Finite Group Databasebeta

← #36365337

Subgroups — #36365337

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#26012307C211yesmaximal, characteristic, complemented
2#26012304C211yesmaximal, characteristic, complemented
3#26012303C211yesmaximal, characteristic, complemented
4#26012300C211yesmaximal, characteristic, complemented
5#26011884C211yesmaximal, characteristic, complemented
6#26011765C211yesmaximal, characteristic, complemented
7#26011757C211yesmaximal, characteristic, complemented
8#26011755C211yesmaximal, characteristic, complemented
9#26011676C211yesmaximal, characteristic, complemented
10#26011672C211yesmaximal, characteristic, complemented
11#26011667C211yesmaximal, characteristic, complemented
12#26011666C211yesmaximal, characteristic, complemented
13#26011612C211yesmaximal, characteristic, complemented
14#26011611C211yesmaximal, characteristic, complemented
15#473×C173C211yesmaximal, characteristic, complemented
16#19149727C2211yescharacteristic
17#19149727C2211yescharacteristic, complemented
18#19149726C2211yescharacteristic
19#19149725C2211yescharacteristic
20#19149712—21no—
21#19149712C2211yescharacteristic
22#19149712C2211yescharacteristic, complemented
23#19149711—21no—
24#19149711—21no—
25#19149711—21no—
26#19149711—21no—
27#19149710C2211yescharacteristic
28#19149710C2211yescharacteristic, complemented
29#19149708—21no—
30#19149707—21no—
31#19149707C2211yescharacteristic
32#19149707C2211yescharacteristic
33#19149706—21no—
34#19149702—21no—
35#19149702—21no—
36#19149701—21no—
37#19149701C2211yescharacteristic
38#19149700—21no—
39#19149699C2211yescharacteristic
40C2×#14583816C2211yescharacteristic, complemented
41Q16×D173—21no—
42Q16×D173—21no—
43C2×#14583815—21no—
44C2×#14583815C2211yescharacteristic
45C2×#14583813C2211yescharacteristic
46SD16×D173—21no—
47#19149687C2211yescharacteristic
48#19149687C2211yescharacteristic, complemented
49#19149685C2211yescharacteristic, complemented
50#19149677C2211yescharacteristic, complemented
51#19149668C2211yescharacteristic
52#19149614C2211yescharacteristic, complemented
53#19149606C2211yescharacteristic
54#19149605C2211yescharacteristic
55#19149605C2211yescharacteristic, complemented
56#19149605C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149595C2211yescharacteristic
59#19149589C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic
61#137×C173C2211yescharacteristic
62C2×Q16×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#101×C173C2211yescharacteristic, complemented
67#14583827—21no—
68#14583827—21no—
69#14583827—21no—
70#14583827—21no—
71#14583827—21no—
72#14583826—21no—
73#14583826—21no—
74#14583826—41no—
75#14583826—41no—
76#14583826C2311yescharacteristic
77#14583826C2311yescharacteristic
78#14583826C2311yescharacteristic
79#14583825—21no—
80#14583825—41no—
81C2×Dic346—21no—
82C2×Dic346C2311yescharacteristic
83Q8×D173—21no—
84Q8×D173—21no—
85Q8×D173—21no—
86Q8×D173—21no—
87Q8×D173—21no—
88C2×D692—21no—
89C2×D692C2311yescharacteristic
90D4×D173—21no—
91D4×D173—41no—
92Dic692—21no—
93Dic692—21no—
94Dic692D411yescharacteristic
95#14583816—21no—
96#14583816—21no—
97#14583816—21no—
98#14583816—21no—
99#14583816D411yescharacteristic
100#14583815—21no—
101#14583815—21no—
102#14583815—21no—
103#14583815—21no—
104#14583815—21no—
105#14583815—21no—
106#14583815D411yescharacteristic
107#14583815D411yescharacteristic
108#14583814D411yescharacteristic
109#14583813—21no—
110#14583813—21no—
111#14583813—21no—
112#14583813—21no—
113#14583813D411yescharacteristic
114D1384—21no—
115D1384—21no—
116D1384D411yescharacteristic
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—
127#14583803—41no—
128C4×Dic173C2311yescharacteristic
129#41×C173C2311yescharacteristic
130C2×Q8×C173C2311yescharacteristic
131C2×Q8×C173C2311yescharacteristic
132Q16×C173—21no—
133Q16×C173—21no—
134Q16×C173—21no—
135Q16×C173—21no—
136SD16×C173—21no—
137SD16×C173—21no—
138M16×C173—21no—
139M16×C173C2311yescharacteristic
140M16×C173C2311yescharacteristic
141M16×C173C2311yescharacteristic
142C2×C8×C173—21no—
143C2×C8×C173C2311yescharacteristic
144C22×D173—21no—
145C22×D173—41no—
146Dic346—21no—
147Dic346—21no—
148Dic346—41no—
149Dic346—41no—
150Dic346—41no—
151Dic346C2×D411yescharacteristic
152Dic346C2×D411yescharacteristic
153Dic346C2×D411yescharacteristic
154D692—21no—
155D692—21no—
156D692—41no—
157D692—41no—
158D692—41no—
159D692C2×D411yescharacteristic
160D692C2×D411yescharacteristic
161D692C2×D411yescharacteristic
162C22⋊D173—21no—
163C22⋊D173—21no—
164C22⋊D173—21no—
165C22⋊D173—41no—
166C22⋊D173—41no—
167C2×Dic173—21no—
168C2×Dic173—21no—
169C2×Dic173—41no—
170C4×D173—21no—
171C4×D173—21no—
172C4×D173—21no—
173C4×D173—41no—
174C4×D173—41no—
175C4×D173—41no—
176C4×D173—41no—
177C4×D173—41no—
178C4×D173—41no—
179#13063436—21no—
180#13063436C2×D411yescharacteristic
181#13063436C2×D411yescharacteristic
182Q8×C173—21no—
183Q8×C173—21no—
184Q8×C173—21no—
185Q8×C173—21no—
186Q8×C173C2×D411yescharacteristic
187D4×C173—21no—
188D4×C173C2×D411yescharacteristic
189C2×C4×C173—21no—
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×C173H(5,𝔽2)11yescharacteristic
200C2×D173—21no—
201C2×D173—21no—
202C2×D173—21no—
203C2×D173—41no—
204C2×D173—81no—
205C2×D173—161no—
206C4×C173—21no—
207C4×C173—41no—
208C4×C173—41no—
209C4×C173C22×D411yescharacteristic
210C4×C173C22×D411yescharacteristic
211Dic173—21no—
212Dic173—21no—
213Dic173—21no—
214Dic173—41no—
215Dic173—41no—
216C2×C173—21no—
217C2×C173—41no—
218C2×C173D4211yescharacteristic
219D173—41no—
220D173—41no—
221D173—41no—
222D173—81no—
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#473D17311yescharacteristic, complemented
232#471—1731no—
233#471—1731no—
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#137C2×D17311yescharacteristic
255#137C2×D17311yescharacteristic
256AGL(1,ℤ/8)—3461no—
257AGL(1,ℤ/8)—3461no—
258AGL(1,ℤ/8)—1731no—
259AGL(1,ℤ/8)—1731no—
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×Q16C2×D17311yescharacteristic, complemented
270C2×SD16—3461no—
271C2×SD16—3461no—
272C2×SD16—1731no—
273C2×SD16—1731no—
274#131—1731no—
275#131—1731no—
276#131—1731no—
277#131C2×D17311yescharacteristic
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#101C2×D17311yescharacteristic, complemented
288#101C2×D17311yescharacteristic, complemented
289#100—1731no—
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×Q8C22×D17311yescharacteristic
314C2×Q8C22×D17311yescharacteristic
315C2×D4—3461no—
316C2×D4—3461no—
317C2×D4—1731no—
318C2×D4—6921no—
319Q16—21no—
320Q16—21no—
321Q16—21no—
322Q16—21no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327Q16—3461no—
328Q16—3461no—
329Q16—1731no—
330Q16—1731no—
331SD16—21no—
332SD16—21no—
333SD16—3461no—
334SD16—3461no—
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—3461no—
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—3461no—
369C23—6921no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—3461no—
375Q8—3461no—
376Q8—1731no—
377Q8—1731no—
378Q8—1731no—
379Q8—6921no—
380Q8—6921no—
381Q8—6921no—
382Q8D4×D17311yescharacteristic
383D4—21no—
384D4—3461no—
385D4—3461no—
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—21no—
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—3461no—
424C22—3461no—
425C22—3461no—
426C22—13841no—
427C22—6921no—
428C22—27681no—
429C22#1914972511yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—3461no—
434C4—3461no—
435C4—3461no—
436C4—6921no—
437C4—6921no—
438C4C2×D4×D17311yescharacteristic
439C4C2×D4×D17311yescharacteristic
440C2—21no—
441C2—41no—
442C2—13841no—
443C2—6921no—
444C2—6921no—
445C2—6921no—
446C2#2601212511yescharacteristic
subgroupG/H
centerC2#26012125
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#473×C173C2
socleC2×C173D42
hypercenter#473D173
hypocenterC173#3289
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #26012307C2semidirect
2. #26012304C2semidirect
3. #26012303C2semidirect
4. #26012300C2semidirect
5. #26011884C2semidirect
6. #26011765C2semidirect
7. #26011757C2semidirect
8. #26011755C2semidirect
9. #26011676C2semidirect
10. #26011672C2semidirect
11. #26011667C2semidirect
12. #26011666C2semidirect
13. #26011612C2semidirect
14. #26011611C2semidirect
15. #473×C173C2semidirect
17. #19149727C22semidirect
22. #19149712C22semidirect
28. #19149710C22semidirect
40. C2×#14583816C22semidirect
48. #19149687C22semidirect
49. #19149685C22semidirect
50. #19149677C22semidirect
52. #19149614C22semidirect
55. #19149605C22semidirect
56. #19149605C22semidirect
57. #19149597C22semidirect
59. #19149589C22semidirect
62. C2×Q16×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #101×C173C22semidirect
223. C173#3289semidirect
231. #473D173semidirect
269. C2×Q16C2×D173semidirect
281. #108C2×D173semidirect
287. #101C2×D173semidirect
288. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123072
#260123042
#260123032
#260123002
#260118842
#260117652
#260117572
#260117552
#260116762
#260116722
#260116672
#260116662
#260116122
#260116112
#473×C1732
#3289173