8—291no—
177C2×D8—292no—
178C2×M16C2×D2911yescharacteristic
179C22×C8—291no—
180C4⋊Q8—292no—
181#127—292no—
182#124—292no—
183#124—292no—
184#123—582no—
185C22⋊Q8—582no—
186#121—582no—
187#120—582no—
188#117—291no—
189#97—292no—
190C4×C8—292no—
191C29#315411yescharacteristic, complemented, Sylow
192C24—581no—
193#41—22no—
194#41—22no—
195#41—582no—
196#41—582no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—291no—
201C2×Q8C22×D2911yescharacteristic
202C2×Q8C22×D2912yes—
203C2×D4—21no—
204C2×D4—291no—
205C2×D4—292no—
206C2×D4—584no—
207C2×D4—581no—
208C2×D4—582no—
209C2×D4—581no—
210C2×D4—582no—
211C2×D4C22×D2911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—581no—
215C22×C4C22×D2911yescharacteristic
216Q16—24no—
217Q16—582no—
218SD16—24no—
219SD16—582no—
220SD16—584no—
221SD16—584no—
222SD16—582no—
223SD16—584no—
224D8—582no—
225D8—584no—
226M16C22⋊D2914yescomplemented
227C2×C8—294no—
228C2×C8—291no—
229C2×C8—291no—
230C2×C8C22×D2912yes—
231C4⋊C4—292no—
232C4⋊C4—584no—
233C22⋊C4—292no—
234C22⋊C4—584no—
235C22⋊C4—584no—
236C42—292no—
237C23—21no—
238C23—581no—
239C23—582no—
240C23—1162no—
241C23—2322no—
242C23D4×D2911yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—581no—
249Q8—581no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—581no—
254D4—582no—
255D4—581no—
256D4—582no—
257D4—1162no—
258D4—1162no—
259D4—2324no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—294no—
265C2×C4—581no—
266C2×C4—1162no—
267C2×C4D4×D2911yescharacteristic
268C2×C4D4×D2912yes—
269C2×C4C2×C22⋊D2912yes—
270C2×C4C23×D2911yescharacteristic
271C8—292no—
272C8—292no—
273C8C2×C22⋊D2914yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—1161no—
278C22—1162no—
279C22—1164no—
280C22—1162no—
281C22—2324no—
282C22#1174530212yes—
283C22C2×D4×D2911yescharacteristic
284C4—41no—
285C4—42no—
286C4—584no—
287C4—1161no—
288C4#1174530212yes—
289C4C2×D4×D2911yescharacteristic
290C4C22×C22⋊D2911yescharacteristic
291C2—22no—
292C2—81no—
293C2—2322no—
294C2—2321no—
295C2C2×#1174530211yescharacteristic
296C2#1427197212yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×#137C2
socleC22×C29C22×D4
hypercenterC2×#137D29
hypocenterC29#3154
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#11745270C2semidirect
2. C2×#11745312C2semidirect
3. C22×#90784C2semidirect
4. #14271737C2semidirect
5. #14271727C2semidirect
6. #14271556C2semidirect
7. #14271549C2semidirect
8. #14271548C2semidirect
9. #14271534C2semidirect
10. #14271121C2semidirect
11. C2×C29×#137C2semidirect
17. C2×Q16×C29C22semidirect
18. C2×SD16×C29C22semidirect
22. #11745309C22semidirect
24. #11745306C22semidirect
25. C2×#90785C22semidirect
29. C2×#90784C22semidirect
32. C2×#90783C22semidirect
33. C2×#90782C22semidirect
41. #11745163C22semidirect
42. C8×Dic29C22semidirect
69. M16×C29D4semidirect
83. C2×#6597D4semidirect
143. C2×#137D29semidirect
168. C2×Q16C2×D29semidirect
175. C2×SD16C2×D29semidirect
191. C29#3154semidirect
226. M16C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452702
C2×#117453122
C22×#907842
#142717372
#142717272
#142715562
#142715492
#142715482
#142715342
#142711212
C2×C29×#1372
#315429