Finite Group Databasebeta

← #37533688

Subgroups — #37533688

Order 23168.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#19153861C211yesmaximal, characteristic
2#26041297C211yesmaximal, characteristic, complemented
3#26041295C211yesmaximal, characteristic, complemented
4#26041284C211yesmaximal, characteristic, complemented
5#26041261C211yesmaximal, characteristic, complemented
6#26041252C211yesmaximal, characteristic, complemented
7#418×C181C211yesmaximal, characteristic, complemented
8#19153861—22no—
9#19153861—22no—
10#19153861—22no—
11#19153861—22no—
12C2×#15668449C2211yescharacteristic
13C2×#15668449C2211yescharacteristic
14C2×#15668448—21no—
15C2×#15668448C412yescomplemented
16C22×C22⋊D181—22no—
17C2×D4×D181—21no—
18C2×D4×D181C412yescomplemented
19C2×#15668433C2211yescharacteristic, complemented
20#19153775C2211yescharacteristic, complemented
21#19153744—22no—
22#19153742—22no—
23#19153737—22no—
24#19153734—21no—
25#19153732—21no—
26#19153730—22no—
27C22×D4×C181C2211yescharacteristic
28C2×M16×C181C2211yescharacteristic, complemented
29#117×C181C2211yescharacteristic, complemented
30#102×C181—22no—
31C23×D181—22no—
32#15668449—21no—
33#15668449—21no—
34#15668449—21no—
35#15668449—21no—
36#15668449—21no—
37#15668449—21no—
38#15668449D412yes—
39#15668449D412yes—
40#15668448—24no—
41#15668448—24no—
42#15668448—24no—
43#15668448—24no—
44C2×Dic362C2×C411yescharacteristic
45C2×Dic362D411yescharacteristic
46C2×D724C2×C411yescharacteristic
47C2×D724D411yescharacteristic
48C2×C22⋊D181—28no—
49C2×C22⋊D181—28no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52C2×C22⋊D181—21no—
53C2×C22⋊D181D412yes—
54D4×D181—24no—
55D4×D181—24no—
56D4×D181—24no—
57D4×D181—24no—
58C22×Dic181—22no—
59C2×C4×D181—21no—
60C2×C4×D181C2×C412yes—
61#15668433—21no—
62#15668433—21no—
63C2×#13089678—21no—
64C4⋊Dic181—21no—
65C22⋊Dic181—21no—
66C4×Dic181—21no—
67C24×C181—21no—
68C2×D4×C181—21no—
69C2×D4×C181—21no—
70C2×D4×C181—21no—
71C2×D4×C181C2×C412yes—
72C2×D4×C181D412yes—
73C2×D4×C181D412yes—
74C22×C4×C181C2311yescharacteristic
75M16×C181—21no—
76M16×C181—21no—
77C2×C8×C181—21no—
78C4⋊C4×C181—21no—
79C22⋊C4×C181—21no—
80C42×C181—21no—
81C22×D181—21no—
82C22×D181—28no—
83C22×D181—84no—
84C22×D181C22⋊C412yes—
85Dic362—21no—
86Dic362—21no—
87Dic362—21no—
88Dic362C22⋊C412yes—
89D724—21no—
90D724—21no—
91D724—21no—
92D724C22⋊C412yes—
93C22⋊D181—28no—
94C22⋊D181—28no—
95C22⋊D181—22no—
96C22⋊D181—22no—
97C22⋊D181—22no—
98C22⋊D181—22no—
99C2×Dic181—28no—
100C2×Dic181—21no—
101C2×Dic181—41no—
102C2×Dic181C22⋊C412yes—
103C4×D181—22no—
104C4×D181—22no—
105C4×D181—22no—
106C4×D181—22no—
107#13089678—41no—
108C23×C181—22no—
109C23×C181—22no—
110C23×C181—22no—
111C23×C181—82no—
112C23×C181C2×D411yescharacteristic
113D4×C181—22no—
114D4×C181—22no—
115D4×C181—22no—
116D4×C181C22⋊C414yes—
117C2×C4×C181—41no—
118C2×C4×C181C22×C411yescharacteristic
119C2×C4×C181C2×D411yescharacteristic
120C2×C4×C181C2×D411yescharacteristic
121C2×C4×C181C2×D411yescharacteristic
122C2×C4×C181C2×D411yescharacteristic
123C2×C4×C181C2×D411yescharacteristic
124C8×C181—41no—
125C22×C181—24no—
126C22×C181—41no—
127C22×C181—44no—
128C22×C181—82no—
129C22×C181—82no—
130C22×C181C2×C22⋊C411yescharacteristic
131C22×C181#12011yescharacteristic
132C22×C181#12011yescharacteristic
133C2×D181—22no—
134C2×D181—22no—
135C2×D181—42no—
136C2×D181—42no—
137C2×D181—88no—
138C4×C181—81no—
139C4×C181C2×C22⋊C411yescharacteristic
140C4×C181C2×C22⋊C411yescharacteristic
141C4×C181#12011yescharacteristic
142C4×C181#12011yescharacteristic
143Dic181—22no—
144Dic181—22no—
145Dic181—81no—
146C2×C181—21no—
147C2×C181—21no—
148C2×C181—21no—
149C2×C181—44no—
150C2×C181#37911yescharacteristic
151D181—42no—
152D181—42no—
153C181#179511yescharacteristic, complemented, Sylow
154#1795—1811nomaximal, Sylow
155C2×H(5,𝔽2)—1811no—
156#421—1811no—
157#421—1811no—
158#418—1811no—
159#418D18111yescharacteristic, complemented
160#413—1811no—
161#410—1811no—
162H(5,𝔽2)—3622no—
163H(5,𝔽2)—3622no—
164H(5,𝔽2)—3622no—
165H(5,𝔽2)—3622no—
166C2×#41—3621no—
167C2×#41—1812no—
168C2×#41—1811no—
169C2×#41—1811no—
170C22×D4—3621no—
171C22×D4—3622no—
172C22×D4—1812no—
173C22×D4C2×D18111yescharacteristic
174C2×M16—1811no—
175C2×M16C2×D18111yescharacteristic, complemented
176#117—1811no—
177#117C2×D18111yescharacteristic, complemented
178#104—3622no—
179#104—3622no—
180#102—22no—
181#102—3622no—
182#101—3621no—
183#100—3621no—
184#99—3622no—
185C24—21no—
186C24—3622no—
187#41—3624no—
188#41—3624no—
189#41—3624no—
190#41—3624no—
191#41—3621no—
192#41—3621no—
193#41—3621no—
194#41—3621no—
195#41—3621no—
196#41—3621no—
197#41—1812no—
198#41—1812no—
199C2×Q8—1811no—
200C2×Q8—1811no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—3624no—
205C2×D4—3624no—
206C2×D4—3624no—
207C2×D4—3628no—
208C2×D4—3628no—
209C2×D4—3622no—
210C2×D4—3621no—
211C2×D4—3624no—
212C2×D4—3622no—
213C2×D4—1812no—
214C2×D4—1811no—
215C2×D4—1811no—
216C2×D4C4×D18112yes—
217C2×D4C22⋊D18112yes—
218C2×D4D72412yes—
219C22×C4—3622no—
220C22×C4—3621no—
221C22×C4—1812no—
222C22×C4C22×D18111yescharacteristic
223M16—21no—
224M16—21no—
225M16—3621no—
226M16—3621no—
227C2×C8—21no—
228C2×C8—3621no—
229C4⋊C4—21no—
230C4⋊C4—3621no—
231C22⋊C4—21no—
232C22⋊C4—3621no—
233C42—21no—
234C42—3621no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—82no—
239C23—3628no—
240C23—3621no—
241C23—14484no—
242C23—1812no—
243C23D4×D18111yescharacteristic
244Q8—3621no—
245Q8—3621no—
246Q8—3621no—
247Q8—1812no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—3628no—
252D4—3628no—
253D4—3622no—
254D4—3622no—
255D4—3622no—
256D4—3622no—
257D4—3621no—
258D4—3621no—
259D4—3621no—
260D4—1812no—
261D4#1566842614yes—
262C2×C4—41no—
263C2×C4—3628no—
264C2×C4—3621no—
265C2×C4—3622no—
266C2×C4—3622no—
267C2×C4—3622no—
268C2×C4—3622no—
269C2×C4—1812no—
270C2×C4—7241no—
271C2×C4C2×C4×D18111yescharacteristic
272C2×C4D4×D18111yescharacteristic
273C2×C4D4×D18111yescharacteristic
274C2×C4D4×D18111yescharacteristic
275C2×C4C2×C22⋊D18111yescharacteristic
276C2×C4C2×D72411yescharacteristic
277C8—41no—
278C8—7241no—
279C22—24no—
280C22—41no—
281C22—44no—
282C22—82no—
283C22—82no—
284C22—3622no—
285C22—3622no—
286C22—14488no—
287C22—7242no—
288C22—7242no—
289C22C22⋊C4×D18111yescharacteristic
290C22#1915378411yescharacteristic
291C22#1915378611yescharacteristic
292C4—81no—
293C4—3622no—
294C4—3622no—
295C4—14481no—
296C4C22⋊C4×D18111yescharacteristic
297C4C2×#1566842611yescharacteristic
298C4#1915378411yescharacteristic
299C4#1915378611yescharacteristic
300C2—21no—
301C2—21no—
302C2—21no—
303C2—44no—
304C2—7242no—
305C2—7242no—
306C2#2604113411yescharacteristic
subgroupG/H
centerC2#26041134
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#418×C181C2
socleC2×C181#379
prime radicalC2×#19153861C2
cosocleC22×C4×C181C23
hypercenter#418D181
hypocenterC181#1795
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #26041297C2semidirect
3. #26041295C2semidirect
4. #26041284C2semidirect
5. #26041261C2semidirect
6. #26041252C2semidirect
7. #418×C181C2semidirect
15. C2×#15668448C4semidirect
18. C2×D4×D181C4semidirect
19. C2×#15668433C22semidirect
20. #19153775C22semidirect
28. C2×M16×C181C22semidirect
29. #117×C181C22semidirect
153. C181#1795semidirect
159. #418D181semidirect
175. C2×M16C2×D181semidirect
177. #117C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538612
#260412972
#260412952
#260412842
#260412612
#260412522
#418×C1812
#1795181