Finite Group Databasebeta

← #16828772

Subgroups — #16828772

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×#11745313C211yesmaximal, characteristic, complemented
3C22×#90773C211yesmaximal, characteristic, complemented
4#14271739C211yesmaximal, characteristic, complemented
5#14271328C214yesmaximal, complemented
6#14271315C214yesmaximal, complemented
7#14271167C211yesmaximal, characteristic, complemented
8#14271129C211yesmaximal, characteristic, complemented
9C29×#416C211yesmaximal, characteristic, complemented
10C2×#41×C29C2211yescharacteristic
11#11745364—24no—
12#11745364—24no—
13C2×#90809C2212yes—
14C2×D4×D29—22no—
15C2×D4×D29C2212yes—
16C2×#90805—21no—
17C2×#90805C2211yescharacteristic
18C22×D116—21no—
19C22×D116C2211yescharacteristic
20C22×Dic58C2211yescharacteristic, complemented
21C22×C8×C29C2211yescharacteristic, complemented
22C29×#117C2211yescharacteristic, complemented
23#11745313—24no—
24C2×#90785C2212yescomplemented
25C2×#90783C2212yescomplemented
26C2×#90777C2211yescharacteristic, complemented
27C2×#90773—24no—
28C2×#90773C2214yescomplemented
29#11745259C2214yescomplemented
30#11745248—22no—
31#11745237—21no—
32#11745231—21no—
33C29×#103C2212yescomplemented
34C29×#102C2212yescomplemented
35#11745168C2214yescomplemented
36#11745157C2214yescomplemented
37C23×D29—21no—
38C23×D29—22no—
39#41×C29—22no—
40#41×C29D414yes—
41C2×Q8×C29C2311yescharacteristic
42C2×D4×C29—21no—
43C2×D4×C29C2311yescharacteristic
44C22×C4×C29—21no—
45C22×C4×C29C2311yescharacteristic
46C2×C22⋊D29—22no—
47C2×C22⋊D29D412yes—
48C22×Dic29—21no—
49#90809—24no—
50#90809—24no—
51D4×D29—216no—
52D4×D29—24no—
53D4×D29—24no—
54#90805—28no—
55#90805—22no—
56#90805—22no—
57C2×D116—28no—
58C2×D116—22no—
59C2×D116—21no—
60C2×D116—22no—
61C2×D116D411yescharacteristic
62C2×D116D414yes—
63C2×D116C2312yes—
64C2×C4×D29—22no—
65C2×C4×D29C2312yes—
66C2×Dic58—22no—
67C2×Dic58—22no—
68C2×Dic58—21no—
69C2×Dic58D411yescharacteristic
70C2×Dic58C2312yes—
71C2×C8×C29—22no—
72C2×C8×C29C2312yes—
73C4⋊C4×C29C2312yes—
74C22⋊C4×C29—21no—
75C42×C29—21no—
76#90785—24no—
77#90783—24no—
78#90781—24no—
79C4⋊Dic29—24no—
80#90777—22no—
81C2×#6597C2312yes—
82#90773—28no—
83C23×C29—21no—
84C23×C29C2×D411yescharacteristic
85C22×D29—22no—
86C22×D29—28no—
87C22×D29—24no—
88C22×D29—84no—
89C22×D29—82no—
90C22×D29C2×D412yes—
91Q8×C29—21no—
92Q8×C29C2×D412yes—
93D4×C29—24no—
94D4×C29—21no—
95D4×C29C2×D412yes—
96C2×C4×C29—21no—
97C2×C4×C29—24no—
98C2×C4×C29—22no—
99C2×C4×C29C2×D412yes—
100C2×C4×C29C2×D411yescharacteristic
101C2×C4×C29C2×D412yes—
102C2×C4×C29C2411yescharacteristic
103C22⋊D29—28no—
104C22⋊D29—24no—
105C2×Dic29—22no—
106C2×Dic29—42no—
107C2×Dic29C2×D412yes—
108D116—28no—
109D116—22no—
110D116—24no—
111D116—22no—
112D116C2×D414yes—
113C4×D29—28no—
114C4×D29—22no—
115C4×D29—22no—
116Dic58—22no—
117Dic58—22no—
118Dic58—22no—
119Dic58—42no—
120C8×C29—22no—
121#6597—22no—
122#3009—291nomaximal, Sylow
123C22×C29—22no—
124C22×C29—22no—
125C22×C29—42no—
126C22×C29#12012yes—
127C22×C29C22×D411yescharacteristic
128C2×D29—22no—
129C2×D29—24no—
130C2×D29—24no—
131C2×D29—44no—
132C2×D29—84no—
133C2×D29—88no—
134C4×C29—22no—
135C4×C29—42no—
136C4×C29#12012yes—
137C4×C29C22×D411yescharacteristic
138C4×C29C22×D411yescharacteristic
139Dic29—22no—
140Dic29—42no—
141C2×H(5,𝔽2)—291no—
142C2×#137—291no—
143C22×SD16—291no—
144#536—291no—
145#452—294no—
146#450—294no—
147#418—291no—
148#416D2911yescharacteristic, complemented
149#408—291no—
150C2×C29—22no—
151C2×C29—42no—
152C2×C29C2×#12011yescharacteristic
153C2×C29#57712yes—
154D29—42no—
155D29—44no—
156H(5,𝔽2)—584no—
157H(5,𝔽2)—584no—
158C2×#41—291no—
159C2×#41—292no—
160C2×#41—581no—
161C2×#41C2×D2911yescharacteristic
162C22×Q8—291no—
163C22×D4—292no—
164C22×D4—291no—
165C22×D4—581no—
166C22×D4—582no—
167#137—584no—
168C2×Q16—292no—
169C2×SD16—292no—
170C2×SD16—294no—
171C2×SD16—584no—
172C2×M16—291no—
173C22×C8C2×D2911yescharacteristic, complemented
174C4⋊Q8—581no—
175#124—581no—
176#123—582no—
177C22⋊Q8—294no—
178#117C2×D2911yescharacteristic, complemented
179#103C2×D2912yescomplemented
180#102—294no—
181#102C2×D2912yescomplemented
182C22⋊C8—294no—
183C29#300911yescharacteristic, complemented, Sylow
184C24—582no—
185C24—581no—
186#41—22no—
187#41—588no—
188#41—584no—
189#41—584no—
190#41—582no—
191#41—582no—
192#41D11614yes—
193C2×Q8—291no—
194C2×Q8—292no—
195C2×Q8—581no—
196C2×Q8—582no—
197C2×Q8—582no—
198C2×Q8C22×D2911yescharacteristic
199C2×D4—21no—
200C2×D4—292no—
201C2×D4—294no—
202C2×D4—291no—
203C2×D4—292no—
204C2×D4—582no—
205C2×D4—588no—
206C2×D4—582no—
207C2×D4—584no—
208C2×D4—5816no—
209C2×D4—584no—
210C2×D4—582no—
211C2×D4—581no—
212C2×D4C22×D2911yescharacteristic
213C22×C4—21no—
214C22×C4—292no—
215C22×C4—582no—
216C22×C4—581no—
217C22×C4C22×D2911yescharacteristic
218Q16—584no—
219SD16—588no—
220SD16—584no—
221M16—582no—
222C2×C8—22no—
223C2×C8—292no—
224C2×C8C22×D2912yes—
225C4⋊C4—584no—
226C4⋊C4C22×D2912yes—
227C22⋊C4—21no—
228C22⋊C4—584no—
229C42—21no—
230C23—21no—
231C23—292no—
232C23—588no—
233C23—582no—
234C23—584no—
235C23—2322no—
236C23—2324no—
237C23D4×D2911yescharacteristic
238Q8—21no—
239Q8—582no—
240Q8—582no—
241Q8—582no—
242Q8—1162no—
243Q8C2×D11612yes—
244D4—24no—
245D4—21no—
246D4—294no—
247D4—588no—
248D4—588no—
249D4—584no—
250D4—582no—
251D4—584no—
252D4—582no—
253D4C2×D11612yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—292no—
258C2×C4—588no—
259C2×C4—582no—
260C2×C4—582no—
261C2×C4—582no—
262C2×C4—1162no—
263C2×C4C2×D11612yes—
264C2×C4D4×D2912yes—
265C2×C4D4×D2911yescharacteristic
266C2×C4C23×D2911yescharacteristic
267C8—22no—
268C8—582no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—584no—
273C22—582no—
274C22—584no—
275C22—1164no—
276C22—2324no—
277C22—2328no—
278C22#1174524412yes—
279C22C2×D4×D2911yescharacteristic
280C4—22no—
281C4—42no—
282C4—582no—
283C4—1162no—
284C4#1174524412yes—
285C4C22×D11611yescharacteristic
286C4C2×D4×D2911yescharacteristic
287C2—22no—
288C2—42no—
289C2—1162no—
290C2—1164no—
291C2C2×#1174524411yescharacteristic
292C2#1427197012yes—
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#416C2
socleC22×C29C22×D4
prime radicalC2×#11745364C2
hypercenter#416D29
hypocenterC29#3009
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#11745313C2semidirect
3. C22×#90773C2semidirect
4. #14271739C2semidirect
5. #14271328C2semidirect
6. #14271315C2semidirect
7. #14271167C2semidirect
8. #14271129C2semidirect
9. C29×#416C2semidirect
20. C22×Dic58C22semidirect
21. C22×C8×C29C22semidirect
22. C29×#117C22semidirect
24. C2×#90785C22semidirect
25. C2×#90783C22semidirect
26. C2×#90777C22semidirect
28. C2×#90773C22semidirect
29. #11745259C22semidirect
33. C29×#103C22semidirect
34. C29×#102C22semidirect
35. #11745168C22semidirect
36. #11745157C22semidirect
148. #416D29semidirect
173. C22×C8C2×D29semidirect
178. #117C2×D29semidirect
179. #103C2×D29semidirect
181. #102C2×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×#117453132
C22×#907732
#142717392
#142713282
#142713152
#142711672
#142711292
C29×#4162
#300929