Finite Group Databasebeta

← #37544952

Subgroups — #37544952

Order 23168.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D181C211yesmaximal, characteristic, complemented
2#26041814C211yesmaximal, characteristic
3#26041689C211yesmaximal, characteristic, complemented
4#26041641C211yesmaximal, characteristic, complemented
5#26041639C211yesmaximal, characteristic, complemented
6#26041616C212yesmaximal, complemented
7#26041599C212yesmaximal
8#26041563C212yesmaximal, complemented
9#26041558C212yesmaximal, complemented
10#26041334C211yesmaximal, characteristic, complemented
11#501×C181C211yesmaximal, characteristic, complemented
12C2×Q8×D181—21no—
13#19153816C2211yescharacteristic
14#19153815—21no—
15#19153815C2212yes—
16#19153814—21no—
17#19153811—21no—
18#19153804C2211yescharacteristic
19#19153801—21no—
20#19153796C2211yescharacteristic
21#19153791C2211yescharacteristic
22#19153781C2211yescharacteristic
23#19153778—21no—
24#19153778C2211yescharacteristic
25#19153774—21no—
26C4⋊C4×D181—21no—
27C4⋊C4×D181C2212yes—
28C42×D181C2211yescharacteristic
29#19153749C2212yes—
30#19153747C2212yes—
31#19153745C2212yes—
32#19153741C2212yes—
33#19153739C2212yes—
34#19153738C2212yes—
35#19153736C2212yes—
36#19153728C2212yes—
37#19153727C2211yescharacteristic
38C4⋊Q8×C181C2211yescharacteristic
39#125×C181C2211yescharacteristic
40#107×C181C2212yes—
41#106×C181C2212yes—
42#97×C181C2211yescharacteristic
43C2×Dic362—22no—
44C2×Dic362—21no—
45C2×Dic362C2311yescharacteristic
46Q8×D181—44no—
47C2×D724—21no—
48C2×D724C2311yescharacteristic
49C2×C4×D181—22no—
50C2×C4×D181—21no—
51C2×C4×D181D411yescharacteristic
52C2×C4×D181D411yescharacteristic
53C2×C4×D181C2311yescharacteristic
54C2×#13089678C2312yes—
55C4⋊Dic181—21no—
56C4⋊Dic181—21no—
57C4⋊Dic181C2312yes—
58#15668428—22no—
59#15668428—22no—
60#15668428—21no—
61#15668426—21no—
62#15668426—21no—
63#15668426—21no—
64#15668426—21no—
65#15668426—21no—
66C4×Dic181—21no—
67C4×Dic181D411yescharacteristic
68C4×Dic181D411yescharacteristic
69C4×Dic181C2311yescharacteristic
70C2×Q8×C181—21no—
71C2×C8×C181C2312yes—
72C4⋊C4×C181—21no—
73C4⋊C4×C181—21no—
74C4⋊C4×C181—21no—
75C4⋊C4×C181C2311yescharacteristic
76C4⋊C4×C181C2312yes—
77C4⋊C4×C181C2311yescharacteristic
78C42×C181C2311yescharacteristic
79C22×D181—21no—
80C22×D181C2×D411yescharacteristic
81Dic362—22no—
82Dic362—42no—
83Dic362—44no—
84D724—22no—
85D724—42no—
86C2×Dic181—21no—
87C2×Dic181—22no—
88C2×Dic181C2×D411yescharacteristic
89C2×Dic181C2×D411yescharacteristic
90C2×Dic181C2×D411yescharacteristic
91C4×D181—22no—
92C4×D181—22no—
93C4×D181—22no—
94C4×D181—44no—
95C4×D181—42no—
96#13089678—22no—
97Q8×C181—42no—
98C2×C4×C181—21no—
99C2×C4×C181—21no—
100C2×C4×C181—22no—
101C2×C4×C181C2×D411yescharacteristic
102C2×C4×C181C2×D411yescharacteristic
103C2×C4×C181C2411yescharacteristic
104C8×C181—22no—
105C22×C181C22×D411yescharacteristic
106C2×D181—21no—
107C2×D181—21no—
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—41no—
111C2×D181#13712yes—
112C4×C181—21no—
113C4×C181—21no—
114C4×C181—42no—
115C4×C181—41no—
116C4×C181—41no—
117C4×C1812−1+412yes—
118Dic181—21no—
119Dic181—21no—
120Dic181—41no—
121Dic181—42no—
122Dic181AGL(1,ℤ/8)12yes—
123C2×C181#53611yescharacteristic
124C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C181C2×#13711yescharacteristic
126D181—22no—
127D181—81no—
128C181#323911yescharacteristic, complemented, Sylow
129#3239—1811nomaximal, Sylow
130C2×C4⋊Q8—1811no—
131#529—1811no—
132#501—1811no—
133#501D18111yescharacteristic, complemented
134#489—1811no—
135#489—1811no—
136#484—1812no—
137#481—1812no—
138#475—1812no—
139#474—1812no—
140#432—1811no—
141C22×Q8—3621no—
142C4⋊Q8—3621no—
143C4⋊Q8—3621no—
144C4⋊Q8—1812no—
145C4⋊Q8—1811no—
146C4⋊Q8C2×D18111yescharacteristic
147#126—3621no—
148#125C2×D18111yescharacteristic
149#124—1811no—
150#123—3621no—
151C22⋊Q8—1811no—
152#121—1811no—
153C4×Q8—1811no—
154C4×D4—3621no—
155C4×D4—1811no—
156#117—3621no—
157C2×C4⋊C4—3621no—
158C2×C4⋊C4—1812no—
159C2×C42—1811no—
160#107—1812no—
161#107C2×D18112yes—
162#106—1812no—
163#106C2×D18112yes—
164C4⋊C8—1812no—
165#103—1812no—
166#103—1812no—
167#102—1812no—
168#102—1812no—
169C22⋊C8—1812no—
170#97—1811no—
171#97C2×D18111yescharacteristic
172C2×Q8—21no—
173C2×Q8—3622no—
174C2×Q8—3621no—
175C2×Q8—1811no—
176C2×Q8—7244no—
177C2×D4—3621no—
178C2×D4—1811no—
179C22×C4—3622no—
180C22×C4—3621no—
181C22×C4—1811no—
182C22×C4—1811no—
183C22×C4—1811no—
184C2×C8—1812no—
185C2×C8C22×D18112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—3622no—
190C4⋊C4—3622no—
191C4⋊C4—3621no—
192C4⋊C4—3621no—
193C4⋊C4—3621no—
194C4⋊C4—1812no—
195C4⋊C4C22×D18112yes—
196C4⋊C4C22×D18111yescharacteristic
197C4⋊C4C22×D18111yescharacteristic
198C22⋊C4—3621no—
199C22⋊C4—3621no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C22⋊C4—3621no—
203C42—3621no—
204C42—1811no—
205C42—1811no—
206C42—1811no—
207C42C22×D18111yescharacteristic
208C23—3621no—
209C23—1811no—
210Q8—42no—
211Q8—3622no—
212Q8—7242no—
213Q8—7244no—
214D4—3622no—
215D4—7242no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—3622no—
220C2×C4—3622no—
221C2×C4—3622no—
222C2×C4—3621no—
223C2×C4—3622no—
224C2×C4—1811no—
225C2×C4—1811no—
226C2×C4—1811no—
227C2×C4—7242no—
228C2×C4—7244no—
229C2×C4D4×D18111yescharacteristic
230C2×C4D4×D18111yescharacteristic
231C2×C4C23×D18111yescharacteristic
232C8—22no—
233C8—3622no—
234C22—3621no—
235C22—3621no—
236C22—1812no—
237C22—7241no—
238C22—7241no—
239C22—7241no—
240C22C2×D4×D18111yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—3621no—
247C4—3621no—
248C4—1812no—
249C4—7242no—
250C4—7241no—
251C4#1915386412yes—
252C2—3622no—
253C2—14481no—
254C2#2604185411yescharacteristic
255C2AGL(1,ℤ/8)×D18111yescharacteristic
256C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#501×C181C2
socleC22×C181C22×D4
prime radical#19153791C22
hypercenter#501D181
hypocenterC181#3239
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D181C2semidirect
3. #26041689C2semidirect
4. #26041641C2semidirect
5. #26041639C2semidirect
6. #26041616C2semidirect
8. #26041563C2semidirect
9. #26041558C2semidirect
10. #26041334C2semidirect
11. #501×C181C2semidirect
128. C181#3239semidirect
133. #501D181semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1812
#260418142
#260416892
#260416412
#260416392
#260416162
#260415992
#260415632
#260415582
#260413342
#501×C1812
#3239181