Finite Group Databasebeta

← #37544581

Subgroups — #37544581

Order 23168.

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

normal subgroups only

This group has 442 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#26041845C212yesmaximal, complemented
2#26041606C212yesmaximal, complemented
3#26041601C212yesmaximal, complemented
4#26041530C212yesmaximal, complemented
5#26041515C211yesmaximal, characteristic, complemented
6#26041414C212yesmaximal, complemented
7#26041411C212yesmaximal, complemented
8#26041245C211yesmaximal, characteristic, complemented
9#466×C181C211yesmaximal, characteristic, complemented
10C2×#15668449C2211yescharacteristic
11C2×D4×D181C2212yescomplemented
12C2×D1448C2211yescharacteristic
13C2×#15668434C2212yes—
14C2×#15668433C2211yescharacteristic, complemented
15#19153812—22no—
16#19153802—22no—
17#19153791C2212yes—
18#19153790C2212yescomplemented
19#19153787C2212yes—
20#19153786—22no—
21#19153784—22no—
22#19153774C2212yescomplemented
23#19153750C2211yescharacteristic, complemented
24#19153747C2212yescomplemented
25#19153738C2212yescomplemented
26#19153737C2212yescomplemented
27#19153736C2212yescomplemented
28#19153728C2212yescomplemented
29#19153728C2212yescomplemented
30C2×D8×C181C2211yescharacteristic
31C22×C8×C181C2211yescharacteristic, complemented
32#121×C181C2212yescomplemented
33#107×C181C2211yescharacteristic, complemented
34#102×C181C2212yescomplemented
35C23×D181—22no—
36#15668449—41no—
37#15668449—41no—
38C2×Dic362D411yescharacteristic
39C2×D724—22no—
40C2×D724D411yescharacteristic
41C2×D724C2312yes—
42C2×C22⋊D181—22no—
43C2×C22⋊D181—22no—
44C2×C22⋊D181—22no—
45C2×C22⋊D181D412yes—
46D4×D181—42no—
47D4×D181—42no—
48C2×C4×D181C2312yes—
49D1448—41no—
50#15668434—42no—
51#15668433—41no—
52C2×#13089678C2312yes—
53C4⋊Dic181C2312yes—
54C22⋊Dic181—22no—
55#15668426—22no—
56#15668426—22no—
57#15668426—22no—
58C4×Dic181—22no—
59C2×D4×C181—22no—
60C2×D4×C181C2312yes—
61C22×C4×C181C2311yescharacteristic
62D8×C181—41no—
63C2×C8×C181—41no—
64C2×C8×C181C2311yescharacteristic
65C2×C8×C181C2311yescharacteristic
66C4⋊C4×C181C2312yes—
67C22⋊C4×C181—22no—
68C22×D181—22no—
69C22×D181—44no—
70C22×D181—82no—
71C22×D181—82no—
72C22×D181C2×D412yes—
73Dic362—41no—
74D724—22no—
75D724—22no—
76D724—41no—
77D724—82no—
78C22⋊D181—42no—
79C22⋊D181—44no—
80C22⋊D181—82no—
81C22⋊D181—82no—
82C2×Dic181—22no—
83C2×Dic181C2×D412yes—
84C4×D181—22no—
85C4×D181—22no—
86#13089678—22no—
87C23×C181—22no—
88C23×C181C2×D411yescharacteristic
89D4×C181—22no—
90D4×C181—22no—
91D4×C181—82no—
92C2×C4×C181—21no—
93C2×C4×C181—21no—
94C2×C4×C181—22no—
95C2×C4×C181C2×D411yescharacteristic
96C2×C4×C181C2411yescharacteristic
97C8×C181—21no—
98C8×C181—21no—
99C22×C181—21no—
100C22×C181—21no—
101C22×C181—21no—
102C22×C181—42no—
103C22×C181—42no—
104C22×C181—42no—
105C22×C181C22×D411yescharacteristic
106C2×D181—22no—
107C2×D181—22no—
108C2×D181—22no—
109C2×D181—42no—
110C2×D181—42no—
111C2×D181—42no—
112C2×D181—88no—
113C4×C181—21no—
114C4×C181—42no—
115C4×C181H(5,𝔽2)11yescharacteristic
116C4×C181H(5,𝔽2)11yescharacteristic
117Dic181—22no—
118Dic181—42no—
119C2×C181—41no—
120C2×C181—82no—
121C2×C181#53411yescharacteristic
122C2×C181#57611yescharacteristic
123C2×C181#57611yescharacteristic
124D181—42no—
125D181—82no—
126C181#319911yescharacteristic, complemented, Sylow
127#3199—1811nomaximal, Sylow
128#535—1812no—
129#482—1812no—
130#482—1812no—
131#469—1812no—
132#466—1811no—
133#466D18111yescharacteristic, complemented
134#447—1812no—
135#447—1812no—
136#408—1811no—
137C2×#41—1811no—
138C22×D4—1812no—
139C2×D8—1812no—
140C2×D8—1811no—
141C2×D8C2×D18111yescharacteristic
142C2×M16—1811no—
143C22×C8C2×D18111yescharacteristic, complemented
144#127—3622no—
145#124—3622no—
146#121—1812no—
147#121—1812no—
148#121—1812no—
149#121C2×D18112yescomplemented
150#120—3622no—
151#120—3622no—
152#117—1812no—
153#107—1811no—
154#107C2×D18111yescharacteristic, complemented
155#106—1812no—
156#102—1812no—
157#102—1812no—
158#102—1812no—
159#102C2×D18112yescomplemented
160C22⋊C8—1812no—
161C22⋊C8—1812no—
162C24—3622no—
163#41—7241no—
164#41—7241no—
165C2×Q8—1811no—
166C2×D4—22no—
167C2×D4—3622no—
168C2×D4—3622no—
169C2×D4—3622no—
170C2×D4—3622no—
171C2×D4—1812no—
172C2×D4—1811no—
173C2×D4—1812no—
174C2×D4—7242no—
175C2×D4—7242no—
176C2×D4C22×D18112yes—
177C22×C4—1812no—
178C22×C4C22×D18111yescharacteristic
179D8—41no—
180D8—7241no—
181D8—7242no—
182M16—7241no—
183C2×C8—41no—
184C2×C8—1812no—
185C2×C8C22×D18111yescharacteristic
186C2×C8C22×D18111yescharacteristic
187C4⋊C4—1812no—
188C4⋊C4C22×D18112yes—
189C22⋊C4—22no—
190C22⋊C4—3622no—
191C22⋊C4—3622no—
192C22⋊C4—3622no—
193C22⋊C4—3622no—
194C42—3622no—
195C23—22no—
196C23—3622no—
197C23—14482no—
198C23—14482no—
199C23—1812no—
200C23—7244no—
201C23D4×D18111yescharacteristic
202Q8—7241no—
203D4—22no—
204D4—22no—
205D4—82no—
206D4—3622no—
207D4—3622no—
208D4—14482no—
209D4—14482no—
210D4—14482no—
211D4—7241no—
212D4—7242no—
213D4—7244no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—3622no—
218C2×C4—3622no—
219C2×C4—3622no—
220C2×C4—1812no—
221C2×C4D4×D18111yescharacteristic
222C2×C4C23×D18111yescharacteristic
223C8—21no—
224C8—21no—
225C8—3622no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—3622no—
233C22—3622no—
234C22—3622no—
235C22—14488no—
236C22—7242no—
237C22—7242no—
238C22—7242no—
239C22C2×D4×D18111yescharacteristic
240C4—21no—
241C4—42no—
242C4—3622no—
243C4—7242no—
244C4#1915386111yescharacteristic
245C4#1915386211yescharacteristic
246C2—41no—
247C2—82no—
248C2—14482no—
249C2—7242no—
250C2#2604183711yescharacteristic
251C2#2604208011yescharacteristic
252C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#466×C181C2
socleC22×C181C22×D4
hypercenter#466D181
hypocenterC181#3199
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26041845C2semidirect
2. #26041606C2semidirect
3. #26041601C2semidirect
4. #26041530C2semidirect
5. #26041515C2semidirect
6. #26041414C2semidirect
7. #26041411C2semidirect
8. #26041245C2semidirect
9. #466×C181C2semidirect
11. C2×D4×D181C22semidirect
14. C2×#15668433C22semidirect
18. #19153790C22semidirect
22. #19153774C22semidirect
23. #19153750C22semidirect
24. #19153747C22semidirect
25. #19153738C22semidirect
26. #19153737C22semidirect
27. #19153736C22semidirect
28. #19153728C22semidirect
29. #19153728C22semidirect
31. C22×C8×C181C22semidirect
32. #121×C181C22semidirect
33. #107×C181C22semidirect
34. #102×C181C22semidirect
126. C181#3199semidirect
133. #466D181semidirect
143. C22×C8C2×D181semidirect
149. #121C2×D181semidirect
154. #107C2×D181semidirect
159. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418452
#260416062
#260416012
#260415302
#260415152
#260414142
#260414112
#260412452
#466×C1812
#3199181