Finite Group Databasebeta

← #36362909

Subgroups — #36362909

Order 22144.

724 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012327C211yesmaximal, characteristic, complemented
2#26012289C211yesmaximal, characteristic, complemented
3#26012037C211yesmaximal, characteristic, complemented
4#26011672C212yesmaximal, complemented
5#26011667C212yesmaximal, complemented
6#26011662C212yesmaximal, complemented
7#26011657C212yesmaximal, complemented
8C2×#19149606C211yesmaximal, characteristic, complemented
9C2×#19149605C211yesmaximal, characteristic, complemented
10#26011473C211yesmaximal, characteristic, complemented
11#575×C173C211yesmaximal, characteristic, complemented
12#19149727—21no—
13#19149726—21no—
14#19149726C2212yes—
15#19149725—21no—
16#19149725C2212yes—
17#19149724—21no—
18#19149724—22no—
19C2×#14583826—22no—
20C2×#14583826C2211yescharacteristic
21C2×#14583826C2211yescharacteristic
22#41×D173—22no—
23#41×D173—22no—
24#19149710C2212yescomplemented
25#19149709—21no—
26#19149705C2212yescomplemented
27#19149704—21no—
28#19149699—22no—
29C2×#14583810C2211yescharacteristic, complemented
30#19149677C2211yescharacteristic, complemented
31#19149675C2211yescharacteristic, complemented
32#19149668C2211yescharacteristic, complemented
33#19149666C2211yescharacteristic, complemented
34#19149651—22no—
35C4×C22⋊D173—22no—
36C2×C4×Dic173C2211yescharacteristic, complemented
37#19149606C2212yescomplemented
38#19149606C2212yescomplemented
39#19149605C2212yescomplemented
40#19149605C2212yescomplemented
41#19149597C2212yescomplemented
42#19149595C2212yescomplemented
43C2×#41×C173C2211yescharacteristic
44C2×#41×C173C2211yescharacteristic, complemented
45#137×C173—21no—
46#137×C173C2212yescomplemented
47AGL(1,ℤ/8)×C173—21no—
48AGL(1,ℤ/8)×C173C2212yescomplemented
49#135×C173—22no—
50C2×M16×C173C2211yescharacteristic, complemented
51#14583827—22no—
52#14583827—22no—
53#14583826—24no—
54#14583826—24no—
55#14583826—22no—
56#14583826—22no—
57#14583826—21no—
58#14583826—21no—
59#14583826—21no—
60#14583826—21no—
61#14583826—21no—
62#14583826—21no—
63#14583826D412yescomplemented
64#14583826C2312yes—
65#14583825—24no—
66#14583825—22no—
67#14583825—24no—
68#14583825—22no—
69C2×Dic346—22no—
70C2×Dic346D411yescharacteristic, complemented
71C2×Dic346C2311yescharacteristic
72Q8×D173—22no—
73Q8×D173—22no—
74C2×D692—22no—
75C2×D692D411yescharacteristic, complemented
76C2×D692C2311yescharacteristic
77C2×C22⋊D173—24no—
78C2×C22⋊D173—22no—
79C2×C22⋊D173—21no—
80C2×C22⋊D173D412yes—
81D4×D173—24no—
82D4×D173—22no—
83D4×D173—24no—
84D4×D173—22no—
85C22×Dic173—21no—
86C2×C4×D173—24no—
87C2×C4×D173—21no—
88C2×C4×D173D412yes—
89#14583816—22no—
90#14583814—22no—
91#14583813—22no—
92#14583811—22no—
93#14583810—21no—
94#14583810C2312yes—
95C2×#13063436—21no—
96#14583805—22no—
97C22⋊Dic173—22no—
98#14583803—22no—
99C4×Dic173—21no—
100C4×Dic173C2311yescharacteristic
101C4×Dic173C2311yescharacteristic
102#41×C173—21no—
103#41×C173—21no—
104#41×C173—21no—
105#41×C173—21no—
106#41×C173—21no—
107#41×C173—21no—
108#41×C173D412yescomplemented
109#41×C173C2312yes—
110C2×Q8×C173D411yescharacteristic, complemented
111C2×Q8×C173C2311yescharacteristic
112C2×D4×C173—21no—
113C2×D4×C173—21no—
114C2×D4×C173D411yescharacteristic, complemented
115C2×D4×C173C2311yescharacteristic
116C22×C4×C173—21no—
117C22×C4×C173—21no—
118C22×C4×C173C2311yescharacteristic
119Q16×C173—22no—
120SD16×C173—22no—
121SD16×C173—22no—
122D8×C173—22no—
123M16×C173—21no—
124M16×C173C2312yes—
125C2×C8×C173—21no—
126C22×D173—24no—
127C22×D173—21no—
128C22×D173C2×D412yes—
129Dic346—24no—
130Dic346—21no—
131Dic346—21no—
132Dic346—21no—
133Dic346C2×D412yes—
134D692—24no—
135D692—21no—
136D692—21no—
137D692—21no—
138D692C2×D412yes—
139C22⋊D173—24no—
140C22⋊D173—24no—
141C22⋊D173—22no—
142C22⋊D173—22no—
143C22⋊D173—22no—
144C22⋊D173—22no—
145C22⋊D173—44no—
146C2×Dic173—24no—
147C2×Dic173—21no—
148C2×Dic173—21no—
149C2×Dic173—21no—
150C2×Dic173—42no—
151C2×Dic173C2×D412yes—
152C4×D173—24no—
153C4×D173—24no—
154C4×D173—22no—
155C4×D173—22no—
156C4×D173—22no—
157C4×D173—22no—
158#13063436—22no—
159C23×C173—21no—
160C23×C173—21no—
161C23×C173C2×D411yescharacteristic
162Q8×C173—21no—
163Q8×C173—21no—
164Q8×C173—21no—
165Q8×C173C2×D412yes—
166D4×C173—22no—
167D4×C173—22no—
168D4×C173—21no—
169D4×C173—21no—
170D4×C173—21no—
171D4×C173—41no—
172D4×C173—41no—
173D4×C173C2×D412yes—
174C2×C4×C173—21no—
175C2×C4×C173—22no—
176C2×C4×C173—22no—
177C2×C4×C173—21no—
178C2×C4×C173—41no—
179C2×C4×C173—41no—
180C2×C4×C173C2×D411yescharacteristic
181C2×C4×C173C2×D411yescharacteristic
182C2×C4×C173C2×D411yescharacteristic
183C2×C4×C173C2×D411yescharacteristic
184C2×C4×C173C2×D411yescharacteristic
185C2×C4×C173C2411yescharacteristic
186C8×C173—22no—
187C22×C173—22no—
188C22×C173—41no—
189C22×C173—42no—
190C22×C173—41no—
191C22×C173—81no—
192C22×C173#12011yescharacteristic
193C22×C173#12011yescharacteristic
194C22×C173C22×D411yescharacteristic
195C2×D173—22no—
196C2×D173—22no—
197C2×D173—42no—
198C2×D173—81no—
199C2×D173—84no—
200C4×C173—22no—
201C4×C173—41no—
202C4×C173#12011yescharacteristic
203C4×C173#12011yescharacteristic
204C4×C173C22×D411yescharacteristic
205C4×C173C22×D411yescharacteristic
206Dic173—22no—
207Dic173—22no—
208Dic173—22no—
209Dic173—41no—
210C2×C173—21no—
211C2×C173—21no—
212C2×C173—21no—
213C2×C173—42no—
214C2×C173—81no—
215C2×C173C2×#12011yescharacteristic
216D173—42no—
217D173—42no—
218C173#301811yescharacteristic, complemented, Sylow
219#3018—1731nomaximal, Sylow
220#585—1731no—
221#575—1731no—
222#575D17311yescharacteristic, complemented
223#532—1731no—
224#456—1732no—
225#455—1732no—
226#454—1732no—
227#453—1732no—
228C2×#104—1731no—
229C2×#104—1731no—
230#413—1731no—
2312−1+4—3461no—
2322−1+4—3461no—
2332−1+4—1732no—
234H(5,𝔽2)—3462no—
235H(5,𝔽2)—3461no—
236H(5,𝔽2)—3461no—
237H(5,𝔽2)—1732no—
238C2×#41—3462no—
239C2×#41—3462no—
240C2×#41—3462no—
241C2×#41—1731no—
242C2×#41—1731no—
243C2×#41C2×D17311yescharacteristic
244C2×#41C2×D17311yescharacteristic, complemented
245#137—21no—
246#137—3461no—
247#137—1732no—
248#137C2×D17312yescomplemented
249AGL(1,ℤ/8)—21no—
250AGL(1,ℤ/8)—3461no—
251AGL(1,ℤ/8)—1732no—
252AGL(1,ℤ/8)C2×D17312yescomplemented
253#135—22no—
254#135—3462no—
255C2×M16—1731no—
256C2×M16C2×D17311yescharacteristic, complemented
257C4⋊Q8—1731no—
258#127—1731no—
259#124—1731no—
260#124—1731no—
261#121—3462no—
262C4×D4—3462no—
263C2×C42—1731no—
264#104—1732no—
265#104—1732no—
266#104—1732no—
267#104—1732no—
268#101—1732no—
269#100—1732no—
270#41—21no—
271#41—21no—
272#41—21no—
273#41—21no—
274#41—21no—
275#41—21no—
276#41—3464no—
277#41—3464no—
278#41—3462no—
279#41—3462no—
280#41—3464no—
281#41—3462no—
282#41—3462no—
283#41—3462no—
284#41—3462no—
285#41—3464no—
286#41—3461no—
287#41—3461no—
288#41—3461no—
289#41—3461no—
290#41—3461no—
291#41—3461no—
292#41—1732no—
293#41—1732no—
294#41C22⋊D17312yescomplemented
295#41C22×D17312yes—
296C2×Q8—3462no—
297C2×Q8—3462no—
298C2×Q8—3462no—
299C2×Q8—1731no—
300C2×Q8—1731no—
301C2×Q8C22⋊D17311yescharacteristic, complemented
302C2×Q8C22×D17311yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—3462no—
306C2×D4—3464no—
307C2×D4—3462no—
308C2×D4—3464no—
309C2×D4—3464no—
310C2×D4—3461no—
311C2×D4—3462no—
312C2×D4—3462no—
313C2×D4—1732no—
314C2×D4—1731no—
315C2×D4—1731no—
316C2×D4C22⋊D17311yescharacteristic, complemented
317C2×D4C22×D17311yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—3464no—
321C22×C4—3461no—
322C22×C4—3461no—
323C22×C4—1732no—
324C22×C4C22×D17311yescharacteristic
325Q16—22no—
326Q16—3462no—
327SD16—22no—
328SD16—22no—
329SD16—3462no—
330SD16—3462no—
331D8—22no—
332D8—3462no—
333M16—21no—
334M16—3461no—
335M16—1732no—
336M16C22×D17312yes—
337C2×C8—21no—
338C2×C8—3461no—
339C4⋊C4—3462no—
340C22⋊C4—3462no—
341C22⋊C4—3462no—
342C42—3461no—
343C42—1731no—
344C42—1731no—
345C23—21no—
346C23—21no—
347C23—3464no—
348C23—3461no—
349C23—1732no—
350C23D4×D17311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—3464no—
355Q8—3461no—
356Q8—3461no—
357Q8—3461no—
358Q8—1732no—
359Q8C2×C22⋊D17312yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—41no—
366D4—41no—
367D4—3464no—
368D4—3464no—
369D4—3464no—
370D4—3462no—
371D4—3462no—
372D4—3462no—
373D4—3462no—
374D4—3461no—
375D4—3461no—
376D4—3461no—
377D4—1732no—
378D4—6924no—
379D4C2×C22⋊D17312yes—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—3464no—
387C2×C4—3464no—
388C2×C4—3464no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3462no—
392C2×C4—3462no—
393C2×C4—3462no—
394C2×C4—3462no—
395C2×C4—3461no—
396C2×C4—1732no—
397C2×C4—6922no—
398C2×C4D4×D17311yescharacteristic
399C2×C4D4×D17311yescharacteristic
400C2×C4D4×D17311yescharacteristic
401C2×C4C2×C22⋊D17311yescharacteristic
402C2×C4C2×C22⋊D17311yescharacteristic
403C2×C4C23×D17311yescharacteristic
404C8—22no—
405C8—3462no—
406C22—22no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—81no—
411C22—3462no—
412C22—3462no—
413C22—13841no—
414C22—13844no—
415C22—6922no—
416C22#1914964711yescharacteristic
417C22#1914964711yescharacteristic
418C22C2×D4×D17311yescharacteristic
419C4—22no—
420C4—41no—
421C4—3462no—
422C4—3462no—
423C4—3462no—
424C4—6921no—
425C4#1914964711yescharacteristic
426C4#1914964711yescharacteristic
427C4C2×D4×D17311yescharacteristic
428C4C22×C22⋊D17311yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—42no—
433C2—81no—
434C2—6922no—
435C2—6922no—
436C2C2×#1914964711yescharacteristic
subgroupG/H
centerC4#19149647
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#575×C173C2
socleC2×C173C2×#120
hypercenter#575D173
hypocenterC173#3018
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012289C2semidirect
3. #26012037C2semidirect
4. #26011672C2semidirect
5. #26011667C2semidirect
6. #26011662C2semidirect
7. #26011657C2semidirect
8. C2×#19149606C2semidirect
9. C2×#19149605C2semidirect
10. #26011473C2semidirect
11. #575×C173C2semidirect
24. #19149710C22semidirect
26. #19149705C22semidirect
29. C2×#14583810C22semidirect
30. #19149677C22semidirect
31. #19149675C22semidirect
32. #19149668C22semidirect
33. #19149666C22semidirect
36. C2×C4×Dic173C22semidirect
37. #19149606C22semidirect
38. #19149606C22semidirect
39. #19149605C22semidirect
40. #19149605C22semidirect
41. #19149597C22semidirect
42. #19149595C22semidirect
44. C2×#41×C173C22semidirect
46. #137×C173C22semidirect
48. AGL(1,ℤ/8)×C173C22semidirect
50. C2×M16×C173C22semidirect
63. #14583826D4semidirect
70. C2×Dic346D4semidirect
75. C2×D692D4semidirect
108. #41×C173D4semidirect
110. C2×Q8×C173D4semidirect
114. C2×D4×C173D4semidirect
218. C173#3018semidirect
222. #575D173semidirect
244. C2×#41C2×D173semidirect
248. #137C2×D173semidirect
252. AGL(1,ℤ/8)C2×D173semidirect
256. C2×M16C2×D173semidirect
294. #41C22⋊D173semidirect
301. C2×Q8C22⋊D173semidirect
316. C2×D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260122892
#260120372
#260116722
#260116672
#260116622
#260116572
C2×#191496062
C2×#191496052
#260114732
#575×C1732
#3018173