Finite Group Databasebeta

← #37544567

Subgroups — #37544567

Order 23168.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041843C211yesmaximal, characteristic, complemented
2#26041617C211yesmaximal, characteristic, complemented
3#26041600C211yesmaximal, characteristic, complemented
4#26041597C211yesmaximal, characteristic, complemented
5#26041592C211yesmaximal, characteristic, complemented
6#26041530C211yesmaximal, characteristic, complemented
7#26041527C211yesmaximal, characteristic, complemented
8#26041513C211yesmaximal, characteristic, complemented
9#26041510C211yesmaximal, characteristic, complemented
10#26041429C211yesmaximal, characteristic, complemented
11#26041424C211yesmaximal, characteristic, complemented
12#26041424C211yesmaximal, characteristic, complemented
13#26041412C211yesmaximal, characteristic, complemented
14#26041246C211yesmaximal, characteristic, complemented
15#537×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668434C2211yescharacteristic
20C2×#15668434C2211yescharacteristic
21C2×#15668433C2211yescharacteristic, complemented
22C22×#13089678C2211yescharacteristic, complemented
23#19153812—21no—
24#19153805—21no—
25#19153790C2211yescharacteristic
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153790C2211yescharacteristic
29#19153784—21no—
30#19153784—21no—
31#19153775C2211yescharacteristic
32#19153749C2211yescharacteristic, complemented
33#19153749C2211yescharacteristic, complemented
34#19153747C2211yescharacteristic, complemented
35#19153747C2211yescharacteristic, complemented
36#19153740C2211yescharacteristic, complemented
37#19153740C2211yescharacteristic, complemented
38#19153737C2211yescharacteristic, complemented
39#19153737C2211yescharacteristic, complemented
40#19153736C2211yescharacteristic
41#19153736C2211yescharacteristic
42#19153736C2211yescharacteristic
43#19153736C2211yescharacteristic
44C22⋊#13089678C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic, complemented
47C22⋊#13089678C2211yescharacteristic, complemented
48C2×#41×C181C2211yescharacteristic
49C2×#41×C181C2211yescharacteristic, complemented
50C22⋊Q8×C181—21no—
51C22⋊Q8×C181C2211yescharacteristic
52C22⋊Q8×C181C2211yescharacteristic, complemented
53#121×C181—21no—
54#121×C181—21no—
55#121×C181—21no—
56#121×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic, complemented
58C2×C4⋊C4×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C2×D724—21no—
61C2×D724—41no—
62C2×D724—41no—
63C2×D724C2311yescharacteristic
64C2×D724C2311yescharacteristic
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69#15668436—41no—
70#15668436—41no—
71#15668434—41no—
72#15668434—41no—
73#15668433—41no—
74C2×#13089678—41no—
75C2×#13089678C2311yescharacteristic
76C2×#13089678C2311yescharacteristic
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81C22⋊Dic181—21no—
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85#15668426—21no—
86C4×Dic181—21no—
87#41×C181—41no—
88#41×C181—41no—
89#41×C181—41no—
90#41×C181—41no—
91C2×Q8×C181D411yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×D4×C181—21no—
94C2×D4×C181—21no—
95C2×D4×C181—21no—
96C2×D4×C181D411yescharacteristic
97C2×D4×C181D411yescharacteristic
98C2×D4×C181D411yescharacteristic
99C2×D4×C181C2311yescharacteristic
100C22×C4×C181—21no—
101C22×C4×C181—21no—
102C22×C4×C181C2311yescharacteristic
103C22×C4×C181C2311yescharacteristic
104C22×C4×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181—21no—
107C4⋊C4×C181—21no—
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22⋊C4×C181—21no—
112C22⋊C4×C181—21no—
113C22⋊C4×C181—21no—
114C22×D181—21no—
115C22×D181—21no—
116C22×D181—42no—
117C22×D181—81no—
118C22×D181—81no—
119D724—21no—
120D724—21no—
121D724—21no—
122D724—21no—
123D724—42no—
124C22⋊D181—81no—
125C22⋊D181—81no—
126C22⋊D181—81no—
127C22⋊D181—81no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130#13089678—21no—
131#13089678—21no—
132#13089678—21no—
133#13089678—21no—
134C23×C181—21no—
135C23×C181C2×D411yescharacteristic
136C23×C181C2×D411yescharacteristic
137C23×C181C2×D411yescharacteristic
138Q8×C181—21no—
139Q8×C181—21no—
140Q8×C181—41no—
141D4×C181—21no—
142D4×C181—21no—
143D4×C181—41no—
144D4×C181—41no—
145D4×C181—41no—
146D4×C181—42no—
147D4×C181—81no—
148D4×C181—81no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181—21no—
155C2×C4×C181—21no—
156C2×C4×C181—21no—
157C2×C4×C181—21no—
158C2×C4×C181—42no—
159C2×C4×C181—81no—
160C2×C4×C181C2×D411yescharacteristic
161C2×C4×C181C2×D411yescharacteristic
162C2×C4×C181C2×D411yescharacteristic
163C2×C4×C181C2411yescharacteristic
164C22×C181—21no—
165C22×C181—21no—
166C22×C181—21no—
167C22×C181—21no—
168C22×C181—21no—
169C22×C181—21no—
170C22×C181—21no—
171C22×C181—21no—
172C22×C181—21no—
173C22×C181—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181C22×D411yescharacteristic
177C2×D181—41no—
178C2×D181—41no—
179C2×D181—41no—
180C2×D181—41no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—84no—
184C4×C181—21no—
185C4×C181—21no—
186C4×C181—21no—
187C4×C181—41no—
188C4×C181—41no—
189C4×C181—41no—
190C4×C181H(5,𝔽2)11yescharacteristic
191C4×C181H(5,𝔽2)11yescharacteristic
192Dic181—41no—
193Dic181—41no—
194C2×C181—41no—
195C2×C181—41no—
196C2×C181—41no—
197C2×C181—81no—
198C2×C181#53411yescharacteristic
199C2×C181#57611yescharacteristic
200C2×C181#57711yescharacteristic
201D181—81no—
202D181—81no—
203C181#319811yescharacteristic, complemented, Sylow
204#3198—1811nomaximal, Sylow
205#537D18111yescharacteristic, complemented
206#535—1811no—
207#485—1811no—
208#482—1811no—
209#481—1811no—
210#480—1811no—
211#469—1811no—
212#468—1811no—
213#466—1811no—
214#465—1811no—
215#450—1811no—
216#449—1811no—
217#449—1811no—
218#447—1811no—
219#408—1811no—
220C2×#41C2×D18111yescharacteristic
221C2×#41C2×D18111yescharacteristic, complemented
222C22×D4—1811no—
223C2×SD16—1811no—
224C2×SD16—1811no—
225C2×D8—1811no—
226C2×D8—1811no—
227C2×M16—1811no—
228C22×C8—1811no—
229#127—3621no—
230#124—3621no—
231C22⋊Q8—21no—
232C22⋊Q8C2×D18111yescharacteristic
233C22⋊Q8C2×D18111yescharacteristic, complemented
234#121—21no—
235#121—21no—
236#121—21no—
237#121—1811no—
238#121—1811no—
239#121—1811no—
240#121—1811no—
241#121C2×D18111yescharacteristic
242#121C2×D18111yescharacteristic, complemented
243#120—3621no—
244#120—3621no—
245#117—1811no—
246C2×C4⋊C4C2×D18111yescharacteristic, complemented
247#107—1811no—
248#107—1811no—
249#106—1811no—
250#106—1811no—
251#103—1811no—
252#103—1811no—
253#102—1811no—
254#102—1811no—
255#102—1811no—
256#102—1811no—
257#102—1811no—
258#102—1811no—
259C22⋊C8—1811no—
260C22⋊C8—1811no—
261C22⋊C8—1811no—
262C22⋊C8—1811no—
263C24—3621no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8C22⋊D18111yescharacteristic
269C2×Q8C22×D18111yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—3621no—
274C2×D4—3621no—
275C2×D4—3621no—
276C2×D4—3621no—
277C2×D4—3621no—
278C2×D4—1811no—
279C2×D4—1811no—
280C2×D4—7241no—
281C2×D4—7241no—
282C2×D4C22⋊D18111yescharacteristic
283C2×D4C22⋊D18111yescharacteristic
284C2×D4C22⋊D18111yescharacteristic
285C2×D4C22×D18111yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C22×D18111yescharacteristic
289C22×C4C22×D18111yescharacteristic
290C22×C4C22×D18111yescharacteristic
291SD16—7241no—
292SD16—7241no—
293D8—7241no—
294D8—7241no—
295M16—7241no—
296C2×C8—1811no—
297C2×C8—1811no—
298C2×C8—1811no—
299C2×C8—1811no—
300C2×C8—7241no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—1811no—
305C4⋊C4—1811no—
306C4⋊C4C22×D18111yescharacteristic
307C4⋊C4C22×D18111yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3621no—
313C22⋊C4—3621no—
314C22⋊C4—3621no—
315C22⋊C4—3621no—
316C22⋊C4—3621no—
317C42—3621no—
318C23—21no—
319C23—3621no—
320C23—3621no—
321C23—14481no—
322C23—14481no—
323C23—7242no—
324C23C2×C22⋊D18111yescharacteristic
325C23C2×C22⋊D18111yescharacteristic
326C23C2×C22⋊D18111yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—41no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—81no—
338D4—3621no—
339D4—3621no—
340D4—3621no—
341D4—3621no—
342D4—14481no—
343D4—14481no—
344D4—14481no—
345D4—14481no—
346D4—7242no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—3621no—
359C2×C4—3621no—
360C2×C4C2×C22⋊D18111yescharacteristic
361C2×C4C2×C22⋊D18111yescharacteristic
362C2×C4C2×C22⋊D18111yescharacteristic
363C2×C4C23×D18111yescharacteristic
364C8—3621no—
365C8—3621no—
366C8—3621no—
367C8—3621no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—14484no—
381C22—7241no—
382C22—7241no—
383C22—7241no—
384C22—7241no—
385C22—7241no—
386C22—7241no—
387C22C22×C22⋊D18111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—7241no—
395C4—7241no—
396C4#1915386111yescharacteristic
397C4#1915386111yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—14481no—
403C2—14481no—
404C2#2604183811yescharacteristic
405C2#2604208211yescharacteristic
406C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#537×C181C2
socleC22×C181C22×D4
hypercenter#537D181
hypocenterC181#3198
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26041843C2semidirect
2. #26041617C2semidirect
3. #26041600C2semidirect
4. #26041597C2semidirect
5. #26041592C2semidirect
6. #26041530C2semidirect
7. #26041527C2semidirect
8. #26041513C2semidirect
9. #26041510C2semidirect
10. #26041429C2semidirect
11. #26041424C2semidirect
12. #26041424C2semidirect
13. #26041412C2semidirect
14. #26041246C2semidirect
15. #537×C181C2semidirect
21. C2×#15668433C22semidirect
22. C22×#13089678C22semidirect
32. #19153749C22semidirect
33. #19153749C22semidirect
34. #19153747C22semidirect
35. #19153747C22semidirect
36. #19153740C22semidirect
37. #19153740C22semidirect
38. #19153737C22semidirect
39. #19153737C22semidirect
46. C22⋊#13089678C22semidirect
47. C22⋊#13089678C22semidirect
49. C2×#41×C181C22semidirect
52. C22⋊Q8×C181C22semidirect
57. #121×C181C22semidirect
58. C2×C4⋊C4×C181C22semidirect
203. C181#3198semidirect
205. #537D181semidirect
221. C2×#41C2×D181semidirect
233. C22⋊Q8C2×D181semidirect
242. #121C2×D181semidirect
246. C2×C4⋊C4C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260418432
#260416172
#260416002
#260415972
#260415922
#260415302
#260415272
#260415132
#260415102
#260414292
#260414242
#260414122
#260412462
#537×C1812
#3198181