Finite Group Databasebeta

← #37544589

Subgroups — #37544589

Order 23168.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041852C211yesmaximal, characteristic, complemented
2#26041845C211yesmaximal, characteristic
3#26041622C211yesmaximal, characteristic, complemented
4#26041619C211yesmaximal, characteristic, complemented
5#26041606C211yesmaximal, characteristic, complemented
6#26041601C211yesmaximal, characteristic, complemented
7#26041525C211yesmaximal, characteristic, complemented
8#26041510C211yesmaximal, characteristic, complemented
9#26041506C211yesmaximal, characteristic, complemented
10#26041431C211yesmaximal, characteristic, complemented
11#26041428C211yesmaximal, characteristic, complemented
12#26041419C211yesmaximal, characteristic, complemented
13#26041416C211yesmaximal, characteristic, complemented
14#26041247C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic, complemented
18C2×D4×D181C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668437C2211yescharacteristic
21C2×#15668436C2211yescharacteristic
22C22×#13089678C2211yescharacteristic, complemented
23#19153815—21no—
24#19153812—21no—
25#19153802—21no—
26#19153802—21no—
27#19153801—21no—
28#19153798—21no—
29#19153796C2211yescharacteristic
30#19153795C2211yescharacteristic, complemented
31#19153792C2211yescharacteristic
32#19153791C2211yescharacteristic
33#19153790C2211yescharacteristic
34#19153787C2211yescharacteristic
35#19153786—21no—
36#19153784—21no—
37#19153774C2211yescharacteristic
38#19153774C2211yescharacteristic, complemented
39#19153750C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153747C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153739C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153728C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic, complemented
51#19153728C2211yescharacteristic, complemented
52C2×SD16×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic, complemented
54C22⋊Q8×C181C2211yescharacteristic, complemented
55#121×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58#102×C181C2211yescharacteristic
59C23×D181—21no—
60#15668449—41no—
61#15668449—41no—
62C2×Dic362—21no—
63C2×Dic362D411yescharacteristic
64C2×Dic362C2311yescharacteristic
65Q8×D181—41no—
66Q8×D181—41no—
67C2×D724—21no—
68C2×D724D411yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181D411yescharacteristic
74C2×C22⋊D181D411yescharacteristic
75D4×D181—41no—
76D4×D181—41no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79C2×C4×D181C2311yescharacteristic
80#15668438—41no—
81#15668437—41no—
82#15668436—41no—
83C2×#13089678—41no—
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C4⋊Dic181—21no—
87C4⋊Dic181C2311yescharacteristic
88C4⋊Dic181C2311yescharacteristic
89#15668428—21no—
90#15668428—21no—
91C22⋊Dic181—21no—
92C22⋊Dic181—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99C4×Dic181—21no—
100C4×Dic181—21no—
101C2×Q8×C181C2311yescharacteristic
102C2×D4×C181—21no—
103C2×D4×C181C2311yescharacteristic
104C22×C4×C181C2311yescharacteristic
105SD16×C181—41no—
106M16×C181—41no—
107C2×C8×C181C2311yescharacteristic
108C2×C8×C181C2311yescharacteristic
109C4⋊C4×C181—21no—
110C4⋊C4×C181C2311yescharacteristic
111C4⋊C4×C181C2311yescharacteristic
112C22⋊C4×C181—21no—
113C22⋊C4×C181—21no—
114C22×D181—21no—
115C22×D181—42no—
116C22×D181—81no—
117C22×D181—81no—
118C22×D181C2×D411yescharacteristic
119C22×D181C2×D411yescharacteristic
120Dic362—21no—
121Dic362—21no—
122Dic362—41no—
123Dic362—42no—
124D724—21no—
125D724—21no—
126D724—41no—
127D724—81no—
128C22⋊D181—41no—
129C22⋊D181—41no—
130C22⋊D181—42no—
131C22⋊D181—81no—
132C22⋊D181—81no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—21no—
136C2×Dic181C2×D411yescharacteristic
137C2×Dic181C2×D411yescharacteristic
138C4×D181—21no—
139C4×D181—21no—
140C4×D181—21no—
141C4×D181—21no—
142C4×D181—42no—
143#13089678—21no—
144#13089678—21no—
145C23×C181—21no—
146C23×C181C2×D411yescharacteristic
147Q8×C181—21no—
148Q8×C181—21no—
149D4×C181—21no—
150D4×C181—21no—
151D4×C181—81no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181C2×D411yescharacteristic
158C2×C4×C181C2411yescharacteristic
159C8×C181—21no—
160C8×C181—21no—
161C22×C181—21no—
162C22×C181—21no—
163C22×C181—21no—
164C22×C181—41no—
165C22×C181—41no—
166C22×C181—41no—
167C22×C181C22×D411yescharacteristic
168C2×D181—21no—
169C2×D181—21no—
170C2×D181—21no—
171C2×D181—21no—
172C2×D181—21no—
173C2×D181—21no—
174C2×D181—41no—
175C2×D181—41no—
176C2×D181—41no—
177C2×D181—84no—
178C4×C181—21no—
179C4×C181—41no—
180C4×C181—41no—
181C4×C181—41no—
182C4×C181H(5,𝔽2)11yescharacteristic
183C4×C181H(5,𝔽2)11yescharacteristic
184Dic181—21no—
185Dic181—21no—
186Dic181—41no—
187Dic181—41no—
188Dic181—41no—
189C2×C181—41no—
190C2×C181—81no—
191C2×C181#53411yescharacteristic
192C2×C181#57711yescharacteristic
193C2×C181#57711yescharacteristic
194D181—41no—
195D181—41no—
196D181—81no—
197C181#320011yescharacteristic, complemented, Sylow
198#3200—1811nomaximal, Sylow
199#536—1811no—
200#535—1811no—
201#485—1811no—
202#485—1811no—
203#482—1811no—
204#482—1811no—
205#468—1811no—
206#468D18111yescharacteristic, complemented
207#465—1811no—
208#465—1811no—
209#450—1811no—
210#450—1811no—
211#448—1811no—
212#448—1811no—
213#408—1811no—
214C2×#41—1811no—
215C22×Q8—1811no—
216C22×D4—1811no—
217C2×SD16—1811no—
218C2×SD16—1811no—
219C2×SD16—1811no—
220C2×SD16C2×D18111yescharacteristic
221C2×M16C2×D18111yescharacteristic, complemented
222C22×C8—1811no—
223C4⋊Q8—3621no—
224#127—3621no—
225#124—3621no—
226#124—3621no—
227#123—3621no—
228#123—3621no—
229C22⋊Q8—1811no—
230C22⋊Q8—1811no—
231C22⋊Q8—1811no—
232C22⋊Q8C2×D18111yescharacteristic, complemented
233#121—1811no—
234#121—1811no—
235#121—1811no—
236#121C2×D18111yescharacteristic
237#120—3621no—
238#120—3621no—
239#117—1811no—
240#117—1811no—
241#107—1811no—
242#107C2×D18111yescharacteristic, complemented
243#106—1811no—
244#106—1811no—
245#103—1811no—
246#103—1811no—
247#103—1811no—
248#103C2×D18111yescharacteristic, complemented
249#102—1811no—
250#102—1811no—
251#102—1811no—
252#102C2×D18111yescharacteristic
253C22⋊C8—1811no—
254C22⋊C8—1811no—
255C22⋊C8—1811no—
256C22⋊C8—1811no—
257C24—3621no—
258#41—7241no—
259#41—7241no—
260C2×Q8—3621no—
261C2×Q8—1811no—
262C2×Q8—1811no—
263C2×Q8—7241no—
264C2×Q8—7241no—
265C2×Q8C22×D18111yescharacteristic
266C2×D4—21no—
267C2×D4—3621no—
268C2×D4—3621no—
269C2×D4—3621no—
270C2×D4—3621no—
271C2×D4—1811no—
272C2×D4—1811no—
273C2×D4—1811no—
274C2×D4—1811no—
275C2×D4—7241no—
276C2×D4—7241no—
277C2×D4C22×D18111yescharacteristic
278C22×C4—3621no—
279C22×C4—1811no—
280C22×C4—1811no—
281C22×C4C22×D18111yescharacteristic
282SD16—41no—
283SD16—7241no—
284SD16—7241no—
285SD16—7241no—
286M16—41no—
287C2×C8—1811no—
288C2×C8—1811no—
289C2×C8—7241no—
290C2×C8C22×D18111yescharacteristic
291C2×C8C22×D18111yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—3621no—
294C4⋊C4—3621no—
295C4⋊C4—3621no—
296C4⋊C4—1811no—
297C4⋊C4—1811no—
298C4⋊C4C22×D18111yescharacteristic
299C4⋊C4C22×D18111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3621no—
303C22⋊C4—3621no—
304C22⋊C4—3621no—
305C22⋊C4—3621no—
306C22⋊C4—3621no—
307C22⋊C4—3621no—
308C22⋊C4—3621no—
309C22⋊C4—3621no—
310C42—3621no—
311C42—3621no—
312C23—21no—
313C23—3621no—
314C23—14481no—
315C23—14481no—
316C23—1811no—
317C23—1811no—
318C23—7242no—
319C23D4×D18111yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3621no—
323Q8—3621no—
324Q8—7241no—
325Q8—7242no—
326D4—21no—
327D4—21no—
328D4—81no—
329D4—3621no—
330D4—3621no—
331D4—14481no—
332D4—14481no—
333D4—14481no—
334D4—7241no—
335D4—7241no—
336D4—7241no—
337D4—7242no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—3621no—
344C2×C4—3621no—
345C2×C4—3621no—
346C2×C4—3621no—
347C2×C4—3621no—
348C2×C4—3621no—
349C2×C4—3621no—
350C2×C4—1811no—
351C2×C4—1811no—
352C2×C4—7242no—
353C2×C4D4×D18111yescharacteristic
354C2×C4C23×D18111yescharacteristic
355C8—21no—
356C8—21no—
357C8—3621no—
358C8—3621no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3621no—
366C22—3621no—
367C22—3621no—
368C22—3621no—
369C22—3621no—
370C22—3621no—
371C22—14484no—
372C22—7241no—
373C22—7241no—
374C22—7241no—
375C22C2×D4×D18111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3621no—
381C4—3621no—
382C4—7241no—
383C4—7241no—
384C4—7241no—
385C4#1915386111yescharacteristic
386C4#1915386211yescharacteristic
387C2—41no—
388C2—81no—
389C2—14481no—
390C2—7241no—
391C2—7241no—
392C2#2604183711yescharacteristic
393C2#2604208811yescharacteristic
394C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#26041845C2
hypercenter#468D181
hypocenterC181#3200
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. #26041852C2semidirect
3. #26041622C2semidirect
4. #26041619C2semidirect
5. #26041606C2semidirect
6. #26041601C2semidirect
7. #26041525C2semidirect
8. #26041510C2semidirect
9. #26041506C2semidirect
10. #26041431C2semidirect
11. #26041428C2semidirect
12. #26041419C2semidirect
13. #26041416C2semidirect
14. #26041247C2semidirect
15. #468×C181C2semidirect
17. C2×Q8×D181C22semidirect
22. C22×#13089678C22semidirect
30. #19153795C22semidirect
38. #19153774C22semidirect
39. #19153750C22semidirect
40. #19153747C22semidirect
41. #19153747C22semidirect
42. #19153741C22semidirect
43. #19153740C22semidirect
44. #19153739C22semidirect
50. #19153728C22semidirect
51. #19153728C22semidirect
53. C2×M16×C181C22semidirect
54. C22⋊Q8×C181C22semidirect
56. #107×C181C22semidirect
57. #103×C181C22semidirect
197. C181#3200semidirect
206. #468D181semidirect
221. C2×M16C2×D181semidirect
232. C22⋊Q8C2×D181semidirect
242. #107C2×D181semidirect
248. #103C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418522
#260418452
#260416222
#260416192
#260416062
#260416012
#260415252
#260415102
#260415062
#260414312
#260414282
#260414192
#260414162
#260412472
#468×C1812
#3200181