Finite Group Databasebeta

← #36365355

Subgroups — #36365355

Order 22144.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012303C211yesmaximal, characteristic, complemented
2#26012301C211yesmaximal, characteristic, complemented
3#26012300C211yesmaximal, characteristic, complemented
4#26012297C211yesmaximal, characteristic, complemented
5#26011862C211yesmaximal, characteristic, complemented
6#26011757C211yesmaximal, characteristic, complemented
7#26011755C211yesmaximal, characteristic, complemented
8#26011753C211yesmaximal, characteristic, complemented
9#26011676C211yesmaximal, characteristic, complemented
10#26011672C211yesmaximal, characteristic, complemented
11#26011661C211yesmaximal, characteristic, complemented
12#26011657C211yesmaximal, characteristic, complemented
13#26011607C211yesmaximal, characteristic, complemented
14#26011606C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16#19149727C2211yescharacteristic
17#19149726C2211yescharacteristic
18#19149725C2211yescharacteristic, complemented
19#19149724C2211yescharacteristic, complemented
20#19149712—21no—
21#19149712C2211yescharacteristic, complemented
22#19149712C2211yescharacteristic, complemented
23#19149711—21no—
24#19149710C2211yescharacteristic, complemented
25#19149708—21no—
26#19149708—21no—
27#19149707—21no—
28#19149707C2211yescharacteristic
29#19149707C2211yescharacteristic
30#19149706—21no—
31#19149706—21no—
32#19149705C2211yescharacteristic
33#19149703—21no—
34#19149702—21no—
35#19149701—21no—
36#19149701C2211yescharacteristic
37#19149700—21no—
38#19149699C2211yescharacteristic
39#19149698—21no—
40C2×#14583815—21no—
41C2×#14583815C2211yescharacteristic, complemented
42C2×#14583814C2211yescharacteristic, complemented
43C2×#14583813C2211yescharacteristic, complemented
44SD16×D173—21no—
45SD16×D173—21no—
46SD16×D173—21no—
47#19149687C2211yescharacteristic, complemented
48#19149687C2211yescharacteristic, complemented
49#19149685C2211yescharacteristic, complemented
50#19149677C2211yescharacteristic, complemented
51#19149675C2211yescharacteristic
52#19149614C2211yescharacteristic, complemented
53#19149606C2211yescharacteristic, complemented
54#19149605C2211yescharacteristic, complemented
55#19149605C2211yescharacteristic, complemented
56#19149605C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic, complemented
59C8×Dic173C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62C2×SD16×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic, complemented
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583826—21no—
71#14583826—21no—
72#14583826—41no—
73#14583826C2311yescharacteristic
74#14583826C2311yescharacteristic
75#14583826C2311yescharacteristic
76#14583825—21no—
77#14583825—21no—
78#14583825—21no—
79#14583825—41no—
80C2×Dic346—21no—
81C2×Dic346C2311yescharacteristic
82Q8×D173—21no—
83Q8×D173—21no—
84Q8×D173—21no—
85C2×D692—21no—
86C2×D692C2311yescharacteristic
87C2×C22⋊D173—41no—
88C2×C22⋊D173—41no—
89D4×D173—21no—
90D4×D173—21no—
91D4×D173—21no—
92D4×D173—41no—
93Dic692—21no—
94Dic692—21no—
95Dic692D411yescharacteristic, complemented
96#14583816—21no—
97#14583816D411yescharacteristic, complemented
98#14583815—21no—
99#14583815—21no—
100#14583815—21no—
101#14583815—21no—
102#14583815—21no—
103#14583815—21no—
104#14583815D411yescharacteristic, complemented
105#14583815D411yescharacteristic, complemented
106#14583814—21no—
107#14583814—21no—
108#14583814—21no—
109#14583814D411yescharacteristic, complemented
110#14583813—21no—
111#14583813—21no—
112#14583813—21no—
113#14583813D411yescharacteristic, complemented
114D1384—21no—
115D1384—21no—
116D1384D411yescharacteristic, complemented
117#14583811—21no—
118#14583811D411yescharacteristic, complemented
119#14583810C2311yescharacteristic
120#14583809—21no—
121#14583809—21no—
122#14583809—21no—
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126C8×D173—21no—
127#14583805—41no—
128C4×Dic173C2311yescharacteristic
129#41×C173C2311yescharacteristic
130C2×Q8×C173C2311yescharacteristic
131C2×D4×C173C2311yescharacteristic
132Q16×C173—21no—
133SD16×C173—21no—
134SD16×C173—21no—
135SD16×C173—21no—
136SD16×C173—21no—
137D8×C173—21no—
138M16×C173—21no—
139M16×C173C2311yescharacteristic
140M16×C173C2311yescharacteristic
141M16×C173C2311yescharacteristic
142C2×C8×C173—21no—
143C2×C8×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146C22×D173—41no—
147Dic346—21no—
148Dic346—21no—
149Dic346—41no—
150Dic346—41no—
151Dic346C2×D411yescharacteristic
152Dic346C2×D411yescharacteristic
153Dic346C2×D411yescharacteristic
154D692—21no—
155D692—21no—
156D692—41no—
157D692—41no—
158D692C2×D411yescharacteristic
159D692C2×D411yescharacteristic
160D692C2×D411yescharacteristic
161C22⋊D173—21no—
162C22⋊D173—21no—
163C22⋊D173—21no—
164C22⋊D173—41no—
165C22⋊D173—41no—
166C22⋊D173—41no—
167C22⋊D173—41no—
168C22⋊D173—82no—
169C2×Dic173—21no—
170C2×Dic173—21no—
171C2×Dic173—41no—
172C2×Dic173—41no—
173C4×D173—21no—
174C4×D173—21no—
175C4×D173—21no—
176C4×D173—41no—
177C4×D173—41no—
178C4×D173—41no—
179C4×D173—41no—
180#13063436—21no—
181#13063436C2×D411yescharacteristic
182#13063436C2×D411yescharacteristic
183C23×C173—21no—
184Q8×C173—21no—
185Q8×C173—21no—
186Q8×C173C2×D411yescharacteristic
187D4×C173—21no—
188D4×C173—21no—
189D4×C173—21no—
190D4×C173C2×D411yescharacteristic
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173C2411yescharacteristic
194C8×C173—21no—
195C8×C173—21no—
196C8×C173—21no—
197C8×C173C2×D411yescharacteristic
198C8×C173C2×D411yescharacteristic
199C22×C173—21no—
200C22×C173—41no—
201C22×C173—81no—
202C22×C173H(5,𝔽2)11yescharacteristic
203C2×D173—21no—
204C2×D173—21no—
205C2×D173—21no—
206C2×D173—41no—
207C2×D173—81no—
208C2×D173—161no—
209C2×D173—161no—
210C4×C173—21no—
211C4×C173—41no—
212C4×C173C22×D411yescharacteristic
213C4×C173C22×D411yescharacteristic
214Dic173—22no—
215Dic173—21no—
216Dic173—21no—
217Dic173—21no—
218Dic173—41no—
219C2×C173—21no—
220C2×C173—41no—
221C2×C173—81no—
222C2×C173D4211yescharacteristic
223D173—41no—
224D173—41no—
225D173—41no—
226D173—81no—
227C173#329111yescharacteristic, complemented, Sylow
228#3291—1731nomaximal, Sylow
229#577—1731no—
230#577—1731no—
231#577—1731no—
232#577—1731no—
233#492—1731no—
234#471—1731no—
235#471—1731no—
236#471—1731no—
237#471D17311yescharacteristic, complemented
238#456—1731no—
239#456—1731no—
240#453—1731no—
241#453—1731no—
242#443—1731no—
243#443—1731no—
2442−1+4—1731no—
2452−1+4—1731no—
246H(5,𝔽2)—1731no—
247H(5,𝔽2)—1731no—
248#137—3461no—
249#137—3461no—
250#137—3461no—
251#137—3461no—
252#137—1731no—
253#137—1731no—
254#137—1731no—
255#137C2×D17311yescharacteristic, complemented
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—3461no—
259AGL(1,ℤ/8)—3461no—
260AGL(1,ℤ/8)—1731no—
261AGL(1,ℤ/8)—1731no—
262AGL(1,ℤ/8)—1731no—
263AGL(1,ℤ/8)C2×D17311yescharacteristic
264#135—3461no—
265#135—3461no—
266#135—3461no—
267#135—3461no—
268#135—1731no—
269#135—1731no—
270C2×SD16—3461no—
271C2×SD16—3461no—
272C2×SD16—3461no—
273C2×SD16—3461no—
274C2×SD16—1731no—
275C2×SD16—1731no—
276C2×SD16—1731no—
277C2×SD16C2×D17311yescharacteristic, complemented
278#131—1731no—
279#131—1731no—
280#131—1731no—
281#131C2×D17311yescharacteristic, complemented
282C4⋊Q8—1731no—
283#127—1731no—
284#108—1731no—
285#108C2×D17311yescharacteristic, complemented
286#104—1731no—
287#104—1731no—
288#104—1731no—
289#104—1731no—
290#101—1731no—
291#101C2×D17311yescharacteristic, complemented
292#100—1731no—
293#100C2×D17311yescharacteristic, complemented
294C4×C8—1731no—
295#41—3461no—
296#41—3461no—
297#41—3461no—
298#41—3461no—
299#41—3461no—
300#41—3461no—
301#41—3461no—
302#41—3461no—
303#41—1731no—
304#41—1731no—
305#41—1731no—
306#41—6921no—
307#41—6921no—
308#41C22×D17311yescharacteristic
309C2×Q8—3461no—
310C2×Q8—3461no—
311C2×Q8—3461no—
312C2×Q8—3461no—
313C2×Q8—1731no—
314C2×Q8C22×D17311yescharacteristic
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—3461no—
318C2×D4—3461no—
319C2×D4—1731no—
320C2×D4—6921no—
321C2×D4—6921no—
322C2×D4—6921no—
323C2×D4C22×D17311yescharacteristic
324Q16—21no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—1731no—
329Q16—1731no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—3461no—
344SD16—3461no—
345SD16—3461no—
346SD16—1731no—
347SD16—1731no—
348SD16—1731no—
349SD16—1731no—
350D8—21no—
351D8—3461no—
352D8—3461no—
353D8—3461no—
354D8—1731no—
355D8—1731no—
356M16—21no—
357M16—3461no—
358M16—3461no—
359M16—3461no—
360M16—1731no—
361M16C22×D17311yescharacteristic
362M16C22×D17311yescharacteristic
363M16C22×D17311yescharacteristic
364C2×C8—21no—
365C2×C8—3461no—
366C2×C8—3461no—
367C2×C8—3461no—
368C2×C8—1731no—
369C2×C8C22×D17311yescharacteristic
370C4⋊C4—6921no—
371C42—1731no—
372C23—21no—
373C23—3461no—
374C23—6921no—
375C23—6921no—
376Q8—21no—
377Q8—21no—
378Q8—3461no—
379Q8—3461no—
380Q8—1731no—
381Q8—1731no—
382Q8—1731no—
383Q8—6921no—
384Q8—6921no—
385Q8D4×D17311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—3461no—
390D4—3461no—
391D4—3461no—
392D4—3461no—
393D4—3461no—
394D4—13842no—
395D4—1731no—
396D4—1731no—
397D4—1731no—
398D4—6921no—
399D4—6921no—
400D4—6921no—
401D4—6921no—
402D4—6921no—
403D4—6921no—
404D4D4×D17311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—3461no—
411C2×C4—3461no—
412C2×C4—6921no—
413C2×C4—6921no—
414C2×C4—6921no—
415C2×C4—6921no—
416C2×C4—6921no—
417C2×C4—6921no—
418C2×C4C23×D17311yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—3461no—
423C8—1731no—
424C8—1731no—
425C8D4×D17311yescharacteristic
426C8D4×D17311yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—3461no—
431C22—3461no—
432C22—3461no—
433C22—13841no—
434C22—6921no—
435C22—27681no—
436C22—27681no—
437C22#1914972511yescharacteristic
438C4—21no—
439C4—41no—
440C4—3462no—
441C4—3461no—
442C4—3461no—
443C4—3461no—
444C4—6921no—
445C4C2×D4×D17311yescharacteristic
446C4C2×D4×D17311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—13841no—
451C2—6921no—
452C2—6921no—
453C2—6921no—
454C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173D42
hypercenter#471D173
hypocenterC173#3291
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #26012303C2semidirect
2. #26012301C2semidirect
3. #26012300C2semidirect
4. #26012297C2semidirect
5. #26011862C2semidirect
6. #26011757C2semidirect
7. #26011755C2semidirect
8. #26011753C2semidirect
9. #26011676C2semidirect
10. #26011672C2semidirect
11. #26011661C2semidirect
12. #26011657C2semidirect
13. #26011607C2semidirect
14. #26011606C2semidirect
15. #471×C173C2semidirect
18. #19149725C22semidirect
19. #19149724C22semidirect
21. #19149712C22semidirect
22. #19149712C22semidirect
24. #19149710C22semidirect
41. C2×#14583815C22semidirect
42. C2×#14583814C22semidirect
43. C2×#14583813C22semidirect
47. #19149687C22semidirect
48. #19149687C22semidirect
49. #19149685C22semidirect
50. #19149677C22semidirect
52. #19149614C22semidirect
53. #19149606C22semidirect
54. #19149605C22semidirect
55. #19149605C22semidirect
56. #19149605C22semidirect
57. #19149597C22semidirect
58. #19149595C22semidirect
59. C8×Dic173C22semidirect
60. #137×C173C22semidirect
62. C2×SD16×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #100×C173C22semidirect
95. Dic692D4semidirect
97. #14583816D4semidirect
104. #14583815D4semidirect
105. #14583815D4semidirect
109. #14583814D4semidirect
113. #14583813D4semidirect
116. D1384D4semidirect
118. #14583811D4semidirect
227. C173#3291semidirect
237. #471D173semidirect
255. #137C2×D173semidirect
277. C2×SD16C2×D173semidirect
281. #131C2×D173semidirect
285. #108C2×D173semidirect
291. #101C2×D173semidirect
293. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123032
#260123012
#260123002
#260122972
#260118622
#260117572
#260117552
#260117532
#260116762
#260116722
#260116612
#260116572
#260116072
#260116062
#471×C1732
#3291173