Finite Group Databasebeta

← #15671733

Subgroups — #15671733

Order 2944.

316 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 77 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×C4×C22⋊D23C211yesmaximal, characteristic
2#13092381C211yesmaximal, characteristic, complemented
3#13092142C212yesmaximal, complemented
4#13092140C212yesmaximal, complemented
5C23×#407C211yesmaximal, characteristic, complemented
6C23×C4×C23C2211yescharacteristic
7C22×C22⋊D23C411yescharacteristic
8C22×C4×D23C411yescharacteristic
9C2×M16×C23—21no—
10C2×C22⋊Dic23C411yescharacteristic
11C4×C22⋊D23—21no—
12C4×C22⋊D23—21no—
13C4×C22⋊D23—21no—
14C4×C22⋊D23—21no—
15C2×(C2×D23)⋊C4C2211yescharacteristic
16C2×Dic23⋊C4C2211yescharacteristic
17C2×C4×Dic23C411yescharacteristic
18C2×#67267—21no—
19C23×C22⋊C8—21no—
20C23×C22⋊C8C2212yes—
21C22⋊(C23⋊C8)—21no—
22C22⋊(C23⋊C8)C2212yes—
23C24×C23C2×C411yescharacteristic
24C23×D23C2×C411yescharacteristic
25C22×C4×C23—42no—
26C22×C4×C23C2×C411yescharacteristic
27C22×C4×C23D411yescharacteristic
28C22×C4×C23D411yescharacteristic
29C22×C4×C23D411yescharacteristic
30C22×C4×C23D411yescharacteristic
31C22×C4×C23C2311yescharacteristic
32C2×C22⋊D23—21no—
33C2×C22⋊D23—21no—
34C2×C22⋊D23—22no—
35C2×C22⋊D23—22no—
36C22×Dic23C2×C411yescharacteristic
37C22×Dic23C2×C411yescharacteristic
38C22×Dic23C2×C411yescharacteristic
39C2×C4×D23—21no—
40C2×C4×D23—21no—
41C2×C4×D23—42no—
42M16×C23—42no—
43C2×C8×C23—22no—
44C22⋊Dic23—21no—
45C22⋊Dic23—21no—
46(C2×D23)⋊C4—21no—
47(C2×D23)⋊C4—21no—
48Dic23⋊C4—21no—
49Dic23⋊C4—21no—
50C4×Dic23—21no—
51C4×Dic23—21no—
52#67267—42no—
53C2×C23⋊C8—22no—
54C23×C23—21no—
55C23×C23—21no—
56C23×C23—41no—
57C23×C23—41no—
58C23×C23C22⋊C411yescharacteristic
59C23×C23C22⋊C411yescharacteristic
60C23×C23C22×C411yescharacteristic
61C22×D23—21no—
62C22×D23—21no—
63C22×D23—41no—
64C22×D23—41no—
65C22×D23M1612yescomplemented
66C2×C4×C23—22no—
67C2×C4×C23—22no—
68C2×C4×C23—22no—
69C2×C4×C23—41no—
70C2×C4×C23—41no—
71C2×C4×C23—41no—
72C2×C4×C23C22⋊C411yescharacteristic
73C2×C4×C23C22⋊C411yescharacteristic
74C2×C4×C23C2×D411yescharacteristic
75C2×C4×C23C2×D411yescharacteristic
76C22⋊D23—24no—
77C22⋊D23—24no—
78C2×Dic23—21no—
79C2×Dic23—21no—
80C2×Dic23—21no—
81C2×Dic23—21no—
82C2×Dic23—41no—
83C2×Dic23—41no—
84C2×Dic23M1612yescomplemented
85C4×D23—42no—
86C4×D23—42no—
87C8×C23—42no—
88C23⋊C8—42no—
89#1465—231nomaximal, Sylow
90C22×C23—21no—
91C22×C23—21no—
92C22×C23—21no—
93C22×C23—21no—
94C22×C23—21no—
95C22×C23—21no—
96C22×C23—21no—
97C22×C23—21no—
98C22×C23—42no—
99C22×C23—42no—
100C22×C23C2×C22⋊C411yescharacteristic
101C22×C23C2×M1611yescharacteristic
102C22×C23C2×M1611yescharacteristic
103C2×D23—22no—
104C2×D23—42no—
105C2×D23—41no—
106C2×D23—41no—
107C2×D23—42no—
108C4×C23—21no—
109C4×C23—21no—
110C4×C23—21no—
111C4×C23#9912yes—
112Dic23—22no—
113Dic23—41no—
114Dic23—41no—
115C2×C4×D4—231no—
116#407—231no—
117#407D2311yescharacteristic, complemented
118#324—232no—
119#323—232no—
120C2×C23—21no—
121C2×C23—21no—
122C2×C23—41no—
123C2×C23—41no—
124C2×C23#40711yescharacteristic
125C2×C23C2×#9911yescharacteristic
126C2×C23#41311yescharacteristic
127D23—42no—
128C22×D4—231no—
129C23×C4—231no—
130C23×C4C2×D2311yescharacteristic
131C2×M16—21no—
132C2×M16—461no—
133C4×D4—461no—
134C4×D4—461no—
135C4×D4—461no—
136C4×D4—461no—
137C2×C4⋊C4—231no—
138C2×C22⋊C4—231no—
139C2×C22⋊C4—231no—
140C2×C42—231no—
141C22⋊C8—21no—
142C22⋊C8—232no—
143C22⋊C8—461no—
144C22⋊C8C2×D2312yes—
145C23#146511yescharacteristic, complemented, Sylow
146C24—231no—
147C24C4×D2311yescharacteristic
148C2×D4—462no—
149C2×D4—462no—
150C2×D4—461no—
151C2×D4—461no—
152C22×C4—42no—
153C22×C4—231no—
154C22×C4—231no—
155C22×C4—231no—
156C22×C4—461no—
157C22×C4—461no—
158C22×C4—922no—
159C22×C4C4×D2311yescharacteristic
160C22×C4D9211yescharacteristic
161C22×C4D9211yescharacteristic
162C22×C4C22⋊D2311yescharacteristic
163C22×C4C22⋊D2311yescharacteristic
164C22×C4C22×D2311yescharacteristic
165M16—42no—
166M16—922no—
167C2×C8—22no—
168C2×C8—462no—
169C4⋊C4—461no—
170C4⋊C4—461no—
171C22⋊C4—461no—
172C22⋊C4—461no—
173C22⋊C4—461no—
174C22⋊C4—461no—
175C42—461no—
176C42—461no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—232no—
182C23—461no—
183C23—461no—
184C23—921no—
185C23—921no—
186C23(C2×D23)⋊C411yescharacteristic
187C23(C2×D23)⋊C411yescharacteristic
188C23C2×C4×D2311yescharacteristic
189D4—464no—
190D4—464no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—232no—
198C2×C4—461no—
199C2×C4—461no—
200C2×C4—461no—
201C2×C4—461no—
202C2×C4—922no—
203C2×C4—922no—
204C2×C4—921no—
205C2×C4—921no—
206C2×C4(C2×D23)⋊C411yescharacteristic
207C2×C4(C2×D23)⋊C411yescharacteristic
208C2×C4C2×D9211yescharacteristic
209C2×C4C2×C22⋊D2311yescharacteristic
210C8—42no—
211C8—922no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—462no—
223C22—922no—
224C22—922no—
225C22—921no—
226C22—921no—
227C22M16×D2311yescharacteristic
228C22M16×D2311yescharacteristic
229C22C2×(C2×D23)⋊C411yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—462no—
234C4—921no—
235C4—921no—
236C4#1062498812yes—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2—922no—
242C2#1309238011yescharacteristic
243C2C2×#1062498811yescharacteristic
244C2#1309239911yescharacteristic
subgroupG/H
centerC22C2×(C2×D23)⋊C4
commutatorC23×C23C22×C4
FrattiniC22×C4C22×D23
FittingC23×#407C2
socleC22×C23C2×C22⋊C4
prime radicalC22×C22⋊D23C4
cosocleC22×C4×C23C23
hypercenter#407D23
hypocenterC23#1465
LCSG ▷ C23×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #13092381C2semidirect
3. #13092142C2semidirect
4. #13092140C2semidirect
5. C23×#407C2semidirect
65. C22×D23M16semidirect
84. C2×Dic23M16semidirect
117. #407D23semidirect
145. C23#1465semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D232
#130923812
#130921422
#130921402
C23×#4072
#146523