Finite Group Databasebeta

← #37545957

Subgroups — #37545957

Order 23168.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D4×D724C211yesmaximal, characteristic, complemented
2#26041627C211yesmaximal, characteristic, complemented
3#26041603C212yesmaximal, complemented
4#26041566C211yesmaximal, characteristic, complemented
5#26041530C212yesmaximal, complemented
6#26041474C211yesmaximal, characteristic, complemented
7#26041412C212yesmaximal, complemented
8C4×#15668434C211yesmaximal, characteristic, complemented
9#26041357C211yesmaximal, characteristic, complemented
10#26041283C212yesmaximal, complemented
11#555×C181C211yesmaximal, characteristic, complemented
12C22×D724C2212yes—
13C2×D4×D181—21no—
14C2×#15668434C2211yescharacteristic
15C2×#15668434C2212yescomplemented
16C2×#15668433C2212yescomplemented
17#19153813C2211yescharacteristic
18#19153791—21no—
19#19153790C2212yes—
20#19153786—22no—
21C4×D724C2211yescharacteristic
22#19153749C2211yescharacteristic, complemented
23#19153747C2212yescomplemented
24C4⋊#13089678C2211yescharacteristic, complemented
25#19153737C2211yescharacteristic, complemented
26#19153736C2211yescharacteristic
27#19153736C2212yes—
28#19153736C2212yescomplemented
29#19153736C2212yescomplemented
30#19153736C2212yescomplemented
31C22⋊#13089678C2212yescomplemented
32C4×#13089678C2211yescharacteristic, complemented
33C4⋊Q8×C181C2211yescharacteristic, complemented
34#124×C181—22no—
35#121×C181—21no—
36#121×C181C2212yescomplemented
37C4×D4×C181C2211yescharacteristic
38C4×D4×C181C2211yescharacteristic, complemented
39#117×C181—21no—
40#117×C181C2212yescomplemented
41C23×D181—22no—
42C2×D724—24no—
43C2×D724—22no—
44C2×D724—21no—
45C2×D724—21no—
46C2×D724—42no—
47C2×D724C2312yes—
48C2×D724C2311yescharacteristic
49C2×C22⋊D181—22no—
50D4×D181—44no—
51C2×C4×D181—21no—
52#15668434—22no—
53#15668434—42no—
54#15668433—42no—
55C2×#13089678C2311yescharacteristic
56C2×#13089678C2311yescharacteristic
57C2×#13089678C2312yes—
58C4⋊Dic181C2311yescharacteristic
59#15668426—21no—
60#15668426—22no—
61C2×Q8×C181—21no—
62C2×D4×C181—22no—
63C2×D4×C181D411yescharacteristic
64C2×D4×C181C2311yescharacteristic
65C22×C4×C181—21no—
66C22×C4×C181C2312yes—
67C4⋊C4×C181—21no—
68C4⋊C4×C181D411yescharacteristic
69C4⋊C4×C181C2311yescharacteristic
70C4⋊C4×C181C2312yes—
71C22⋊C4×C181—22no—
72C22⋊C4×C181—22no—
73C22⋊C4×C181—21no—
74C22⋊C4×C181D412yes—
75C42×C181—22no—
76C42×C181C2311yescharacteristic
77C22×D181—21no—
78C22×D181—22no—
79C22×D181—44no—
80C22×D181—44no—
81C22×D181—82no—
82D724—22no—
83D724—21no—
84D724—22no—
85D724—44no—
86D724—42no—
87D724—81no—
88D724#4112yes—
89C22⋊D181—44no—
90C2×Dic181—21no—
91C4×D181—42no—
92#13089678—21no—
93#13089678—21no—
94#13089678—22no—
95C23×C181—21no—
96C23×C181C2×D412yes—
97Q8×C181—81no—
98D4×C181—22no—
99D4×C181—21no—
100D4×C181—21no—
101D4×C181—82no—
102C2×C4×C181—22no—
103C2×C4×C181—22no—
104C2×C4×C181—21no—
105C2×C4×C181—22no—
106C2×C4×C181—42no—
107C2×C4×C181C2×D412yes—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×D411yescharacteristic
110C2×C4×C181C2411yescharacteristic
111C22×C181—22no—
112C22×C181—22no—
113C22×C181—22no—
114C22×C181—41no—
115C22×C181—41no—
116C22×C181—41no—
117C22×C181C22×D411yescharacteristic
118C2×D181—22no—
119C2×D181—41no—
120C2×D181—41no—
121C2×D181—42no—
122C2×D181—42no—
123C2×D181—42no—
124C2×D181—88no—
125C4×C181—22no—
126C4×C181—21no—
127C4×C181—41no—
128C4×C181—42no—
129C4×C181AGL(1,ℤ/8)12yes—
130C4×C181C2×#4111yescharacteristic
131C4×C181H(5,𝔽2)11yescharacteristic
132Dic181—41no—
133C2×C181—42no—
134C2×C181—81no—
135C2×C181#54611yescharacteristic
136C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C181#57611yescharacteristic
138D181—42no—
139D181—82no—
140C181#332511yescharacteristic, complemented, Sylow
141#3325—1811nomaximal, Sylow
142#555D18111yescharacteristic, complemented
143D42—1811no—
144#486—1811no—
145#482—1812no—
146#476—1811no—
147#469—1812no—
148#459—1811no—
149#447—1812no—
150C4×D8—1811no—
151#436—1811no—
152#418—1812no—
153C22×D4—3621no—
154C22×D4—1812no—
155C2×D8—1811no—
156C2×D8—1812no—
157C2×M16—1812no—
158C4⋊Q8C2×D18111yescharacteristic, complemented
159#127—1811no—
160#124—22no—
161#121—21no—
162#121—3621no—
163#121—1812no—
164#121C2×D18112yescomplemented
165#120—3622no—
166C4×D4—1811no—
167C4×D4C2×D18111yescharacteristic
168C4×D4C2×D18111yescharacteristic, complemented
169#117—21no—
170#117C2×D18112yescomplemented
171#107—1811no—
172#106—1812no—
173C4⋊C8—1811no—
174#102—1811no—
175#102—1811no—
176#102—1812no—
177#102—1812no—
178#102—1812no—
179#102—1812no—
180C22⋊C8—1812no—
181C4×C8—1811no—
182C24—3622no—
183C2×Q8—21no—
184C2×D4—22no—
185C2×D4—3624no—
186C2×D4—3622no—
187C2×D4—3621no—
188C2×D4—3622no—
189C2×D4—3621no—
190C2×D4—1812no—
191C2×D4—1811no—
192C2×D4—7242no—
193C2×D4—7244no—
194C2×D4C22⋊D18111yescharacteristic
195C2×D4C22×D18111yescharacteristic
196C22×C4—21no—
197C22×C4—3621no—
198C22×C4C22×D18112yes—
199D8—3622no—
200D8—7242no—
201M16—7242no—
202C2×C8—1811no—
203C2×C8—1811no—
204C2×C8—1812no—
205C4⋊C4—21no—
206C4⋊C4—1811no—
207C4⋊C4C22⋊D18111yescharacteristic
208C4⋊C4C22×D18112yes—
209C4⋊C4C22×D18111yescharacteristic
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—21no—
213C22⋊C4—3621no—
214C22⋊C4—3622no—
215C22⋊C4C22⋊D18112yes—
216C42—22no—
217C42C22×D18111yescharacteristic
218C23—21no—
219C23—3621no—
220C23—3622no—
221C23—14482no—
222C23—7244no—
223C23—7244no—
224C23C2×C22⋊D18112yes—
225Q8—81no—
226D4—22no—
227D4—21no—
228D4—21no—
229D4—82no—
230D4—3622no—
231D4—3621no—
232D4—3622no—
233D4—14481no—
234D4—1812no—
235D4—7244no—
236D4—7244no—
237D4—7242no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—3621no—
244C2×C4—7242no—
245C2×C4C2×C22⋊D18112yes—
246C2×C4C2×C22⋊D18111yescharacteristic
247C2×C4C2×C22⋊D18111yescharacteristic
248C2×C4C23×D18111yescharacteristic
249C8—3621no—
250C8—3621no—
251C8—3622no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—3622no—
259C22—14488no—
260C22—7242no—
261C22—7242no—
262C22—7242no—
263C22—7241no—
264C22—7241no—
265C22C22×C22⋊D18111yescharacteristic
266C4—22no—
267C4—21no—
268C4—41no—
269C4—42no—
270C4—7241no—
271C4#1915384112yes—
272C4#41×D18111yescharacteristic
273C4#1915386211yescharacteristic
274C2—42no—
275C2—81no—
276C2—14482no—
277C2—7242no—
278C2#2604191711yescharacteristic
279C2C2×#1915384111yescharacteristic
280C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#555×C181C2
socleC22×C181C22×D4
hypercenter#555D181
hypocenterC181#3325
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. D4×D724C2semidirect
2. #26041627C2semidirect
3. #26041603C2semidirect
4. #26041566C2semidirect
5. #26041530C2semidirect
6. #26041474C2semidirect
7. #26041412C2semidirect
8. C4×#15668434C2semidirect
9. #26041357C2semidirect
10. #26041283C2semidirect
11. #555×C181C2semidirect
15. C2×#15668434C22semidirect
16. C2×#15668433C22semidirect
22. #19153749C22semidirect
23. #19153747C22semidirect
24. C4⋊#13089678C22semidirect
25. #19153737C22semidirect
28. #19153736C22semidirect
29. #19153736C22semidirect
30. #19153736C22semidirect
31. C22⋊#13089678C22semidirect
32. C4×#13089678C22semidirect
33. C4⋊Q8×C181C22semidirect
36. #121×C181C22semidirect
38. C4×D4×C181C22semidirect
40. #117×C181C22semidirect
140. C181#3325semidirect
142. #555D181semidirect
158. C4⋊Q8C2×D181semidirect
164. #121C2×D181semidirect
168. C4×D4C2×D181semidirect
170. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
D4×D7242
#260416272
#260416032
#260415662
#260415302
#260414742
#260414122
C4×#156684342
#260413572
#260412832
#555×C1812
#3325181