Finite Group Databasebeta

← #35322230

Subgroups — #35322230

Order 21248.

324 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127531C211yesmaximal, characteristic, complemented
2C2×#19127509C211yesmaximal, characteristic
3#24909171C211yesmaximal, characteristic
4#24908351C211yesmaximal, characteristic
5#24907342C211yesmaximal, characteristic, complemented
6#24905559C211yesmaximal, characteristic, complemented
7C83×#1374C211yesmaximal, characteristic, complemented
8C2×#14581814C2211yescharacteristic
9C2×#14581728C2211yescharacteristic
10C2×#14581728C2211yescharacteristic
11#19127739C2211yescharacteristic
12C2×#14581720—21no—
13#19127667—21no—
14#19127656—21no—
15#19127531—22no—
16#19127509—22no—
17#19127506C411yescharacteristic, complemented
18#19127506C411yescharacteristic, complemented
19#19127503C411yescharacteristic, complemented
20#19127503C411yescharacteristic, complemented
21#19127461—21no—
22C22×C16×C83C2211yescharacteristic
23C22×C16×C83C2211yescharacteristic
24C2×C4×C8×C83C2211yescharacteristic
25C2×#13050590—21no—
26#14581814—21no—
27#14581814—21no—
28#14581814—21no—
29#14581814—21no—
30C2×Dic332C2×C411yescharacteristic
31C2×#13050579—21no—
32C2×D664C2×C411yescharacteristic
33C2×#13050574—21no—
34C2×#13050574—21no—
35#14581751—21no—
36#14581728—21no—
37#14581728—21no—
38#14581728—21no—
39#14581728D412yes—
40#14581726C2×C411yescharacteristic
41#14581726C2×C411yescharacteristic
42#14581724—21no—
43#14581720—42no—
44(C2×D83)⋊C8—41no—
45(C2×D83)⋊C8—41no—
46#14581702—41no—
47C8×Dic83—41no—
48C22×C8×C83D411yescharacteristic
49C22×C8×C83C2311yescharacteristic
50C2×C42×C83D411yescharacteristic
51C2×C16×C83—21no—
52C2×C16×C83—21no—
53C2×C16×C83—21no—
54C2×C16×C83—21no—
55C2×C16×C83—21no—
56C2×C16×C83C2×C411yescharacteristic
57C2×C16×C83C2×C411yescharacteristic
58C4×C8×C83—21no—
59C4×C8×C83—21no—
60#13050590—41no—
61#13050590—41no—
62#13050590—41no—
63#13050590—41no—
64C2×Dic166—21no—
65C2×D332—21no—
66C2×C22⋊D83—41no—
67C2×C4×D83—41no—
68Dic332—21no—
69Dic332—21no—
70#13050579—41no—
71#13050579—41no—
72D664—21no—
73D664—21no—
74#13050574—22no—
75#13050574—41no—
76#13050574—41no—
77#13050574—41no—
78C2×C83⋊C8—41no—
79C2×C83⋊C8—41no—
80C4⋊Dic83—21no—
81C4⋊Dic83—21no—
82C22⋊Dic83—41no—
83C4×Dic83—41no—
84C22×C4×C83—21no—
85C22×C4×C83C2×D411yescharacteristic
86C2×C8×C83—21no—
87C2×C8×C83—21no—
88C2×C8×C83D811yescharacteristic
89C2×C8×C83D811yescharacteristic
90C2×C8×C83C22×C411yescharacteristic
91C2×C8×C83C2×D411yescharacteristic
92C2×C8×C83#4111yescharacteristic
93C2×C8×C83#4111yescharacteristic
94C2×C8×C83#4111yescharacteristic
95C2×C8×C83#4111yescharacteristic
96C42×C83—21no—
97C42×C83—21no—
98C16×C83—21no—
99C16×C83—21no—
100C16×C83—41no—
101C23×C83#13511yescharacteristic
102C22×D83—41no—
103C2×C4×C83—22no—
104C2×C4×C83—41no—
105C2×C4×C83—41no—
106C2×C4×C83C4×D411yescharacteristic
107C2×C4×C83#12111yescharacteristic
108C2×C4×C83#12311yescharacteristic
109C2×C4×C83C2×D811yescharacteristic
110C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C83#13711yescharacteristic
112C22⋊D83—81no—
113C22⋊D83—81no—
114C2×Dic83—41no—
115C2×Dic83—41no—
116C2×Dic83—41no—
117D332—41no—
118D332—41no—
119C4×D83—81no—
120C4×D83—81no—
121Dic166—41no—
122Dic166—41no—
123C8×C83—21no—
124C8×C83—21no—
125C8×C83#11711yescharacteristic
126C8×C83C4×D411yescharacteristic
127C8×C83#12411yescharacteristic
128C8×C83#12611yescharacteristic
129C83⋊C8—42no—
130C83⋊C8—81no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83C4×D811yescharacteristic
134C22×C83#46411yescharacteristic
135C22×C83#48211yescharacteristic
136C2×D83—81no—
137C2×D83—81no—
138C2×D83—81no—
139C4×C83—42no—
140C4×C83#38811yescharacteristic
141C4×C83#44011yescharacteristic
142C4×C83#45911yescharacteristic
143C4×C83#48011yescharacteristic
144Dic83—81no—
145Dic83—81no—
146Dic83—81no—
147#13134—831nomaximal, Sylow
148C2×C83—21no—
149C2×C83—21no—
150C2×C83#192511yescharacteristic
151C2×C83#217811yescharacteristic
152C2×C83#217911yescharacteristic
153D83—161no—
154C2×#368—831no—
155C2×#359—831no—
156#2138—831no—
157#1974—831no—
158#1772—831no—
159#1382—831no—
160#1374D8311yescharacteristic, complemented
161C83#1313411yescharacteristic, complemented, Sylow
162C2×#135—831no—
163C22×C16C2×D8311yescharacteristic
164C22×C16C2×D8311yescharacteristic
165C2×#108—831no—
166C2×#108—831no—
167#427—831no—
168C2×#104—1661no—
169#408—1661no—
170#405—1661no—
171C2×C4×C8C2×D8311yescharacteristic
172#368—1662no—
173#359—1662no—
174#358—831no—
175#358—831no—
176#357—831no—
177#357—831no—
178#339—1661no—
179C2×#41—1661no—
180#135—1661no—
181#135—1661no—
182#135—1661no—
183#135—1661no—
184C2×Q16—831no—
185C2×SD16—1661no—
186C2×D8—831no—
187C2×M16—1661no—
188C2×M16—1661no—
189C22×C8D33211yescharacteristic
190C22×C8C22×D8311yescharacteristic
191#117—1661no—
192C2×C42D33211yescharacteristic
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16C4×D8311yescharacteristic
199C2×C16C4×D8311yescharacteristic
200#108—832no—
201#108—1661no—
202#108—1661no—
203#108—1661no—
204#107—831no—
205#107—831no—
206#106—1661no—
207#104—3322no—
208C22⋊C8—3321no—
209C22⋊C8—3321no—
210#97—3321no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—3321no—
214#41—3321no—
215#41—3321no—
216#41—3321no—
217#41—3321no—
218C2×Q8—1661no—
219C2×D4—3321no—
220C2×D4—1661no—
221C22×C4—21no—
222C22×C4—3321no—
223C22×C4C2×D33211yescharacteristic
224Q16—1661no—
225Q16—1661no—
226SD16—3321no—
227SD16—3321no—
228D8—1661no—
229D8—1661no—
230M16—3321no—
231M16—3321no—
232M16—3321no—
233M16—1662no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—3321no—
237C2×C8—3321no—
238C2×C8D66411yescharacteristic
239C2×C8D66411yescharacteristic
240C2×C8C2×C4×D8311yescharacteristic
241C2×C8D4×D8311yescharacteristic
242C2×C8#1305058911yescharacteristic
243C2×C8#1305058911yescharacteristic
244C2×C8#1305059011yescharacteristic
245C2×C8#1305059111yescharacteristic
246C4⋊C4—1661no—
247C4⋊C4—1661no—
248C22⋊C4—3321no—
249C42—21no—
250C42—21no—
251C42—3321no—
252C16—21no—
253C16—21no—
254C16—41no—
255C23—3321no—
256C23#1458181411yescharacteristic
257Q8—3321no—
258Q8—3321no—
259D4—3321no—
260D4—3321no—
261D4—6641no—
262D4—6641no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3321no—
267C2×C4—3321no—
268C2×C4—3321no—
269C2×C4—6641no—
270C2×C4—6641no—
271C2×C4C4×D33211yescharacteristic
272C2×C4#1458176711yescharacteristic
273C2×C4#1458177611yescharacteristic
274C2×C4C2×D66411yescharacteristic
275C2×C4#1458182011yescharacteristic
276C2×C4#1458182511yescharacteristic
277C8—21no—
278C8—21no—
279C8—3322no—
280C8—6641no—
281C8#1458175011yescharacteristic
282C8(C22⋊D83)⋊C411yescharacteristic
283C8#1458177811yescharacteristic
284C8#1458178711yescharacteristic
285C22—21no—
286C22—21no—
287C22—6641no—
288C22—6641no—
289C22—6641no—
290C22C4×D66411yescharacteristic
291C22#1912792411yescharacteristic
292C22#1912802611yescharacteristic
293C4—42no—
294C4—6641no—
295C4—6641no—
296C4—6641no—
297C4#1912759311yescharacteristic
298C4#1912780311yescharacteristic
299C4#1912789711yescharacteristic
300C4#1912801511yescharacteristic
301C2—21no—
302C2—21no—
303C2—13281no—
304C2#2490808411yescharacteristic
305C2#2490934511yescharacteristic
306C2#2490935011yescharacteristic
subgroupG/H
centerC2×C4C2×D664
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1374C2
socleC22×C83C4×D8
prime radicalC2×#14581814C22
cosocleC22×C8×C83C23
hypercenter#1374D83
hypocenterC83#13134
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1374

Complements

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

NH ≅ G/Ntype
1. C2×#19127531C2semidirect
5. #24907342C2semidirect
6. #24905559C2semidirect
7. C83×#1374C2semidirect
17. #19127506C4semidirect
18. #19127506C4semidirect
19. #19127503C4semidirect
20. #19127503C4semidirect
160. #1374D83semidirect
161. C83#13134semidirect

Maximal subgroups

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

subgroupindex
C2×#191275312
C2×#191275092
#249091712
#249083512
#249073422
#249055592
C83×#13742
#1313483