Finite Group Databasebeta

← #37534178

Subgroups — #37534178

Order 23168.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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×#19153800C211yesmaximal, characteristic
2C2×#19153772C211yesmaximal, characteristic, complemented
3C2×#19153730C211yesmaximal, characteristic
4C2×#19153730C211yesmaximal, characteristic
5#26041140C211yesmaximal, characteristic, complemented
6#26041046C211yesmaximal, characteristic, complemented
7#342×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2211yescharacteristic
9C23×Dic181C2211yescharacteristic
10#19153800—22no—
11#19153800—21no—
12#19153800—21no—
13#19153772—22no—
14#19153772—22no—
15C2×C4⋊Dic181C412yescomplemented
16C2×#15668428—21no—
17C2×C22⋊Dic181C2211yescharacteristic
18C2×C22⋊Dic181C2211yescharacteristic
19C2×#15668426C412yescomplemented
20C2×C4×Dic181—21no—
21#19153730—22no—
22#19153730—22no—
23#19153723—22no—
24C2×C22⋊C4×C181C2211yescharacteristic
25C2×C22⋊C4×C181C2211yescharacteristic
26C2×C22⋊C4×C181C2211yescharacteristic
27C23×D181C2×C411yescharacteristic
28C2×C22⋊D181—21no—
29C2×C22⋊D181—21no—
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181—21no—
34C22×Dic181—21no—
35C22×Dic181—21no—
36C22×Dic181—21no—
37C22×Dic181—21no—
38C22×Dic181—21no—
39C22×Dic181—21no—
40C22×Dic181C2×C411yescharacteristic
41C22×Dic181C2×C412yes—
42C22×Dic181D412yes—
43C22×Dic181D411yescharacteristic
44C22×Dic181D411yescharacteristic
45C4⋊Dic181—22no—
46C4⋊Dic181—22no—
47#15668428—24no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181—21no—
51C22⋊Dic181D412yes—
52#15668426—22no—
53#15668426—22no—
54C4×Dic181—24no—
55C24×C181C2311yescharacteristic
56C22×C4×C181—21no—
57C22×C4×C181—21no—
58C22×C4×C181C2×C412yes—
59C22⋊C4×C181—21no—
60C22⋊C4×C181—21no—
61C22⋊C4×C181—21no—
62C22⋊C4×C181—21no—
63C22⋊C4×C181—21no—
64C22⋊C4×C181D412yes—
65C22×D181—21no—
66C22×D181—21no—
67C22×D181—21no—
68C22×D181—42no—
69C22⋊D181—41no—
70C22⋊D181—41no—
71C22⋊D181—41no—
72C22⋊D181—41no—
73C2×Dic181—22no—
74C2×Dic181—22no—
75C2×Dic181—21no—
76C2×Dic181—21no—
77C2×Dic181—22no—
78C2×Dic181—22no—
79C2×Dic181—21no—
80C2×Dic181—21no—
81C2×Dic181—21no—
82C2×Dic181—21no—
83C2×Dic181—21no—
84C2×Dic181—22no—
85C2×Dic181—42no—
86C2×Dic181—42no—
87C2×Dic181—41no—
88C2×Dic181C22⋊C412yes—
89C2×Dic181C22⋊C412yes—
90C23×C181—42no—
91C23×C181C22×C411yescharacteristic
92C23×C181C2×D411yescharacteristic
93C23×C181C2×D411yescharacteristic
94C23×C181C2×D411yescharacteristic
95C23×C181C2×D411yescharacteristic
96C23×C181#4111yescharacteristic
97C23×C181#4111yescharacteristic
98C2×C4×C181—22no—
99C2×C4×C181—22no—
100C2×C4×C181—42no—
101C2×C4×C181—42no—
102C2×C4×C181—42no—
103C2×C4×C181—41no—
104C22×C181—22no—
105C22×C181—22no—
106C22×C181—22no—
107C22×C181—41no—
108C22×C181—41no—
109C22×C181—41no—
110C22×C181—41no—
111C22×C181C2×C22⋊C411yescharacteristic
112C22×C181C4×D411yescharacteristic
113C22×C181C4×D411yescharacteristic
114C22×C181#12011yescharacteristic
115C22×C181#12111yescharacteristic
116C22×C181#12111yescharacteristic
117C22×C181#12311yescharacteristic
118C2×D181—41no—
119C2×D181—42no—
120C2×D181—81no—
121C2×D181—81no—
122C4×C181—42no—
123C4×C181—42no—
124C4×C181—81no—
125Dic181—22no—
126Dic181—22no—
127Dic181—42no—
128Dic181—41no—
129Dic181—81no—
130C2×C181—21no—
131C2×C181—21no—
132C2×C181—21no—
133C2×C181—21no—
134C2×C181—21no—
135C2×C181—21no—
136C2×C181#38611yescharacteristic
137C2×C181#41012yes—
138D181—81no—
139C181#189911yescharacteristic, complemented, Sylow
140#1899—1811nomaximal, Sylow
141C2×#123—1811no—
142C2×#117—1811no—
143C2×#99—1811no—
144C2×#99—1811no—
145#381—1811no—
146#342—1811no—
147#342D18111yescharacteristic, complemented
148C22×D4—1811no—
149C23×C4—1811no—
150#123—3622no—
151#123—3621no—
152#123—3621no—
153#117—3622no—
154#117—3622no—
155C2×C4⋊C4—3621no—
156C2×C4⋊C4—1812no—
157C2×C22⋊C4—1811no—
158C2×C22⋊C4—1812no—
159C2×C22⋊C4—1811no—
160C2×C22⋊C4C2×D18111yescharacteristic
161C2×C22⋊C4C2×D18111yescharacteristic
162C2×C22⋊C4C2×D18111yescharacteristic
163C2×C42—3621no—
164#99—3622no—
165#99—3622no—
166#95—3622no—
167C24—1811no—
168C24C22×D18111yescharacteristic
169C2×D4—3621no—
170C2×D4—3621no—
171C2×D4—3621no—
172C2×D4—3621no—
173C2×D4—3621no—
174C2×D4—3621no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—3621no—
178C22×C4—3621no—
179C22×C4—3621no—
180C22×C4—3621no—
181C22×C4—3621no—
182C22×C4—3621no—
183C22×C4—1811no—
184C22×C4—1812no—
185C22×C4—1812no—
186C22×C4—1811no—
187C22×C4—1811no—
188C22×C4C4×D18112yes—
189C4⋊C4—3624no—
190C4⋊C4—3622no—
191C4⋊C4—3622no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—3622no—
198C22⋊C4—3622no—
199C22⋊C4—3621no—
200C22⋊C4—3621no—
201C22⋊C4—3621no—
202C22⋊C4—1812no—
203C22⋊C4D72412yes—
204C42—3624no—
205C23—42no—
206C23—3621no—
207C23—3621no—
208C23—3621no—
209C23—7242no—
210C23C2×C4×D18111yescharacteristic
211C23D4×D18111yescharacteristic
212C23D4×D18111yescharacteristic
213C23D4×D18111yescharacteristic
214C23C2×D72411yescharacteristic
215C23#1566844911yescharacteristic
216C23#1566845011yescharacteristic
217D4—7241no—
218D4—7241no—
219D4—7241no—
220D4—7241no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—3622no—
228C2×C4—3622no—
229C2×C4—3621no—
230C2×C4—3621no—
231C2×C4—3622no—
232C2×C4—3622no—
233C2×C4—3621no—
234C2×C4—3621no—
235C2×C4—3621no—
236C2×C4—3621no—
237C2×C4—3621no—
238C2×C4—3622no—
239C2×C4—1812no—
240C2×C4—1812no—
241C2×C4—7242no—
242C2×C4—7241no—
243C2×C4—7242no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—14481no—
252C22—14481no—
253C22—7241no—
254C22—7242no—
255C22C22⋊C4×D18111yescharacteristic
256C22#1915377811yescharacteristic
257C22C4×D72411yescharacteristic
258C22#1915378611yescharacteristic
259C22#1915378911yescharacteristic
260C22#1915379111yescharacteristic
261C22#1915380111yescharacteristic
262C4—42no—
263C4—42no—
264C4—81no—
265C4—3622no—
266C4—3622no—
267C4—14481no—
268C4—7242no—
269C4—7241no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—14481no—
277C2#2604115711yescharacteristic
278C2#2604125112yes—
subgroupG/H
centerC22C22⋊C4×D181
commutatorC23×C181C22×C4
FrattiniC24C22×D181
Fitting#342×C181C2
socleC22×C181C2×C22⋊C4
prime radicalC22×C22⋊D181C22
cosocleC24×C181C23
hypercenter#342D181
hypocenterC181#1899
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#19153772C2semidirect
5. #26041140C2semidirect
6. #26041046C2semidirect
7. #342×C181C2semidirect
15. C2×C4⋊Dic181C4semidirect
19. C2×#15668426C4semidirect
139. C181#1899semidirect
147. #342D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538002
C2×#191537722
C2×#191537302
#260411402
#260410462
#342×C1812
#1899181