Finite Group Databasebeta

← #37545496

Subgroups — #37545496

Order 23168.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#26042092C212yesmaximal, complemented
2#26042085C212yesmaximal, complemented
3#26041551C212yesmaximal, complemented
4#26041539C212yesmaximal, complemented
5#26041461C212yesmaximal, complemented
6#26041456C211yesmaximal, characteristic, complemented
7#26041451C211yesmaximal, characteristic, complemented
8#26041402C212yesmaximal, complemented
9#497×C181C211yesmaximal, characteristic, complemented
10#19153864C2212yes—
11#19153864C2211yescharacteristic, complemented
12#19153861C2211yescharacteristic
13#19153848—22no—
14#19153848—22no—
15#19153847C2212yes—
16#19153847C2212yes—
17#19153847C2212yescomplemented
18#19153845—22no—
19#19153843—22no—
20#19153842C2212yes—
21#19153839—22no—
22#19153838C2212yes—
23#19153837—22no—
24Q16×D181—22no—
25SD16×D181—22no—
26#19153822C2212yes—
27#19153822C2212yescomplemented
28#19153752C2212yescomplemented
29#19153744C2212yes—
30#19153744C2212yescomplemented
31#19153735C2211yescharacteristic, complemented
32#19153735C2212yescomplemented
33#19153733C2211yescharacteristic
34C2×Q16×C181C2212yescomplemented
35C2×SD16×C181C2212yes—
36C4⋊Q8×C181C2211yescharacteristic, complemented
37#124×C181C2211yescharacteristic
38#97×C181C2211yescharacteristic, complemented
39#15668450—22no—
40#15668450—22no—
41#15668450—22no—
42#15668449—42no—
43#15668449—41no—
44#15668449C2312yes—
45#15668449C2312yes—
46#15668448—22no—
47Q8×D181—22no—
48Q8×D181—22no—
49Q8×D181—22no—
50C2×C22⋊D181—41no—
51D4×D181—22no—
52Dic724D412yes—
53#15668439—22no—
54#15668439—22no—
55#15668439—22no—
56#15668438D412yes—
57#15668438D412yes—
58#15668437—22no—
59#15668436—22no—
60#15668436—22no—
61#15668436—22no—
62D1448D412yes—
63#15668434—22no—
64#15668433C2312yes—
65#15668433C2312yes—
66#15668432—22no—
67#15668432—22no—
68C8×D181—22no—
69C8×D181—22no—
70C2×Q8×C181C2311yescharacteristic
71C2×Q8×C181C2312yes—
72C2×D4×C181C2311yescharacteristic
73Q16×C181—22no—
74Q16×C181—22no—
75SD16×C181—22no—
76SD16×C181—22no—
77C2×C8×C181C2312yes—
78C4⋊C4×C181—41no—
79C22⋊C4×C181—41no—
80C42×C181C2311yescharacteristic
81C22×D181—41no—
82Dic362—41no—
83Dic362—42no—
84Dic362C2×D412yes—
85Dic362C2×D412yes—
86D724—41no—
87D724—42no—
88D724C2×D412yes—
89D724C2×D412yes—
90C22⋊D181—22no—
91C22⋊D181—22no—
92C22⋊D181—42no—
93C2×Dic181—41no—
94C4×D181—22no—
95C4×D181—22no—
96C4×D181—42no—
97C4×D181—42no—
98C4×D181—42no—
99#13089678—22no—
100#13089678—22no—
101C23×C181—21no—
102Q8×C181—22no—
103Q8×C181—22no—
104Q8×C181—22no—
105D4×C181—22no—
106C2×C4×C181—21no—
107C2×C4×C181—21no—
108C2×C4×C181—22no—
109C2×C4×C181C2411yescharacteristic
110C8×C181C2×D412yes—
111C8×C181C2×D412yes—
112C22×C181—41no—
113C22×C181—81no—
114C22×C181H(5,𝔽2)11yescharacteristic
115C2×D181—22no—
116C2×D181—22no—
117C2×D181—161no—
118C4×C181—41no—
119C4×C181—41no—
120C4×C181—42no—
121C4×C181C22×D412yes—
122Dic181—22no—
123Dic181—22no—
124C2×C181—21no—
125C2×C181—81no—
126C2×C181D4211yescharacteristic
127D181—42no—
128D181—42no—
129C181#328911yescharacteristic, complemented, Sylow
130#3289—1811nomaximal, Sylow
131#578—1812no—
132#577—1812no—
133#497D18111yescharacteristic, complemented
134#473—1812no—
135#471—1812no—
136#456—1812no—
137#455—1811no—
138#454—1811no—
139#444—1812no—
1402−1+4—1812no—
1412−1+4—1811no—
142H(5,𝔽2)—1811no—
143#137—3622no—
144#137—3622no—
145#137—3622no—
146#137—1812no—
147#137—1812no—
148#137—1812no—
149AGL(1,ℤ/8)—3622no—
150AGL(1,ℤ/8)—1812no—
151#135—3622no—
152#135—3622no—
153#135—1812no—
154C2×Q16—3622no—
155C2×Q16C2×D18112yescomplemented
156C2×SD16—3622no—
157C2×SD16C2×D18112yes—
158#131—1812no—
159#131—1812no—
160C4⋊Q8C2×D18111yescharacteristic, complemented
161#124C2×D18111yescharacteristic
162#108—1812no—
163#104—1812no—
164#104—1812no—
165#101—1811no—
166#101—1812no—
167#100—1811no—
168#97C2×D18111yescharacteristic, complemented
169#41—3622no—
170#41—3622no—
171#41—3622no—
172#41—3622no—
173#41—1812no—
174#41—1812no—
175#41—7241no—
176#41—7242no—
177C2×Q8—3622no—
178C2×Q8—3622no—
179C2×Q8—3622no—
180C2×Q8C22×D18111yescharacteristic
181C2×Q8C22×D18112yes—
182C2×D4—3622no—
183C2×D4—7241no—
184C2×D4C22×D18111yescharacteristic
185Q16—22no—
186Q16—22no—
187Q16—3622no—
188Q16—3622no—
189Q16—3622no—
190Q16—1812no—
191SD16—22no—
192SD16—22no—
193SD16—3622no—
194SD16—3622no—
195SD16—3622no—
196SD16—3622no—
197SD16—1812no—
198SD16—1812no—
199D8—3622no—
200D8—1812no—
201M16—3622no—
202M16—3622no—
203M16—1812no—
204M16—1812no—
205C2×C8—3622no—
206C2×C8—3622no—
207C2×C8C22×D18112yes—
208C4⋊C4—41no—
209C22⋊C4—41no—
210C42C22×D18111yescharacteristic
211C23—21no—
212C23—7241no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—1812no—
217Q8—1812no—
218Q8—7242no—
219Q8—7241no—
220D4—22no—
221D4—3622no—
222D4—3622no—
223D4—1812no—
224D4—1812no—
225D4—7241no—
226D4—7242no—
227D4—7242no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—3622no—
232C2×C4—3622no—
233C2×C4—7242no—
234C2×C4—7242no—
235C2×C4—7242no—
236C2×C4—7241no—
237C2×C4C23×D18111yescharacteristic
238C8—3622no—
239C8—3622no—
240C8D4×D18112yes—
241C8D4×D18112yes—
242C22—41no—
243C22—81no—
244C22—28961no—
245C22—3622no—
246C22—3622no—
247C22#1915386111yescharacteristic
248C4—41no—
249C4—41no—
250C4—42no—
251C4—3622no—
252C4—3622no—
253C4C2×D4×D18112yes—
254C2—21no—
255C2—81no—
256C2—7242no—
257C2—7242no—
258C2#2604191411yescharacteristic
subgroupG/H
centerC2#26041914
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#497×C181C2
socleC2×C181D42
hypercenter#497D181
hypocenterC181#3289
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #26042092C2semidirect
2. #26042085C2semidirect
3. #26041551C2semidirect
4. #26041539C2semidirect
5. #26041461C2semidirect
6. #26041456C2semidirect
7. #26041451C2semidirect
8. #26041402C2semidirect
9. #497×C181C2semidirect
11. #19153864C22semidirect
17. #19153847C22semidirect
27. #19153822C22semidirect
28. #19153752C22semidirect
30. #19153744C22semidirect
31. #19153735C22semidirect
32. #19153735C22semidirect
34. C2×Q16×C181C22semidirect
36. C4⋊Q8×C181C22semidirect
38. #97×C181C22semidirect
129. C181#3289semidirect
133. #497D181semidirect
155. C2×Q16C2×D181semidirect
160. C4⋊Q8C2×D181semidirect
168. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420922
#260420852
#260415512
#260415392
#260414612
#260414562
#260414512
#260414022
#497×C1812
#3289181