Finite Group Databasebeta

← #35322283

Subgroups — #35322283

Order 21248.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127523C211yesmaximal, characteristic, complemented
2C2×#19127515C211yesmaximal, characteristic
3#24909178C211yesmaximal, characteristic
4#24908337C211yesmaximal, characteristic
5#24907349C211yesmaximal, characteristic, complemented
6#24905542C211yesmaximal, characteristic, complemented
7C83×#1386C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C2×#14581732C2211yescharacteristic
10C2×#14581728C2211yescharacteristic
11C2×#14581712—21no—
12#19127657—21no—
13C2×#14581702C2211yescharacteristic
14#19127602—21no—
15#19127523—22no—
16#19127515—22no—
17#19127501C414yescomplemented
18#19127460—21no—
19C2×M32×C83C2211yescharacteristic
20#427×C83C2211yescharacteristic
21C2×#106×C83C2211yescharacteristic
22C22×D332—21no—
23C2×D664—21no—
24C2×D664—21no—
25C2×D664—21no—
26C2×D664—21no—
27C2×D664—21no—
28C2×D664C2×C412yes—
29C2×#13050574—21no—
30C22×C83⋊C8D411yescharacteristic
31C2×(C2×D83)⋊C4—42no—
32C2×C4×Dic83D411yescharacteristic
33#14581732—21no—
34#14581732—21no—
35C2×C83⋊C16C2×C412yes—
36#14581728—21no—
37#14581728D412yes—
38#14581712—42no—
39#14581703—41no—
40#14581702—21no—
41#14581702—21no—
42C4×C83⋊C8—41no—
43C22×C8×C83C2311yescharacteristic
44#117×C83—21no—
45C2×C4⋊C4×C83—21no—
46M32×C83—21no—
47M32×C83—21no—
48C2×C16×C83—21no—
49#107×C83C2×C412yes—
50#106×C83—21no—
51#106×C83—21no—
52#106×C83—21no—
53C23×D83—41no—
54C2×D332—22no—
55C2×D332—41no—
56C2×D332—41no—
57C2×D332—41no—
58C2×D332—41no—
59C2×D332—41no—
60C22×Dic83—21no—
61D664—22no—
62D664—22no—
63D664—41no—
64D664—41no—
65#13050574—22no—
66#13050574—41no—
67C2×C83⋊C8—21no—
68C2×C83⋊C8—21no—
69C2×C83⋊C8—41no—
70C2×C83⋊C8SD1612yes—
71(C2×D83)⋊C4—84no—
72C4×Dic83—21no—
73C4×Dic83—21no—
74C83⋊C16—22no—
75C22×C4×C83—41no—
76C22×C4×C83C2×D411yescharacteristic
77C2×C8×C83C22×C411yescharacteristic
78C2×C8×C83C2×D411yescharacteristic
79C2×C8×C83#4111yescharacteristic
80C2×C8×C83#4111yescharacteristic
81C2×C8×C83#4111yescharacteristic
82C2×C8×C83#4111yescharacteristic
83C4⋊C4×C83—22no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—41no—
86C22⋊C4×C83—41no—
87C42×C83—41no—
88C16×C83—41no—
89C23×C83AGL(1,ℤ/8)11yescharacteristic
90C22×D83—42no—
91C22×D83—82no—
92C22×D83—82no—
93C22×D83—81no—
94C22×D83—81no—
95C2×C4×C83—42no—
96C2×C4×C83—82no—
97C2×C4×C83—81no—
98C2×C4×C83C4×D411yescharacteristic
99C2×C4×C83#12111yescharacteristic
100C2×C4×C83#12311yescharacteristic
101C2×C4×C83C2×SD1611yescharacteristic
102C2×C4×C83#13511yescharacteristic
103C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
104C2×Dic83—42no—
105C2×Dic83—41no—
106D332—42no—
107D332—42no—
108D332—81no—
109D332—81no—
110C8×C83#11711yescharacteristic
111C8×C83C4×D411yescharacteristic
112C8×C83#12411yescharacteristic
113C8×C83#12611yescharacteristic
114C83⋊C8—22no—
115C83⋊C8—42no—
116C22×C83—22no—
117C22×C83#44211yescharacteristic
118C22×C83#46011yescharacteristic
119C22×C83#48211yescharacteristic
120C2×D83—84no—
121C2×D83—82no—
122C2×D83—162no—
123C2×D83—161no—
124C2×D83—161no—
125C4×C83—82no—
126C4×C83—161no—
127C4×C83#38811yescharacteristic
128C4×C83C4×SD1611yescharacteristic
129C4×C83#46311yescharacteristic
130C4×C83#48111yescharacteristic
131Dic83—81no—
132#13142—831nomaximal, Sylow
133C2×C83—21no—
134C2×C83—21no—
135C2×C83#192611yescharacteristic
136C2×C83AGL(1,ℤ/16)12yes—
137D83—162no—
138C2×#365—831no—
139C2×#361—831no—
140#2140—831no—
141#1972—831no—
142#1773—831no—
143#1386D8311yescharacteristic, complemented
144#1375—831no—
145C83#1314211yescharacteristic, complemented, Sylow
146C22×D8—831no—
147C2×M32—831no—
148C2×M32C2×D8311yescharacteristic
149C2×#108—831no—
150#427C2×D8311yescharacteristic
151C2×#106C2×D8311yescharacteristic
152C2×#102—1661no—
153#405—1661no—
154C2×#97—831no—
155#390—1661no—
156#365—1662no—
157#361—1662no—
158#357—834no—
159#339—1661no—
160C22×D4—1661no—
161C2×D8—832no—
162C2×D8—1661no—
163C2×D8—1661no—
164C2×D8—1661no—
165C2×D8—1661no—
166C2×D8—1661no—
167C2×M16—1661no—
168C22×C8—831no—
169C22×C8C22×D8311yescharacteristic
170#117—21no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—3322no—
173C2×C42—831no—
174M32—21no—
175M32—21no—
176M32—1661no—
177M32—1661no—
178C2×C16—21no—
179C2×C16—832no—
180#108—832no—
181#108—1661no—
182#107C4×D8312yes—
183#106—21no—
184#106—21no—
185#106—21no—
186#102—3322no—
187#97—3321no—
188#97—1661no—
189#97—1661no—
190C4×C8—3321no—
191C24—3321no—
192C2×D4—3321no—
193C2×D4—3321no—
194C2×D4—3321no—
195C2×D4—3321no—
196C2×D4—3321no—
197C2×D4—1662no—
198C22×C4—41no—
199C22×C4—1661no—
200C22×C4D4×D8311yescharacteristic
201D8—3321no—
202D8—3321no—
203D8—1662no—
204D8—1662no—
205M16—3321no—
206M16—1662no—
207C2×C8—3321no—
208C2×C8—832no—
209C2×C8—1661no—
210C2×C8—1661no—
211C2×C8C2×C4×D8311yescharacteristic
212C2×C8D4×D8311yescharacteristic
213C2×C8#1305058911yescharacteristic
214C2×C8#1305059011yescharacteristic
215C2×C8#1305059011yescharacteristic
216C2×C8#1305059111yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—6644no—
222C42—41no—
223C42—1661no—
224C42—1661no—
225C16—41no—
226C16—1662no—
227C23—3322no—
228C23—6642no—
229C23—6642no—
230C23—6641no—
231C23—6641no—
232C23#1458181611yescharacteristic
233D4—3322no—
234D4—3322no—
235D4—6641no—
236D4—6641no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—3322no—
241C2×C4—3321no—
242C2×C4(C22⋊D83)⋊C411yescharacteristic
243C2×C4#1458176511yescharacteristic
244C2×C4#1458177711yescharacteristic
245C2×C4SD16×D8311yescharacteristic
246C2×C4#1458181511yescharacteristic
247C2×C4#1458181911yescharacteristic
248C8—3322no—
249C8—1662no—
250C8#1458174911yescharacteristic
251C8#1458175411yescharacteristic
252C8#1458177811yescharacteristic
253C8#1458178711yescharacteristic
254C22—22no—
255C22—13281no—
256C22—13281no—
257C22—13282no—
258C22—6644no—
259C22—6642no—
260C22#1912781011yescharacteristic
261C22#1912790011yescharacteristic
262C22#1912802811yescharacteristic
263C4—82no—
264C4—161no—
265C4—6641no—
266C4#1912759011yescharacteristic
267C4#1912778611yescharacteristic
268C4#1912791911yescharacteristic
269C4#1912802111yescharacteristic
270C2—21no—
271C2—21no—
272C2—13282no—
273C2#2490808811yescharacteristic
274C2#2490935812yes—
subgroupG/H
centerC22#19127810
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#1386C2
socleC22×C83#442
prime radicalC22×D664C22
cosocleC22×C8×C83C23
hypercenter#1386D83
hypocenterC83#13142
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
1. C2×#19127523C2semidirect
5. #24907349C2semidirect
6. #24905542C2semidirect
7. C83×#1386C2semidirect
17. #19127501C4semidirect
143. #1386D83semidirect
145. C83#13142semidirect

Maximal subgroups

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

subgroupindex
C2×#191275232
C2×#191275152
#249091782
#249083372
#249073492
#249055422
C83×#13862
#1314283