Finite Group Databasebeta

← #37545946

Subgroups — #37545946

Order 23168.

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

normal subgroups only

This group has 448 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
2#26041627C211yesmaximal, characteristic, complemented
3#26041603C212yesmaximal, complemented
4#26041556C211yesmaximal, characteristic, complemented
5#26041527C212yesmaximal, complemented
6#26041479C211yesmaximal, characteristic, complemented
7#26041429C212yesmaximal, complemented
8C4×#15668436C211yesmaximal, characteristic, complemented
9#26041357C211yesmaximal, characteristic, complemented
10#26041283C212yesmaximal, complemented
11#556×C181C211yesmaximal, characteristic, complemented
12C22×D724C2212yes—
13C2×D4×D181—21no—
14C2×#15668436C2211yescharacteristic
15C2×#15668436C2212yescomplemented
16C2×#15668433C2212yescomplemented
17#19153813C2211yescharacteristic
18#19153791—21no—
19#19153790C2212yes—
20#19153786—22no—
21C4×D724C2211yescharacteristic
22#19153749C2212yescomplemented
23#19153747C2211yescharacteristic, complemented
24C4⋊#13089678C2211yescharacteristic, complemented
25#19153740C2211yescharacteristic, complemented
26#19153736C2211yescharacteristic
27#19153736C2212yes—
28#19153736C2212yescomplemented
29#19153736C2212yescomplemented
30#19153736C2212yescomplemented
31C22⋊#13089678C2212yescomplemented
32C4×#13089678C2211yescharacteristic, complemented
33C4⋊Q8×C181C2211yescharacteristic, complemented
34#126×C181—22no—
35#124×C181—21no—
36C22⋊Q8×C181C2212yescomplemented
37C4×Q8×C181—21no—
38C4×Q8×C181C2211yescharacteristic, complemented
39C4×D4×C181C2211yescharacteristic
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#15668436—22no—
53#15668436—42no—
54#15668433—42no—
55C2×#13089678C2311yescharacteristic
56C2×#13089678C2311yescharacteristic
57C2×#13089678C2312yes—
58C4⋊Dic181C2311yescharacteristic
59#15668426—21no—
60#15668426—22no—
61C2×Q8×C181—21no—
62C2×Q8×C181C2311yescharacteristic
63C2×D4×C181D411yescharacteristic
64C22×C4×C181C2312yes—
65C4⋊C4×C181—22no—
66C4⋊C4×C181—21no—
67C4⋊C4×C181—21no—
68C4⋊C4×C181D411yescharacteristic
69C4⋊C4×C181C2311yescharacteristic
70C4⋊C4×C181C2312yes—
71C22⋊C4×C181—22no—
72C22⋊C4×C181—22no—
73C22⋊C4×C181D412yes—
74C42×C181—22no—
75C42×C181—21no—
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×C181C2×D412yes—
96Q8×C181—21no—
97Q8×C181—21no—
98Q8×C181—42no—
99D4×C181—22no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—21no—
103C2×C4×C181—21no—
104C2×C4×C181—22no—
105C2×C4×C181C2×D412yes—
106C2×C4×C181C2×D411yescharacteristic
107C2×C4×C181C2×D411yescharacteristic
108C2×C4×C181C2411yescharacteristic
109C22×C181—22no—
110C22×C181—22no—
111C22×C181—22no—
112C22×C181C22×D411yescharacteristic
113C2×D181—22no—
114C2×D181—41no—
115C2×D181—41no—
116C2×D181—42no—
117C2×D181—42no—
118C2×D181—42no—
119C2×D181—88no—
120C4×C181—22no—
121C4×C181—21no—
122C4×C181—41no—
123C4×C181—42no—
124C4×C181—41no—
125C4×C181AGL(1,ℤ/8)12yes—
126C4×C181C2×#4111yescharacteristic
127C4×C181H(5,𝔽2)11yescharacteristic
128Dic181—41no—
129C2×C181—42no—
130C2×C181#54611yescharacteristic
131C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C181#57711yescharacteristic
133D181—42no—
134D181—82no—
135C181#332411yescharacteristic, complemented, Sylow
136#3324—1811nomaximal, Sylow
137#556D18111yescharacteristic, complemented
138D42—1811no—
139#486—1811no—
140#482—1812no—
141#474—1811no—
142#468—1812no—
143#460—1811no—
144#450—1812no—
145C4×SD16—1811no—
146#436—1811no—
147#418—1812no—
148C22×D4—3621no—
149C22×D4—1812no—
150C2×SD16—1811no—
151C2×SD16—1812no—
152C2×M16—1812no—
153C4⋊Q8C2×D18111yescharacteristic, complemented
154#127—1811no—
155#126—22no—
156#124—21no—
157C22⋊Q8C2×D18112yescomplemented
158#121—3621no—
159#121—1812no—
160#120—3622no—
161C4×Q8—21no—
162C4×Q8C2×D18111yescharacteristic, complemented
163C4×D4—1811no—
164C4×D4C2×D18111yescharacteristic
165#117C2×D18112yescomplemented
166#107—1812no—
167#106—1811no—
168C4⋊C8—1811no—
169#103—1811no—
170#102—1811no—
171#102—1812no—
172#102—1812no—
173#102—1812no—
174#102—1812no—
175C22⋊C8—1812no—
176C4×C8—1811no—
177C24—3622no—
178C2×Q8—21no—
179C2×Q8C22×D18111yescharacteristic
180C2×D4—3624no—
181C2×D4—3622no—
182C2×D4—3621no—
183C2×D4—3622no—
184C2×D4—3621no—
185C2×D4—1812no—
186C2×D4—1811no—
187C2×D4—7242no—
188C2×D4—7244no—
189C2×D4C22⋊D18111yescharacteristic
190C22×C4—3621no—
191C22×C4C22×D18112yes—
192SD16—3622no—
193SD16—7242no—
194M16—7242no—
195C2×C8—1811no—
196C2×C8—1811no—
197C2×C8—1812no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—1811no—
202C4⋊C4C22⋊D18111yescharacteristic
203C4⋊C4C22×D18111yescharacteristic
204C4⋊C4C22×D18112yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—3621no—
208C22⋊C4—3622no—
209C22⋊C4C22⋊D18112yes—
210C42—22no—
211C42—21no—
212C42C22×D18111yescharacteristic
213C23—3621no—
214C23—3622no—
215C23—14482no—
216C23—7244no—
217C23—7244no—
218C23C2×C22⋊D18112yes—
219Q8—21no—
220Q8—21no—
221Q8—42no—
222D4—22no—
223D4—3622no—
224D4—3621no—
225D4—3622no—
226D4—14481no—
227D4—1812no—
228D4—7244no—
229D4—7244no—
230D4—7242no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—3621no—
237C2×C4—7242no—
238C2×C4C2×C22⋊D18112yes—
239C2×C4C2×C22⋊D18111yescharacteristic
240C2×C4C2×C22⋊D18111yescharacteristic
241C2×C4C23×D18111yescharacteristic
242C8—3621no—
243C8—3621no—
244C8—3622no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—3622no—
249C22—14488no—
250C22—7242no—
251C22—7242no—
252C22—7242no—
253C22—7241no—
254C22—7241no—
255C22C22×C22⋊D18111yescharacteristic
256C4—22no—
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—7241no—
262C4#1915384112yes—
263C4#41×D18111yescharacteristic
264C4#1915386211yescharacteristic
265C2—42no—
266C2—14482no—
267C2—7242no—
268C2#2604191711yescharacteristic
269C2C2×#1915384111yescharacteristic
270C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#556×C181C2
socleC22×C181C22×D4
prime radicalD4×D724C2
hypercenter#556D181
hypocenterC181#3324
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #26041627C2semidirect
3. #26041603C2semidirect
4. #26041556C2semidirect
5. #26041527C2semidirect
6. #26041479C2semidirect
7. #26041429C2semidirect
8. C4×#15668436C2semidirect
9. #26041357C2semidirect
10. #26041283C2semidirect
11. #556×C181C2semidirect
15. C2×#15668436C22semidirect
16. C2×#15668433C22semidirect
22. #19153749C22semidirect
23. #19153747C22semidirect
24. C4⋊#13089678C22semidirect
25. #19153740C22semidirect
28. #19153736C22semidirect
29. #19153736C22semidirect
30. #19153736C22semidirect
31. C22⋊#13089678C22semidirect
32. C4×#13089678C22semidirect
33. C4⋊Q8×C181C22semidirect
36. C22⋊Q8×C181C22semidirect
38. C4×Q8×C181C22semidirect
40. #117×C181C22semidirect
135. C181#3324semidirect
137. #556D181semidirect
153. C4⋊Q8C2×D181semidirect
157. C22⋊Q8C2×D181semidirect
162. C4×Q8C2×D181semidirect
165. #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
#260415562
#260415272
#260414792
#260414292
C4×#156684362
#260413572
#260412832
#556×C1812
#3324181