Finite Group Databasebeta

← #37544564

Subgroups — #37544564

Order 23168.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#26041792C212yesmaximal, complemented
2#26041618C212yesmaximal, complemented
3#26041605C212yesmaximal, complemented
4#26041532C211yesmaximal, characteristic, complemented
5#26041527C212yesmaximal, complemented
6#26041440C212yesmaximal, complemented
7#26041427C212yesmaximal, complemented
8#26041242C211yesmaximal, characteristic, complemented
9#470×C181C211yesmaximal, characteristic, complemented
10C2×#15668450C2212yes—
11C22×C4×D181C2211yescharacteristic
12C2×#15668436C2212yescomplemented
13C2×D1448C2211yescharacteristic, complemented
14C2×#15668433C2211yescharacteristic
15#19153801—22no—
16#19153792C2212yes—
17#19153791C2212yes—
18#19153790C2212yes—
19#19153786—22no—
20#19153778—22no—
21#19153776—22no—
22#19153774C2212yes—
23#19153749C2212yescomplemented
24#19153748C2211yescharacteristic, complemented
25#19153740C2212yescomplemented
26#19153738C2212yescomplemented
27#19153736C2212yescomplemented
28#19153728C2212yes—
29#19153728C2212yes—
30C2×Q16×C181C2211yescharacteristic, complemented
31C2×M16×C181C2211yescharacteristic
32C22⋊Q8×C181C2212yes—
33#106×C181C2211yescharacteristic, complemented
34#103×C181C2212yescomplemented
35C23×D181D411yescharacteristic
36#15668450—42no—
37#15668450—42no—
38C2×D724—22no—
39C2×D724—22no—
40C2×D724C2312yes—
41C2×C22⋊D181—22no—
42C22×Dic181D411yescharacteristic
43C2×C4×D181—22no—
44C2×C4×D181—41no—
45C2×C4×D181—41no—
46C2×C4×D181D412yes—
47C2×C4×D181C2312yes—
48#15668436—42no—
49D1448—41no—
50#15668433—41no—
51C2×#13089678C2312yes—
52C4⋊Dic181C2312yes—
53#15668428—22no—
54#15668426—22no—
55#15668426—22no—
56#15668426—22no—
57#15668426—22no—
58C4×Dic181—22no—
59C2×Q8×C181C2312yes—
60C22×C4×C181C2311yescharacteristic
61Q16×C181—41no—
62M16×C181—41no—
63C2×C8×C181C2311yescharacteristic
64C2×C8×C181C2311yescharacteristic
65C4⋊C4×C181—22no—
66C4⋊C4×C181C2312yes—
67C22⋊C4×C181—22no—
68C22×D181—22no—
69C22×D181—41no—
70C22×D181—42no—
71C22×D181C2×D412yes—
72D724—22no—
73D724—22no—
74D724—44no—
75D724—44no—
76C22⋊D181—44no—
77C2×Dic181—22no—
78C2×Dic181—41no—
79C2×Dic181C2×D412yes—
80C4×D181—22no—
81C4×D181—22no—
82C4×D181—42no—
83C4×D181—44no—
84#13089678—22no—
85C23×C181C2×D411yescharacteristic
86Q8×C181—22no—
87Q8×C181—22no—
88C2×C4×C181—21no—
89C2×C4×C181—21no—
90C2×C4×C181—22no—
91C2×C4×C181—22no—
92C2×C4×C181C2×D411yescharacteristic
93C2×C4×C181C2411yescharacteristic
94C8×C181—21no—
95C8×C181—21no—
96C22×C181—21no—
97C22×C181—22no—
98C22×C181C22×D411yescharacteristic
99C2×D181—22no—
100C2×D181—22no—
101C2×D181—22no—
102C2×D181—44no—
103C2×D181—42no—
104C2×D181—42no—
105C2×D181—42no—
106C4×C181—21no—
107C4×C181—42no—
108C4×C181—42no—
109C4×C181H(5,𝔽2)11yescharacteristic
110C4×C181H(5,𝔽2)11yescharacteristic
111Dic181—22no—
112Dic181—42no—
113C2×C181—41no—
114C2×C181#53411yescharacteristic
115C2×C181#57512yes—
116D181—42no—
117D181—82no—
118C181#319711yescharacteristic, complemented, Sylow
119#3197—1811nomaximal, Sylow
120#525—1812no—
121#485—1812no—
122#482—1812no—
123#470D18111yescharacteristic, complemented
124#469—1811no—
125#468—1812no—
126#452—1812no—
127#449—1812no—
128#407—1811no—
129C2×#41—1812no—
130C23×C4—1811no—
131C2×Q16C2×D18111yescharacteristic, complemented
132C2×SD16—1812no—
133C2×D8—1811no—
134C2×M16—1811no—
135C2×M16C2×D18111yescharacteristic
136#123—3622no—
137C22⋊Q8—1812no—
138C22⋊Q8C2×D18112yes—
139#121—1812no—
140#121—1812no—
141#120—3622no—
142C4×D4—3622no—
143C4×D4—3622no—
144#117—1812no—
145#107—1812no—
146#106—1811no—
147#106C2×D18111yescharacteristic, complemented
148#103—1812no—
149#103C2×D18112yescomplemented
150#102—1812no—
151#102—1812no—
152C22⋊C8—1812no—
153C22⋊C8—1812no—
154C24—1811no—
155#41—7242no—
156#41—7242no—
157C2×Q8C22×D18112yes—
158C2×D4—3622no—
159C2×D4—3622no—
160C2×D4—3622no—
161C2×D4—1812no—
162C22×C4—3622no—
163C22×C4—1812no—
164C22×C4—1811no—
165C22×C4—1812no—
166C22×C4—7241no—
167C22×C4—7241no—
168C22×C4C22×D18111yescharacteristic
169Q16—41no—
170SD16—7242no—
171D8—7241no—
172M16—41no—
173M16—7241no—
174C2×C8—1812no—
175C2×C8C22×D18111yescharacteristic
176C2×C8C22×D18111yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—3622no—
179C4⋊C4—1812no—
180C4⋊C4C22×D18112yes—
181C22⋊C4—22no—
182C22⋊C4—3622no—
183C22⋊C4—3622no—
184C22⋊C4—3622no—
185C22⋊C4—3622no—
186C42—3622no—
187C23—3622no—
188C23—1812no—
189C23—7241no—
190C23—7242no—
191C23D4×D18111yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—3622no—
195D4—3622no—
196D4—7244no—
197D4—7244no—
198D4—7244no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—3622no—
204C2×C4—3622no—
205C2×C4—3622no—
206C2×C4—1812no—
207C2×C4—7242no—
208C2×C4—7244no—
209C2×C4—7241no—
210C2×C4D4×D18111yescharacteristic
211C2×C4C23×D18111yescharacteristic
212C8—21no—
213C8—21no—
214C8—3622no—
215C22—22no—
216C22—21no—
217C22—3622no—
218C22—3622no—
219C22—3622no—
220C22—7244no—
221C22—7242no—
222C22—7242no—
223C22—7242no—
224C22C2×D4×D18111yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—3622no—
229C4—7242no—
230C4#1915386111yescharacteristic
231C4#1915386211yescharacteristic
232C2—41no—
233C2—14482no—
234C2—7242no—
235C2#2604183711yescharacteristic
236C2#2604207812yes—
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
hypercenter#470D181
hypocenterC181#3197
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #26041792C2semidirect
2. #26041618C2semidirect
3. #26041605C2semidirect
4. #26041532C2semidirect
5. #26041527C2semidirect
6. #26041440C2semidirect
7. #26041427C2semidirect
8. #26041242C2semidirect
9. #470×C181C2semidirect
12. C2×#15668436C22semidirect
13. C2×D1448C22semidirect
23. #19153749C22semidirect
24. #19153748C22semidirect
25. #19153740C22semidirect
26. #19153738C22semidirect
27. #19153736C22semidirect
30. C2×Q16×C181C22semidirect
33. #106×C181C22semidirect
34. #103×C181C22semidirect
118. C181#3197semidirect
123. #470D181semidirect
131. C2×Q16C2×D181semidirect
147. #106C2×D181semidirect
149. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260417922
#260416182
#260416052
#260415322
#260415272
#260414402
#260414272
#260412422
#470×C1812
#3197181