Finite Group Databasebeta

← #37546251

Subgroups — #37546251

Order 23168.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041999C211yesmaximal, characteristic
2#26041689C211yesmaximal, characteristic, complemented
3#26041620C211yesmaximal, characteristic, complemented
4#26041617C211yesmaximal, characteristic, complemented
5#26041617C211yesmaximal, characteristic, complemented
6#26041614C211yesmaximal, characteristic, complemented
7#26041583C211yesmaximal, characteristic, complemented
8#26041574C211yesmaximal, characteristic, complemented
9#26041535C211yesmaximal, characteristic, complemented
10#26041518C211yesmaximal, characteristic, complemented
11#26041384C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic
13#26041319C211yesmaximal, characteristic, complemented
14#26041315C211yesmaximal, characteristic, complemented
15#552×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
17C2×#15668433C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153809—21no—
21#19153809—21no—
22#19153803—21no—
23#19153795C2211yescharacteristic
24#19153795C2211yescharacteristic
25C4×Dic362C2211yescharacteristic
26#19153781—21no—
27#19153775C2211yescharacteristic
28#19153775C2211yescharacteristic
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153749C2211yescharacteristic
33#19153749C2211yescharacteristic
34#19153747C2211yescharacteristic
35#19153747C2211yescharacteristic
36#19153747C2211yescharacteristic
37#19153747C2211yescharacteristic
38C4⋊#13089678C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153739C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153739C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48#126×C181—21no—
49#125×C181—21no—
50#125×C181C2211yescharacteristic
51#123×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53C22⋊Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55C4×D4×C181C2211yescharacteristic
56#117×C181—21no—
57#117×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61#15668439—22no—
62#15668433—22no—
63C2×#13089678—22no—
64C2×#13089678C2311yescharacteristic
65C2×#13089678C2311yescharacteristic
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C4⋊Dic181—21no—
69C4⋊Dic181C2311yescharacteristic
70C4⋊Dic181C2311yescharacteristic
71C4⋊Dic181C2311yescharacteristic
72#15668428—21no—
73#15668428—21no—
74#15668428—21no—
75C22⋊Dic181—21no—
76C22⋊Dic181—21no—
77C22⋊Dic181—21no—
78C22⋊Dic181—21no—
79C4×Dic181—21no—
80C4×Dic181—21no—
81C4×Dic181—21no—
82C2×Q8×C181C2311yescharacteristic
83C2×D4×C181D411yescharacteristic
84C22×C4×C181—21no—
85C22×C4×C181C2311yescharacteristic
86C22×C4×C181C2311yescharacteristic
87C4⋊C4×C181—21no—
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
91C4⋊C4×C181—21no—
92C4⋊C4×C181—21no—
93C4⋊C4×C181D411yescharacteristic
94C4⋊C4×C181C2311yescharacteristic
95C4⋊C4×C181C2311yescharacteristic
96C4⋊C4×C181C2311yescharacteristic
97C22⋊C4×C181—21no—
98C22⋊C4×C181—21no—
99C22⋊C4×C181—21no—
100C22⋊C4×C181D411yescharacteristic
101C22⋊C4×C181D411yescharacteristic
102C42×C181—21no—
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105Dic362—21no—
106Dic362—21no—
107Dic362—42no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112#13089678—21no—
113#13089678—21no—
114#13089678—21no—
115#13089678#4112yes—
116C23×C181C2×D411yescharacteristic
117C23×C181C2×D411yescharacteristic
118Q8×C181—21no—
119Q8×C181—21no—
120D4×C181—41no—
121C2×C4×C181—21no—
122C2×C4×C181—21no—
123C2×C4×C181—21no—
124C2×C4×C181—21no—
125C2×C4×C181—21no—
126C2×C4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—42no—
130C2×C4×C181C2×D411yescharacteristic
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2411yescharacteristic
135C22×C181—21no—
136C22×C181—21no—
137C22×C181—21no—
138C22×C181—21no—
139C22×C181—21no—
140C22×C181—21no—
141C22×C181C22×D411yescharacteristic
142C4×C181—21no—
143C4×C181—21no—
144C4×C181—21no—
145C4×C181—21no—
146C4×C181—41no—
147C4×C181—41no—
148C4×C181—41no—
149C4×C181—41no—
150C4×C181C2×#4111yescharacteristic
151C4×C1812−1+411yescharacteristic
152Dic181—41no—
153Dic181—41no—
154Dic181—41no—
155Dic181—41no—
156C2×C181—41no—
157C2×C181—41no—
158C2×C181#54711yescharacteristic
159C2×C181#57511yescharacteristic
160C2×C181#57811yescharacteristic
161C181#334911yescharacteristic, complemented, Sylow
162#3349—1811nomaximal, Sylow
163#556—1811no—
164#552D18111yescharacteristic
165#501—1811no—
166#485—1811no—
167#485—1811no—
168#485—1811no—
169#484—1811no—
170#479—1811no—
171#477—1811no—
172#470—1811no—
173#467—1811no—
174#441—1811no—
175#435—1811no—
176#428—1811no—
177#427—1811no—
178C2×Q16—1811no—
179C2×M16—1811no—
180C22×C8—1811no—
181C4⋊Q8—1811no—
182#126—21no—
183#126—3621no—
184#126—3621no—
185#125—21no—
186#125C2×D18111yescharacteristic
187#124—3621no—
188#123—21no—
189C22⋊Q8—1811no—
190C22⋊Q8—1811no—
191C22⋊Q8C2×D18111yescharacteristic
192C22⋊Q8C2×D18111yescharacteristic
193C4×Q8—3621no—
194C4×Q8—1811no—
195C4×Q8C2×D18111yescharacteristic
196C4×D4C2×D18111yescharacteristic
197#117—21no—
198#117—1811no—
199#117—1811no—
200#117C2×D18111yescharacteristic
201C2×C4⋊C4C2×D18111yescharacteristic
202#107—1811no—
203#107—1811no—
204#107—1811no—
205#107—1811no—
206#107—1811no—
207#106—1811no—
208#106—1811no—
209#106—1811no—
210#106—1811no—
211C4⋊C8—1811no—
212#103—1811no—
213#103—1811no—
214#103—1811no—
215#103—1811no—
216#103—1811no—
217#103—1811no—
218C22⋊C8—1811no—
219C22⋊C8—1811no—
220#97—1811no—
221C2×Q8—3621no—
222C2×Q8—1811no—
223C2×Q8C22×D18111yescharacteristic
224C2×D4C22⋊D18111yescharacteristic
225C22×C4—21no—
226C22×C4C22×D18111yescharacteristic
227C22×C4C22×D18111yescharacteristic
228Q16—3622no—
229M16—3622no—
230C2×C8—3622no—
231C2×C8—1811no—
232C2×C8—1811no—
233C2×C8—1811no—
234C2×C8—1811no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—1811no—
246C4⋊C4—1811no—
247C4⋊C4—1811no—
248C4⋊C4C22⋊D18111yescharacteristic
249C4⋊C4C22×D18111yescharacteristic
250C4⋊C4C22×D18111yescharacteristic
251C4⋊C4C22×D18111yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—3621no—
256C22⋊C4—3621no—
257C22⋊C4—3621no—
258C22⋊C4—3621no—
259C22⋊C4C22⋊D18111yescharacteristic
260C22⋊C4C22⋊D18111yescharacteristic
261C42—21no—
262C42—21no—
263C42—3621no—
264C42—3621no—
265C42—3621no—
266C42C22×D18111yescharacteristic
267C23C2×C22⋊D18111yescharacteristic
268C23C2×C22⋊D18111yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—3621no—
272Q8—3621no—
273Q8—7242no—
274D4—41no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—3621no—
285C2×C4—3621no—
286C2×C4—3621no—
287C2×C4—3621no—
288C2×C4C2×C22⋊D18111yescharacteristic
289C2×C4C2×C22⋊D18111yescharacteristic
290C2×C4C2×C22⋊D18111yescharacteristic
291C2×C4C2×C22⋊D18111yescharacteristic
292C2×C4C23×D18111yescharacteristic
293C8—3621no—
294C8—3621no—
295C8—3621no—
296C8—1812no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22C22×C22⋊D18111yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—7241no—
313C4—7241no—
314C4—7241no—
315C4—7241no—
316C4#41×D18111yescharacteristic
317C4#1915386311yescharacteristic
318C2—41no—
319C2—41no—
320C2#2604192811yescharacteristic
321C2#2604207711yescharacteristic
322C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#552×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181D4
hypercenter#552D181
hypocenterC181#3349
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #26041689C2semidirect
3. #26041620C2semidirect
4. #26041617C2semidirect
5. #26041617C2semidirect
6. #26041614C2semidirect
7. #26041583C2semidirect
8. #26041574C2semidirect
9. #26041535C2semidirect
10. #26041518C2semidirect
11. #26041384C2semidirect
13. #26041319C2semidirect
14. #26041315C2semidirect
161. C181#3349semidirect

Maximal subgroups

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

subgroupindex
#260419992
#260416892
#260416202
#260416172
#260416142
#260415832
#260415742
#260415352
#260415182
#260413842
#260413512
#260413192
#260413152
#552×C1812
#3349181