Finite Group Databasebeta

← #36365335

Subgroups — #36365335

Order 22144.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012308C211yesmaximal, characteristic, complemented
2#26012307C211yesmaximal, characteristic, complemented
3#26012301C211yesmaximal, characteristic, complemented
4#26012297C211yesmaximal, characteristic, complemented
5#26011882C211yesmaximal, characteristic, complemented
6#26011767C211yesmaximal, characteristic, complemented
7#26011765C211yesmaximal, characteristic, complemented
8#26011753C211yesmaximal, characteristic, complemented
9#26011676C211yesmaximal, characteristic, complemented
10#26011672C211yesmaximal, characteristic, complemented
11#26011671C211yesmaximal, characteristic, complemented
12#26011662C211yesmaximal, characteristic, complemented
13#26011612C211yesmaximal, characteristic, complemented
14#26011611C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16#19149727C2211yescharacteristic
17#19149726C2211yescharacteristic
18#19149726C2211yescharacteristic, complemented
19#19149724C2211yescharacteristic
20#19149712—21no—
21#19149712—21no—
22#19149712C2211yescharacteristic
23#19149712C2211yescharacteristic
24#19149712C2211yescharacteristic
25#19149712C2211yescharacteristic, complemented
26#19149711—21no—
27#19149710C2211yescharacteristic
28#19149708—21no—
29#19149708—21no—
30#19149708—21no—
31#19149706—21no—
32#19149705C2211yescharacteristic
33#19149703—21no—
34#19149701—21no—
35#19149701C2211yescharacteristic
36#19149700—21no—
37#19149700—21no—
38#19149699C2211yescharacteristic
39#19149698—21no—
40C2×Dic692—21no—
41C2×Dic692C2211yescharacteristic, complemented
42C2×#14583816C2211yescharacteristic, complemented
43Q16×D173—21no—
44C2×#14583814C2211yescharacteristic
45SD16×D173—21no—
46SD16×D173—21no—
47#19149687C2211yescharacteristic
48#19149687C2211yescharacteristic, complemented
49#19149685C2211yescharacteristic
50#19149677C2211yescharacteristic, complemented
51#19149666C2211yescharacteristic
52#19149614C2211yescharacteristic, complemented
53#19149606C2211yescharacteristic, complemented
54#19149605C2211yescharacteristic
55#19149605C2211yescharacteristic
56#19149605C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149597C2211yescharacteristic, complemented
59#19149589C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62C2×SD16×C173C2211yescharacteristic
63#131×C173C2211yescharacteristic, complemented
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic
67#14583827—21no—
68#14583827—21no—
69#14583826—21no—
70#14583826—21no—
71#14583826—41no—
72#14583826C2311yescharacteristic
73#14583826C2311yescharacteristic
74#14583826C2311yescharacteristic
75#14583825—21no—
76#14583825—21no—
77#14583825—21no—
78#14583825—21no—
79#14583825—41no—
80#14583825—41no—
81C2×Dic346—21no—
82C2×Dic346—21no—
83C2×Dic346C2311yescharacteristic
84C2×Dic346C2311yescharacteristic
85Q8×D173—21no—
86Q8×D173—21no—
87Q8×D173—21no—
88Q8×D173—21no—
89C2×C22⋊D173—41no—
90D4×D173—21no—
91D4×D173—21no—
92Dic692—21no—
93Dic692—21no—
94Dic692—21no—
95Dic692—21no—
96Dic692—21no—
97Dic692—21no—
98Dic692D411yescharacteristic
99#14583816—21no—
100#14583816—21no—
101#14583816—21no—
102#14583816D411yescharacteristic
103#14583815—21no—
104#14583815—21no—
105#14583815—21no—
106#14583815—21no—
107#14583815D411yescharacteristic
108#14583815D411yescharacteristic
109#14583814—21no—
110#14583814—21no—
111#14583814—21no—
112#14583814D411yescharacteristic
113#14583813—21no—
114#14583813D411yescharacteristic
115D1384D411yescharacteristic
116#14583811—21no—
117#14583811D411yescharacteristic
118#14583810C2311yescharacteristic
119#14583809—21no—
120#14583809—21no—
121#14583809—21no—
122C2×#13063436C2311yescharacteristic
123C8×D173—21no—
124C8×D173—21no—
125C8×D173—21no—
126#14583805—41no—
127C22⋊Dic173—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—41no—
145Dic346—21no—
146Dic346—21no—
147Dic346—21no—
148Dic346—21no—
149Dic346—41no—
150Dic346—41no—
151Dic346—41no—
152Dic346C2×D411yescharacteristic
153Dic346C2×D411yescharacteristic
154Dic346C2×D411yescharacteristic
155D692—41no—
156D692C2×D411yescharacteristic
157D692C2×D411yescharacteristic
158D692C2×D411yescharacteristic
159C22⋊D173—21no—
160C22⋊D173—21no—
161C22⋊D173—21no—
162C22⋊D173—41no—
163C22⋊D173—41no—
164C22⋊D173—41no—
165C22⋊D173—41no—
166C2×Dic173—21no—
167C2×Dic173—21no—
168C2×Dic173—21no—
169C2×Dic173—41no—
170C2×Dic173—41no—
171C2×Dic173—41no—
172C4×D173—21no—
173C4×D173—21no—
174C4×D173—21no—
175C4×D173—41no—
176C4×D173—41no—
177C4×D173—41no—
178C4×D173—41no—
179#13063436—21no—
180#13063436C2×D411yescharacteristic
181#13063436C2×D411yescharacteristic
182C23×C173—21no—
183Q8×C173—21no—
184Q8×C173—21no—
185Q8×C173C2×D411yescharacteristic
186D4×C173—21no—
187D4×C173—21no—
188D4×C173—21no—
189D4×C173C2×D411yescharacteristic
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173C2411yescharacteristic
193C8×C173—21no—
194C8×C173—21no—
195C8×C173—21no—
196C8×C173C2×D411yescharacteristic
197C8×C173C2×D411yescharacteristic
198C22×C173—21no—
199C22×C173—41no—
200C22×C173—81no—
201C22×C173H(5,𝔽2)11yescharacteristic
202C2×D173—21no—
203C2×D173—21no—
204C2×D173—21no—
205C2×D173—161no—
206C4×C173—21no—
207C4×C173—41no—
208C4×C173C22×D411yescharacteristic
209C4×C173C22×D411yescharacteristic
210Dic173—21no—
211Dic173—21no—
212Dic173—21no—
213Dic173—41no—
214Dic173—41no—
215Dic173—41no—
216C2×C173—21no—
217C2×C173—41no—
218C2×C173—81no—
219C2×C173D4211yescharacteristic
220D173—41no—
221D173—41no—
222D173—41no—
223C173#328911yescharacteristic, complemented, Sylow
224#3289—1731nomaximal, Sylow
225#578—1731no—
226#578—1731no—
227#577—1731no—
228#577—1731no—
229#497—1731no—
230#473—1731no—
231#473—1731no—
232#471—1731no—
233#471D17311yescharacteristic, complemented
234#456—1731no—
235#456—1731no—
236#455—1731no—
237#454—1731no—
238#444—1731no—
239#444—1731no—
2402−1+4—1731no—
2412−1+4—1731no—
2422−1+4—1731no—
243H(5,𝔽2)—1731no—
244#137—3461no—
245#137—3461no—
246#137—3461no—
247#137—3461no—
248#137—3461no—
249#137—3461no—
250#137—1731no—
251#137—1731no—
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)—1731no—
259AGL(1,ℤ/8)C2×D17311yescharacteristic
260#135—3461no—
261#135—3461no—
262#135—3461no—
263#135—3461no—
264#135—1731no—
265#135—1731no—
266C2×Q16—3461no—
267C2×Q16—3461no—
268C2×Q16—1731no—
269C2×Q16—1731no—
270C2×SD16—3461no—
271C2×SD16—3461no—
272C2×SD16—1731no—
273C2×SD16C2×D17311yescharacteristic
274#131—1731no—
275#131—1731no—
276#131—1731no—
277#131C2×D17311yescharacteristic, complemented
278C4⋊Q8—1731no—
279#124—1731no—
280#108—1731no—
281#108C2×D17311yescharacteristic, complemented
282#104—1731no—
283#104—1731no—
284#104—1731no—
285#104—1731no—
286#101—1731no—
287#101—1731no—
288#101C2×D17311yescharacteristic, complemented
289#100C2×D17311yescharacteristic
290#97—1731no—
291#41—3461no—
292#41—3461no—
293#41—3461no—
294#41—3461no—
295#41—3461no—
296#41—3461no—
297#41—3461no—
298#41—3461no—
299#41—1731no—
300#41—1731no—
301#41—1731no—
302#41—6921no—
303#41—6921no—
304#41—6921no—
305#41C22×D17311yescharacteristic
306C2×Q8—3461no—
307C2×Q8—3461no—
308C2×Q8—3461no—
309C2×Q8—3461no—
310C2×Q8—3461no—
311C2×Q8—3461no—
312C2×Q8—1731no—
313C2×Q8—1731no—
314C2×Q8C22×D17311yescharacteristic
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—6921no—
318C2×D4C22×D17311yescharacteristic
319Q16—21no—
320Q16—3461no—
321Q16—3461no—
322Q16—3461no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16—1731no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339SD16—3461no—
340SD16—3461no—
341SD16—3461no—
342SD16—3461no—
343SD16—1731no—
344SD16—1731no—
345SD16—1731no—
346SD16—1731no—
347D8—21no—
348D8—3461no—
349D8—1731no—
350D8—1731no—
351M16—21no—
352M16—3461no—
353M16—3461no—
354M16—3461no—
355M16—1731no—
356M16C22×D17311yescharacteristic
357M16C22×D17311yescharacteristic
358M16C22×D17311yescharacteristic
359C2×C8—21no—
360C2×C8—3461no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8—1731no—
364C2×C8C22×D17311yescharacteristic
365C4⋊C4—6921no—
366C22⋊C4—6921no—
367C42—1731no—
368C23—21no—
369C23—6921no—
370Q8—21no—
371Q8—21no—
372Q8—3461no—
373Q8—3461no—
374Q8—3461no—
375Q8—3461no—
376Q8—1731no—
377Q8—1731no—
378Q8—1731no—
379Q8—6921no—
380Q8—6921no—
381Q8—6921no—
382Q8D4×D17311yescharacteristic
383D4—21no—
384D4—21no—
385D4—21no—
386D4—3461no—
387D4—3461no—
388D4—3461no—
389D4—1731no—
390D4—1731no—
391D4—1731no—
392D4—6921no—
393D4—6921no—
394D4—6921no—
395D4—6921no—
396D4—6921no—
397D4D4×D17311yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—3461no—
401C2×C4—3461no—
402C2×C4—3461no—
403C2×C4—3461no—
404C2×C4—3461no—
405C2×C4—3461no—
406C2×C4—6921no—
407C2×C4—6921no—
408C2×C4—6921no—
409C2×C4—6921no—
410C2×C4—6921no—
411C2×C4—6921no—
412C2×C4—6921no—
413C2×C4C23×D17311yescharacteristic
414C8—21no—
415C8—21no—
416C8—21no—
417C8—3461no—
418C8—1731no—
419C8—1731no—
420C8D4×D17311yescharacteristic
421C8D4×D17311yescharacteristic
422C22—21no—
423C22—41no—
424C22—81no—
425C22—3461no—
426C22—3461no—
427C22—3461no—
428C22—27681no—
429C22#1914972511yescharacteristic
430C4—21no—
431C4—41no—
432C4—3461no—
433C4—3461no—
434C4—3461no—
435C4—6921no—
436C4—6921no—
437C4—6921no—
438C4C2×D4×D17311yescharacteristic
439C4C2×D4×D17311yescharacteristic
440C2—21no—
441C2—41no—
442C2—81no—
443C2—6921no—
444C2—6921no—
445C2—6921no—
446C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173D42
hypercenter#471D173
hypocenterC173#3289
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. #26012308C2semidirect
2. #26012307C2semidirect
3. #26012301C2semidirect
4. #26012297C2semidirect
5. #26011882C2semidirect
6. #26011767C2semidirect
7. #26011765C2semidirect
8. #26011753C2semidirect
9. #26011676C2semidirect
10. #26011672C2semidirect
11. #26011671C2semidirect
12. #26011662C2semidirect
13. #26011612C2semidirect
14. #26011611C2semidirect
15. #471×C173C2semidirect
18. #19149726C22semidirect
25. #19149712C22semidirect
41. C2×Dic692C22semidirect
42. C2×#14583816C22semidirect
48. #19149687C22semidirect
50. #19149677C22semidirect
52. #19149614C22semidirect
53. #19149606C22semidirect
56. #19149605C22semidirect
57. #19149597C22semidirect
58. #19149597C22semidirect
59. #19149589C22semidirect
60. #137×C173C22semidirect
63. #131×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
223. C173#3289semidirect
233. #471D173semidirect
255. #137C2×D173semidirect
277. #131C2×D173semidirect
281. #108C2×D173semidirect
288. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123082
#260123072
#260123012
#260122972
#260118822
#260117672
#260117652
#260117532
#260116762
#260116722
#260116712
#260116622
#260116122
#260116112
#471×C1732
#3289173