Finite Group Databasebeta

← #36364651

Subgroups — #36364651

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012060C211yesmaximal, characteristic, complemented
2#26011881C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011853C211yesmaximal, characteristic, complemented
5#26011839C211yesmaximal, characteristic, complemented
6#26011713C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011691C211yesmaximal, characteristic, complemented
9#26011686C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011641C211yesmaximal, characteristic, complemented
12#26011634C211yesmaximal, characteristic, complemented
13#26011624C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#542×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583814C2211yescharacteristic, complemented
20C2×#14583813C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×#14583813C2211yescharacteristic, complemented
23C2×#14583811C2211yescharacteristic
24C2×#14583811C2211yescharacteristic
25C2×#14583811C2211yescharacteristic, complemented
26#19149676C2211yescharacteristic
27#19149676C2211yescharacteristic
28#19149667C2211yescharacteristic
29#19149667C2211yescharacteristic
30#19149654—21no—
31#19149654—21no—
32#19149649—21no—
33#19149649—21no—
34C4⋊#13063436C2211yescharacteristic, complemented
35C4⋊#13063436C2211yescharacteristic, complemented
36#19149602C2211yescharacteristic, complemented
37#19149602C2211yescharacteristic, complemented
38#19149599C2211yescharacteristic
39#19149599C2211yescharacteristic
40#19149599C2211yescharacteristic
41#19149599C2211yescharacteristic
42#19149599C2211yescharacteristic, complemented
43#19149599C2211yescharacteristic, complemented
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic, complemented
47C4×#13063436C2211yescharacteristic, complemented
48C4⋊Q8×C173C2211yescharacteristic, complemented
49#126×C173—21no—
50#124×C173—21no—
51#124×C173C2211yescharacteristic, complemented
52#123×C173—21no—
53C22⋊Q8×C173—21no—
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56C4×Q8×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic, complemented
58#117×C173C2211yescharacteristic
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—41no—
69C2×D692—41no—
70C2×D692C2311yescharacteristic
71C2×D692C2311yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C4×D173—21no—
75#14583816—41no—
76#14583814—41no—
77#14583813—41no—
78#14583813—41no—
79#14583813—41no—
80#14583811—41no—
81#14583811—41no—
82#14583811—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91C2×Q8×C173—21no—
92C2×Q8×C173C2311yescharacteristic
93C2×D4×C173—21no—
94C2×D4×C173C2311yescharacteristic
95C22×C4×C173—21no—
96C22×C4×C173C2311yescharacteristic
97C4⋊C4×C173—21no—
98C4⋊C4×C173—21no—
99C4⋊C4×C173—21no—
100C4⋊C4×C173D411yescharacteristic
101C4⋊C4×C173D411yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173—21no—
112C42×C173C2311yescharacteristic
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—42no—
118C22×D173—81no—
119C22×D173—81no—
120Dic346—21no—
121Dic346—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—21no—
127D692—21no—
128D692—42no—
129D692—81no—
130D692—81no—
131D692—81no—
132D692—81no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138#13063436—21no—
139#13063436—21no—
140C23×C173—21no—
141C23×C173C2×D411yescharacteristic
142Q8×C173—21no—
143Q8×C173—21no—
144Q8×C173—81no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—81no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—81no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2411yescharacteristic
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173C22×D411yescharacteristic
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—84no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173—41no—
183C4×C173—41no—
184C4×C173—41no—
185C4×C173H(5,𝔽2)11yescharacteristic
186C4×C173H(5,𝔽2)11yescharacteristic
187Dic173—41no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173#53511yescharacteristic
191C2×C173#57611yescharacteristic
192C2×C173#57711yescharacteristic
193D173—81no—
194D173—81no—
195D173—81no—
196C173#322411yescharacteristic, complemented, Sylow
197#3224—1731nomaximal, Sylow
198#542D17311yescharacteristic, complemented
199#535—1731no—
200#496—1731no—
201#495—1731no—
202#489—1731no—
203#486—1731no—
204#464—1731no—
205#463—1731no—
206#460—1731no—
207#459—1731no—
208#452—1731no—
209#450—1731no—
210#449—1731no—
211#447—1731no—
212#433—1731no—
213C2×#41—1731no—
214C22×D4—1731no—
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×D8—1731no—
221C2×D8—1731no—
222C2×D8—1731no—
223C4⋊Q8C2×D17311yescharacteristic, complemented
224#127—1731no—
225#127—1731no—
226#126—21no—
227#124—21no—
228#124—1731no—
229#124—1731no—
230#124C2×D17311yescharacteristic, complemented
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8C2×D17311yescharacteristic
234#121—3461no—
235#121—3461no—
236#121C2×D17311yescharacteristic
237#120—3461no—
238#120—3461no—
239C4×Q8C2×D17311yescharacteristic, complemented
240C4×D4C2×D17311yescharacteristic, complemented
241#117C2×D17311yescharacteristic
242C4⋊C8—1731no—
243C4⋊C8—1731no—
244#103—1731no—
245#103—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C4×C8—1731no—
256C24—3461no—
257#41—6921no—
258#41—6921no—
259C2×Q8—21no—
260C2×Q8—1731no—
261C2×Q8C22×D17311yescharacteristic
262C2×D4—21no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—1731no—
270C2×D4—1731no—
271C2×D4—1731no—
272C2×D4—6921no—
273C2×D4—6921no—
274C2×D4C22×D17311yescharacteristic
275C22×C4—21no—
276C22×C4—3461no—
277C22×C4C22×D17311yescharacteristic
278Q16—6921no—
279SD16—6921no—
280SD16—6921no—
281SD16—6921no—
282SD16—6921no—
283D8—6921no—
284D8—6921no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8—1731no—
289C2×C8—1731no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22⋊D17311yescharacteristic
294C4⋊C4C22⋊D17311yescharacteristic
295C4⋊C4C22×D17311yescharacteristic
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4C22⋊D17311yescharacteristic
307C22⋊C4C22⋊D17311yescharacteristic
308C42—21no—
309C42C22×D17311yescharacteristic
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—3461no—
313C23—3461no—
314C23—3461no—
315C23—13841no—
316C23—13841no—
317C23—6922no—
318C23C2×C22⋊D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—3461no—
323Q8—3461no—
324D4—21no—
325D4—21no—
326D4—81no—
327D4—3461no—
328D4—3461no—
329D4—3461no—
330D4—3461no—
331D4—3461no—
332D4—3461no—
333D4—13841no—
334D4—13841no—
335D4—13841no—
336D4—13841no—
337D4—6922no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—3461no—
346C2×C4—6922no—
347C2×C4C2×C22⋊D17311yescharacteristic
348C2×C4C2×C22⋊D17311yescharacteristic
349C2×C4C2×C22⋊D17311yescharacteristic
350C2×C4C2×C22⋊D17311yescharacteristic
351C2×C4C2×C22⋊D17311yescharacteristic
352C2×C4C23×D17311yescharacteristic
353C8—3461no—
354C8—3461no—
355C8—3461no—
356C8—3461no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—13844no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22—6921no—
368C22—6921no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C22×C22⋊D17311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—6921no—
383C4#1914972511yescharacteristic
384C4#1914972511yescharacteristic
385C2—41no—
386C2—81no—
387C2—13841no—
388C2—13841no—
389C2—13841no—
390C2#2601205511yescharacteristic
391C2#2601229411yescharacteristic
392C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#542×C173C2
socleC22×C173C22×D4
hypercenter#542D173
hypocenterC173#3224
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #26012060C2semidirect
2. #26011881C2semidirect
3. #26011873C2semidirect
4. #26011853C2semidirect
5. #26011839C2semidirect
6. #26011713C2semidirect
7. #26011706C2semidirect
8. #26011691C2semidirect
9. #26011686C2semidirect
10. #26011651C2semidirect
11. #26011641C2semidirect
12. #26011634C2semidirect
13. #26011624C2semidirect
14. #26011552C2semidirect
15. #542×C173C2semidirect
18. C2×#14583816C22semidirect
19. C2×#14583814C22semidirect
22. C2×#14583813C22semidirect
25. C2×#14583811C22semidirect
34. C4⋊#13063436C22semidirect
35. C4⋊#13063436C22semidirect
36. #19149602C22semidirect
37. #19149602C22semidirect
42. #19149599C22semidirect
43. #19149599C22semidirect
46. #19149590C22semidirect
47. C4×#13063436C22semidirect
48. C4⋊Q8×C173C22semidirect
51. #124×C173C22semidirect
56. C4×Q8×C173C22semidirect
57. C4×D4×C173C22semidirect
196. C173#3224semidirect
198. #542D173semidirect
223. C4⋊Q8C2×D173semidirect
230. #124C2×D173semidirect
239. C4×Q8C2×D173semidirect
240. C4×D4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120602
#260118812
#260118732
#260118532
#260118392
#260117132
#260117062
#260116912
#260116862
#260116512
#260116412
#260116342
#260116242
#260115522
#542×C1732
#3224173