Finite Group Databasebeta

← #16829970

Subgroups — #16829970

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×#11745304C211yesmaximal, characteristic, complemented
2C2×#11745311C211yesmaximal, characteristic, complemented
3C2×#11745312C211yesmaximal, characteristic, complemented
4C22×#90784C211yesmaximal, characteristic, complemented
5#14271732C211yesmaximal, characteristic, complemented
6#14271555C212yesmaximal, complemented
7#14271553C212yesmaximal, complemented
8#14271545C212yesmaximal, complemented
9#14271537C212yesmaximal, complemented
10#14271119C211yesmaximal, characteristic, complemented
11C29×#536C211yesmaximal, characteristic, complemented
12C2×#41×C29C2211yescharacteristic
13C22×Q8×C29C2211yescharacteristic
14C2×#90805C2211yescharacteristic
15C22×D116C2211yescharacteristic
16C29×C4⋊Q8C2212yescomplemented
17C29×#124C2212yescomplemented
18C29×#123—22no—
19C29×C22⋊Q8—22no—
20C29×#117C2211yescharacteristic
21#11745312—24no—
22#11745311—24no—
23C2×#90785C2212yescomplemented
24C2×#90785C2211yescharacteristic, complemented
25#11745304—24no—
26C2×#90784—24no—
27C2×#90784C2212yes—
28C2×#90784C2212yes—
29C2×#90784C2212yes—
30C2×#90784C2211yescharacteristic, complemented
31C2×#90784C2212yescomplemented
32C2×#90783C2211yescharacteristic, complemented
33C2×#90782C2212yes—
34C2×#90782C2211yescharacteristic, complemented
35C2×#90777C2211yescharacteristic
36C22×#6597C2211yescharacteristic
37#11745257—22no—
38#11745244—22no—
39#11745237C2212yes—
40#11745236C2212yes—
41#11745151C2212yescomplemented
42C4×#6597C2212yescomplemented
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—
57#90805—22no—
58#90805—22no—
59C2×D116—22no—
60C2×D116—22no—
61C2×D116—21no—
62C2×D116—24no—
63C2×D116C2312yes—
64C2×D116C2311yescharacteristic
65C2×C4×D29—21no—
66C2×Dic58C2311yescharacteristic
67C4⋊C4×C29—24no—
68C4⋊C4×C29D412yes—
69C22⋊C4×C29—24no—
70C22⋊C4×C29D412yes—
71C42×C29C2312yes—
72#90785—24no—
73#90785—22no—
74#90784—24no—
75#90784—24no—
76#90784—24no—
77#90784—24no—
78#90784—22no—
79#90783—22no—
80#90782—24no—
81#90782—22no—
82#90781—24no—
83#90777D414yescomplemented
84C2×#6597D414yescomplemented
85C2×#6597C2311yescharacteristic
86C2×#6597C2311yescharacteristic
87C2×#6597C2312yes—
88C23×C29—21no—
89C23×C29C2×D411yescharacteristic
90C22×D29—22no—
91C22×D29—21no—
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×D412yes—
107C2×C4×C29C2×D412yes—
108C2×C4×C29C2×D414yes—
109C2×C4×C29C2×D411yescharacteristic
110C2×C4×C29C2411yescharacteristic
111C22⋊D29—42no—
112C2×Dic29—21no—
113D116—22no—
114D116—21no—
115D116—22no—
116D116—21no—
117D116—42no—
118D116—84no—
119C4×D29—42no—
120Dic58—21no—
121Dic58—21no—
122#6597C2×D412yes—
123#6597C2×D412yes—
124#6597C2×D414yes—
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—42no—
134C2×D29—44no—
135C2×D29—84no—
136C4×C29—24no—
137C4×C29—41no—
138C4×C29—42no—
139C4×C29#12712yes—
140C4×C29C22×D411yescharacteristic
141C4×C29C22×D411yescharacteristic
142Dic29—41no—
143C2×#137—291no—
144C2×AGL(1,ℤ/8)—291no—
145C2×#135—291no—
146C22×SD16—291no—
147#536D2911yescharacteristic, complemented
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—584no—
165AGL(1,ℤ/8)—584no—
166#135—584no—
167C2×Q16—291no—
168C2×Q16—292no—
169C2×SD16—292no—
170C2×SD16—292no—
171C2×SD16—292no—
172C2×SD16—291no—
173C2×SD16—292no—
174C2×SD16—291no—
175C2×SD16—584no—
176C2×D8—292no—
177C2×D8—291no—
178C2×M16—291no—
179C22×C8—291no—
180C4⋊Q8C2×D2912yescomplemented
181#127—292no—
182#124—292no—
183#124C2×D2912yescomplemented
184#123—22no—
185C22⋊Q8—22no—
186#121—582no—
187#120—582no—
188#117C2×D2911yescharacteristic
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×D2912yes—
202C2×Q8C22×D2911yescharacteristic
203C2×D4—21no—
204C2×D4—292no—
205C2×D4—291no—
206C2×D4—581no—
207C2×D4—582no—
208C2×D4—582no—
209C2×D4—581no—
210C2×D4—584no—
211C2×D4C22×D2911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—581no—
215C22×C4C22×D2911yescharacteristic
216Q16—582no—
217Q16—584no—
218SD16—582no—
219SD16—584no—
220SD16—584no—
221SD16—582no—
222SD16—584no—
223SD16—584no—
224D8—582no—
225D8—584no—
226M16—294no—
227C2×C8—294no—
228C2×C8—291no—
229C2×C8—292no—
230C2×C8—291no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D2912yes—
233C22⋊C4—24no—
234C22⋊C4—584no—
235C22⋊C4C22⋊D2912yes—
236C42C22×D2912yes—
237C23—21no—
238C23—582no—
239C23—581no—
240C23—1162no—
241C23—2322no—
242C23C2×C22⋊D2911yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—581no—
249Q8—581no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—582no—
254D4—581no—
255D4—582no—
256D4—581no—
257D4—1162no—
258D4—1162no—
259D4—2324no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—581no—
265C2×C4—1162no—
266C2×C4D4×D2912yes—
267C2×C4D4×D2912yes—
268C2×C4C2×C22⋊D2914yes—
269C2×C4C2×C22⋊D2911yescharacteristic
270C2×C4C23×D2911yescharacteristic
271C8—292no—
272C8—292no—
273C8—294no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—1161no—
278C22—1162no—
279C22—1162no—
280C22—1164no—
281C22—2324no—
282C22#1174530212yes—
283C22C22×C22⋊D2911yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—1161no—
288C4#1174530212yes—
289C4C2×D4×D2911yescharacteristic
290C4C2×D4×D2911yescharacteristic
291C2—22no—
292C2—81no—
293C2—2322no—
294C2—2321no—
295C2C2×#1174530211yescharacteristic
296C2#1427197112yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#536C2
socleC22×C29C22×D4
hypercenter#536D29
hypocenterC29#3154
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#11745304C2semidirect
2. C2×#11745311C2semidirect
3. C2×#11745312C2semidirect
4. C22×#90784C2semidirect
5. #14271732C2semidirect
6. #14271555C2semidirect
7. #14271553C2semidirect
8. #14271545C2semidirect
9. #14271537C2semidirect
10. #14271119C2semidirect
11. C29×#536C2semidirect
16. C29×C4⋊Q8C22semidirect
17. C29×#124C22semidirect
23. C2×#90785C22semidirect
24. C2×#90785C22semidirect
30. C2×#90784C22semidirect
31. C2×#90784C22semidirect
32. C2×#90783C22semidirect
34. C2×#90782C22semidirect
41. #11745151C22semidirect
42. C4×#6597C22semidirect
83. #90777D4semidirect
84. C2×#6597D4semidirect
147. #536D29semidirect
180. C4⋊Q8C2×D29semidirect
183. #124C2×D29semidirect
191. C29#3154semidirect

Maximal subgroups

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

subgroupindex
C2×#117453042
C2×#117453112
C2×#117453122
C22×#907842
#142717322
#142715552
#142715532
#142715452
#142715372
#142711192
C29×#5362
#315429