Finite Group Databasebeta

← #37534849

Subgroups — #37534849

Order 23168.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#26041838C211yesmaximal, characteristic
2C2×#19153799C211yesmaximal, characteristic, complemented
3C2×#19153731C211yesmaximal, characteristic, complemented
4C2×#19153730C211yesmaximal, characteristic, complemented
5#26041142C211yesmaximal, characteristic, complemented
6#26041045C211yesmaximal, characteristic, complemented
7C2×#123×C181C211yesmaximal, characteristic, complemented
8C22×C22⋊D181C2211yescharacteristic
9#19153799—22no—
10#19153799—21no—
11#19153799—22no—
12#19153797—21no—
13#19153788—22no—
14#19153785—21no—
15#19153784—21no—
16C2×#15668428—21no—
17C2×C22⋊Dic181C2211yescharacteristic
18C2×C22⋊Dic181C2211yescharacteristic
19C2×C22⋊Dic181C2211yescharacteristic
20C2×#15668426—21no—
21#19153731—22no—
22#19153730—22no—
23#19153723—22no—
24C22×D4×C181C2211yescharacteristic
25C23×C4×C181C2211yescharacteristic
26#123×C181—22no—
27#123×C181—22no—
28C2×C4⋊C4×C181—21no—
29C2×C22⋊C4×C181—21no—
30C2×C22⋊C4×C181C2211yescharacteristic
31C23×D181D411yescharacteristic
32C2×C22⋊D181—22no—
33C2×C22⋊D181—21no—
34C2×C22⋊D181—21no—
35C2×C22⋊D181—21no—
36C2×C22⋊D181—21no—
37C2×C22⋊D181—21no—
38C2×C22⋊D181—21no—
39C22×Dic181—21no—
40C22×Dic181—21no—
41C22×Dic181D411yescharacteristic
42C22×Dic181D412yes—
43#15668428—24no—
44#15668428—22no—
45C22⋊Dic181—22no—
46C22⋊Dic181—21no—
47C22⋊Dic181—21no—
48C22⋊Dic181—21no—
49C22⋊Dic181—21no—
50C22⋊Dic181—21no—
51C22⋊Dic181D412yescomplemented
52#15668426—24no—
53#15668426—22no—
54C24×C181D411yescharacteristic
55C24×C181C2311yescharacteristic
56C2×D4×C181—21no—
57C2×D4×C181—21no—
58C2×D4×C181—21no—
59C2×D4×C181—21no—
60C2×D4×C181—21no—
61C2×D4×C181—21no—
62C22×C4×C181—21no—
63C22×C4×C181—21no—
64C22×C4×C181—21no—
65C22×C4×C181—21no—
66C22×C4×C181—21no—
67C22×C4×C181—21no—
68C22×C4×C181—21no—
69C22×C4×C181D411yescharacteristic
70C22×C4×C181D412yescomplemented
71C4⋊C4×C181—24no—
72C22⋊C4×C181—24no—
73C22⋊C4×C181—21no—
74C22⋊C4×C181D412yescomplemented
75C22×D181—21no—
76C22×D181—21no—
77C22×D181—21no—
78C22×D181—42no—
79C22⋊D181—41no—
80C22⋊D181—41no—
81C22⋊D181—41no—
82C22⋊D181—41no—
83C22⋊D181—82no—
84C2×Dic181—22no—
85C2×Dic181—22no—
86C2×Dic181—21no—
87C2×Dic181—21no—
88C2×Dic181—21no—
89C2×Dic181—42no—
90C2×Dic181—42no—
91C2×Dic181—42no—
92C2×Dic181—41no—
93C23×C181—21no—
94C23×C181—21no—
95C23×C181—21no—
96C23×C181—42no—
97C23×C181—42no—
98C23×C181C2×D411yescharacteristic
99C23×C181C2×D411yescharacteristic
100C23×C181C2×D411yescharacteristic
101C23×C181C2×D411yescharacteristic
102C23×C181C2×D411yescharacteristic
103C23×C181C2×D411yescharacteristic
104C23×C181#4111yescharacteristic
105D4×C181—41no—
106D4×C181—41no—
107D4×C181—41no—
108D4×C181—41no—
109C2×C4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181—21no—
112C2×C4×C181—22no—
113C2×C4×C181—21no—
114C2×C4×C181—22no—
115C2×C4×C181—21no—
116C2×C4×C181—42no—
117C2×C4×C181—41no—
118C2×C4×C181—41no—
119C2×C4×C181—41no—
120C2×C4×C181—41no—
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—41no—
125C22×C181—41no—
126C22×C181—41no—
127C22×C181—41no—
128C22×C181—42no—
129C22×C181—41no—
130C22×C181—42no—
131C22×C181—81no—
132C22×C181#12011yescharacteristic
133C22×C181#12011yescharacteristic
134C22×C181#12011yescharacteristic
135C22×C181#12111yescharacteristic
136C22×C181#12111yescharacteristic
137C22×C181#12111yescharacteristic
138C22×C181#12711yescharacteristic
139C2×D181—42no—
140C2×D181—41no—
141C2×D181—42no—
142C2×D181—81no—
143C4×C181—42no—
144C4×C181—42no—
145C4×C181—41no—
146Dic181—42no—
147Dic181—41no—
148Dic181—42no—
149Dic181—81no—
150C2×C181—21no—
151C2×C181—21no—
152C2×C181—21no—
153C2×C181—21no—
154C2×C181—21no—
155C2×C181—21no—
156C2×C181—81no—
157C2×C181#39211yescharacteristic
158C2×C181#45812yes—
159D181—81no—
160C181#202211yescharacteristic, complemented, Sylow
161#2022—1811nomaximal, Sylow
162#534—1811no—
163C2×#123—1811no—
164C2×#123D18111yescharacteristic, complemented
165C2×#99—1811no—
166C2×#99—1811no—
167#381—1811no—
168#342—1811no—
169C22×D4—1811no—
170C22×D4C2×D18111yescharacteristic
171C23×C4C2×D18111yescharacteristic
172#123—22no—
173#123—22no—
174#123—3621no—
175#123—3621no—
176#123—3622no—
177#123—3622no—
178#121—3622no—
179#120—3621no—
180#120—3621no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—3621no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—3621no—
185C2×C22⋊C4—1811no—
186C2×C22⋊C4—1811no—
187C2×C22⋊C4—1811no—
188C2×C22⋊C4C2×D18111yescharacteristic
189#99—3622no—
190#99—3622no—
191#95—3622no—
192C24—1811no—
193C24C22⋊D18111yescharacteristic
194C24C22×D18111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3622no—
202C2×D4—3621no—
203C2×D4—3621no—
204C2×D4—3621no—
205C2×D4—3621no—
206C2×D4—3621no—
207C2×D4—3621no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3621no—
216C22×C4—3621no—
217C22×C4—1811no—
218C22×C4—1812no—
219C22×C4C22⋊D18111yescharacteristic
220C22×C4C22⋊D18112yescomplemented
221C4⋊C4—24no—
222C4⋊C4—3624no—
223C4⋊C4—3622no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—3624no—
227C22⋊C4—3622no—
228C22⋊C4—3622no—
229C22⋊C4—3621no—
230C22⋊C4—3621no—
231C22⋊C4—3621no—
232C22⋊C4—3621no—
233C22⋊C4—3621no—
234C22⋊C4—1812no—
235C22⋊C4C22⋊D18112yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—3621no—
242C23—3621no—
243C23—3621no—
244C23—7242no—
245C23D4×D18111yescharacteristic
246C23D4×D18111yescharacteristic
247C23D4×D18111yescharacteristic
248C23C2×C22⋊D18111yescharacteristic
249C23C2×C22⋊D18111yescharacteristic
250C23C2×C22⋊D18111yescharacteristic
251C23#1566844811yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—14482no—
257D4—7241no—
258D4—7241no—
259D4—7241no—
260D4—7241no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—3622no—
274C2×C4—3621no—
275C2×C4—3622no—
276C2×C4—3621no—
277C2×C4—3621no—
278C2×C4—7242no—
279C2×C4—7241no—
280C2×C4—7242no—
281C2×C4—7242no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—14481no—
294C22—7242no—
295C22—7241no—
296C22—7242no—
297C22#1915378411yescharacteristic
298C22#1915378411yescharacteristic
299C22#1915378511yescharacteristic
300C22#1915378711yescharacteristic
301C22#1915378811yescharacteristic
302C22#1915378811yescharacteristic
303C22#1915381211yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—14481no—
308C4—7242no—
309C4—7241no—
310C4—7242no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—14481no—
319C2#2604118711yescharacteristic
320C2#2604146812yes—
subgroupG/H
centerC22#19153784
commutator, cosocleC24×C181C23
FrattiniC24C22×D181
FittingC2×#123×C181C2
socleC22×C181#120
prime radical#26041838C2
hypercenterC2×#123D181
hypocenterC181#2022
LCSG ▷ C24×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#19153799C2semidirect
3. C2×#19153731C2semidirect
4. C2×#19153730C2semidirect
5. #26041142C2semidirect
6. #26041045C2semidirect
7. C2×#123×C181C2semidirect
51. C22⋊Dic181D4semidirect
70. C22×C4×C181D4semidirect
74. C22⋊C4×C181D4semidirect
160. C181#2022semidirect
164. C2×#123D181semidirect
220. C22×C4C22⋊D181semidirect
235. C22⋊C4C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
#260418382
C2×#191537992
C2×#191537312
C2×#191537302
#260411422
#260410452
C2×#123×C1812
#2022181