Finite Group Databasebeta

← #37545103

Subgroups — #37545103

Order 23168.

386 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041831C211yesmaximal, characteristic, complemented
2#26041824C211yesmaximal, characteristic
3#26041573C211yesmaximal, characteristic, complemented
4#26041572C211yesmaximal, characteristic, complemented
5#26041558C211yesmaximal, characteristic, complemented
6#26041557C211yesmaximal, characteristic, complemented
7#26041527C211yesmaximal, characteristic, complemented
8#26041525C211yesmaximal, characteristic, complemented
9#26041523C211yesmaximal, characteristic, complemented
10#26041483C211yesmaximal, characteristic, complemented
11#26041482C211yesmaximal, characteristic, complemented
12#26041478C211yesmaximal, characteristic, complemented
13#26041477C211yesmaximal, characteristic, complemented
14#26041302C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668438C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668433C2211yescharacteristic
21#19153815C2211yescharacteristic
22#19153815C2211yescharacteristic
23#19153814C2211yescharacteristic
24#19153812C2211yescharacteristic
25#19153806—21no—
26#19153802—21no—
27#19153795C2211yescharacteristic
28#19153793—21no—
29#19153790C2211yescharacteristic
30#19153788—21no—
31Q8×Dic181C2211yescharacteristic
32#19153781—21no—
33#19153781C2211yescharacteristic
34D4×Dic181C2211yescharacteristic
35#19153778C2211yescharacteristic
36#19153776—21no—
37#19153772—21no—
38C2×C4×Dic181C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153749C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153745C2211yescharacteristic
44#19153745C2211yescharacteristic
45#19153745C2211yescharacteristic
46#19153741C2211yescharacteristic
47#19153740C2211yescharacteristic
48#19153739C2211yescharacteristic
49#19153738C2211yescharacteristic
50#19153737C2211yescharacteristic
51#19153736C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C22⋊Q8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59#15668448—44no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63C2×D724C2311yescharacteristic
64C2×C22⋊D181—21no—
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C22×Dic181—21no—
68C22×Dic181D411yescharacteristic
69C22×Dic181D411yescharacteristic
70C2×C4×D181—21no—
71#15668438—41no—
72#15668437—41no—
73#15668436—41no—
74#15668433—41no—
75C2×#13089678C2311yescharacteristic
76C2×#13089678C2311yescharacteristic
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80#15668428—21no—
81#15668428—21no—
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181—21no—
88#15668426—21no—
89#15668426—21no—
90C4×Dic181—21no—
91C4×Dic181—21no—
92C4×Dic181D411yescharacteristic
93C4×Dic181D411yescharacteristic
94C4×Dic181C2311yescharacteristic
95C4×Dic181C2311yescharacteristic
96C2×Q8×C181C2311yescharacteristic
97C2×D4×C181—21no—
98C2×D4×C181C2311yescharacteristic
99C22×C4×C181C2311yescharacteristic
100SD16×C181—41no—
101M16×C181—41no—
102C2×C8×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C4⋊C4×C181—21no—
105C4⋊C4×C181C2311yescharacteristic
106C4⋊C4×C181C2311yescharacteristic
107C22⋊C4×C181—21no—
108C22⋊C4×C181—21no—
109C22×D181—21no—
110Dic362—21no—
111Dic362—21no—
112Dic362—42no—
113Dic362—42no—
114D724—21no—
115D724—21no—
116C22⋊D181—42no—
117C22⋊D181—42no—
118C22⋊D181—42no—
119C2×Dic181—22no—
120C2×Dic181—22no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—42no—
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C4×D181—42no—
130#13089678—21no—
131#13089678—21no—
132C23×C181—21no—
133C23×C181C2×D411yescharacteristic
134Q8×C181—21no—
135Q8×C181—21no—
136D4×C181—21no—
137D4×C181—21no—
138D4×C181—42no—
139C2×C4×C181—21no—
140C2×C4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2411yescharacteristic
146C8×C181—21no—
147C8×C181—21no—
148C22×C181—21no—
149C22×C181—21no—
150C22×C181—21no—
151C22×C181—41no—
152C22×C181—41no—
153C22×C181—41no—
154C22×C181C22×D411yescharacteristic
155C2×D181—41no—
156C2×D181—41no—
157C2×D181—41no—
158C4×C181—21no—
159C4×C181—41no—
160C4×C181—41no—
161C4×C181—41no—
162C4×C181H(5,𝔽2)11yescharacteristic
163C4×C1812−1+411yescharacteristic
164Dic181—21no—
165Dic181—21no—
166Dic181—41no—
167Dic181—41no—
168Dic181—41no—
169Dic181AGL(1,ℤ/8)12yes—
170Dic181#13712yes—
171C2×C181—41no—
172C2×C181—81no—
173C2×C181#53711yescharacteristic
174C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C181C2×#13711yescharacteristic
176D181—81no—
177C181#325411yescharacteristic, complemented, Sylow
178#3254—1811nomaximal, Sylow
179#533—1811no—
180#532—1811no—
181#477—1811no—
182#477—1811no—
183#474—1811no—
184#474—1811no—
185#468—1811no—
186#468—1811no—
187#468—1811no—
188#468D18111yescharacteristic, complemented
189#461—1811no—
190#461—1811no—
191#460—1811no—
192#460—1811no—
193#423—1811no—
194C2×#41—3621no—
195C2×SD16—1811no—
196C2×SD16—1811no—
197C2×SD16—1811no—
198C2×SD16C2×D18111yescharacteristic
199C2×M16—1811no—
200C2×M16C2×D18111yescharacteristic
201C4⋊Q8—1811no—
202C4⋊Q8—1811no—
203C4⋊Q8—1811no—
204#127—1811no—
205#125—3621no—
206#124—3621no—
207C22⋊Q8—3621no—
208C22⋊Q8—1811no—
209C22⋊Q8C2×D18111yescharacteristic
210#121—3621no—
211#121—1811no—
212#121C2×D18111yescharacteristic
213C4×Q8—3621no—
214C4×Q8—1811no—
215C4×Q8—1811no—
216C4×D4—3621no—
217C4×D4—1811no—
218C4×D4—1811no—
219#117—3621no—
220C2×C42—1811no—
221#107—1811no—
222#107—1811no—
223#107—1811no—
224#107C2×D18111yescharacteristic
225C4⋊C8—1811no—
226C4⋊C8—1811no—
227C4⋊C8—1811no—
228C4⋊C8—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#103C2×D18111yescharacteristic
233#102—1811no—
234#102—1811no—
235#102—1811no—
236#102C2×D18111yescharacteristic
237#41—7244no—
238C2×Q8—3621no—
239C2×Q8—3621no—
240C2×Q8—1811no—
241C2×Q8C22×D18111yescharacteristic
242C2×D4—21no—
243C2×D4—3621no—
244C2×D4—3621no—
245C2×D4—3621no—
246C2×D4—1811no—
247C2×D4C22×D18111yescharacteristic
248C22×C4—3621no—
249C22×C4—3621no—
250C22×C4—1811no—
251C22×C4—1811no—
252C22×C4C22×D18111yescharacteristic
253SD16—41no—
254SD16—7241no—
255SD16—7241no—
256SD16—7241no—
257M16—41no—
258M16—7241no—
259C2×C8—1811no—
260C2×C8—1811no—
261C2×C8C22×D18111yescharacteristic
262C2×C8C22×D18111yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3621no—
265C4⋊C4—3621no—
266C4⋊C4—3621no—
267C4⋊C4—3621no—
268C4⋊C4—3621no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—1811no—
272C4⋊C4—1811no—
273C4⋊C4C22×D18111yescharacteristic
274C4⋊C4C22×D18111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3621no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C42—3621no—
282C42—3621no—
283C42—1811no—
284C42—1811no—
285C42—1811no—
286C42—1811no—
287C23—21no—
288C23—3621no—
289C23D4×D18111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3621no—
293Q8—3621no—
294Q8—7242no—
295Q8—7242no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—3621no—
300D4—3621no—
301D4—7242no—
302D4—7242no—
303D4—7242no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3622no—
310C2×C4—3622no—
311C2×C4—3621no—
312C2×C4—3621no—
313C2×C4—3621no—
314C2×C4—1811no—
315C2×C4—1811no—
316C2×C4—1811no—
317C2×C4—1811no—
318C2×C4—7242no—
319C2×C4—7242no—
320C2×C4D4×D18111yescharacteristic
321C2×C4C23×D18111yescharacteristic
322C8—21no—
323C8—21no—
324C8—3621no—
325C8—3621no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—7241no—
333C22—7241no—
334C22—7241no—
335C22C2×D4×D18111yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—3621no—
341C4—3621no—
342C4—1812no—
343C4—1812no—
344C4—7241no—
345C4—7241no—
346C4—7241no—
347C4#1915386111yescharacteristic
348C4#1915386311yescharacteristic
349C2—41no—
350C2—81no—
351C2—14481no—
352C2#2604185611yescharacteristic
353C2AGL(1,ℤ/8)×D18111yescharacteristic
354C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#26041824C2
hypercenter#468D181
hypocenterC181#3254
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
1. #26041831C2semidirect
3. #26041573C2semidirect
4. #26041572C2semidirect
5. #26041558C2semidirect
6. #26041557C2semidirect
7. #26041527C2semidirect
8. #26041525C2semidirect
9. #26041523C2semidirect
10. #26041483C2semidirect
11. #26041482C2semidirect
12. #26041478C2semidirect
13. #26041477C2semidirect
14. #26041302C2semidirect
15. #468×C181C2semidirect
177. C181#3254semidirect
188. #468D181semidirect

Maximal subgroups

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

subgroupindex
#260418312
#260418242
#260415732
#260415722
#260415582
#260415572
#260415272
#260415252
#260415232
#260414832
#260414822
#260414782
#260414772
#260413022
#468×C1812
#3254181