Finite Group Databasebeta

← #16822756

Subgroups — #16822756

Order 3712.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#11745255C211yesmaximal, characteristic
2C2×#11745254C211yesmaximal, characteristic, complemented
3#14271102C211yesmaximal, characteristic, complemented
4#14271100C211yesmaximal, characteristic, complemented
5#14271090C212yesmaximal, complemented
6#14271089C211yesmaximal, characteristic, complemented
7#14271080C212yesmaximal
8C4⋊C4×Dic29C212yesmaximal, complemented
9#14271010C211yesmaximal, characteristic, complemented
10#14271010C212yesmaximal, complemented
11C29×#398C211yesmaximal, characteristic, complemented
12C22×D116C2211yescharacteristic
13C22×C4×D29C2211yescharacteristic
14C22×C4×D29C2211yescharacteristic
15C2×C4⋊C4×C29C2212yes—
16C2×C4⋊C4×C29C2211yescharacteristic
17C2×C4⋊C4×C29C2211yescharacteristic
18C2×#90781C2212yes—
19C2×#90781C2212yes—
20C2×#90781C2212yes—
21C2×#90781C2212yes—
22C2×#90781C2211yescharacteristic
23C2×#90781C2211yescharacteristic
24C2×C4⋊Dic29C2212yes—
25C2×C4⋊Dic29C2211yescharacteristic
26C2×C4×Dic29C2212yes—
27C2×C4×Dic29C2212yes—
28C2×C4×Dic29C2211yescharacteristic
29#11745255—24no—
30#11745254—24no—
31C29×#95C2212yes—
32C29×#95C2211yescharacteristic
33#11745179C2211yescharacteristic
34#11745179C2212yes—
35#11745179C2212yes—
36#11745179C2212yes—
37C23×D29C2311yescharacteristic
38C23×D29C2311yescharacteristic
39C22×C4×C29C2312yes—
40C22×C4×C29C2311yescharacteristic
41C22×C4×C29C2311yescharacteristic
42C22×C4×C29C2312yes—
43C22×C4×C29C2311yescharacteristic
44C22×Dic29C2312yes—
45C22×Dic29C2311yescharacteristic
46C22×Dic29C2312yes—
47C22×Dic29C2311yescharacteristic
48C2×D116—22no—
49C2×D116—22no—
50C2×D116—22no—
51C2×C4×D29—24no—
52C2×C4×D29—22no—
53C2×C4×D29—24no—
54C2×C4×D29D414yes—
55C4⋊C4×C29—22no—
56C4⋊C4×C29—22no—
57C4⋊C4×C29—24no—
58#90781—24no—
59#90781—22no—
60#90781—24no—
61#90781—22no—
62#90781—24no—
63#90781—24no—
64C4⋊Dic29—22no—
65C4⋊Dic29—24no—
66C4×Dic29—24no—
67C4×Dic29—22no—
68C4×Dic29—24no—
69C23×C29C2411yescharacteristic
70C22×D29—21no—
71C22×D29—22no—
72C22×D29—22no—
73C22×D29—22no—
74C22×D29—22no—
75C22×D29—21no—
76C22×D29—21no—
77C22×D29—21no—
78C22×D29C2×D412yes—
79C22×D29#4112yes—
80C2×C4×C29—21no—
81C2×C4×C29—22no—
82C2×C4×C29—22no—
83C2×C4×C29—22no—
84C2×C4×C29—21no—
85C2×C4×C29—22no—
86C2×C4×C29—21no—
87C2×C4×C29—22no—
88C2×C4×C29—22no—
89C2×C4×C29—22no—
90C2×C4×C29C2×D412yes—
91C2×C4×C29#4112yes—
92C2×C4×C29#4114yes—
93C2×Dic29—22no—
94C2×Dic29—21no—
95C2×Dic29—22no—
96C2×Dic29—22no—
97C2×Dic29—22no—
98C2×Dic29—22no—
99C2×Dic29—22no—
100C2×Dic29—22no—
101C2×Dic29C2×D412yes—
102C2×Dic29#4114yes—
103D116—28no—
104C4×D29—28no—
105C4×D29—44no—
106#2395—291nomaximal, Sylow
107C22×C29C22×D411yescharacteristic
108C22×C29C2×#4111yescharacteristic
109C22×C29C2×#4112yes—
110C22×C29C2×#4112yes—
111C22×C29C2×#4111yescharacteristic
112C2×D29—22no—
113C2×D29—22no—
114C2×D29—22no—
115C2×D29—24no—
116C2×D29—22no—
117C2×D29—41no—
118C2×D29—42no—
119C2×D29—42no—
120C2×D29—42no—
121C2×D29—41no—
122C4×C29—22no—
123C4×C29—22no—
124C4×C29—24no—
125C4×C29—41no—
126C4×C29—42no—
127Dic29—42no—
128Dic29—42no—
129Dic29—41no—
130Dic29#12314yes—
131C2×C4×D4—291no—
132C2×#117—291no—
133#400—291no—
134#399—291no—
135#398D2911yescharacteristic, complemented
136#397—291no—
137#397—292no—
138#394—292no—
139C4×C4⋊C4—292no—
140C4×C22⋊C4—292no—
141C4×C22⋊C4—291no—
142C2×C29C2×#12311yescharacteristic
143C2×C29#52511yescharacteristic
144C2×C29#52912yes—
145C2×C29#52912yes—
146C2×C29#53211yescharacteristic
147D29—42no—
148D29—42no—
149C22×D4—291no—
150C23×C4—291no—
151C23×C4—291no—
152C4×D4—584no—
153#117—584no—
154C2×C4⋊C4—291no—
155C2×C4⋊C4—292no—
156C2×C4⋊C4C2×D2911yescharacteristic
157C2×C4⋊C4C2×D2911yescharacteristic
158C2×C4⋊C4C2×D2912yes—
159C2×C22⋊C4—292no—
160C2×C22⋊C4—291no—
161C2×C22⋊C4—291no—
162C2×C22⋊C4—292no—
163C2×C22⋊C4—292no—
164C2×C22⋊C4—292no—
165C2×C42—292no—
166C2×C42—291no—
167C2×C42—292no—
168#95—291no—
169#95—292no—
170#95—292no—
171#95—292no—
172#95C2×D2912yes—
173#95C2×D2911yescharacteristic
174C29#239511yescharacteristic, complemented, Sylow
175C24—291no—
176C24—291no—
177C2×D4—582no—
178C2×D4—582no—
179C2×D4—582no—
180C22×C4—291no—
181C22×C4—294no—
182C22×C4—292no—
183C22×C4—291no—
184C22×C4—292no—
185C22×C4—584no—
186C22×C4—582no—
187C22×C4—584no—
188C22×C4C22×D2911yescharacteristic
189C22×C4C22×D2912yes—
190C22×C4C22×D2911yescharacteristic
191C22×C4C22×D2911yescharacteristic
192C22×C4C22×D2912yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—582no—
197C4⋊C4—584no—
198C22⋊C4—584no—
199C22⋊C4—582no—
200C22⋊C4—584no—
201C22⋊C4—582no—
202C22⋊C4—584no—
203C22⋊C4—584no—
204C42—584no—
205C42—582no—
206C42—584no—
207C23—292no—
208C23—292no—
209C23—581no—
210C23—581no—
211C23—581no—
212C23—582no—
213C23—582no—
214C23—582no—
215C23—582no—
216C23—581no—
217C23C23×D2911yescharacteristic
218D4—588no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—292no—
230C2×C4—294no—
231C2×C4—588no—
232C2×C4—581no—
233C2×C4—582no—
234C2×C4—582no—
235C2×C4—582no—
236C2×C4—582no—
237C2×C4—582no—
238C2×C4—582no—
239C2×C4—582no—
240C2×C4—1164no—
241C2×C4D4×D2912yes—
242C2×C4#9080712yes—
243C2×C4#9080914yes—
244C22—582no—
245C22—582no—
246C22—582no—
247C22—584no—
248C22—582no—
249C22—1161no—
250C22—1162no—
251C22—1162no—
252C22—1162no—
253C22—1161no—
254C22C2×D4×D2911yescharacteristic
255C22C2×#9080711yescharacteristic
256C22C2×#9080912yes—
257C22#41×D2912yes—
258C22#41×D2911yescharacteristic
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—294no—
265C4—1162no—
266C4—1161no—
267C4—1162no—
268C2—1162no—
269C2—1162no—
270C2#123×D2911yescharacteristic
271C2#1427167311yescharacteristic
272C2#1427169812yes—
273C2#1427169912yes—
274C2#1427171411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#398C2
prime radicalC22×D116C22
hypercenter#398D29
hypocenterC29#2395
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#11745254C2semidirect
3. #14271102C2semidirect
4. #14271100C2semidirect
5. #14271090C2semidirect
6. #14271089C2semidirect
8. C4⋊C4×Dic29C2semidirect
9. #14271010C2semidirect
10. #14271010C2semidirect
11. C29×#398C2semidirect
135. #398D29semidirect
174. C29#2395semidirect

Maximal subgroups

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

subgroupindex
C2×#117452552
C2×#117452542
#142711022
#142711002
#142710902
#142710892
#142710802
C4⋊C4×Dic292
#142710102
C29×#3982
#239529