Finite Group Databasebeta

← #37545148

Subgroups — #37545148

Order 23168.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041896C211yesmaximal, characteristic
2#26041889C211yesmaximal, characteristic, complemented
3#26041585C211yesmaximal, characteristic, complemented
4#26041581C211yesmaximal, characteristic, complemented
5#26041573C211yesmaximal, characteristic, complemented
6#26041562C211yesmaximal, characteristic, complemented
7#26041537C211yesmaximal, characteristic, complemented
8#26041518C211yesmaximal, characteristic, complemented
9#26041506C211yesmaximal, characteristic
10#26041500C211yesmaximal, characteristic, complemented
11#26041499C211yesmaximal, characteristic, complemented
12#26041488C211yesmaximal, characteristic, complemented
13#26041482C211yesmaximal, characteristic, complemented
14#26041307C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic
16C2×Dic724C2211yescharacteristic
17C2×#15668439C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C22×#13089678C2211yescharacteristic
20#19153815C2211yescharacteristic
21#19153814C2211yescharacteristic
22#19153809—21no—
23#19153809—21no—
24#19153809—21no—
25#19153807—21no—
26#19153807C2211yescharacteristic
27#19153806—21no—
28#19153803—21no—
29#19153803C2211yescharacteristic
30#19153797—21no—
31#19153795C2211yescharacteristic
32#19153795C2211yescharacteristic
33#19153793—21no—
34Q8×Dic181C2211yescharacteristic
35#19153781C2211yescharacteristic
36#19153781C2211yescharacteristic
37D4×Dic181C2211yescharacteristic
38#19153775C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153748C2211yescharacteristic
41#19153747C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153745C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153741C2211yescharacteristic
48#19153740C2211yescharacteristic
49#19153739C2211yescharacteristic
50#19153739C2211yescharacteristic
51#19153737C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C22⋊Q8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362C2311yescharacteristic
61C2×Dic362C2311yescharacteristic
62C22×Dic181—21no—
63Dic724—41no—
64#15668439—41no—
65#15668437—41no—
66C2×#13089678—41no—
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C4⋊Dic181—21no—
70C4⋊Dic181—21no—
71C4⋊Dic181D411yescharacteristic
72C4⋊Dic181D411yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88C22⋊Dic181D411yescharacteristic
89C22⋊Dic181D411yescharacteristic
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181—21no—
93C4×Dic181C2311yescharacteristic
94C4×Dic181C2311yescharacteristic
95C2×Q8×C181C2311yescharacteristic
96C2×D4×C181—21no—
97C2×D4×C181C2311yescharacteristic
98C22×C4×C181C2311yescharacteristic
99SD16×C181—41no—
100M16×C181—41no—
101C2×C8×C181C2311yescharacteristic
102C2×C8×C181C2311yescharacteristic
103C4⋊C4×C181—21no—
104C4⋊C4×C181C2311yescharacteristic
105C4⋊C4×C181C2311yescharacteristic
106C22⋊C4×C181—21no—
107C22⋊C4×C181—21no—
108Dic362—21no—
109Dic362—21no—
110Dic362—21no—
111Dic362—21no—
112Dic362—81no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—21no—
116C2×Dic181—21no—
117C2×Dic181—81no—
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120C2×Dic181C2×D411yescharacteristic
121C2×Dic181C2×D411yescharacteristic
122#13089678—21no—
123#13089678—21no—
124C23×C181—21no—
125C23×C181C2×D411yescharacteristic
126Q8×C181—21no—
127Q8×C181—21no—
128D4×C181—21no—
129D4×C181—21no—
130D4×C181—81no—
131C2×C4×C181—21no—
132C2×C4×C181—21no—
133C2×C4×C181—21no—
134C2×C4×C181—21no—
135C2×C4×C181—21no—
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C8×C181—21no—
139C8×C181—21no—
140C22×C181—21no—
141C22×C181—21no—
142C22×C181—21no—
143C22×C181—41no—
144C22×C181—41no—
145C22×C181—41no—
146C22×C181C22×D411yescharacteristic
147C4×C181—21no—
148C4×C181—41no—
149C4×C181—41no—
150C4×C181—41no—
151C4×C181H(5,𝔽2)11yescharacteristic
152C4×C1812−1+411yescharacteristic
153Dic181—21no—
154Dic181—21no—
155Dic181—21no—
156Dic181—21no—
157Dic181—41no—
158Dic181—41no—
159Dic181—41no—
160Dic181—41no—
161C2×C181—41no—
162C2×C181—81no—
163C2×C181#53711yescharacteristic
164C2×C181#57711yescharacteristic
165C2×C181#57811yescharacteristic
166C181#325811yescharacteristic, complemented, Sylow
167#3258—1811nomaximal, Sylow
168#542—1811no—
169#541—1811no—
170#479—1811no—
171#478—1811no—
172#477—1811no—
173#475—1811no—
174#470—1811no—
175#468D18111yescharacteristic
176#467—1811no—
177#465—1811no—
178#464—1811no—
179#464—1811no—
180#462—1811no—
181#461—1811no—
182#424—1811no—
183C2×Q16—1811no—
184C2×Q16—1811no—
185C2×SD16—1811no—
186C2×SD16C2×D18111yescharacteristic
187C2×M16C2×D18111yescharacteristic
188C22×C8—1811no—
189C4⋊Q8—1811no—
190C4⋊Q8—1811no—
191#126—3621no—
192#126—3621no—
193#126—3621no—
194#125—3621no—
195#125—3621no—
196#125—1811no—
197#124—3621no—
198#124—1811no—
199#123—3621no—
200C22⋊Q8—3621no—
201C22⋊Q8—1811no—
202C22⋊Q8—1811no—
203C22⋊Q8C2×D18111yescharacteristic
204#121C2×D18111yescharacteristic
205C4×Q8—1811no—
206C4×Q8—1811no—
207C4×Q8—1811no—
208C4×D4—1811no—
209#117—1811no—
210#107—1811no—
211#107C2×D18111yescharacteristic
212#106—1811no—
213#106—1811no—
214C4⋊C8—1811no—
215C4⋊C8—1811no—
216C4⋊C8—1811no—
217C4⋊C8—1811no—
218#103—1811no—
219#103—1811no—
220#103—1811no—
221#103—1811no—
222#103—1811no—
223#103C2×D18111yescharacteristic
224#102—1811no—
225#102C2×D18111yescharacteristic
226C2×Q8—3621no—
227C2×Q8—1811no—
228C2×Q8—1811no—
229C2×Q8C22×D18111yescharacteristic
230C2×D4—21no—
231C2×D4C22×D18111yescharacteristic
232C22×C4—3621no—
233C22×C4C22×D18111yescharacteristic
234Q16—7241no—
235Q16—7241no—
236SD16—41no—
237SD16—7241no—
238M16—41no—
239C2×C8—1811no—
240C2×C8—1811no—
241C2×C8—7241no—
242C2×C8C22×D18111yescharacteristic
243C2×C8C22×D18111yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—3621no—
251C4⋊C4—3621no—
252C4⋊C4—3621no—
253C4⋊C4—3621no—
254C4⋊C4—3621no—
255C4⋊C4—1811no—
256C4⋊C4—1811no—
257C4⋊C4—1811no—
258C4⋊C4—1811no—
259C4⋊C4C22×D18111yescharacteristic
260C4⋊C4C22×D18111yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C22⋊C4—3621no—
266C22⋊C4—3621no—
267C22⋊C4—3621no—
268C22⋊C4—1811no—
269C22⋊C4—1811no—
270C42—3621no—
271C42—3621no—
272C42—3621no—
273C42—1811no—
274C42—1811no—
275C23—21no—
276C23D4×D18111yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—3621no—
280Q8—3621no—
281Q8—3621no—
282Q8—3621no—
283Q8—14481no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—3621no—
293C2×C4—3621no—
294C2×C4—3621no—
295C2×C4—3621no—
296C2×C4—14481no—
297C2×C4—1811no—
298C2×C4—1811no—
299C2×C4—1811no—
300C2×C4—1811no—
301C2×C4D4×D18111yescharacteristic
302C2×C4C23×D18111yescharacteristic
303C8—21no—
304C8—21no—
305C8—3621no—
306C8—3621no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C2×D4×D18111yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3621no—
319C4—3621no—
320C4—3621no—
321C4—3621no—
322C4—7241no—
323C4—7241no—
324C4—7241no—
325C4—7241no—
326C4#1915386111yescharacteristic
327C4#1915386311yescharacteristic
328C2—41no—
329C2—81no—
330C2#2604185611yescharacteristic
331C2#2604208911yescharacteristic
332C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#121×C181C22
hypercenter#468D181
hypocenterC181#3258
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #26041889C2semidirect
3. #26041585C2semidirect
4. #26041581C2semidirect
5. #26041573C2semidirect
6. #26041562C2semidirect
7. #26041537C2semidirect
8. #26041518C2semidirect
10. #26041500C2semidirect
11. #26041499C2semidirect
12. #26041488C2semidirect
13. #26041482C2semidirect
14. #26041307C2semidirect
166. C181#3258semidirect

Maximal subgroups

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

subgroupindex
#260418962
#260418892
#260415852
#260415812
#260415732
#260415622
#260415372
#260415182
#260415062
#260415002
#260414992
#260414882
#260414822
#260413072
#468×C1812
#3258181