Finite Group Databasebeta

← #16828767

Subgroups — #16828767

Order 3712.

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

normal subgroups only

This group has 752 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×#11745364C211yesmaximal, characteristic
2C2×#11745304C211yesmaximal, characteristic, complemented
3C22×#90784C211yesmaximal, characteristic, complemented
4#14271323C214yesmaximal, complemented
5#14271313C214yesmaximal, complemented
6#14271167C211yesmaximal, characteristic, complemented
7#14271157C211yesmaximal, characteristic, complemented
8#14271130C211yesmaximal, characteristic, complemented
9C29×#536C211yesmaximal, characteristic, complemented
10C2×#41×C29C2211yescharacteristic
11C22×Q8×C29C2211yescharacteristic, complemented
12#11745364—24no—
13#11745364—24no—
14C2×#90809—21no—
15C2×D4×D29—22no—
16C2×D4×D29—21no—
17C2×#90805C2212yes—
18C2×#90805C2211yescharacteristic
19C22×D116C2212yes—
20C22×D116C2211yescharacteristic
21C29×C4⋊Q8—21no—
22C29×#124—21no—
23C29×#123—22no—
24C29×C22⋊Q8C2214yescomplemented
25C29×#117C2211yescharacteristic, complemented
26C2×#90785C2212yescomplemented
27#11745304—24no—
28C2×#90784—24no—
29C2×#90784C2212yescomplemented
30C2×#90784C2214yescomplemented
31C2×#90777C2211yescharacteristic, complemented
32C22×#6597C2211yescharacteristic, complemented
33C22⋊#6597C2214yescomplemented
34#11745158C2212yescomplemented
35#11745157C2212yescomplemented
36#11745157C2214yescomplemented
37C23×D29—22no—
38C23×D29—21no—
39#41×C29—22no—
40#41×C29—22no—
41C2×Q8×C29—21no—
42C2×Q8×C29—22no—
43C2×Q8×C29—22no—
44C2×Q8×C29D411yescharacteristic
45C2×Q8×C29C2312yes—
46C2×D4×C29D412yes—
47C2×D4×C29D411yescharacteristic
48C22×C4×C29—21no—
49C22×C4×C29C2312yes—
50C22×C4×C29C2311yescharacteristic
51C2×C22⋊D29—22no—
52C2×C22⋊D29—21no—
53#90809—28no—
54D4×D29—216no—
55D4×D29—28no—
56#90805—22no—
57#90805—24no—
58#90805—24no—
59#90805D414yes—
60C2×D116—22no—
61C2×D116—21no—
62C2×D116—22no—
63C2×D116—24no—
64C2×D116—24no—
65C2×D116D414yes—
66C2×D116C2311yescharacteristic
67C2×D116C2312yes—
68C2×C4×D29—22no—
69C2×C4×D29—21no—
70C2×Dic58C2311yescharacteristic
71C4⋊C4×C29—24no—
72C4⋊C4×C29C2312yes—
73C22⋊C4×C29—21no—
74C22⋊C4×C29—24no—
75C42×C29—21no—
76#90785—24no—
77#90784—28no—
78#90784—24no—
79#90777—22no—
80C2×#6597—22no—
81C2×#6597C2312yes—
82C2×#6597C2312yes—
83C23×C29C2×D412yes—
84C23×C29C2×D411yescharacteristic
85C22×D29—22no—
86C22×D29—21no—
87C22×D29—28no—
88C22×D29—24no—
89C22×D29—82no—
90C22×D29—84no—
91Q8×C29—22no—
92Q8×C29—22no—
93Q8×C29—22no—
94Q8×C29—42no—
95D4×C29—24no—
96D4×C29—22no—
97C2×C4×C29—22no—
98C2×C4×C29—22no—
99C2×C4×C29—22no—
100C2×C4×C29—22no—
101C2×C4×C29—42no—
102C2×C4×C29C2×D412yes—
103C2×C4×C29C2×D412yes—
104C2×C4×C29C2×D412yes—
105C2×C4×C29C2×D411yescharacteristic
106C2×C4×C29C2411yescharacteristic
107C22⋊D29—28no—
108C22⋊D29—24no—
109C2×Dic29—21no—
110D116—28no—
111D116—24no—
112D116—21no—
113D116—22no—
114D116C2×D412yes—
115D116C2×D414yes—
116C4×D29—28no—
117C4×D29—24no—
118Dic58—21no—
119Dic58C2×D412yes—
120#6597—22no—
121#6597—22no—
122#3009—291nomaximal, Sylow
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—24no—
126C22×C29#12012yes—
127C22×C29C22×D411yescharacteristic
128C2×D29—24no—
129C2×D29—22no—
130C2×D29—44no—
131C2×D29—42no—
132C2×D29—84no—
133C2×D29—88no—
134C4×C29—22no—
135C4×C29—42no—
136C4×C29—42no—
137C4×C29#12012yes—
138C4×C29C22×D411yescharacteristic
139C4×C29C22×D411yescharacteristic
140Dic29—22no—
141C2×H(5,𝔽2)—291no—
142C2×#137—291no—
143C22×SD16—291no—
144#536D2911yescharacteristic, complemented
145#452—294no—
146#450—294no—
147#418—291no—
148#416—291no—
149#408—291no—
150C2×C29—22no—
151C2×C29—42no—
152C2×C29C2×#12011yescharacteristic
153C2×C29#57712yes—
154D29—44no—
155D29—42no—
156H(5,𝔽2)—584no—
157H(5,𝔽2)—584no—
158C2×#41—291no—
159C2×#41—292no—
160C2×#41—581no—
161C2×#41C2×D2911yescharacteristic
162C22×Q8C2×D2911yescharacteristic, complemented
163C22×D4—291no—
164C22×D4—292no—
165C22×D4—582no—
166C22×D4—581no—
167#137—584no—
168C2×Q16—292no—
169C2×SD16—294no—
170C2×SD16—292no—
171C2×SD16—584no—
172C2×M16—291no—
173C22×C8—291no—
174C4⋊Q8—21no—
175#124—21no—
176#123—22no—
177C22⋊Q8C2×D2914yescomplemented
178#117C2×D2911yescharacteristic, complemented
179#103—292no—
180#102—292no—
181#102—294no—
182C22⋊C8—294no—
183C29#300911yescharacteristic, complemented, Sylow
184C24—582no—
185C24—581no—
186#41—22no—
187#41—22no—
188#41—294no—
189#41—584no—
190#41—584no—
191#41—588no—
192#41—582no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—291no—
197C2×Q8C22⋊D2911yescharacteristic
198C2×Q8C22×D2912yes—
199C2×D4—294no—
200C2×D4—292no—
201C2×D4—291no—
202C2×D4—584no—
203C2×D4—584no—
204C2×D4—5816no—
205C2×D4—588no—
206C2×D4—582no—
207C2×D4—581no—
208C2×D4—582no—
209C2×D4—582no—
210C2×D4—581no—
211C2×D4C22⋊D2912yes—
212C2×D4C22⋊D2911yescharacteristic
213C22×C4—21no—
214C22×C4—582no—
215C22×C4—581no—
216C22×C4C22×D2912yes—
217C22×C4C22×D2911yescharacteristic
218Q16—584no—
219SD16—588no—
220SD16—584no—
221M16—582no—
222C2×C8—292no—
223C2×C8—292no—
224C2×C8—582no—
225C4⋊C4—24no—
226C4⋊C4C22×D2912yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—588no—
231C23—582no—
232C23—581no—
233C23—584no—
234C23—2322no—
235C23—2324no—
236C23C2×C22⋊D2912yes—
237C23C2×C22⋊D2911yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—292no—
243Q8—581no—
244D4—24no—
245D4—22no—
246D4—294no—
247D4—292no—
248D4—588no—
249D4—588no—
250D4—584no—
251D4—584no—
252D4—581no—
253D4—582no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—588no—
260C2×C4—581no—
261C2×C4—584no—
262C2×C4D4×D2912yes—
263C2×C4D4×D2912yes—
264C2×C4C2×C22⋊D2912yes—
265C2×C4C2×C22⋊D2911yescharacteristic
266C2×C4C23×D2911yescharacteristic
267C8—582no—
268C8—582no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—584no—
273C22—582no—
274C22—1164no—
275C22—1162no—
276C22—2324no—
277C22—2328no—
278C22#1174529912yes—
279C22C22×C22⋊D2911yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—582no—
284C4#1174529912yes—
285C4C2×D4×D2911yescharacteristic
286C4C2×D4×D2911yescharacteristic
287C2—22no—
288C2—42no—
289C2—1164no—
290C2—1162no—
291C2C2×#1174529911yescharacteristic
292C2#1427197112yes—
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#536C2
socleC22×C29C22×D4
prime radicalC2×#11745364C2
hypercenter#536D29
hypocenterC29#3009
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#11745304C2semidirect
3. C22×#90784C2semidirect
4. #14271323C2semidirect
5. #14271313C2semidirect
6. #14271167C2semidirect
7. #14271157C2semidirect
8. #14271130C2semidirect
9. C29×#536C2semidirect
11. C22×Q8×C29C22semidirect
24. C29×C22⋊Q8C22semidirect
25. C29×#117C22semidirect
26. C2×#90785C22semidirect
29. C2×#90784C22semidirect
30. C2×#90784C22semidirect
31. C2×#90777C22semidirect
32. C22×#6597C22semidirect
33. C22⋊#6597C22semidirect
34. #11745158C22semidirect
35. #11745157C22semidirect
36. #11745157C22semidirect
144. #536D29semidirect
162. C22×Q8C2×D29semidirect
177. C22⋊Q8C2×D29semidirect
178. #117C2×D29semidirect
183. C29#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#117453642
C2×#117453042
C22×#907842
#142713232
#142713132
#142711672
#142711572
#142711302
C29×#5362
#300929