Finite Group Databasebeta

← #16821131

Subgroups — #16821131

Order 3712.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#14271670C211yesmaximal, characteristic
2#14271147C211yesmaximal, characteristic, complemented
3#14270952C212yesmaximal, complemented
4#14270949C212yesmaximal, complemented
5C29×#410C211yesmaximal, characteristic, complemented
6C2×#41×C29C2211yescharacteristic
7C22×C4×D29C411yescharacteristic, complemented
8C29×#117—21no—
9#11745308—21no—
10#11745300—21no—
11#11745299C411yescharacteristic, complemented
12#11745299C2211yescharacteristic
13#11745297C411yescharacteristic, complemented
14#11745297C2211yescharacteristic
15C4×C22⋊D29—21no—
16C4×C22⋊D29—21no—
17#11745288C411yescharacteristic, complemented
18C2×#90777—21no—
19C29×#99—21no—
20C29×#99C2212yes—
21#11745184—21no—
22#11745181C2212yes—
23C23×D29C2×C411yescharacteristic
24#41×C29—41no—
25#41×C29—41no—
26C2×Q8×C29D411yescharacteristic
27C2×D4×C29C2×C411yescharacteristic
28C2×D4×C29D411yescharacteristic
29C2×D4×C29C2311yescharacteristic
30C22×C4×C29C2×C411yescharacteristic
31C22×C4×C29D411yescharacteristic
32C22×C4×C29D411yescharacteristic
33C2×C22⋊D29—21no—
34C2×C22⋊D29—21no—
35C22×Dic29C2×C411yescharacteristic
36C2×C4×D29—21no—
37C2×C4×D29—21no—
38C2×C4×D29—41no—
39C2×C4×D29—41no—
40C4⋊C4×C29—41no—
41C22⋊C4×C29—22no—
42C42×C29—41no—
43C22⋊Dic29—21no—
44C22⋊Dic29—21no—
45C22⋊Dic29C2×C411yescharacteristic
46C22⋊Dic29C2×C411yescharacteristic
47#90781—21no—
48#90781—21no—
49C4⋊Dic29—21no—
50#90779—21no—
51#90779—21no—
52C4×Dic29—21no—
53#90777—22no—
54#90777—41no—
55C2×#6597—41no—
56C23×C29C22⋊C411yescharacteristic
57C23×C29C22×C411yescharacteristic
58C23×C29C2×D411yescharacteristic
59C22×D29—21no—
60C22×D29—42no—
61C22×D29—41no—
62Q8×C29—41no—
63D4×C29—41no—
64D4×C29—41no—
65D4×C29—41no—
66C2×C4×C29—21no—
67C2×C4×C29—21no—
68C2×C4×C29—21no—
69C2×C4×C29—21no—
70C2×C4×C29—21no—
71C2×C4×C29—21no—
72C2×C4×C29—42no—
73C2×C4×C29C22⋊C411yescharacteristic
74C2×C4×C29C22⋊C411yescharacteristic
75C2×C4×C29C22⋊C411yescharacteristic
76C2×C4×C29C2×D411yescharacteristic
77C22⋊D29—42no—
78C22⋊D29—42no—
79C2×Dic29—21no—
80C2×Dic29—21no—
81C2×Dic29—21no—
82C2×Dic29—41no—
83C4×D29—42no—
84C4×D29—41no—
85C4×D29—41no—
86#6597—42no—
87#2120—291nomaximal, Sylow
88C22×C29—21no—
89C22×C29—21no—
90C22×C29—21no—
91C22×C29—41no—
92C22×C29—41no—
93C22×C29—41no—
94C22×C29C2×C22⋊C411yescharacteristic
95C2×D29—22no—
96C2×D29—41no—
97C2×D29—42no—
98C2×D29—41no—
99C2×D29—42no—
100C4×C29—21no—
101C4×C29—21no—
102C4×C29—21no—
103C4×C29—82no—
104C4×C29#9911yescharacteristic
105C4×C29#9911yescharacteristic
106Dic29—41no—
107Dic29—41no—
108Dic29—41no—
109#525—291no—
110#413—291no—
111#410D2911yescharacteristic, complemented
112#352—292no—
113AΣL(1,𝔽16)—292no—
114C2×C29—21no—
115C2×C29—41no—
116C2×C29—41no—
117C2×C29—41no—
118C2×C29C2×#9911yescharacteristic
119D29—42no—
120C2×#41C2×D2911yescharacteristic
121C23×C4—291no—
122C2×M16—581no—
123#123—291no—
124#123—291no—
125C22⋊Q8—581no—
126#121—581no—
127#120—291no—
128#120—291no—
129C4×D4—581no—
130C4×D4—581no—
131#117—21no—
132#117—291no—
133#101—581no—
134#100—292no—
135#99—21no—
136#99C2×D2912yes—
137C29#212011yescharacteristic, complemented, Sylow
138C24—291no—
139#41—41no—
140#41—41no—
141C2×Q8D11611yescharacteristic
142C2×D4—581no—
143C2×D4—581no—
144C2×D4C4×D2911yescharacteristic
145C2×D4C22⋊D2911yescharacteristic
146C2×D4C22×D2911yescharacteristic
147C22×C4—291no—
148C22×C4—581no—
149C22×C4—581no—
150C22×C4—1161no—
151C22×C4—1161no—
152C22×C4C4×D2911yescharacteristic
153C22×C4D11611yescharacteristic
154C22×C4C22⋊D2911yescharacteristic
155M16—582no—
156M16—1161no—
157C2×C8—1161no—
158C4⋊C4—41no—
159C4⋊C4—581no—
160C4⋊C4—581no—
161C4⋊C4—581no—
162C22⋊C4—22no—
163C22⋊C4—291no—
164C22⋊C4—291no—
165C22⋊C4—581no—
166C22⋊C4—581no—
167C22⋊C4—581no—
168C22⋊C4—581no—
169C42—41no—
170C42—581no—
171C23—581no—
172C23—1161no—
173C23—1162no—
174C23#9078111yescharacteristic
175C23C2×C4×D2911yescharacteristic
176C23C2×C22⋊D2911yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—1162no—
182D4—1162no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—581no—
191C2×C4—581no—
192C2×C4—581no—
193C2×C4—1162no—
194C2×C4—1161no—
195C2×C4—1161no—
196C2×C4—1161no—
197C2×C4#9078111yescharacteristic
198C2×C4#9078111yescharacteristic
199C2×C4#9078111yescharacteristic
200C2×C4C2×D11611yescharacteristic
201C8—1162no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—582no—
209C22—1161no—
210C22—1162no—
211C22—1162no—
212C22—1161no—
213C22C2×#9078111yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—1161no—
219C4—1161no—
220C4—1161no—
221C4#1174515411yescharacteristic
222C4#1174515411yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—1162no—
228C2C2×#1174515411yescharacteristic
subgroupG/H
centerC4#11745154
commutatorC23×C29C22×C4
FrattiniC2×D4C22×D29
FittingC29×#410C2
socleC2×C29C2×#99
prime radical#14271670C2
cosocleC2×D4×C29C23
hypercenter#410D29
hypocenterC29#2120
LCSG ▷ C23×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #14271147C2semidirect
3. #14270952C2semidirect
4. #14270949C2semidirect
5. C29×#410C2semidirect
7. C22×C4×D29C4semidirect
11. #11745299C4semidirect
13. #11745297C4semidirect
17. #11745288C4semidirect
111. #410D29semidirect
137. C29#2120semidirect

Maximal subgroups

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

subgroupindex
#142716702
#142711472
#142709522
#142709492
C29×#4102
#212029