Finite Group Databasebeta

← #36364426

Subgroups — #36364426

Order 22144.

442 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012058C211yesmaximal, characteristic, complemented
2#26012057C211yesmaximal, characteristic, complemented
3#26011816C211yesmaximal, characteristic, complemented
4#26011814C211yesmaximal, characteristic, complemented
5#26011813C211yesmaximal, characteristic, complemented
6#26011812C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic, complemented
8#26011741C211yesmaximal, characteristic, complemented
9#26011726C211yesmaximal, characteristic, complemented
10#26011724C211yesmaximal, characteristic, complemented
11#26011625C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011623C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic, complemented
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2211yescharacteristic, complemented
21C2×C8×D173C2211yescharacteristic, complemented
22#19149675—21no—
23#19149675—21no—
24#19149666—21no—
25#19149665—21no—
26#19149654C2211yescharacteristic, complemented
27#19149653C2211yescharacteristic
28#19149650C2211yescharacteristic
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic, complemented
31#19149650C2211yescharacteristic, complemented
32#19149650C2211yescharacteristic, complemented
33#19149649—21no—
34#19149648—21no—
35#19149648—21no—
36#19149647—21no—
37#19149638C2211yescharacteristic, complemented
38#19149637C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic, complemented
53C2×D8×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62C2×Dic346D411yescharacteristic
63C2×D692—21no—
64C2×D692D411yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173D411yescharacteristic
74C2×C22⋊D173D411yescharacteristic
75D4×D173—41no—
76D4×D173—41no—
77C2×C4×D173C2311yescharacteristic
78C2×C4×D173C2311yescharacteristic
79#14583811—41no—
80#14583811—41no—
81#14583809—41no—
82C2×#13063436C2311yescharacteristic
83C2×#13063436C2311yescharacteristic
84C8×D173—41no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C4×Dic173—21no—
99C24×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173—41no—
103C2×D4×C173—41no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108D8×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173—81no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120Dic346—41no—
121D692—21no—
122D692—21no—
123D692—41no—
124D692—81no—
125C22⋊D173—41no—
126C22⋊D173—41no—
127C22⋊D173—42no—
128C22⋊D173—81no—
129C22⋊D173—81no—
130C22⋊D173—81no—
131C22⋊D173—81no—
132C22⋊D173—81no—
133C22⋊D173—81no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—21no—
143#13063436—21no—
144#13063436—21no—
145C23×C173—21no—
146C23×C173—21no—
147C23×C173—21no—
148C23×C173—42no—
149C23×C173—81no—
150C23×C173—81no—
151C23×C173C2×D411yescharacteristic
152D4×C173—21no—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—42no—
159D4×C173—81no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2411yescharacteristic
165C8×C173—21no—
166C8×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—84no—
180C22×C173C22×D411yescharacteristic
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—41no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—84no—
191C4×C173—21no—
192C4×C173—41no—
193C4×C173H(5,𝔽2)11yescharacteristic
194C4×C173H(5,𝔽2)11yescharacteristic
195Dic173—21no—
196Dic173—21no—
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—41no—
201C2×C173—81no—
202C2×C173—81no—
203C2×C173—81no—
204C2×C173#53411yescharacteristic
205C2×C173#57611yescharacteristic
206C2×C173#57611yescharacteristic
207D173—41no—
208D173—41no—
209D173—81no—
210C173#319911yescharacteristic, complemented, Sylow
211#3199—1731nomaximal, Sylow
212#535—1731no—
213#535—1731no—
214#482—1731no—
215#482—1731no—
216#482—1731no—
217#482—1731no—
218#469—1731no—
219#469—1731no—
220#466—1731no—
221#466—1731no—
222#447—1731no—
223#447—1731no—
224#447—1731no—
225#447D17311yescharacteristic, complemented
226#408—1731no—
227C2×#41—1731no—
228C22×D4—1731no—
229C22×D4C2×D17311yescharacteristic, complemented
230C2×D8—1731no—
231C2×D8—1731no—
232C2×D8C2×D17311yescharacteristic
233C2×D8C2×D17311yescharacteristic
234C2×M16—1731no—
235C22×C8—1731no—
236#127—3461no—
237#127—3461no—
238#124—3461no—
239#124—3461no—
240#121—1731no—
241#121—1731no—
242#121—1731no—
243#121—1731no—
244#121—1731no—
245#121—1731no—
246#121—1731no—
247#121C2×D17311yescharacteristic
248#120—3461no—
249#120—3461no—
250#120—3461no—
251#120—3461no—
252#117—1731no—
253#117—1731no—
254#107—1731no—
255#107—1731no—
256#106—1731no—
257#106—1731no—
258#102—1731no—
259#102—1731no—
260#102—1731no—
261#102—1731no—
262#102—1731no—
263#102—1731no—
264#102C2×D17311yescharacteristic, complemented
265#102C2×D17311yescharacteristic, complemented
266C22⋊C8—1731no—
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269C22⋊C8C2×D17311yescharacteristic, complemented
270C24—21no—
271C24—3461no—
272#41—6921no—
273#41—6921no—
274C2×Q8—1731no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—3461no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—1731no—
288C2×D4—1731no—
289C2×D4—1731no—
290C2×D4—1731no—
291C2×D4—6921no—
292C2×D4—6921no—
293C2×D4C22×D17311yescharacteristic
294C2×D4C22×D17311yescharacteristic
295C2×D4C22×D17311yescharacteristic
296C22×C4—1731no—
297C22×C4—1731no—
298C22×C4C22×D17311yescharacteristic
299D8—41no—
300D8—41no—
301D8—6921no—
302D8—6921no—
303M16—6921no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8—6921no—
307C2×C8C22×D17311yescharacteristic
308C2×C8C22×D17311yescharacteristic
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C4⋊C4C22×D17311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C42—3461no—
324C42—3461no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—3461no—
332C23—13841no—
333C23—13841no—
334C23—1731no—
335C23—1731no—
336C23—6922no—
337C23D4×D17311yescharacteristic
338Q8—6921no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—81no—
347D4—3461no—
348D4—3461no—
349D4—13841no—
350D4—13841no—
351D4—13841no—
352D4—13841no—
353D4—13841no—
354D4—13841no—
355D4—13841no—
356D4—6921no—
357D4—6921no—
358D4—6921no—
359D4—6922no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—3461no—
370C2×C4—1731no—
371C2×C4—1731no—
372C2×C4D4×D17311yescharacteristic
373C2×C4C23×D17311yescharacteristic
374C8—21no—
375C8—21no—
376C8—3461no—
377C8—3461no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—84no—
391C22—3461no—
392C22—3461no—
393C22—3461no—
394C22—3461no—
395C22—3461no—
396C22—3461no—
397C22—13844no—
398C22—6921no—
399C22—6921no—
400C22—6921no—
401C22C2×D4×D17311yescharacteristic
402C4—21no—
403C4—41no—
404C4—3461no—
405C4—3461no—
406C4—6921no—
407C4—6921no—
408C4—6921no—
409C4#1914972411yescharacteristic
410C4#1914972411yescharacteristic
411C2—41no—
412C2—81no—
413C2—81no—
414C2—81no—
415C2—13841no—
416C2—6921no—
417C2—6921no—
418C2#2601205111yescharacteristic
419C2#2601229211yescharacteristic
420C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3199
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26012058C2semidirect
2. #26012057C2semidirect
3. #26011816C2semidirect
4. #26011814C2semidirect
5. #26011813C2semidirect
6. #26011812C2semidirect
7. #26011743C2semidirect
8. #26011741C2semidirect
9. #26011726C2semidirect
10. #26011724C2semidirect
11. #26011625C2semidirect
12. #26011623C2semidirect
13. #26011623C2semidirect
14. #26011456C2semidirect
15. #447×C173C2semidirect
17. C2×D4×D173C22semidirect
20. C2×#14583809C22semidirect
21. C2×C8×D173C22semidirect
26. #19149654C22semidirect
30. #19149650C22semidirect
31. #19149650C22semidirect
32. #19149650C22semidirect
37. #19149638C22semidirect
38. #19149637C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. #19149601C22semidirect
44. #19149601C22semidirect
45. #19149600C22semidirect
46. #19149600C22semidirect
47. #19149600C22semidirect
48. #19149600C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
52. C22×D4×C173C22semidirect
56. #102×C173C22semidirect
57. #102×C173C22semidirect
58. C22⋊C8×C173C22semidirect
210. C173#3199semidirect
225. #447D173semidirect
229. C22×D4C2×D173semidirect
264. #102C2×D173semidirect
265. #102C2×D173semidirect
269. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120582
#260120572
#260118162
#260118142
#260118132
#260118122
#260117432
#260117412
#260117262
#260117242
#260116252
#260116232
#260114562
#447×C1732
#3199173