Finite Group Databasebeta

← #36363295

Subgroups — #36363295

Order 22144.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012287C211yesmaximal, characteristic
2C2×#19149708C211yesmaximal, characteristic, complemented
3#137×D173C211yesmaximal, characteristic, complemented
4#26012256C211yesmaximal, characteristic, complemented
5#26011767C211yesmaximal, characteristic, complemented
6#26011766C211yesmaximal, characteristic, complemented
7#26011765C211yesmaximal, characteristic, complemented
8#26011764C211yesmaximal, characteristic, complemented
9#26011753C211yesmaximal, characteristic, complemented
10#26011752C211yesmaximal, characteristic, complemented
11#26011751C211yesmaximal, characteristic, complemented
12#26011538C211yesmaximal, characteristic, complemented
13#26011472C211yesmaximal, characteristic, complemented
14#101×D173C211yesmaximal, characteristic, complemented
15#471×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17#41×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149712C2211yescharacteristic
20#19149712C2211yescharacteristic
21#19149711—21no—
22#19149710C2211yescharacteristic
23#19149709C2211yescharacteristic
24#19149709C2211yescharacteristic
25#19149708—21no—
26#19149708—21no—
27#19149708—21no—
28#19149708—21no—
29#19149708—21no—
30#19149708—21no—
31#19149705C2211yescharacteristic
32#19149703—21no—
33#19149700—21no—
34#19149698—21no—
35C2×Dic692C2211yescharacteristic, complemented
36C2×#14583816C2211yescharacteristic, complemented
37Q16×D173—21no—
38C2×#14583814C2211yescharacteristic, complemented
39SD16×D173—21no—
40#19149687—21no—
41#19149687C2211yescharacteristic
42#19149686C2211yescharacteristic
43#19149685—21no—
44#19149685C2211yescharacteristic
45C2×#14583809—21no—
46C2×#14583809C2211yescharacteristic
47M16×D173—21no—
48M16×D173C2211yescharacteristic
49M16×D173C2211yescharacteristic
50M16×D173C2211yescharacteristic
51#19149615C2211yescharacteristic, complemented
52#19149614C2211yescharacteristic, complemented
53#19149614C2211yescharacteristic, complemented
54#19149598C2211yescharacteristic, complemented
55#19149597C2211yescharacteristic, complemented
56#19149597C2211yescharacteristic, complemented
57#19149597C2211yescharacteristic, complemented
58#19149597C2211yescharacteristic, complemented
59#19149596C2211yescharacteristic, complemented
60#137×C173C2211yescharacteristic
61AGL(1,ℤ/8)×C173C2211yescharacteristic
62C2×SD16×C173C2211yescharacteristic, complemented
63#131×C173C2211yescharacteristic
64#108×C173C2211yescharacteristic, complemented
65#101×C173C2211yescharacteristic, complemented
66#100×C173C2211yescharacteristic, complemented
67#14583827D411yescharacteristic
68#14583826—21no—
69#14583826C2311yescharacteristic
70#14583825—21no—
71#14583825—21no—
72#14583825—21no—
73#14583825—21no—
74#14583825—21no—
75#14583825D411yescharacteristic
76C2×Dic346—21no—
77C2×Dic346C2311yescharacteristic
78C2×Dic346C2311yescharacteristic
79Q8×D173—21no—
80Q8×D173—21no—
81Q8×D173—21no—
82Q8×D173—21no—
83Q8×D173D411yescharacteristic
84C2×C22⋊D173—21no—
85D4×D173—21no—
86D4×D173D411yescharacteristic
87C22×Dic173—21no—
88C2×C4×D173—21no—
89C2×C4×D173—21no—
90C2×C4×D173C2311yescharacteristic
91Dic692—21no—
92Dic692—21no—
93Dic692—21no—
94Dic692—21no—
95#14583816—21no—
96#14583816—21no—
97#14583816—21no—
98#14583816—21no—
99#14583816—21no—
100#14583815—21no—
101#14583815—21no—
102#14583814—21no—
103#14583814—21no—
104#14583814—21no—
105#14583814—21no—
106#14583814—21no—
107#14583813—21no—
108#14583811—21no—
109#14583810—21no—
110#14583810C2311yescharacteristic
111#14583810C2311yescharacteristic
112#14583810C2311yescharacteristic
113#14583809—21no—
114#14583809—21no—
115#14583809—21no—
116#14583809—21no—
117#14583809—21no—
118#14583809D411yescharacteristic
119#14583809D411yescharacteristic
120#14583809D411yescharacteristic
121#14583809D411yescharacteristic
122C2×#13063436—21no—
123C2×#13063436C2311yescharacteristic
124C8×D173—21no—
125C8×D173—21no—
126C8×D173—21no—
127C8×D173—21no—
128C8×D173—21no—
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#4111yescharacteristic
146Dic346—21no—
147Dic346—21no—
148Dic346—21no—
149Dic346—21no—
150Dic346—21no—
151Dic346—41no—
152Dic346—41no—
153Dic346C2×D411yescharacteristic
154D692—21no—
155D692C2×D411yescharacteristic
156C22⋊D173—21no—
157C22⋊D173—21no—
158C22⋊D173—21no—
159C22⋊D173—41no—
160C22⋊D173—41no—
161C2×Dic173—21no—
162C2×Dic173—21no—
163C2×Dic173—21no—
164C2×Dic173—41no—
165C2×Dic173—41no—
166C2×Dic173#4111yescharacteristic
167C4×D173—21no—
168C4×D173—21no—
169C4×D173—21no—
170C4×D173—41no—
171C4×D173—41no—
172C4×D173C2×D411yescharacteristic
173C4×D173C2×D411yescharacteristic
174C4×D173C2×D411yescharacteristic
175C4×D173C2×D411yescharacteristic
176#13063436—21no—
177#13063436—21no—
178#13063436—21no—
179#13063436C2×D411yescharacteristic
180#13063436C2×D411yescharacteristic
181C23×C173—21no—
182Q8×C173—21no—
183Q8×C173—21no—
184Q8×C173C2×D411yescharacteristic
185D4×C173—21no—
186D4×C173—21no—
187D4×C173—21no—
188D4×C173C2×D411yescharacteristic
189C2×C4×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173C2411yescharacteristic
192C8×C173—21no—
193C8×C173—21no—
194C8×C173—21no—
195C8×C173C2×D411yescharacteristic
196C8×C173C2×D411yescharacteristic
197C22×C173—21no—
198C22×C173—41no—
199C22×C173—81no—
200C22×C173C2×#4111yescharacteristic
201C2×D173—21no—
202C2×D173—41no—
203C2×D173—81no—
204C2×D173#12111yescharacteristic
205C2×D173#12111yescharacteristic
206C4×C173—21no—
207C4×C173—41no—
208C4×C173C22×D411yescharacteristic
209C4×C173C22×D411yescharacteristic
210Dic173—21no—
211Dic173—41no—
212Dic173—41no—
213Dic173#12111yescharacteristic
214Dic173#12111yescharacteristic
215C2×C173—21no—
216C2×C173—41no—
217C2×C173—81no—
218C2×C173C2×#12111yescharacteristic
219D173—21no—
220D173—21no—
221D173—41no—
222C173#306911yescharacteristic, complemented, Sylow
223#3069—1731nomaximal, Sylow
224#575—1731no—
225C2×#137—1731no—
226C2×#137—1731no—
227#568—1731no—
228#473—1731no—
229#473—1731no—
230#473—1731no—
231#473—1731no—
232#471—1731no—
233#471—1731no—
234#471—1731no—
235#471D17311yescharacteristic, complemented
236#430—1731no—
237#413—1731no—
238C2×#101—1731no—
239C2×#41—1731no—
240C2×#41—1731no—
241C22×Q8—1731no—
242#137—3461no—
243#137—3461no—
244#137—3461no—
245#137—3461no—
246#137—3461no—
247#137—3461no—
248#137—3461no—
249#137—1731no—
250#137—1731no—
251#137—1731no—
252#137—1731no—
253#137—1731no—
254#137C2×D17311yescharacteristic
255AGL(1,ℤ/8)—3461no—
256AGL(1,ℤ/8)—1731no—
257AGL(1,ℤ/8)C2×D17311yescharacteristic
258#135—3461no—
259#135—3461no—
260C2×Q16—3461no—
261C2×Q16—1731no—
262C2×Q16—1731no—
263C2×SD16—3461no—
264C2×SD16—1731no—
265C2×SD16C2×D17311yescharacteristic, complemented
266#131—3461no—
267#131—3461no—
268#131—1731no—
269#131—1731no—
270#131—1731no—
271#131C2×D17311yescharacteristic
272C2×M16—3461no—
273C2×M16—3461no—
274C2×M16—1731no—
275C2×M16—1731no—
276C2×M16—1731no—
277C2×M16—1731no—
278#108—1731no—
279#108—1731no—
280#108—1731no—
281#108C2×D17311yescharacteristic, complemented
282#101—1731no—
283#101—1731no—
284#101—1731no—
285#101—1731no—
286#101—1731no—
287#101C2×D17311yescharacteristic, complemented
288#100—1731no—
289#100C2×D17311yescharacteristic, complemented
290#41—3461no—
291#41—3461no—
292#41—3461no—
293#41—3461no—
294#41—3461no—
295#41—3461no—
296#41—1731no—
297#41—1731no—
298#41—1731no—
299#41C22×D17311yescharacteristic
300C2×Q8—3461no—
301C2×Q8—3461no—
302C2×Q8—3461no—
303C2×Q8—3461no—
304C2×Q8—3461no—
305C2×Q8—1731no—
306C2×Q8—1731no—
307C2×Q8—1731no—
308C2×Q8C22×D17311yescharacteristic
309C2×D4—3461no—
310C2×D4—3461no—
311C2×D4—1731no—
312C2×D4C22×D17311yescharacteristic
313C22×C4—3461no—
314C22×C4—3461no—
315C22×C4—3461no—
316C22×C4—1731no—
317Q16—21no—
318Q16—3461no—
319Q16—3461no—
320Q16—3461no—
321Q16—3461no—
322Q16—3461no—
323Q16—3461no—
324Q16—3461no—
325Q16—3461no—
326Q16—3461no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—21no—
331SD16—3461no—
332SD16—3461no—
333SD16—3461no—
334SD16—3461no—
335SD16—3461no—
336SD16—3461no—
337SD16—3461no—
338SD16—3461no—
339D8—21no—
340D8—3461no—
341M16—21no—
342M16—3461no—
343M16—3461no—
344M16—3461no—
345M16—3461no—
346M16—3461no—
347M16—3461no—
348M16—1731no—
349M16—1731no—
350M16—1731no—
351M16—1731no—
352M16—1731no—
353M16—1731no—
354M16—1731no—
355M16C22×D17311yescharacteristic
356M16C22×D17311yescharacteristic
357M16C22×D17311yescharacteristic
358C2×C8—21no—
359C2×C8—3461no—
360C2×C8—3461no—
361C2×C8—3461no—
362C2×C8—3461no—
363C2×C8—3461no—
364C2×C8—3461no—
365C2×C8—1731no—
366C2×C8C22×D17311yescharacteristic
367C23—21no—
368C23—3461no—
369C23—1731no—
370Q8—21no—
371Q8—21no—
372Q8—3461no—
373Q8—3461no—
374Q8—3461no—
375Q8—3461no—
376Q8—3461no—
377Q8—1731no—
378Q8—6921no—
379Q8—6921no—
380Q8D4×D17311yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—3461no—
385D4—3461no—
386D4—3461no—
387D4—3461no—
388D4—1731no—
389D4—6921no—
390D4—6921no—
391D4D4×D17311yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—3461no—
397C2×C4—3461no—
398C2×C4—3461no—
399C2×C4—3461no—
400C2×C4—1731no—
401C2×C4—1731no—
402C2×C4—1731no—
403C2×C4—1731no—
404C2×C4—1731no—
405C2×C4—6921no—
406C2×C4—6921no—
407C2×C4—6921no—
408C2×C4—6921no—
409C2×C4C23×D17311yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—3461no—
414C8—3461no—
415C8—3461no—
416C8—1731no—
417C8—1731no—
418C8D4×D17311yescharacteristic
419C8D4×D17311yescharacteristic
420C22—21no—
421C22—41no—
422C22—81no—
423C22—3461no—
424C22—13841no—
425C22—1731no—
426C22—1731no—
427C22—6921no—
428C22#41×D17311yescharacteristic
429C4—21no—
430C4—41no—
431C4—3461no—
432C4—1731no—
433C4—1731no—
434C4—6921no—
435C4—6921no—
436C4C2×D4×D17311yescharacteristic
437C4C2×D4×D17311yescharacteristic
438C2—21no—
439C2—41no—
440C2—81no—
441C2—3461no—
442C2—3461no—
443C2—6921no—
444C2#121×D17311yescharacteristic
subgroupG/H
centerC2#121×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#471×C173C2
socleC2×C173C2×#121
prime radical#26012287C2
hypercenter#471D173
hypocenterC173#3069
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
2. C2×#19149708C2semidirect
3. #137×D173C2semidirect
4. #26012256C2semidirect
5. #26011767C2semidirect
6. #26011766C2semidirect
7. #26011765C2semidirect
8. #26011764C2semidirect
9. #26011753C2semidirect
10. #26011752C2semidirect
11. #26011751C2semidirect
12. #26011538C2semidirect
13. #26011472C2semidirect
14. #101×D173C2semidirect
15. #471×C173C2semidirect
35. C2×Dic692C22semidirect
36. C2×#14583816C22semidirect
38. C2×#14583814C22semidirect
51. #19149615C22semidirect
52. #19149614C22semidirect
53. #19149614C22semidirect
54. #19149598C22semidirect
55. #19149597C22semidirect
56. #19149597C22semidirect
57. #19149597C22semidirect
58. #19149597C22semidirect
59. #19149596C22semidirect
62. C2×SD16×C173C22semidirect
64. #108×C173C22semidirect
65. #101×C173C22semidirect
66. #100×C173C22semidirect
222. C173#3069semidirect
235. #471D173semidirect
265. C2×SD16C2×D173semidirect
281. #108C2×D173semidirect
287. #101C2×D173semidirect
289. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122872
C2×#191497082
#137×D1732
#260122562
#260117672
#260117662
#260117652
#260117642
#260117532
#260117522
#260117512
#260115382
#260114722
#101×D1732
#471×C1732
#3069173