Finite Group Databasebeta

← #37549702

Subgroups — #37549702

Order 23168.

376 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C22×#15668459C211yesmaximal, characteristic, complemented
2C2×#19153881C212yesmaximal, complemented
3C4×C22⋊#10624382C211yesmaximal, characteristic, complemented
4#26042149C212yesmaximal, complemented
5#26041792C211yesmaximal, characteristic
6C2×#15668459—24no—
7C2×#15668459—21no—
8C2×#15668459C2212yes—
9C22×#13089685—21no—
10C2×C22⋊#10624382—21no—
11C2×C4×#10624382C2212yes—
12#19153881—24no—
13(C2×#10624382)⋊C4—21no—
14C2×#15668450C2211yescharacteristic
15C22×C4×D181C2211yescharacteristic
16#19153801—21no—
17#19153792C411yescharacteristic, complemented
18#19153791C411yescharacteristic, complemented
19#19153790C411yescharacteristic, complemented
20#19153786—21no—
21#19153778—21no—
22#19153776—21no—
23#19153774C2211yescharacteristic
24C22⋊Q8×C181C411yescharacteristic, complemented
25C22×#10624382—22no—
26#15668459—24no—
27#15668459—22no—
28#15668459D414yescomplemented
29C2×#13089685—24no—
30C2×#13089685—22no—
31C22⋊#10624382—24no—
32C4×#10624382—22no—
33C4×#10624382—22no—
34C23×D181D411yescharacteristic
35#15668450—22no—
36#15668450—22no—
37C2×D724—21no—
38C2×D724—21no—
39C2×D724C2×C411yescharacteristic
40C2×C22⋊D181—21no—
41C22×Dic181D411yescharacteristic
42C2×C4×D181—22no—
43C2×C4×D181—22no—
44C2×C4×D181—21no—
45C2×C4×D181C2×C411yescharacteristic
46C2×C4×D181D411yescharacteristic
47C2×C4×D181D411yescharacteristic
48C2×C4×D181C2311yescharacteristic
49C4⋊Dic181C2×C411yescharacteristic
50#15668428—21no—
51#15668426—21no—
52#15668426—21no—
53#15668426—21no—
54#15668426—21no—
55C4×Dic181—21no—
56C2×Q8×C181C2×C411yescharacteristic
57C22×C4×C181C2×C411yescharacteristic
58C4⋊C4×C181—21no—
59C4⋊C4×C181C2×C411yescharacteristic
60C22⋊C4×C181—21no—
61C2×#10624382—24no—
62C2×#10624382—42no—
63C2×#10624382—42no—
64#13089685—24no—
65C22×D181—22no—
66C22×D181—22no—
67C22×D181—21no—
68C22×D181—41no—
69C22×D181C22⋊C411yescharacteristic
70C22×D181C2×D411yescharacteristic
71D724—22no—
72D724—42no—
73D724—42no—
74C22⋊D181—42no—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—21no—
78C2×Dic181C22⋊C411yescharacteristic
79C2×Dic181C2×D411yescharacteristic
80C4×D181—22no—
81C4×D181—22no—
82C4×D181—21no—
83C4×D181—21no—
84C4×D181—41no—
85C4×D181—41no—
86C4×D181C2×D412yes—
87C4×D181#4112yes—
88C23×C181C22⋊C411yescharacteristic
89Q8×C181—22no—
90C2×C4×C181—22no—
91C2×C4×C181—21no—
92C2×C4×C181—21no—
93C2×C4×C181C22⋊C411yescharacteristic
94C2×C4×C181C22×C411yescharacteristic
95C22×C181—21no—
96C22×C181—21no—
97C22×C181#10412yescomplemented
98C22×C181C2×C22⋊C411yescharacteristic
99C2×D181—21no—
100C2×D181—21no—
101C2×D181—21no—
102C2×D181—21no—
103C2×D181—21no—
104C2×D181—44no—
105C2×D181—41no—
106C2×D181—41no—
107C2×D181—41no—
108C2×D181#12112yes—
109#10624382—44no—
110C4×C181—21no—
111C4×C181—41no—
112C4×C181—41no—
113C4×C181C4×D412yes—
114Dic181—21no—
115Dic181—41no—
116Dic181#12011yescharacteristic
117Dic181#12311yescharacteristic
118C2×C181—22no—
119C2×C181#38611yescharacteristic
120C2×C181C2×#10411yescharacteristic
121C2×C181#42111yescharacteristic
122D181—22no—
123D181—41no—
124D181—81no—
125C181#187111yescharacteristic, complemented, Sylow
126#1871—1811nomaximal, Sylow
127C22×M16—1811no—
128#525—1811no—
129C2×#104—1812no—
130C4×C22⋊C4—1811no—
131#339—1812no—
132C2×#41—1811no—
133C23×C4—1811no—
134C2×M16—3621no—
135C2×M16—3624no—
136C2×M16—1812no—
137C22×C8—3621no—
138#123—3621no—
139C22⋊Q8—1811no—
140C22⋊Q8#1062438211yescharacteristic, complemented
141#121—1811no—
142#121—1811no—
143#120—3621no—
144C4×D4—3621no—
145C4×D4—3621no—
146#117—1811no—
147C2×C22⋊C4—3621no—
148C2×C42—1812no—
149#104—3624no—
150#95—3621no—
151C24—1811no—
152#41—3622no—
153#41—3622no—
154C2×Q8C2×#1062438211yescharacteristic
155C2×D4—3621no—
156C2×D4—3621no—
157C2×D4—3621no—
158C2×D4—1811no—
159C22×C4—3622no—
160C22×C4—3622no—
161C22×C4—3621no—
162C22×C4—3622no—
163C22×C4—1811no—
164C22×C4—1811no—
165C22×C4—1811no—
166C22×C4—1811no—
167C22×C4—1811no—
168C22×C4C2×#1062438211yescharacteristic
169M16—3624no—
170M16—3622no—
171M16—1814no—
172C2×C8—3624no—
173C2×C8—3622no—
174C4⋊C4—21no—
175C4⋊C4—3621no—
176C4⋊C4—1811no—
177C4⋊C4C2×#1062438211yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—3624no—
183C22⋊C4—3621no—
184C42—3621no—
185C42—3622no—
186C42—3622no—
187C23—3622no—
188C23—3622no—
189C23—3621no—
190C23—1811no—
191C23—1811no—
192C23—7241no—
193C23C22⋊#1062438211yescharacteristic
194Q8—22no—
195D4—3622no—
196D4—7242no—
197D4—7242no—
198D4—7242no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—3622no—
203C2×C4—3621no—
204C2×C4—3621no—
205C2×C4—3621no—
206C2×C4—3621no—
207C2×C4—3622no—
208C2×C4—3624no—
209C2×C4—3621no—
210C2×C4—1811no—
211C2×C4—1811no—
212C2×C4—1812no—
213C2×C4—1812no—
214C2×C4—7241no—
215C2×C4—7241no—
216C2×C4—7242no—
217C2×C4—7242no—
218C2×C4C22⋊#1062438211yescharacteristic
219C2×C4C22×#1062438211yescharacteristic
220C8—3624no—
221C22—21no—
222C22—21no—
223C22—3621no—
224C22—3621no—
225C22—3621no—
226C22—3621no—
227C22—3621no—
228C22—1812no—
229C22—7244no—
230C22—7241no—
231C22—7241no—
232C22—7241no—
233C22#1915388112yescomplemented
234C22C2×C22⋊#1062438211yescharacteristic
235C4—21no—
236C4—41no—
237C4—41no—
238C4—3621no—
239C4—1811no—
240C4—1811no—
241C4—7244no—
242C4—7241no—
243C4D4×#1062438212yes—
244C2—22no—
245C2—3622no—
246C2—14481no—
247C2—7241no—
248C2#2604222711yescharacteristic
249C2C2×#1915388111yescharacteristic
250C2#2604227611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC22⋊Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radical#26041792C2
cosocleC2×C4×D181C23
hypercenterC22⋊Q8#10624382
hypocenterC181#1871
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#15668459C2semidirect
2. C2×#19153881C2semidirect
3. C4×C22⋊#10624382C2semidirect
4. #26042149C2semidirect
17. #19153792C4semidirect
18. #19153791C4semidirect
19. #19153790C4semidirect
24. C22⋊Q8×C181C4semidirect
28. #15668459D4semidirect
97. C22×C181#104semidirect
125. C181#1871semidirect
140. C22⋊Q8#10624382semidirect
233. C22#19153881semidirect

Maximal subgroups

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

subgroupindex
C22×#156684592
C2×#191538812
C4×C22⋊#106243822
#260421492
#260417922
#1871181