Finite Group Databasebeta

← #16835502

Subgroups — #16835502

Order 3712.

386 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 68 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
1#14272230C211yesmaximal, characteristic
2D29⋊#99C211yesmaximal, characteristic
3#14272106C212yesmaximal, complemented
4#14272040C212yesmaximal, complemented
5#123×D29C211yesmaximal, characteristic
6C2×D4×D29C2211yescharacteristic
7C22×C4×D29C2211yescharacteristic
8C2×C22⋊#1181—21no—
9C2×C22⋊#1181C2212yes—
10C2×C22⋊#1181C2212yes—
11D4×#1181—24no—
12#11745345—21no—
13C29×#123C411yescharacteristic, complemented
14#11745297C411yescharacteristic, complemented
15#11745291C411yescharacteristic, complemented
16#11745256—21no—
17C4⋊C4×D29—21no—
18#11745248C411yescharacteristic, complemented
19#11745245—21no—
20C22⋊C4×D29—21no—
21C22⋊C4×D29C2211yescharacteristic
22#11745227—21no—
23(C2×#1181)⋊C4—24no—
24#11745215—21no—
25C23×D29D411yescharacteristic
26C23×D29C2311yescharacteristic
27C22×#1181—22no—
28C22×#1181—22no—
29C22×#1181D414yes—
30C2×D4×C29C2×C411yescharacteristic
31C22×C4×C29C2×C411yescharacteristic, complemented
32C2×C22⋊D29C2×C411yescharacteristic
33C2×C22⋊D29D411yescharacteristic
34C22×Dic29C2×C411yescharacteristic, complemented
35D4×D29—22no—
36D4×D29—22no—
37C2×D116D411yescharacteristic
38C2×C4×D29—22no—
39C2×C4×D29—21no—
40C2×C4×D29—21no—
41C2×C4×D29—21no—
42C2×C4×D29—22no—
43C2×C4×D29D411yescharacteristic
44C22⋊#1181—22no—
45C22⋊#1181—22no—
46C22⋊#1181—22no—
47C22⋊#1181—22no—
48C22⋊#1181—22no—
49C4⋊#1181—22no—
50C4×#1181—22no—
51C4⋊C4×C29—21no—
52C22⋊C4×C29—21no—
53C22⋊C4×C29C2×C411yescharacteristic, complemented
54C22⋊Dic29—21no—
55C22⋊Dic29C2×C411yescharacteristic, complemented
56#90781—21no—
57#90781—21no—
58#90781—21no—
59C4⋊Dic29—21no—
60#90779—21no—
61#90779—21no—
62C23×C29C22⋊C411yescharacteristic
63C23×C29C22×C411yescharacteristic
64C22×D29—21no—
65C22×D29—21no—
66C22×D29—42no—
67C22×D29—42no—
68C22×D29C22⋊C411yescharacteristic
69C22×D29C2×D412yes—
70C22×D29C2×D411yescharacteristic
71C22×D29C2×D411yescharacteristic
72C22×D29#4112yes—
73C2×#1181—24no—
74C2×#1181—44no—
75C2×#1181—44no—
76C2×#1181—42no—
77C2×#1181—82no—
78D4×C29—22no—
79C2×C4×C29—21no—
80C2×C4×C29—21no—
81C2×C4×C29—22no—
82C2×C4×C29C22⋊C411yescharacteristic
83C22⋊D29—22no—
84C22⋊D29—22no—
85C2×Dic29—21no—
86C2×Dic29—21no—
87C2×Dic29—22no—
88C2×Dic29C22⋊C411yescharacteristic
89D116—22no—
90C4×D29—21no—
91C4×D29—21no—
92C4×D29—42no—
93C4×D29—41no—
94C4×D29—42no—
95C4×D29—41no—
96#1890—291nomaximal, Sylow
97C22×C29—21no—
98C22×C29—41no—
99C22×C29—41no—
100C22×C29C2×C22⋊C411yescharacteristic
101C22×C29C4×D412yes—
102C2×D29—22no—
103C2×D29—24no—
104C2×D29—21no—
105C2×D29—41no—
106C2×D29—42no—
107C2×D29—42no—
108C2×D29—41no—
109C2×D29—81no—
110C2×D29#12011yescharacteristic
111C2×D29#12112yes—
112C2×D29#12311yescharacteristic
113#1181—44no—
114#1181—82no—
115C4×C29—21no—
116C4×C29—41no—
117C4×C29—41no—
118Dic29—21no—
119Dic29—41no—
120Dic29—41no—
121C2×#123—291no—
122#518—291no—
123C2×#99—291no—
124#381—292no—
125#342—292no—
126C2×C29—21no—
127C2×C29—22no—
128C2×C29—41no—
129C2×C29#38611yescharacteristic
130D29—22no—
131D29—21no—
132D29—41no—
133D29#45812yes—
134C22×D4—291no—
135C23×C4—291no—
136#123—291no—
137#123—291no—
138#123—291no—
139#123—581no—
140#123—581no—
141#123#118111yescharacteristic, complemented
142C4×D4—584no—
143#117—581no—
144C2×C4⋊C4—581no—
145C2×C22⋊C4—291no—
146C2×C22⋊C4—292no—
147C2×C22⋊C4—292no—
148C2×C22⋊C4—581no—
149C2×C22⋊C4—581no—
150#99—581no—
151#99—581no—
152#95—584no—
153C29#189011yescharacteristic, complemented, Sylow
154C24—291no—
155C24—291no—
156C2×D4—291no—
157C2×D4—291no—
158C2×D4—291no—
159C2×D4—582no—
160C2×D4—582no—
161C2×D4C2×#118111yescharacteristic
162C22×C4—291no—
163C22×C4—291no—
164C22×C4—294no—
165C22×C4—582no—
166C22×C4—581no—
167C22×C4—581no—
168C22×C4—581no—
169C22×C4—582no—
170C22×C4—582no—
171C22×C4—582no—
172C22×C4C2×#118111yescharacteristic, complemented
173C4⋊C4—21no—
174C4⋊C4—581no—
175C4⋊C4—581no—
176C4⋊C4—582no—
177C4⋊C4—581no—
178C22⋊C4—21no—
179C22⋊C4—291no—
180C22⋊C4—581no—
181C22⋊C4—581no—
182C22⋊C4—581no—
183C22⋊C4—582no—
184C22⋊C4—582no—
185C22⋊C4—582no—
186C22⋊C4—581no—
187C22⋊C4—582no—
188C22⋊C4—582no—
189C22⋊C4C2×#118111yescharacteristic, complemented
190C42—582no—
191C23—292no—
192C23—291no—
193C23—292no—
194C23—291no—
195C23—291no—
196C23—581no—
197C23—581no—
198C23—1162no—
199C23—1162no—
200C23C22⋊#118111yescharacteristic
201C23C22×#118111yescharacteristic
202D4—22no—
203D4—582no—
204D4—582no—
205D4—582no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—291no—
210C2×C4—581no—
211C2×C4—581no—
212C2×C4—584no—
213C2×C4—582no—
214C2×C4—581no—
215C2×C4—581no—
216C2×C4—1161no—
217C2×C4—1161no—
218C2×C4—1162no—
219C2×C4—1162no—
220C2×C4—1164no—
221C2×C4—1162no—
222C2×C4—1164no—
223C2×C4—2322no—
224C2×C4C22⋊#118111yescharacteristic
225C22—21no—
226C22—41no—
227C22—41no—
228C22—291no—
229C22—291no—
230C22—292no—
231C22—584no—
232C22—582no—
233C22—581no—
234C22—1161no—
235C22—1162no—
236C22—1161no—
237C22—1162no—
238C22—2321no—
239C22D4×#118112yes—
240C22C2×C22⋊#118111yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—581no—
245C4—1161no—
246C4—1164no—
247C4—1161no—
248C4—2322no—
249C2—21no—
250C2—22no—
251C2—41no—
252C2—292no—
253C2—582no—
254C2—581no—
255C2—1161no—
256C2#1427211111yescharacteristic
subgroupG/H
centerC2#14272111
commutatorC23×C29C22×C4
FrattiniC23C22×#1181
FittingC29×#123C4
socleC2×C29#386
prime radicalC2×D4×D29C22
cosocleC23×D29C23
hypercenter#123#1181
hypocenterC29#1890
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #14272106C2semidirect
4. #14272040C2semidirect
13. C29×#123C4semidirect
14. #11745297C4semidirect
15. #11745291C4semidirect
18. #11745248C4semidirect
31. C22×C4×C29C2×C4semidirect
34. C22×Dic29C2×C4semidirect
53. C22⋊C4×C29C2×C4semidirect
55. C22⋊Dic29C2×C4semidirect
141. #123#1181semidirect
153. C29#1890semidirect
172. C22×C4C2×#1181semidirect
189. C22⋊C4C2×#1181semidirect

Maximal subgroups

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

subgroupindex
#142722302
D29⋊#992
#142721062
#142720402
#123×D292
#189029