Finite Group Databasebeta

← #16835745

Subgroups — #16835745

Order 3712.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#14272149C211yesmaximal, characteristic, complemented
2#14272142C211yesmaximal, characteristic, complemented
3#14272066C212yesmaximal, complemented
4#14272064C212yesmaximal, complemented
5#14271675C211yesmaximal, characteristic
6C2×#90805C2211yescharacteristic
7C22×C4×D29C411yescharacteristic, complemented
8#11745345—21no—
9D29⋊M16—21no—
10C29×#117C411yescharacteristic, complemented
11#11745259—21no—
12#11745257—21no—
13#11745248C411yescharacteristic, complemented
14#11745248C2211yescharacteristic
15#11745244C411yescharacteristic, complemented
16#11745244C2211yescharacteristic
17C4×D116—21no—
18C4×D116—21no—
19#11745229C2212yes—
20#11745227C2212yes—
21#11745221—21no—
22#11745215—21no—
23C23×D29C2×C411yescharacteristic
24C22×C4×C29C2×C411yescharacteristic
25C2×C22⋊D29C2×C411yescharacteristic
26C2×C22⋊D29C2311yescharacteristic
27C22×Dic29C2×C411yescharacteristic
28#90805—41no—
29#90805—41no—
30C2×D116—21no—
31C2×D116—21no—
32C2×D116D411yescharacteristic
33C2×C4×D29—21no—
34C2×C4×D29—21no—
35C2×C4×D29—41no—
36C2×C4×D29—41no—
37C2×C4×D29D411yescharacteristic
38C2×C4×D29D411yescharacteristic
39C2×Dic58D411yescharacteristic
40C22⋊#1181—22no—
41#90800—22no—
42C4⋊#1181—41no—
43C4×#1181—41no—
44#90796—41no—
45D29⋊C8—41no—
46C4⋊C4×C29—21no—
47C22⋊C4×C29C2×C411yescharacteristic
48C22⋊C4×C29C2×C411yescharacteristic
49C42×C29—21no—
50#90781—21no—
51#90781—21no—
52#90781—21no—
53#90781—21no—
54C4⋊Dic29—21no—
55C4⋊Dic29—21no—
56C23×C29C22×C411yescharacteristic
57C22×D29—21no—
58C22×D29—42no—
59C22×D29—41no—
60C22×D29C22⋊C411yescharacteristic
61C22×D29C2×D411yescharacteristic
62C2×#1181—42no—
63C2×C4×C29—21no—
64C2×C4×C29—21no—
65C2×C4×C29—21no—
66C2×C4×C29—21no—
67C2×C4×C29C22⋊C411yescharacteristic
68C2×C4×C29C22⋊C411yescharacteristic
69C22⋊D29—41no—
70C22⋊D29—41no—
71C2×Dic29—21no—
72C2×Dic29—41no—
73C2×Dic29C22⋊C411yescharacteristic
74C2×Dic29C2×D411yescharacteristic
75D116—41no—
76D116—42no—
77D116—42no—
78C4×D29—21no—
79C4×D29—21no—
80C4×D29—21no—
81C4×D29—21no—
82C4×D29—42no—
83C4×D29—41no—
84C4×D29—41no—
85Dic58—41no—
86#6599—42no—
87#2120—291nomaximal, Sylow
88C22×C29—21no—
89C22×C29—41no—
90C22×C29C2×C22⋊C411yescharacteristic
91C2×D29—22no—
92C2×D29—21no—
93C2×D29—21no—
94C2×D29—41no—
95C2×D29—42no—
96C2×D29—41no—
97C2×D29—41no—
98C2×D29—41no—
99C2×D29—42no—
100#1181—82no—
101C4×C29—21no—
102C4×C29—41no—
103C4×C29—41no—
104C4×C29#9911yescharacteristic
105C4×C29#9911yescharacteristic
106Dic29—21no—
107Dic29—21no—
108Dic29—41no—
109#525—291no—
110#413—291no—
111#410—291no—
112#352—292no—
113AΣL(1,𝔽16)—292no—
114C2×C29—21no—
115C2×C29—41no—
116C2×C29C2×#9911yescharacteristic
117D29—41no—
118D29—41no—
119D29—42no—
120C2×#41—291no—
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—581no—
132#117#118111yescharacteristic, complemented
133#101—581no—
134#100—292no—
135#99—292no—
136#99—581no—
137C29#212011yescharacteristic, complemented, Sylow
138C24—291no—
139#41—1161no—
140#41—1161no—
141C2×Q8—291no—
142C2×D4—291no—
143C2×D4—291no—
144C2×D4—291no—
145C2×D4—581no—
146C2×D4—581no—
147C22×C4—291no—
148C22×C4—291no—
149C22×C4—291no—
150C22×C4—581no—
151C22×C4—581no—
152C22×C4—1161no—
153C22×C4—1161no—
154C22×C4C2×#118111yescharacteristic
155M16—582no—
156M16—1161no—
157C2×C8—1161no—
158C4⋊C4—21no—
159C4⋊C4—581no—
160C4⋊C4—581no—
161C4⋊C4—1161no—
162C22⋊C4—581no—
163C22⋊C4—581no—
164C22⋊C4—581no—
165C22⋊C4—581no—
166C22⋊C4—582no—
167C22⋊C4C2×#118111yescharacteristic
168C22⋊C4C2×#118111yescharacteristic
169C42—21no—
170C42—1161no—
171C23—291no—
172C23—291no—
173C23—581no—
174C23—1161no—
175C23—1162no—
176C23C22×#118111yescharacteristic
177Q8—1161no—
178D4—1161no—
179D4—1162no—
180D4—1162no—
181D4—1161no—
182D4—1161no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—291no—
188C2×C4—291no—
189C2×C4—581no—
190C2×C4—581no—
191C2×C4—581no—
192C2×C4—581no—
193C2×C4—581no—
194C2×C4—1162no—
195C2×C4—1161no—
196C2×C4—1161no—
197C2×C4—1161no—
198C2×C4—1162no—
199C2×C4C22⋊#118111yescharacteristic
200C2×C4C22⋊#118111yescharacteristic
201C8—1162no—
202C22—21no—
203C22—41no—
204C22—582no—
205C22—581no—
206C22—581no—
207C22—1161no—
208C22—1161no—
209C22—1161no—
210C22—1162no—
211C22—1162no—
212C22—1161no—
213C22C2×C22⋊#118111yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—581no—
218C4—581no—
219C4—1161no—
220C4—2322no—
221C4#1174521511yescharacteristic
222C4#1174521511yescharacteristic
223C2—21no—
224C2—41no—
225C2—1161no—
226C2—1162no—
227C2—1161no—
228C2C2×#1174521511yescharacteristic
subgroupG/H
centerC4#11745215
commutatorC23×C29C22×C4
FrattiniC23C22×#1181
FittingC29×#117C4
socleC2×C29C2×#99
prime radical#14271675C2
cosocleC2×C22⋊D29C23
hypercenter#117#1181
hypocenterC29#2120
LCSG ▷ C23×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #14272149C2semidirect
2. #14272142C2semidirect
3. #14272066C2semidirect
4. #14272064C2semidirect
7. C22×C4×D29C4semidirect
10. C29×#117C4semidirect
13. #11745248C4semidirect
15. #11745244C4semidirect
132. #117#1181semidirect
137. C29#2120semidirect

Maximal subgroups

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

subgroupindex
#142721492
#142721422
#142720662
#142720642
#142716752
#212029