Finite Group Databasebeta

← #37546412

Subgroups — #37546412

Order 23168.

348 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042011C211yesmaximal, characteristic
2#26042005C211yesmaximal, characteristic, complemented
3#26041635C211yesmaximal, characteristic, complemented
4#26041633C211yesmaximal, characteristic, complemented
5#26041624C211yesmaximal, characteristic
6#26041580C211yesmaximal, characteristic, complemented
7#26041577C211yesmaximal, characteristic, complemented
8#26041562C211yesmaximal, characteristic, complemented
9#26041503C211yesmaximal, characteristic, complemented
10#26041477C211yesmaximal, characteristic
11#26041403C211yesmaximal, characteristic, complemented
12Q16×Dic181C211yesmaximal, characteristic, complemented
13SD16×Dic181C211yesmaximal, characteristic, complemented
14C4×#15668436C211yesmaximal, characteristic, complemented
15#464×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17#19153815—21no—
18#19153815C2211yescharacteristic
19#19153812C2211yescharacteristic
20#19153808—21no—
21#19153808C2211yescharacteristic
22#19153807—21no—
23#19153807C2211yescharacteristic
24#19153805C2211yescharacteristic
25#19153802—21no—
26#19153800—21no—
27#19153798—21no—
28Q8×Dic181C2211yescharacteristic
29Q8×Dic181C2211yescharacteristic
30Q8×Dic181C2211yescharacteristic
31C4×Dic362—21no—
32C4×Dic362C2211yescharacteristic
33C4×D724C2211yescharacteristic
34D4×Dic181C2211yescharacteristic
35#19153776—21no—
36#19153750C2211yescharacteristic
37#19153748C2211yescharacteristic
38#19153747C2211yescharacteristic
39#19153745C2211yescharacteristic
40#19153745C2211yescharacteristic
41#19153740C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic
49C4×#13089678C2211yescharacteristic
50C8×Dic181C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54#124×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362D411yescharacteristic
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C22×Dic181—21no—
65C2×C4×D181—21no—
66#15668436—22no—
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C4⋊Dic181—21no—
70C4⋊Dic181—21no—
71C4⋊Dic181—21no—
72C4⋊Dic181—21no—
73C4⋊Dic181D411yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76C4⋊Dic181C2311yescharacteristic
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82#15668428D411yescharacteristic
83C22⋊Dic181—21no—
84#15668426—21no—
85#15668426—21no—
86#15668426—21no—
87C4×Dic181—21no—
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181C2311yescharacteristic
91C4×Dic181C2311yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×Q8×C181C2311yescharacteristic
94C2×D4×C181C2311yescharacteristic
95Q16×C181—22no—
96SD16×C181—22no—
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105C22×D181—21no—
106Dic362—22no—
107Dic362—42no—
108D724—21no—
109D724—21no—
110C22⋊D181—42no—
111C22⋊D181—81no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—81no—
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C4×D181—81no—
121#13089678—21no—
122#13089678—21no—
123C23×C181—21no—
124Q8×C181—21no—
125Q8×C181—21no—
126Q8×C181—21no—
127Q8×C181#4112yes—
128D4×C181—21no—
129D4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2411yescharacteristic
136C8×C181—21no—
137C8×C181—21no—
138C22×C181—41no—
139C22×C181—41no—
140C22×C181—41no—
141C22×C181C22×D411yescharacteristic
142C2×D181—41no—
143C2×D181—41no—
144C2×D181—41no—
145C4×C181—22no—
146C4×C181—21no—
147C4×C181—21no—
148C4×C181—41no—
149C4×C181—41no—
150C4×C181C2×#4111yescharacteristic
151C4×C1812−1+411yescharacteristic
152Dic181—21no—
153Dic181—21no—
154Dic181—21no—
155Dic181—41no—
156Dic181—41no—
157Dic181—41no—
158Dic181#13512yes—
159C2×C181—81no—
160C2×C181#54811yescharacteristic
161C2×C181C2×#13511yescharacteristic
162C2×C181#57711yescharacteristic
163D181—81no—
164C181#336511yescharacteristic, complemented, Sylow
165#3365—1811nomaximal, Sylow
166#559—1811no—
167#557—1811no—
168#488—1811no—
169#488—1811no—
170#486—1811no—
171#478—1811no—
172#478—1811no—
173#475—1811no—
174#464—1811no—
175#464D18111yescharacteristic, complemented
176#460—1811no—
177C8×Q8—1811no—
178C4×Q16—1811no—
179C4×SD16—1811no—
180C4×SD16—1811no—
181C2×Q16C2×D18111yescharacteristic
182C2×SD16—1811no—
183C2×SD16C2×D18111yescharacteristic
184C4⋊Q8—3621no—
185C4⋊Q8—1811no—
186#127—1811no—
187#125—3621no—
188#125—3621no—
189#125—1811no—
190#125—1811no—
191#124—3621no—
192#124—1811no—
193#124C2×D18111yescharacteristic
194#123—3621no—
195#123—3621no—
196C4×Q8—3621no—
197C4×Q8—1811no—
198C4×Q8—1811no—
199C4×Q8—1811no—
200C4×Q8—1811no—
201C4×Q8C2×D18111yescharacteristic
202C4×D4—3621no—
203C4×D4—1811no—
204C4×D4—1811no—
205#107—1811no—
206#106—1811no—
207#106—1811no—
208C4⋊C8—1811no—
209C4⋊C8—1811no—
210C4⋊C8C2×D18111yescharacteristic
211#103—1811no—
212#103—1811no—
213#103—1811no—
214#103C2×D18111yescharacteristic
215#102—1811no—
216#102—1811no—
217#102—1811no—
218#102—1811no—
219#102—1811no—
220#102C2×D18111yescharacteristic
221C4×C8—1811no—
222C4×C8—1811no—
223C4×C8—1811no—
224C2×Q8—3621no—
225C2×Q8—1811no—
226C2×Q8C22×D18111yescharacteristic
227C2×Q8C22×D18111yescharacteristic
228C2×D4—3621no—
229C2×D4—3621no—
230C2×D4—1811no—
231C2×D4C22×D18111yescharacteristic
232C22×C4—3621no—
233C22×C4—3621no—
234Q16—22no—
235SD16—22no—
236SD16—3622no—
237C2×C8—1811no—
238C2×C8—1811no—
239C2×C8C22×D18111yescharacteristic
240C2×C8C22×D18111yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—1811no—
251C4⋊C4—1811no—
252C4⋊C4—1811no—
253C4⋊C4—1811no—
254C4⋊C4—1811no—
255C4⋊C4—1811no—
256C4⋊C4C22×D18111yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—3621no—
260C22⋊C4—3621no—
261C22⋊C4—3621no—
262C22⋊C4—3621no—
263C42—21no—
264C42—3621no—
265C42—3621no—
266C42—3621no—
267C42—1811no—
268C42—1811no—
269C42C22×D18111yescharacteristic
270C23—21no—
271C23—3621no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—3622no—
276Q8—7242no—
277Q8#1566844812yes—
278D4—21no—
279D4—21no—
280D4—3621no—
281D4—3621no—
282D4—14481no—
283D4—7242no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—3621no—
288C2×C4—3621no—
289C2×C4—3621no—
290C2×C4—14481no—
291C2×C4—14481no—
292C2×C4—1811no—
293C2×C4—1811no—
294C2×C4—1811no—
295C2×C4—1811no—
296C2×C4D4×D18111yescharacteristic
297C2×C4D4×D18111yescharacteristic
298C2×C4C23×D18111yescharacteristic
299C8—21no—
300C8—21no—
301C8—3621no—
302C8—3621no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—7241no—
307C22—7241no—
308C22—7241no—
309C22C2×D4×D18111yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—3621no—
316C4—3621no—
317C4—3621no—
318C4—1812no—
319C4—7241no—
320C4—7241no—
321C4—7241no—
322C4C2×#1566844811yescharacteristic
323C4#1915386311yescharacteristic
324C2—81no—
325C2—14481no—
326C2#2604193511yescharacteristic
327C2#135×D18111yescharacteristic
328C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radical#19153812C22
hypercenter#464D181
hypocenterC181#3365
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #26042005C2semidirect
3. #26041635C2semidirect
4. #26041633C2semidirect
6. #26041580C2semidirect
7. #26041577C2semidirect
8. #26041562C2semidirect
9. #26041503C2semidirect
11. #26041403C2semidirect
12. Q16×Dic181C2semidirect
13. SD16×Dic181C2semidirect
14. C4×#15668436C2semidirect
15. #464×C181C2semidirect
164. C181#3365semidirect
175. #464D181semidirect

Maximal subgroups

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

subgroupindex
#260420112
#260420052
#260416352
#260416332
#260416242
#260415802
#260415772
#260415622
#260415032
#260414772
#260414032
Q16×Dic1812
SD16×Dic1812
C4×#156684362
#464×C1812
#3365181