Finite Group Databasebeta

← #37546170

Subgroups — #37546170

Order 23168.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041969C211yesmaximal, characteristic
2#26041685C211yesmaximal, characteristic, complemented
3#26041617C211yesmaximal, characteristic, complemented
4#26041614C211yesmaximal, characteristic, complemented
5#26041609C211yesmaximal, characteristic, complemented
6#26041607C211yesmaximal, characteristic, complemented
7#26041586C211yesmaximal, characteristic, complemented
8#26041561C211yesmaximal, characteristic, complemented
9#26041535C211yesmaximal, characteristic, complemented
10#26041518C211yesmaximal, characteristic, complemented
11#26041384C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic
13#26041319C211yesmaximal, characteristic, complemented
14C2×#19153747C211yesmaximal, characteristic, complemented
15#554×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
17C2×#15668433C2211yescharacteristic
18C22×#13089678C2211yescharacteristic
19#19153809—21no—
20#19153808C2211yescharacteristic
21#19153806—21no—
22#19153797—21no—
23#19153795C2211yescharacteristic
24#19153795C2211yescharacteristic
25C4×Dic362C2211yescharacteristic
26#19153775C2211yescharacteristic
27#19153772—21no—
28C2×C4⋊Dic181C2211yescharacteristic
29#19153749C2211yescharacteristic
30#19153749C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153747C2211yescharacteristic
33#19153747C2211yescharacteristic
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
48C4⋊Q8×C181C2211yescharacteristic
49#125×C181—21no—
50C22⋊Q8×C181—21no—
51C22⋊Q8×C181—21no—
52C22⋊Q8×C181C2211yescharacteristic
53C22⋊Q8×C181C2211yescharacteristic
54C4×Q8×C181C2211yescharacteristic
55C4×D4×C181—21no—
56C4×D4×C181C2211yescharacteristic
57C2×C4⋊C4×C181C2211yescharacteristic
58C2×C4⋊C4×C181C2211yescharacteristic
59C2×Dic362C2311yescharacteristic
60C22×Dic181—21no—
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—
75#15668428—21no—
76#15668428—21no—
77C22⋊Dic181—21no—
78C22⋊Dic181—21no—
79C22⋊Dic181—21no—
80C4×Dic181—21no—
81C4×Dic181—21no—
82C2×Q8×C181—21no—
83C2×Q8×C181C2311yescharacteristic
84C2×D4×C181D411yescharacteristic
85C22×C4×C181—21no—
86C22×C4×C181—21no—
87C22×C4×C181C2311yescharacteristic
88C22×C4×C181C2311yescharacteristic
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
91C4⋊C4×C181—21no—
92C4⋊C4×C181—21no—
93C4⋊C4×C181—21no—
94C4⋊C4×C181—21no—
95C4⋊C4×C181D411yescharacteristic
96C4⋊C4×C181C2311yescharacteristic
97C4⋊C4×C181C2311yescharacteristic
98C4⋊C4×C181C2311yescharacteristic
99C22⋊C4×C181—21no—
100C22⋊C4×C181—21no—
101C22⋊C4×C181D411yescharacteristic
102C22⋊C4×C181D411yescharacteristic
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105Dic362—21no—
106Dic362—21no—
107C2×Dic181—21no—
108C2×Dic181—21no—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—42no—
112#13089678—21no—
113#13089678—21no—
114#13089678—21no—
115#13089678#4112yes—
116C23×C181C2×D411yescharacteristic
117C23×C181C2×D411yescharacteristic
118Q8×C181—21no—
119Q8×C181—21no—
120Q8×C181—81no—
121D4×C181—22no—
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—21no—
130C2×C4×C181—81no—
131C2×C4×C181—81no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2411yescharacteristic
137C22×C181—21no—
138C22×C181—21no—
139C22×C181—21no—
140C22×C181—21no—
141C22×C181—21no—
142C22×C181#13712yes—
143C22×C181C22×D411yescharacteristic
144C4×C181—21no—
145C4×C181—21no—
146C4×C181—21no—
147C4×C181—21no—
148C4×C181—41no—
149C4×C181—41no—
150C4×C181—41no—
151C4×C181—41no—
152C4×C181C2×#4111yescharacteristic
153C4×C1812−1+411yescharacteristic
154Dic181—41no—
155Dic181—41no—
156Dic181—41no—
157Dic181—41no—
158C2×C181—22no—
159C2×C181—41no—
160C2×C181#54711yescharacteristic
161C2×C181C2×#13711yescharacteristic
162C2×C181#57711yescharacteristic
163C181#334411yescharacteristic, complemented, Sylow
164#3344—1811nomaximal, Sylow
165#554D18111yescharacteristic
166#552—1811no—
167#501—1811no—
168#485—1811no—
169#484—1811no—
170#483—1811no—
171#483—1811no—
172#479—1811no—
173#475—1811no—
174#470—1811no—
175#467—1811no—
176#441—1811no—
177#435—1811no—
178#428—1811no—
179C2×#106—1811no—
180C2×Q16—1811no—
181C2×M16—1811no—
182C22×C8—1811no—
183C4⋊Q8C2×D18111yescharacteristic
184#126—3621no—
185#125—21no—
186#125—3621no—
187#125—1811no—
188#123—3621no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—1811no—
192C22⋊Q8—1811no—
193C22⋊Q8C2×D18111yescharacteristic
194C22⋊Q8C2×D18111yescharacteristic
195C4×Q8—1811no—
196C4×Q8C2×D18111yescharacteristic
197C4×D4—21no—
198C4×D4C2×D18111yescharacteristic
199#117—3621no—
200#117—1811no—
201C2×C4⋊C4—1811no—
202C2×C4⋊C4C2×D18111yescharacteristic
203C2×C4⋊C4C2×D18111yescharacteristic
204#107—1811no—
205#107—1811no—
206#107—1811no—
207#106—1811no—
208#106—1811no—
209#106—1811no—
210#106—1811no—
211#106—1811no—
212#106—1811no—
213C4⋊C8—1811no—
214#103—1811no—
215#103—1811no—
216#103—1811no—
217#103—1811no—
218#103—1811no—
219#103—1811no—
220C22⋊C8—1811no—
221C22⋊C8—1811no—
222#97—1811no—
223C2×Q8—21no—
224C2×Q8—1811no—
225C2×Q8C22×D18111yescharacteristic
226C2×D4C22⋊D18111yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—3621no—
230C22×C4C22×D18111yescharacteristic
231C22×C4C22×D18111yescharacteristic
232Q16—3622no—
233M16—3622no—
234C2×C8—3622no—
235C2×C8—1811no—
236C2×C8—1811no—
237C2×C8—1811no—
238C2×C8—1811no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—1811no—
252C4⋊C4—1811no—
253C4⋊C4—1811no—
254C4⋊C4C22⋊D18111yescharacteristic
255C4⋊C4C22×D18111yescharacteristic
256C4⋊C4C22×D18111yescharacteristic
257C4⋊C4C22×D18111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—3621no—
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C22⋊C4C22⋊D18111yescharacteristic
264C22⋊C4C22⋊D18111yescharacteristic
265C42—21no—
266C42—3621no—
267C42—3621no—
268C42C22×D18111yescharacteristic
269C23C2×C22⋊D18111yescharacteristic
270C23C2×C22⋊D18111yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—81no—
274Q8—3621no—
275Q8—3621no—
276D4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—3621no—
288C2×C4—3621no—
289C2×C4—3621no—
290C2×C4—3621no—
291C2×C4—7242no—
292C2×C4C2×C22⋊D18111yescharacteristic
293C2×C4C2×C22⋊D18111yescharacteristic
294C2×C4C2×C22⋊D18111yescharacteristic
295C2×C4C2×C22⋊D18111yescharacteristic
296C2×C4C23×D18111yescharacteristic
297C8—3621no—
298C8—3621no—
299C8—3621no—
300C8—1812no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22#1915384712yes—
307C22C22×C22⋊D18111yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—7241no—
317C4—7241no—
318C4—7241no—
319C4—7241no—
320C4#41×D18111yescharacteristic
321C4#1915386311yescharacteristic
322C2—22no—
323C2—41no—
324C2#2604192811yescharacteristic
325C2C2×#1915384711yescharacteristic
326C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#554×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181D4
hypercenter#554D181
hypocenterC181#3344
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #26041685C2semidirect
3. #26041617C2semidirect
4. #26041614C2semidirect
5. #26041609C2semidirect
6. #26041607C2semidirect
7. #26041586C2semidirect
8. #26041561C2semidirect
9. #26041535C2semidirect
10. #26041518C2semidirect
11. #26041384C2semidirect
13. #26041319C2semidirect
14. C2×#19153747C2semidirect
163. C181#3344semidirect

Maximal subgroups

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

subgroupindex
#260419692
#260416852
#260416172
#260416142
#260416092
#260416072
#260415862
#260415612
#260415352
#260415182
#260413842
#260413512
#260413192
C2×#191537472
#554×C1812
#3344181