Finite Group Databasebeta

← #37543517

Subgroups — #37543517

Order 23168.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#26041832C211yesmaximal, characteristic, complemented
2C2×#19153778C211yesmaximal, characteristic
3#26041582C212yesmaximal, complemented
4#26041578C212yesmaximal, complemented
5#26041568C212yesmaximal, complemented
6#26041558C212yesmaximal, complemented
7#26041319C211yesmaximal, characteristic, complemented
8#26041302C211yesmaximal, characteristic, complemented
9#26041285C211yesmaximal, characteristic
10#26041283C211yesmaximal, characteristic
11#428×C181C211yesmaximal, characteristic, complemented
12C22×D724C2211yescharacteristic
13C22×C4×D181—21no—
14C2×#15668433C2211yescharacteristic
15#19153815C2211yescharacteristic, complemented
16#19153815C2211yescharacteristic, complemented
17#19153807C2211yescharacteristic, complemented
18#19153807C2211yescharacteristic, complemented
19#19153793—21no—
20#19153793—21no—
21C4×Dic362—21no—
22C4×Dic362—21no—
23#19153778—21no—
24#19153778—21no—
25#19153778C2212yes—
26#19153778C2212yes—
27#19153775C2211yescharacteristic
28C2×#15668426—21no—
29C2×C4×Dic181C2211yescharacteristic
30#19153749C2212yescomplemented
31#19153747C2212yescomplemented
32#19153745C2212yescomplemented
33#19153745C2212yescomplemented
34#19153738C2212yes—
35#19153738C2212yes—
36#19153736C2212yes—
37#19153736C2212yes—
38C2×M16×C181C2211yescharacteristic
39#117×C181C2211yescharacteristic
40C2×C4⋊C4×C181C2211yescharacteristic
41#107×C181C2212yescomplemented
42#106×C181C2212yescomplemented
43C23×D181—21no—
44C2×Dic362—21no—
45C2×Dic362—21no—
46C2×D724—21no—
47C2×D724—22no—
48C2×D724Q814yescomplemented
49C2×D724C2312yes—
50C22×Dic181D411yescharacteristic
51C22×Dic181D411yescharacteristic
52C2×C4×D181—22no—
53C2×C4×D181—22no—
54C2×C4×D181—44no—
55#15668433—22no—
56C2×#13089678C2312yes—
57C4⋊Dic181—21no—
58C4⋊Dic181—21no—
59C4⋊Dic181C2311yescharacteristic
60C4⋊Dic181C2311yescharacteristic
61#15668428—21no—
62#15668428—21no—
63#15668428—21no—
64#15668428—21no—
65C22⋊Dic181—21no—
66#15668426—22no—
67#15668426—22no—
68C4×Dic181—21no—
69C4×Dic181D411yescharacteristic
70C4×Dic181D411yescharacteristic
71C4×Dic181C2311yescharacteristic
72C4×Dic181C2311yescharacteristic
73C22×C4×C181—21no—
74C22×C4×C181C2311yescharacteristic
75M16×C181—22no—
76C2×C8×C181C2312yes—
77C4⋊C4×C181—21no—
78C4⋊C4×C181C2311yescharacteristic
79C4⋊C4×C181C2312yes—
80C4⋊C4×C181C2311yescharacteristic
81C22⋊C4×C181—21no—
82C42×C181—21no—
83C22×D181—24no—
84C22×D181—22no—
85C22×D181—42no—
86C22×D181—42no—
87Dic362—42no—
88Dic362—42no—
89D724—24no—
90D724—22no—
91D724C2×Q814yes—
92C2×Dic181—22no—
93C2×Dic181—21no—
94C2×Dic181—21no—
95C2×Dic181—41no—
96C2×Dic181C2×D411yescharacteristic
97C2×Dic181C2×D411yescharacteristic
98C2×Dic181C2×D411yescharacteristic
99C2×Dic181C2×D411yescharacteristic
100C4×D181—44no—
101C4×D181—44no—
102#13089678—22no—
103C23×C181C2×D411yescharacteristic
104C2×C4×C181—21no—
105C2×C4×C181—21no—
106C2×C4×C181—22no—
107C2×C4×C181—42no—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2×Q812yes—
110C2×C4×C181#4112yes—
111C2×C4×C181C2411yescharacteristic
112C8×C181—22no—
113C22×C181—21no—
114C22×C181—21no—
115C22×C181C22⋊Q812yes—
116C22×C181C22×D411yescharacteristic
117C2×D181—24no—
118C2×D181—48no—
119C2×D181—42no—
120C2×D181—42no—
121C4×C181—42no—
122C4×C181—41no—
123C4×C181—41no—
124C4×C181C22⋊Q812yes—
125C4×C181C22×Q811yescharacteristic
126C4×C181C2×#4111yescharacteristic
127Dic181—21no—
128Dic181—21no—
129Dic181—21no—
130Dic181—41no—
131Dic181—41no—
132Dic181AGL(1,ℤ/8)12yes—
133C2×C181—22no—
134C2×C181C2×C22⋊Q811yescharacteristic
135C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C181#57511yescharacteristic
137D181—44no—
138C181#307711yescharacteristic, complemented, Sylow
139#3077—1811nomaximal, Sylow
140#533—1811no—
141C2×C4×D4—1811no—
142#478—1812no—
143#478—1812no—
144#476—1812no—
145#474—1812no—
146#428—1811no—
147#428D18111yescharacteristic, complemented
148#423—1811no—
149#418—1811no—
150#418—1811no—
151C22×D4—1811no—
152C23×C4—3621no—
153C2×M16—1811no—
154C2×M16C2×D18111yescharacteristic
155C4⋊Q8—1811no—
156C4⋊Q8—1811no—
157#125—1811no—
158#125—1811no—
159C22⋊Q8—3621no—
160C22⋊Q8—3621no—
161C4×Q8—3621no—
162C4×Q8—3621no—
163C4×D4—3621no—
164C4×D4—3621no—
165C4×D4—1812no—
166C4×D4—1812no—
167#117—1811no—
168#117C2×D18111yescharacteristic
169C2×C4⋊C4C2×D18111yescharacteristic
170C2×C22⋊C4—3621no—
171C2×C42—1811no—
172#107—1812no—
173#107C2×D18112yescomplemented
174#106—1812no—
175#106C2×D18112yescomplemented
176C4⋊C8—1812no—
177C4⋊C8—1812no—
178#102—1812no—
179#102—1812no—
180#102—1812no—
181#102—1812no—
182C24—3621no—
183C2×Q8—3621no—
184C2×Q8—3621no—
185C2×D4—3621no—
186C2×D4—3622no—
187C2×D4—1814no—
188C2×D4—1812no—
189C22×C4—21no—
190C22×C4—3622no—
191C22×C4—3622no—
192C22×C4—1811no—
193C22×C4—1811no—
194C22×C4—7244no—
195C22×C4C22×D18111yescharacteristic
196M16—22no—
197M16—3622no—
198C2×C8—1812no—
199C2×C8C22×D18112yes—
200C4⋊C4—21no—
201C4⋊C4—3621no—
202C4⋊C4—3621no—
203C4⋊C4—3621no—
204C4⋊C4—3621no—
205C4⋊C4—3621no—
206C4⋊C4—3621no—
207C4⋊C4—1811no—
208C4⋊C4—1811no—
209C4⋊C4C22×D18111yescharacteristic
210C4⋊C4C22×D18111yescharacteristic
211C4⋊C4C22×D18112yes—
212C22⋊C4—21no—
213C22⋊C4—3622no—
214C22⋊C4—3622no—
215C22⋊C4—3621no—
216C42—21no—
217C42—3621no—
218C42—1811no—
219C42—1811no—
220C42—1811no—
221C42—1811no—
222C23—3624no—
223C23—3622no—
224C23—7242no—
225C23—7242no—
226C23D4×D18111yescharacteristic
227Q8—7242no—
228Q8—7242no—
229D4—3624no—
230D4—3622no—
231D4—1814no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—3622no—
237C2×C4—3621no—
238C2×C4—3621no—
239C2×C4—1811no—
240C2×C4—1811no—
241C2×C4—1811no—
242C2×C4—1811no—
243C2×C4—7244no—
244C2×C4—7244no—
245C2×C4—7241no—
246C2×C4D4×D18111yescharacteristic
247C2×C4Q8×D18112yes—
248C2×C4#1566844912yes—
249C2×C4C23×D18111yescharacteristic
250C8—22no—
251C8—3622no—
252C22—21no—
253C22—21no—
254C22—3624no—
255C22—7248no—
256C22—7242no—
257C22—7242no—
258C22#1915379412yes—
259C22C2×D4×D18111yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—3621no—
264C4—3621no—
265C4—3621no—
266C4—1812no—
267C4—7241no—
268C4—7241no—
269C4#1915379412yes—
270C4C2×Q8×D18111yescharacteristic
271C4C2×#1566844911yescharacteristic
272C2—22no—
273C2—7244no—
274C2C2×#1915379411yescharacteristic
275C2AGL(1,ℤ/8)×D18111yescharacteristic
276C2#2604207811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#428×C181C2
socleC22×C181C22×D4
prime radicalC22×D724C22
hypercenter#428D181
hypocenterC181#3077
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #26041832C2semidirect
3. #26041582C2semidirect
4. #26041578C2semidirect
5. #26041568C2semidirect
6. #26041558C2semidirect
7. #26041319C2semidirect
8. #26041302C2semidirect
11. #428×C181C2semidirect
15. #19153815C22semidirect
16. #19153815C22semidirect
17. #19153807C22semidirect
18. #19153807C22semidirect
30. #19153749C22semidirect
31. #19153747C22semidirect
32. #19153745C22semidirect
33. #19153745C22semidirect
41. #107×C181C22semidirect
42. #106×C181C22semidirect
48. C2×D724Q8semidirect
138. C181#3077semidirect
147. #428D181semidirect
173. #107C2×D181semidirect
175. #106C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418322
C2×#191537782
#260415822
#260415782
#260415682
#260415582
#260413192
#260413022
#260412852
#260412832
#428×C1812
#3077181