Finite Group Databasebeta

← #37549862

Subgroups — #37549862

Order 23168.

408 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 72 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×#19153875C212yesmaximal, complemented
2C2×#19153874C212yesmaximal, complemented
3#26042221C211yesmaximal, characteristic, complemented
4C4×C22⋊#10624382C211yesmaximal, characteristic, complemented
5C2×#19153789C211yesmaximal, characteristic
6C2×C4⋊#10624382—21no—
7C2×C22⋊#10624382C2212yes—
8C2×C22⋊#10624382C2212yes—
9C2×C4×#10624382—21no—
10#19153875—24no—
11#19153874—24no—
12(C2×#10624382)⋊C4—21no—
13(C2×#10624382)⋊C4—21no—
14C22×D724C2211yescharacteristic
15C22×C22⋊D181C411yescharacteristic, complemented
16C22×C22⋊D181C2211yescharacteristic
17C22×C4×D181C2211yescharacteristic
18#19153789—21no—
19#19153789—21no—
20#19153789—21no—
21#19153789—21no—
22C2×#15668428C411yescharacteristic, complemented
23C2×#15668426C411yescharacteristic, complemented
24C2×C22⋊C4×C181C411yescharacteristic, complemented
25C22×#10624382—22no—
26C22×#10624382—22no—
27C4⋊#10624382—42no—
28C22⋊#10624382—22no—
29C22⋊#10624382—22no—
30C22⋊#10624382—22no—
31C22⋊#10624382D414yescomplemented
32C4×#10624382—42no—
33C23×D181C2×C411yescharacteristic
34C23×D181C2311yescharacteristic
35C2×D724—21no—
36C2×D724—21no—
37C2×D724—22no—
38C2×D724—22no—
39C2×C22⋊D181—21no—
40C2×C22⋊D181—21no—
41C2×C22⋊D181—22no—
42C2×C22⋊D181—22no—
43C2×C22⋊D181—21no—
44C2×C22⋊D181—21no—
45C2×C22⋊D181—42no—
46C22×Dic181C2×C411yescharacteristic
47C22×Dic181C2×C411yescharacteristic
48C2×C4×D181—42no—
49C2×C4×D181D411yescharacteristic
50C2×C4×D181D411yescharacteristic
51C2×C4×D181D411yescharacteristic
52C2×C4×D181D411yescharacteristic
53#15668428—21no—
54#15668428—21no—
55#15668426—21no—
56#15668426—21no—
57C24×C181C2×C411yescharacteristic
58C22×C4×C181C2×C411yescharacteristic
59C22×C4×C181C2×C411yescharacteristic
60C22⋊C4×C181—21no—
61C22⋊C4×C181—21no—
62C2×#10624382—24no—
63C2×#10624382—42no—
64C2×#10624382—42no—
65C2×#10624382—42no—
66C2×#10624382—42no—
67C2×#10624382—42no—
68C22×D181—21no—
69C22×D181—22no—
70C22×D181—41no—
71C22×D181—41no—
72C22×D181—41no—
73C22×D181—41no—
74C22×D181C2×D411yescharacteristic
75C22×D181C2×D411yescharacteristic
76C22×D181C2×D412yes—
77C22×D181#4112yes—
78D724—42no—
79D724—41no—
80D724—41no—
81C22⋊D181—42no—
82C22⋊D181—42no—
83C22⋊D181—41no—
84C22⋊D181—41no—
85C22⋊D181—42no—
86C2×Dic181—22no—
87C2×Dic181—21no—
88C2×Dic181—22no—
89C2×Dic181C22⋊C411yescharacteristic
90C2×Dic181C22⋊C411yescharacteristic
91C4×D181—22no—
92C4×D181—22no—
93C4×D181—41no—
94C4×D181—41no—
95C23×C181—21no—
96C23×C181—22no—
97C23×C181—41no—
98C23×C181—41no—
99C23×C181C22×C411yescharacteristic
100C2×C4×C181—21no—
101C2×C4×C181—22no—
102C2×C4×C181—41no—
103C2×C4×C181C22⋊C411yescharacteristic
104C2×C4×C181C22⋊C411yescharacteristic
105C22×C181—21no—
106C22×C181—21no—
107C22×C181—41no—
108C22×C181—41no—
109C22×C181—41no—
110C22×C181—82no—
111C22×C181C2×C22⋊C411yescharacteristic
112C22×C181C4×D412yes—
113C2×D181—21no—
114C2×D181—21no—
115C2×D181—21no—
116C2×D181—21no—
117C2×D181—42no—
118C2×D181—42no—
119C2×D181—41no—
120C2×D181—41no—
121C2×D181—41no—
122C2×D181—82no—
123C2×D181#12112yes—
124C2×D181#12411yescharacteristic
125C2×D181#12711yescharacteristic
126#10624382—44no—
127#10624382—82no—
128C4×C181—21no—
129C4×C181—21no—
130C4×C181—41no—
131Dic181—21no—
132Dic181—41no—
133Dic181#9912yescomplemented
134C2×C181—22no—
135C2×C181—81no—
136C2×C181#39011yescharacteristic
137C2×C181C2×#9911yescharacteristic
138C2×C181#41011yescharacteristic
139D181—22no—
140D181—21no—
141D181—21no—
142D181—81no—
143C181#196411yescharacteristic, complemented, Sylow
144#1964—1811nomaximal, Sylow
145C2×#121—1811no—
146C2×#99—1812no—
147C2×#99—1812no—
148#380—1811no—
149C4×C22⋊C4—1811no—
150C22×D4—1811no—
151C22×D4—1811no—
152C22×D4—1811no—
153C23×C4—1811no—
154#121—3621no—
155#121—3621no—
156#121—3621no—
157#121—3621no—
158C2×C4⋊C4—3621no—
159C2×C4⋊C4—1811no—
160C2×C22⋊C4—1812no—
161C2×C22⋊C4—1811no—
162C2×C22⋊C4—1812no—
163C2×C22⋊C4#1062438211yescharacteristic, complemented
164C2×C42—3621no—
165#99—3624no—
166#99—3624no—
167#95—3621no—
168#95—3621no—
169C24—1811no—
170C24—1811no—
171C24C2×#1062438211yescharacteristic
172C2×D4—3622no—
173C2×D4—3621no—
174C2×D4—3621no—
175C2×D4—3622no—
176C2×D4—3622no—
177C2×D4—3621no—
178C2×D4—3621no—
179C2×D4—3622no—
180C2×D4—3621no—
181C2×D4—3621no—
182C2×D4—7242no—
183C22×C4—3622no—
184C22×C4—3622no—
185C22×C4—1811no—
186C22×C4—1811no—
187C22×C4—1811no—
188C22×C4—1811no—
189C22×C4—1811no—
190C22×C4—1811no—
191C22×C4—7242no—
192C22×C4C2×#1062438211yescharacteristic
193C22×C4C2×#1062438211yescharacteristic
194C4⋊C4—3621no—
195C4⋊C4—3621no—
196C4⋊C4—7242no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—3622no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C22⋊C4—3622no—
203C22⋊C4—3622no—
204C22⋊C4—1814no—
205C42—7242no—
206C23—21no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—3622no—
211C23—3621no—
212C23—1811no—
213C23—1811no—
214C23—1812no—
215C23—1812no—
216C23—7241no—
217C23—7241no—
218C23—7241no—
219C23—7241no—
220C23C22×#1062438211yescharacteristic
221D4—7242no—
222D4—7241no—
223D4—7241no—
224D4—7242no—
225D4—7242no—
226D4—7242no—
227D4—7241no—
228D4—7241no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—3622no—
233C2×C4—3622no—
234C2×C4—3622no—
235C2×C4—3624no—
236C2×C4—3621no—
237C2×C4—3622no—
238C2×C4—1811no—
239C2×C4—1811no—
240C2×C4—7241no—
241C2×C4—7241no—
242C2×C4—7242no—
243C2×C4—7242no—
244C2×C4—7242no—
245C2×C4—7242no—
246C2×C4—7242no—
247C2×C4C22⋊#1062438211yescharacteristic
248C2×C4C22⋊#1062438211yescharacteristic
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—82no—
255C22—3621no—
256C22—3621no—
257C22—3621no—
258C22—3621no—
259C22—14482no—
260C22—1812no—
261C22—1811no—
262C22—1811no—
263C22—7242no—
264C22—7242no—
265C22—7241no—
266C22—7241no—
267C22—7241no—
268C22C2×C22⋊#1062438211yescharacteristic
269C22D4×#1062438212yes—
270C4—21no—
271C4—21no—
272C4—41no—
273C4—3621no—
274C4—14482no—
275C4—1812no—
276C4—7244no—
277C4—7241no—
278C2—22no—
279C2—81no—
280C2—3621no—
281C2—3622no—
282C2—3621no—
283C2—14481no—
284C2#2604223111yescharacteristic
285C2D181⋊#9911yescharacteristic
286C2#2604225811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC23×C181C22×C4
FrattiniC23C22×#10624382
FittingC2×C22⋊C4×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC2×#19153789C2
cosocleC23×D181C23
hypercenterC2×C22⋊C4#10624382
hypocenterC181#1964
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19153875C2semidirect
2. C2×#19153874C2semidirect
3. #26042221C2semidirect
4. C4×C22⋊#10624382C2semidirect
15. C22×C22⋊D181C4semidirect
22. C2×#15668428C4semidirect
23. C2×#15668426C4semidirect
24. C2×C22⋊C4×C181C4semidirect
31. C22⋊#10624382D4semidirect
133. Dic181#99semidirect
143. C181#1964semidirect
163. C2×C22⋊C4#10624382semidirect

Maximal subgroups

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

subgroupindex
C2×#191538752
C2×#191538742
#260422212
C4×C22⋊#106243822
C2×#191537892
#1964181