Finite Group Databasebeta

← #37546756

Subgroups — #37546756

Order 23168.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#26042004C211yesmaximal, characteristic, complemented
2#26041999C211yesmaximal, characteristic
3#26041670C212yesmaximal
4#26041658C211yesmaximal, characteristic, complemented
5#26041499C212yesmaximal
6#26041490C211yesmaximal, characteristic, complemented
7#26041488C212yesmaximal, complemented
8#26041406C211yesmaximal, characteristic, complemented
9#26041383C212yesmaximal, complemented
10C4×#15668437C211yesmaximal, characteristic
11#461×C181C211yesmaximal, characteristic
12C2×Dic724C2212yes—
13C2×#15668439C2212yes—
14C2×#15668439C2212yes—
15C2×#15668437C2211yescharacteristic
16#19153816C2211yescharacteristic
17#19153814C2212yes—
18#19153809—22no—
19#19153807—21no—
20#19153806—22no—
21#19153803C2212yes—
22#19153795—21no—
23Q8×Dic181C2212yes—
24C4×Dic362C2211yescharacteristic
25C4×Dic362C2211yescharacteristic
26#19153781—21no—
27#19153781C2212yes—
28#19153775—21no—
29#19153747C2211yescharacteristic
30#19153745C2212yes—
31#19153741C2212yes—
32#19153739C2211yescharacteristic
33#19153737C2211yescharacteristic
34#19153726C2212yes—
35C4×#13089678C2211yescharacteristic
36C2×SD16×C181C2212yes—
37C4⋊Q8×C181C2211yescharacteristic
38C4×D4×C181C2211yescharacteristic
39C4⋊C8×C181C2211yescharacteristic
40#103×C181C2212yes—
41C2×Dic362D411yescharacteristic
42C2×Dic362C2311yescharacteristic
43C2×Dic362C2312yes—
44Dic724—42no—
45#15668439—24no—
46#15668439—42no—
47#15668437—22no—
48C2×#13089678C2311yescharacteristic
49C2×#13089678C2311yescharacteristic
50C4⋊Dic181—22no—
51C4⋊Dic181—21no—
52C4⋊Dic181—21no—
53C4⋊Dic181D411yescharacteristic
54C4⋊Dic181C2311yescharacteristic
55#15668428—22no—
56#15668428—22no—
57#15668428—22no—
58#15668428—21no—
59#15668428D412yes—
60C22⋊Dic181—22no—
61C22⋊Dic181—22no—
62C4×Dic181—22no—
63C4×Dic181—22no—
64C4×Dic181—21no—
65C4×Dic181C2312yes—
66C2×Q8×C181C2312yes—
67C2×D4×C181C2311yescharacteristic
68C22×C4×C181—21no—
69SD16×C181—42no—
70C2×C8×C181C2312yes—
71C4⋊C4×C181—21no—
72C4⋊C4×C181—21no—
73C4⋊C4×C181C2311yescharacteristic
74C22⋊C4×C181—21no—
75C42×C181C2311yescharacteristic
76Dic362—22no—
77Dic362—21no—
78Dic362—22no—
79Dic362—22no—
80Dic362—21no—
81C2×Dic181—21no—
82C2×Dic181—22no—
83C2×Dic181—21no—
84C2×Dic181C2×D412yes—
85C2×Dic181C2×D412yes—
86#13089678—21no—
87#13089678#4112yes—
88C23×C181—21no—
89Q8×C181—22no—
90Q8×C181—22no—
91D4×C181—21no—
92D4×C181—21no—
93C2×C4×C181—21no—
94C2×C4×C181—22no—
95C2×C4×C181—42no—
96C2×C4×C181C2×D411yescharacteristic
97C2×C4×C181C2×D411yescharacteristic
98C2×C4×C181C2411yescharacteristic
99C8×C181—22no—
100C22×C181—41no—
101C22×C181—41no—
102C22×C181—41no—
103C22×C181C22×D411yescharacteristic
104C4×C181—21no—
105C4×C181—42no—
106C4×C181—41no—
107C4×C181#13712yes—
108C4×C181C2×#4111yescharacteristic
109C4×C181H(5,𝔽2)11yescharacteristic
110Dic181—22no—
111Dic181—22no—
112Dic181—42no—
113Dic181—41no—
114Dic181—41no—
115C2×C181—81no—
116C2×C181#55011yescharacteristic
117C2×C181C2×#13711yescharacteristic
118C2×C181#57811yescharacteristic
119C181#339811yescharacteristic, complemented, Sylow
120#3398—1811nomaximal, Sylow
121#557—1811no—
122#556—1811no—
123#497—1812no—
124#494—1811no—
125#464—1812no—
126#462—1811no—
127#462—1812no—
128#461D18111yescharacteristic
129#446—1811no—
130#441—1812no—
131C4×SD16—1811no—
132C2×Q16—1812no—
133C2×Q16—1812no—
134C2×Q16—1812no—
135C2×SD16—1811no—
136C2×SD16C2×D18112yes—
137C4⋊Q8—1812no—
138C4⋊Q8—1811no—
139C4⋊Q8C2×D18111yescharacteristic
140#126—3622no—
141#125—3622no—
142#125—3621no—
143#124—1812no—
144C22⋊Q8—3621no—
145C4×Q8—3621no—
146C4×Q8—1812no—
147C4×Q8—1812no—
148C4×Q8—1811no—
149C4×Q8—1811no—
150C4×D4C2×D18111yescharacteristic
151#117—3621no—
152#106—1811no—
153C4⋊C8—1812no—
154C4⋊C8C2×D18111yescharacteristic
155#103—1812no—
156#103—1811no—
157#103C2×D18112yes—
158#102—1811no—
159#97—1812no—
160C4×C8—1811no—
161C2×Q8—1812no—
162C2×Q8—1811no—
163C2×Q8—1811no—
164C2×Q8C22×D18112yes—
165C2×D4C22×D18111yescharacteristic
166C22×C4—21no—
167Q16—3624no—
168Q16—7242no—
169Q16—7242no—
170SD16—42no—
171SD16—3622no—
172C2×C8—1811no—
173C2×C8—1811no—
174C2×C8C22×D18112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3622no—
178C4⋊C4—3622no—
179C4⋊C4—3622no—
180C4⋊C4—3622no—
181C4⋊C4—3621no—
182C4⋊C4—3621no—
183C4⋊C4—3621no—
184C4⋊C4—1811no—
185C4⋊C4—1811no—
186C4⋊C4—1812no—
187C4⋊C4C22×D18111yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—3622no—
190C22⋊C4—3622no—
191C42—3622no—
192C42—3622no—
193C42—3621no—
194C42—1812no—
195C42C22×D18111yescharacteristic
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—3622no—
200Q8—3621no—
201Q8—3622no—
202Q8—3621no—
203Q8—3622no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3621no—
210C2×C4—3622no—
211C2×C4—3621no—
212C2×C4—1812no—
213C2×C4—1812no—
214C2×C4D4×D18111yescharacteristic
215C2×C4D4×D18111yescharacteristic
216C2×C4C23×D18111yescharacteristic
217C8—22no—
218C8—3621no—
219C8—1812no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C2×D4×D18111yescharacteristic
224C4—21no—
225C4—42no—
226C4—41no—
227C4—3622no—
228C4—3622no—
229C4—7242no—
230C4—7241no—
231C4—7241no—
232C4#1915384512yes—
233C4#41×D18111yescharacteristic
234C4#1915386111yescharacteristic
235C2—81no—
236C2#2604194711yescharacteristic
237C2C2×#1915384511yescharacteristic
238C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#461×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181C23
hypercenter#461D181
hypocenterC181#3398
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #26042004C2semidirect
4. #26041658C2semidirect
6. #26041490C2semidirect
7. #26041488C2semidirect
8. #26041406C2semidirect
9. #26041383C2semidirect
119. C181#3398semidirect

Maximal subgroups

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

subgroupindex
#260420042
#260419992
#260416702
#260416582
#260414992
#260414902
#260414882
#260414062
#260413832
C4×#156684372
#461×C1812
#3398181