Finite Group Databasebeta

← #37534003

Subgroups — #37534003

Order 23168.

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

normal subgroups only

This group has 374 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
1#26041752C211yesmaximal, characteristic
2C2×#19153730C211yesmaximal, characteristic
3#26041156C211yesmaximal, characteristic
4#26041154C211yesmaximal, characteristic, complemented
5#26041151C211yesmaximal, characteristic, complemented
6#26041045C211yesmaximal, characteristic, complemented
7#342×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2211yescharacteristic
9C23×Dic181C2211yescharacteristic
10C4×C22⋊D181—21no—
11C4×C22⋊D181—21no—
12#19153776—21no—
13#19153776—21no—
14#19153772—21no—
15C2×C4⋊Dic181—21no—
16C2×#15668428—21no—
17C2×C22⋊Dic181C2211yescharacteristic
18C2×C22⋊Dic181C2211yescharacteristic
19C2×#15668426—21no—
20C22⋊C4×D181—21no—
21#19153730C414yescomplemented
22#19153723—21no—
23#19153723—21no—
24#19153723—21no—
25C2×C22⋊C4×C181C2211yescharacteristic
26C2×C22⋊C4×C181C2211yescharacteristic
27C2×C22⋊C4×C181C2211yescharacteristic
28C23×D181D411yescharacteristic
29C2×C22⋊D181—21no—
30C2×C22⋊D181—21no—
31C2×C22⋊D181—21no—
32C2×C22⋊D181—21no—
33C2×C22⋊D181C2×C412yes—
34C2×C22⋊D181Q812yes—
35C22×Dic181—21no—
36C22×Dic181—21no—
37C22×Dic181—21no—
38C22×Dic181—21no—
39C22×Dic181—42no—
40C22×Dic181D411yescharacteristic
41C22×Dic181D411yescharacteristic
42C22×Dic181D411yescharacteristic
43C2×C4×D181—21no—
44C2×C4×D181—21no—
45C4⋊Dic181—42no—
46#15668428—21no—
47#15668428—21no—
48#15668428—42no—
49C22⋊Dic181—21no—
50C22⋊Dic181—21no—
51C22⋊Dic181—21no—
52C22⋊Dic181C2×C412yescomplemented
53#15668426—21no—
54#15668426—21no—
55#15668426—42no—
56C4×Dic181—21no—
57C4×Dic181—21no—
58C24×C181C2311yescharacteristic
59C22×C4×C181—21no—
60C22×C4×C181—21no—
61C22×C4×C181D411yescharacteristic
62C22×C4×C181D411yescharacteristic
63C22⋊C4×C181—21no—
64C22⋊C4×C181—21no—
65C22⋊C4×C181—21no—
66C22⋊C4×C181—21no—
67C22⋊C4×C181—21no—
68C22⋊C4×C181C2×C412yescomplemented
69C22×D181—21no—
70C22×D181—21no—
71C22×D181—41no—
72C22×D181—41no—
73C22×D181C4⋊C412yescomplemented
74C22⋊D181—22no—
75C22⋊D181—22no—
76C22⋊D181—22no—
77C22⋊D181—22no—
78C2×Dic181—21no—
79C2×Dic181—21no—
80C2×Dic181—22no—
81C2×Dic181—22no—
82C2×Dic181—22no—
83C2×Dic181—22no—
84C2×Dic181—21no—
85C2×Dic181—21no—
86C2×Dic181—41no—
87C2×Dic181—41no—
88C2×Dic181—41no—
89C2×Dic181—41no—
90C2×Dic181—41no—
91C2×Dic181—41no—
92C2×Dic181—41no—
93C2×Dic181—41no—
94C2×Dic181C4⋊C412yescomplemented
95C4×D181—81no—
96C4×D181—81no—
97C23×C181—41no—
98C23×C181—41no—
99C23×C181C22×C411yescharacteristic
100C23×C181C2×D411yescharacteristic
101C23×C181C2×D411yescharacteristic
102C23×C181C2×D411yescharacteristic
103C23×C181C2×Q811yescharacteristic
104C23×C181#4111yescharacteristic
105C23×C181#4111yescharacteristic
106C2×C4×C181—22no—
107C2×C4×C181—21no—
108C2×C4×C181—21no—
109C2×C4×C181—41no—
110C2×C4×C181—41no—
111C2×C4×C181—41no—
112C2×C4×C181—41no—
113C2×C4×C181—41no—
114C2×C4×C181—41no—
115C2×C4×C181—41no—
116C22×C181—21no—
117C22×C181—21no—
118C22×C181—21no—
119C22×C181—21no—
120C22×C181—21no—
121C22×C181—21no—
122C22×C181—41no—
123C22×C181—41no—
124C22×C181—41no—
125C22×C181—41no—
126C22×C181C2×C4⋊C411yescharacteristic
127C22×C181C4×D411yescharacteristic
128C22×C181C4×D411yescharacteristic
129C22×C181#12011yescharacteristic
130C22×C181C22⋊Q811yescharacteristic
131C22×C181C22⋊Q811yescharacteristic
132C22×C181#12311yescharacteristic
133C2×D181—22no—
134C2×D181—41no—
135C2×D181—41no—
136C2×D181—42no—
137C2×D181—81no—
138C4×C181—42no—
139C4×C181—41no—
140C4×C181—41no—
141C4×C181—81no—
142Dic181—22no—
143Dic181—42no—
144Dic181—41no—
145Dic181—42no—
146Dic181—41no—
147C2×C181—21no—
148C2×C181—21no—
149C2×C181—21no—
150C2×C181—21no—
151C2×C181—21no—
152C2×C181—21no—
153C2×C181#38511yescharacteristic
154C2×C181B(4,𝔽2)11yescharacteristic
155C2×C181#45811yescharacteristic
156D181—42no—
157C181#186711yescharacteristic, complemented, Sylow
158#1867—1811nomaximal, Sylow
159#518—1811no—
160C2×#99—1811no—
161#386—1811no—
162#385—1811no—
163#385—1811no—
164#342—1811no—
165#342D18111yescharacteristic, complemented
166C22×D4—1811no—
167C23×C4—1811no—
168C4×D4—3621no—
169C4×D4—3621no—
170C4×D4—3621no—
171C4×D4—3621no—
172#117—3621no—
173C2×C4⋊C4—3621no—
174C2×C4⋊C4—3621no—
175C2×C22⋊C4—3621no—
176C2×C22⋊C4—3621no—
177C2×C22⋊C4—1811no—
178C2×C22⋊C4—1811no—
179C2×C22⋊C4C2×D18111yescharacteristic
180C2×C22⋊C4C2×D18111yescharacteristic
181C2×C22⋊C4C2×D18111yescharacteristic
182#99—1814no—
183#95—3621no—
184#95—3621no—
185#95—3621no—
186C24—1811no—
187C24C22×D18111yescharacteristic
188C2×D4—3621no—
189C2×D4—3621no—
190C2×D4—3621no—
191C2×D4—3621no—
192C2×D4—1812no—
193C2×D4—1812no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—3621no—
197C22×C4—3621no—
198C22×C4—3621no—
199C22×C4—3621no—
200C22×C4—3621no—
201C22×C4—3621no—
202C22×C4—1811no—
203C22×C4—1811no—
204C22×C4—1811no—
205C22×C4—7242no—
206C22×C4D72411yescharacteristic
207C22×C4D72411yescharacteristic
208C4⋊C4—3621no—
209C4⋊C4—3621no—
210C4⋊C4—7242no—
211C4⋊C4—7242no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—3621no—
218C22⋊C4—3621no—
219C22⋊C4—3621no—
220C22⋊C4—3621no—
221C22⋊C4—3621no—
222C22⋊C4—1812no—
223C22⋊C4—7242no—
224C22⋊C4C4×D18112yescomplemented
225C42—3621no—
226C42—3621no—
227C23—41no—
228C23—41no—
229C23—3621no—
230C23—3621no—
231C23—1812no—
232C23—7241no—
233C23—7241no—
234C23C2×C4×D18111yescharacteristic
235C23D4×D18111yescharacteristic
236C23D4×D18111yescharacteristic
237C23C2×D72411yescharacteristic
238C23Q8×D18111yescharacteristic
239C23#1566844811yescharacteristic
240C23#1566844911yescharacteristic
241D4—3622no—
242D4—3622no—
243D4—3622no—
244D4—3622no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—3621no—
256C2×C4—3621no—
257C2×C4—3622no—
258C2×C4—3622no—
259C2×C4—3622no—
260C2×C4—3622no—
261C2×C4—3621no—
262C2×C4—3621no—
263C2×C4—14481no—
264C2×C4—14481no—
265C2×C4—1812no—
266C2×C4—7241no—
267C2×C4—7241no—
268C2×C4—7241no—
269C2×C4—7241no—
270C2×C4—7241no—
271C2×C4—7241no—
272C2×C4—7241no—
273C2×C4—7241no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—3622no—
285C22—14481no—
286C22—7242no—
287C22—7241no—
288C22—7241no—
289C22C4⋊C4×D18111yescharacteristic
290C22#1915377611yescharacteristic
291C22C4×D72411yescharacteristic
292C22#1915378611yescharacteristic
293C22#1915379411yescharacteristic
294C22#1915379611yescharacteristic
295C22#1915379811yescharacteristic
296C4—42no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—3622no—
301C4—7242no—
302C4—7242no—
303C4—7241no—
304C4—7241no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—7242no—
312C2#2604115311yescharacteristic
313C2#2604146711yescharacteristic
314C2#2604147011yescharacteristic
subgroupG/H
centerC22#19153786
commutatorC23×C181C22×C4
FrattiniC24C22×D181
Fitting#342×C181C2
socleC22×C181#120
prime radicalC22×C22⋊D181C22
cosocleC24×C181C23
hypercenter#342D181
hypocenterC181#1867
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #26041154C2semidirect
5. #26041151C2semidirect
6. #26041045C2semidirect
7. #342×C181C2semidirect
21. #19153730C4semidirect
52. C22⋊Dic181C2×C4semidirect
68. C22⋊C4×C181C2×C4semidirect
73. C22×D181C4⋊C4semidirect
94. C2×Dic181C4⋊C4semidirect
157. C181#1867semidirect
165. #342D181semidirect
224. C22⋊C4C4×D181semidirect

Maximal subgroups

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

subgroupindex
#260417522
C2×#191537302
#260411562
#260411542
#260411512
#260410452
#342×C1812
#1867181