Finite Group Databasebeta

← #37536988

Subgroups — #37536988

Order 23168.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×Dic181C211yesmaximal, characteristic, complemented
2C2×#19153772C211yesmaximal, characteristic
3#26041219C211yesmaximal, characteristic, complemented
4#26041215C211yesmaximal, characteristic
5#26041213C212yesmaximal, complemented
6#26041202C212yesmaximal
7#26041198C211yesmaximal, characteristic, complemented
8#26041130C212yesmaximal, complemented
9#26041130C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic181C212yesmaximal
11#393×C181C211yesmaximal, characteristic
12C23×Dic181C2211yescharacteristic
13Q8×Dic181—24no—
14#19153772—24no—
15C2×C4⋊Dic181C2212yes—
16C2×C4⋊Dic181C2211yescharacteristic
17C2×#15668428C2212yes—
18C2×#15668428C2212yes—
19C2×#15668428C2212yes—
20C2×#15668428C2211yescharacteristic
21C2×#15668428C2211yescharacteristic
22C2×C22⋊Dic181C2212yes—
23C2×C22⋊Dic181C2211yescharacteristic
24C2×C4×Dic181C2212yes—
25C2×C4×Dic181C2212yes—
26C2×C4×Dic181C2211yescharacteristic
27C2×C4×Dic181C2211yescharacteristic
28#19153723C2212yes—
29#19153723C2212yes—
30#19153723C2211yescharacteristic
31#19153723C2212yes—
32C22×Q8×C181C2211yescharacteristic
33C2×C22⋊C4×C181C2212yes—
34C2×C22⋊C4×C181C2211yescharacteristic
35#95×C181C2212yes—
36#95×C181C2211yescharacteristic
37C22×Dic181—22no—
38C22×Dic181—22no—
39C22×Dic181Q814yescomplemented
40C22×Dic181C2311yescharacteristic
41C22×Dic181C2312yes—
42C22×Dic181C2311yescharacteristic
43C22×Dic181C2311yescharacteristic
44C22×Dic181C2312yes—
45C22×Dic181C2311yescharacteristic
46C4⋊Dic181—24no—
47C4⋊Dic181—22no—
48#15668428—22no—
49#15668428—24no—
50#15668428—24no—
51#15668428—22no—
52#15668428—24no—
53C22⋊Dic181—22no—
54C22⋊Dic181—24no—
55C4×Dic181—22no—
56C4×Dic181—24no—
57C4×Dic181—24no—
58C4×Dic181D414yes—
59C24×C181C2311yescharacteristic
60C2×Q8×C181—22no—
61C2×Q8×C181—24no—
62C22×C4×C181C2312yes—
63C22×C4×C181C2311yescharacteristic
64C22×C4×C181C2311yescharacteristic
65C22×C4×C181C2312yes—
66C22⋊C4×C181—22no—
67C22⋊C4×C181—24no—
68C2×Dic181—21no—
69C2×Dic181—22no—
70C2×Dic181—22no—
71C2×Dic181—22no—
72C2×Dic181—21no—
73C2×Dic181—28no—
74C2×Dic181—21no—
75C2×Dic181—22no—
76C2×Dic181—22no—
77C2×Dic181—22no—
78C2×Dic181—22no—
79C2×Dic181C2×D412yes—
80C2×Dic181C2×D412yes—
81C2×Dic181C2×Q812yes—
82C2×Dic181C2×Q812yes—
83C2×Dic181#4114yes—
84C2×Dic181#4112yes—
85C23×C181—21no—
86C23×C181—22no—
87C23×C181—22no—
88C23×C181—21no—
89C23×C181C2×Q812yes—
90C23×C181C2411yescharacteristic
91Q8×C181—28no—
92C2×C4×C181—22no—
93C2×C4×C181—22no—
94C2×C4×C181—22no—
95C2×C4×C181—21no—
96C2×C4×C181—22no—
97C2×C4×C181—22no—
98C2×C4×C181—22no—
99C2×C4×C181—22no—
100C2×C4×C181C2×D412yes—
101C2×C4×C181#4114yes—
102C22×C181—22no—
103C22×C181—24no—
104C22×C181—41no—
105C22×C181—42no—
106C22×C181—41no—
107C22×C181C22×D411yescharacteristic
108C22×C181C22×Q811yescharacteristic
109C22×C181C2×#4111yescharacteristic
110C22×C181C2×#4112yes—
111C22×C181C2×#4112yes—
112C4×C181—22no—
113C4×C181—24no—
114C4×C181—42no—
115C4×C181—41no—
116Dic181—22no—
117Dic181—24no—
118Dic181—22no—
119Dic181—41no—
120Dic181—42no—
121Dic181C22⋊Q814yes—
122C2×C181—42no—
123C2×C181C2×C22⋊Q811yescharacteristic
124C2×C181#52511yescharacteristic
125C2×C181#52912yes—
126C2×C181#53211yescharacteristic
127C2×C181#53312yes—
128C181#239611yescharacteristic, complemented, Sylow
129#2396—1811nomaximal, Sylow
130C2×C4×Q8—1811no—
131C2×#117—1811no—
132#400—1811no—
133#399—1811no—
134#398—1812no—
135#396—1812no—
136#395—1811no—
137#393D18111yescharacteristic
138C4×C4⋊C4—1811no—
139C4×C4⋊C4—1812no—
140C4×C22⋊C4—1812no—
141C22×Q8C2×D18111yescharacteristic
142C23×C4—1811no—
143C4×Q8—3624no—
144#117—3624no—
145C2×C4⋊C4—1811no—
146C2×C4⋊C4—1811no—
147C2×C4⋊C4—1812no—
148C2×C4⋊C4—1811no—
149C2×C4⋊C4—1812no—
150C2×C4⋊C4—1812no—
151C2×C4⋊C4—1812no—
152C2×C22⋊C4—1811no—
153C2×C22⋊C4—1812no—
154C2×C22⋊C4C2×D18111yescharacteristic
155C2×C22⋊C4C2×D18112yes—
156C2×C42—1811no—
157C2×C42—1812no—
158C2×C42—1812no—
159C2×C42—1811no—
160#95—1812no—
161#95—1812no—
162#95—1811no—
163#95—1812no—
164#95C2×D18112yes—
165#95C2×D18111yescharacteristic
166C24C22×D18111yescharacteristic
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—3622no—
170C22×C4—3622no—
171C22×C4—1811no—
172C22×C4—1812no—
173C22×C4—1814no—
174C22×C4—1811no—
175C22×C4—1812no—
176C22×C4—1811no—
177C22×C4—1811no—
178C22×C4C22×D18112yes—
179C22×C4C22×D18111yescharacteristic
180C22×C4C22×D18111yescharacteristic
181C22×C4C22×D18112yes—
182C4⋊C4—3622no—
183C4⋊C4—3624no—
184C4⋊C4—3624no—
185C4⋊C4—3624no—
186C4⋊C4—3622no—
187C4⋊C4—3622no—
188C4⋊C4—3624no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—3622no—
192C22⋊C4—3624no—
193C42—3622no—
194C42—3624no—
195C42—3624no—
196C42—1814no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D18112yes—
202C23C23×D18111yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—3621no—
213C2×C4—3622no—
214C2×C4—3622no—
215C2×C4—3621no—
216C2×C4—3622no—
217C2×C4—3628no—
218C2×C4—3621no—
219C2×C4—3622no—
220C2×C4—3622no—
221C2×C4—3622no—
222C2×C4—3622no—
223C2×C4—1812no—
224C2×C4—1812no—
225C2×C4—1814no—
226C2×C4—1812no—
227C2×C4—1812no—
228C2×C4—1812no—
229C2×C4D4×D18112yes—
230C2×C4#1566844814yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D18111yescharacteristic
237C22C2×Q8×D18111yescharacteristic
238C22#41×D18111yescharacteristic
239C22#41×D18112yes—
240C22C2×#1566844812yes—
241C4—22no—
242C4—24no—
243C4—42no—
244C4—41no—
245C4—3622no—
246C4—3624no—
247C4—3622no—
248C4—1814no—
249C4—7241no—
250C4—7242no—
251C2—42no—
252C2C22⋊Q8×D18111yescharacteristic
253C2#2604178511yescharacteristic
254C2#2604180812yes—
255C2#2604182811yescharacteristic
256C2#2604183112yes—
subgroupG/H
center, FrattiniC23C23×D181
commutator, socle, cosocleC23×C181C24
Fitting#393×C181C2
prime radicalC24×C181C23
hypercenter#393D181
hypocenterC181#2396
LCSG ▷ C23×C181 ▷ C181
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic181C2semidirect
3. #26041219C2semidirect
5. #26041213C2semidirect
7. #26041198C2semidirect
8. #26041130C2semidirect
9. #26041130C2semidirect
39. C22×Dic181Q8semidirect
128. C181#2396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1812
C2×#191537722
#260412192
#260412152
#260412132
#260412022
#260411982
#260411302
C22⋊C4×Dic1812
#393×C1812
#2396181