Finite Group Databasebeta

← #16820034

Subgroups — #16820034

Order 3712.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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#14271722C211yesmaximal, characteristic
2C2×#11745182C211yesmaximal, characteristic, complemented
3C2×#11745154C212yesmaximal, complemented
4#14270929C212yesmaximal, complemented
5C29×#518C211yesmaximal, characteristic, complemented
6C22×D4×C29C2211yescharacteristic
7C22×C22⋊D29C2211yescharacteristic
8C4×D4×C29—24no—
9C29×#117—21no—
10C2×C22⋊C4×C29—21no—
11C2×C22⋊C4×C29C2212yes—
12#11745315—21no—
13C2×C22⋊Dic29C2212yes—
14C2×C22⋊Dic29C2211yescharacteristic
15#11745301—22no—
16#11745299C412yescomplemented
17#11745297C412yescomplemented
18#11745291—21no—
19#11745182—22no—
20#11745154—24no—
21C24×C29D411yescharacteristic
22C24×C29C2311yescharacteristic
23C23×D29C2×C411yescharacteristic, complemented
24C2×D4×C29—22no—
25C2×D4×C29—22no—
26C2×D4×C29C2×C412yes—
27C2×D4×C29D412yes—
28C22×C4×C29—22no—
29C22×C4×C29—22no—
30C22×C4×C29D411yescharacteristic
31C2×C22⋊D29—21no—
32C2×C22⋊D29—21no—
33C2×C22⋊D29—22no—
34C2×C22⋊D29—22no—
35C2×C22⋊D29—22no—
36C22×Dic29—21no—
37C22×Dic29—22no—
38C22×Dic29C2×C411yescharacteristic, complemented
39C4⋊C4×C29—22no—
40C22⋊C4×C29—22no—
41C22⋊C4×C29—22no—
42C22⋊C4×C29D414yescomplemented
43C42×C29—22no—
44C22⋊Dic29—22no—
45C22⋊Dic29—21no—
46C22⋊Dic29—22no—
47C22⋊Dic29—22no—
48C22⋊Dic29C2×C412yescomplemented
49#90781—22no—
50#90779—22no—
51C23×C29—21no—
52C23×C29—21no—
53C23×C29—81no—
54C23×C29—81no—
55C23×C29C22⋊C412yes—
56C23×C29C22×C411yescharacteristic
57C23×C29C2×D411yescharacteristic
58C23×C29C2×D411yescharacteristic
59C23×C29C2×D412yes—
60C23×C29#4112yes—
61C22×D29—21no—
62C22×D29—22no—
63C22×D29—81no—
64D4×C29—24no—
65D4×C29—24no—
66C2×C4×C29—21no—
67C2×C4×C29—21no—
68C2×C4×C29—24no—
69C2×C4×C29—82no—
70C2×C4×C29—82no—
71C2×C4×C29C22⋊C412yes—
72C22⋊D29—41no—
73C22⋊D29—42no—
74C22⋊D29—41no—
75C22⋊D29—82no—
76C2×Dic29—21no—
77C2×Dic29—22no—
78C2×Dic29—22no—
79C2×Dic29—42no—
80C2×Dic29—81no—
81C2×Dic29—82no—
82#1913—291nomaximal, Sylow
83C22×C29—22no—
84C22×C29—41no—
85C22×C29—42no—
86C22×C29—42no—
87C22×C29—41no—
88C22×C29—41no—
89C22×C29—42no—
90C22×C29—42no—
91C22×C29—81no—
92C22×C29C2×C22⋊C411yescharacteristic
93C22×C29C4×D412yes—
94C22×C29#12011yescharacteristic
95C22×C29#12112yes—
96C22×C29#12311yescharacteristic
97C2×D29—42no—
98C2×D29—41no—
99C2×D29—82no—
100C4×C29—22no—
101C4×C29—44no—
102Dic29—42no—
103Dic29—41no—
104Dic29—82no—
105#534—291no—
106#518D2911yescharacteristic, complemented
107C2×#99—291no—
108C2×#99—292no—
109#342—292no—
110C2×C29—21no—
111C2×C29—22no—
112C2×C29—21no—
113C2×C29—22no—
114C2×C29—21no—
115C2×C29—42no—
116C2×C29#38611yescharacteristic
117D29—81no—
118C22×D4—291no—
119C22×D4C2×D2911yescharacteristic
120#123—292no—
121#123—581no—
122#121—582no—
123#120—292no—
124#120—581no—
125C4×D4—24no—
126#117—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—291no—
129C2×C22⋊C4—292no—
130C2×C22⋊C4C2×D2912yes—
131#99—582no—
132#99—584no—
133C29#191311yescharacteristic, complemented, Sylow
134C24—291no—
135C24C22⋊D2911yescharacteristic
136C24C22×D2911yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—581no—
140C2×D4—582no—
141C2×D4—582no—
142C2×D4—581no—
143C2×D4—582no—
144C2×D4C4×D2912yes—
145C2×D4D11612yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—291no—
149C22×C4—581no—
150C22×C4—582no—
151C22×C4C22⋊D2911yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—582no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—292no—
157C22⋊C4—582no—
158C22⋊C4—581no—
159C22⋊C4—582no—
160C22⋊C4—582no—
161C22⋊C4—582no—
162C22⋊C4C22⋊D2914yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—581no—
169C23—582no—
170C23—2321no—
171C23#9078112yes—
172C23C2×C4×D2911yescharacteristic
173C23C2×D11611yescharacteristic
174C23#9080512yes—
175C23C2×C22⋊D2911yescharacteristic
176C23C2×C22⋊D2912yes—
177D4—24no—
178D4—24no—
179D4—1161no—
180D4—1162no—
181D4—1161no—
182D4—2322no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—582no—
189C2×C4—581no—
190C2×C4—582no—
191C2×C4—1162no—
192C2×C4—2321no—
193C2×C4—2322no—
194C2×C4#9078112yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—1162no—
205C22—1161no—
206C22—2322no—
207C22C2×#9078111yescharacteristic
208C22C4×C22⋊D2912yes—
209C22#1174529111yescharacteristic
210C22#1174529212yes—
211C22#1174531511yescharacteristic
212C4—22no—
213C4—44no—
214C4—1162no—
215C4—1161no—
216C4—2322no—
217C2—21no—
218C2—21no—
219C2—22no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—2321no—
224C2#1427104211yescharacteristic
subgroupG/H
centerC2#14271042
commutatorC23×C29C22×C4
FrattiniC24C22×D29
FittingC29×#518C2
socleC2×C29#386
prime radical#14271722C2
cosocleC24×C29C23
hypercenter#518D29
hypocenterC29#1913
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#11745182C2semidirect
3. C2×#11745154C2semidirect
4. #14270929C2semidirect
5. C29×#518C2semidirect
16. #11745299C4semidirect
17. #11745297C4semidirect
23. C23×D29C2×C4semidirect
38. C22×Dic29C2×C4semidirect
42. C22⋊C4×C29D4semidirect
48. C22⋊Dic29C2×C4semidirect
106. #518D29semidirect
133. C29#1913semidirect
162. C22⋊C4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
#142717222
C2×#117451822
C2×#117451542
#142709292
C29×#5182
#191329