Finite Group Databasebeta

← #35307478

Subgroups — #35307478

Order 21248.

400 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#24915735C211yesmaximal, characteristic
2#24908436C212yesmaximal
3#24908282C211yesmaximal, characteristic
4#24905345C212yesmaximal
5C83×#1295C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7C2×#14581715—21no—
8C2×#14581715—21no—
9C2×#14581703C2211yescharacteristic
10#19127643—22no—
11#19127640—21no—
12#19127622—22no—
13#19127621—21no—
14#19127567C2212yes—
15#19127464—22no—
16#19127464—22no—
17C2×#97×C83C2212yes—
18#382×C83C2211yescharacteristic
19C22×Dic166—21no—
20C22×Dic166—21no—
21C22×C83⋊C8—21no—
22#14581792—22no—
23#14581792—21no—
24#14581792Q812yescomplemented
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—22no—
27C2×C4⋊Dic83D412yes—
28C2×C4⋊Dic83D412yes—
29C2×Dic83⋊C4—22no—
30#14581715—42no—
31#14581715—42no—
32#14581703—22no—
33C83⋊#95—21no—
34C83⋊#95—22no—
35C83⋊#95—21no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C22×C8×C83—22no—
39C2×C4⋊C4×C83D411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C42×C83C2311yescharacteristic
42#97×C83—22no—
43#97×C83—22no—
44#95×C83—22no—
45C2×Dic166—22no—
46C2×Dic166—41no—
47C2×Dic166—41no—
48C2×Dic166—41no—
49C2×Dic166—41no—
50C2×Dic166—41no—
51C2×Dic166—81no—
52C2×Dic166—82no—
53C22×Dic83—22no—
54C22×Dic83—22no—
55C22×Dic83—41no—
56C2×C83⋊C8—22no—
57C2×C83⋊C8—41no—
58C2×C83⋊C8—41no—
59C4⋊Dic83—24no—
60C4⋊Dic83—42no—
61C4⋊Dic83—42no—
62C4⋊Dic83—42no—
63C4⋊Dic83—81no—
64Dic83⋊C4—82no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83C2×D412yes—
68C22×C4×C83C2×D411yescharacteristic
69C2×C8×C83—24no—
70C2×C8×C83—42no—
71C2×C8×C83—42no—
72C4⋊C4×C83—41no—
73C4⋊C4×C83—41no—
74C42×C83C2×Q811yescharacteristic
75C42×C83#4111yescharacteristic
76C42×C83#4112yes—
77C23×C83#12011yescharacteristic
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—41no—
83C2×C4×C83—41no—
84C2×C4×C83—42no—
85C2×C4×C83—42no—
86C2×C4×C83C22⋊Q812yes—
87C2×C4×C83C22⋊Q811yescharacteristic
88C2×C4×C83#12412yes—
89C2×C4×C83#12411yescharacteristic
90C2×Dic83—42no—
91C2×Dic83—42no—
92C2×Dic83—42no—
93C2×Dic83—42no—
94C2×Dic83—42no—
95C2×Dic83—42no—
96C2×Dic83—42no—
97C2×Dic83—81no—
98C2×Dic83—81no—
99Dic166—42no—
100Dic166—42no—
101Dic166—81no—
102Dic166—81no—
103Dic166—84no—
104C8×C83—44no—
105C83⋊C8—42no—
106C22×C83#39311yescharacteristic
107C22×C83#45311yescharacteristic
108C22×C83#45411yescharacteristic
109C22×C83#45512yes—
110C22×C83#45612yes—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—22no—
114C4×C83—22no—
115C4×C83—81no—
116C4×C83—81no—
117Dic83—82no—
118Dic83—82no—
119Dic83—82no—
120#10359—831nomaximal, Sylow
121C2×C83#165911yescharacteristic
122C2×C83#166011yescharacteristic
123C2×C83#166412yes—
124C2×C83#202711yescharacteristic
125C2×C83#202812yes—
126#2844—831no—
127#1993—832no—
128#1960—831no—
129#1295—832no—
130#1295D8311yescharacteristic
131C83#1035911yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—831no—
133C2×#103—1661no—
134C2×#103—1661no—
135C2×#97—831no—
136C2×#97C2×D8312yes—
137#400—1662no—
138#400—1661no—
139#395—1662no—
140#395—1661no—
141#382—832no—
142#382C2×D8311yescharacteristic
143#340—1662no—
144#340—1662no—
145C22×Q8—1661no—
146C22×Q8—1661no—
147C22×C8—22no—
148C22×C8—1661no—
149C4⋊Q8—832no—
150C4⋊Q8—1661no—
151C4⋊Q8—1662no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—832no—
154C2×C4⋊C4—1662no—
155C2×C4⋊C4—1662no—
156C2×C4⋊C4—1661no—
157C2×C4⋊C4D33211yescharacteristic
158C2×C4⋊C4D33211yescharacteristic
159C2×C42C22×D8311yescharacteristic
160#103—3322no—
161#103—3322no—
162#97—22no—
163#97—22no—
164#97—1662no—
165#95—22no—
166#95—1662no—
167#95—1661no—
168#95—1661no—
169#95—1662no—
170#95—1662no—
171C2×Q8—3321no—
172C2×Q8—3321no—
173C2×Q8—3321no—
174C2×Q8—3321no—
175C2×Q8—3321no—
176C2×Q8—6641no—
177C2×Q8—6642no—
178C2×Q8—1662no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—3321no—
182C22×C4—1662no—
183C22×C4—1662no—
184C22×C4D4×D8312yes—
185C22×C4C2×D33211yescharacteristic
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—3321no—
190C2×C8—3321no—
191C2×C8—1662no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—3322no—
195C4⋊C4—3322no—
196C4⋊C4—3322no—
197C4⋊C4—6642no—
198C4⋊C4—6641no—
199C4⋊C4—1664no—
200C42Q8×D8311yescharacteristic
201C42#1305058911yescharacteristic
202C42#1305059012yes—
203C23#1458176211yescharacteristic
204Q8—3322no—
205Q8—3322no—
206Q8—6641no—
207Q8—6641no—
208Q8—6644no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—3322no—
218C2×C4—3322no—
219C2×C4—3322no—
220C2×C4—3322no—
221C2×C4—3322no—
222C2×C4—3322no—
223C2×C4—3322no—
224C2×C4—6641no—
225C2×C4—6641no—
226C2×C4#1458177012yes—
227C2×C4#1458177211yescharacteristic
228C2×C4#1458177812yes—
229C2×C4#1458178011yescharacteristic
230C8—44no—
231C8—3322no—
232C22#1912761411yescharacteristic
233C22#1912787111yescharacteristic
234C22#1912787611yescharacteristic
235C22#1912788012yes—
236C22#1912788512yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—81no—
243C4—6642no—
244C4—6642no—
245C4—6642no—
246C2#2490677011yescharacteristic
247C2#2490677511yescharacteristic
248C2#2490679812yes—
249C2#2490862011yescharacteristic
250C2#2490862312yes—
subgroupG/H
centerC23#14581762
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1295C2
socle, prime radicalC23×C83#120
hypercenter#1295D83
hypocenterC83#10359
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
24. #14581792Q8semidirect
131. C83#10359semidirect

Maximal subgroups

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

subgroupindex
#249157352
#249084362
#249082822
#249053452
C83×#12952
#1035983