Finite Group Databasebeta

← #35293724

Subgroups — #35293724

Order 21248.

438 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 83 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#24915409C211yesmaximal, characteristic
2#24909000C212yesmaximal, complemented
3#24908996C211yesmaximal, characteristic
4#24905288C212yesmaximal, complemented
5C83×#1270C211yesmaximal, characteristic
6C2×#14581726—22no—
7C2×#14581723—21no—
8C2×#14581717—22no—
9C2×#14581713—21no—
10#19127640—21no—
11#19127613—22no—
12#19127607C2212yes—
13#19127581—21no—
14#19127576—22no—
15#19127571C2211yescharacteristic
16#19127456C2211yescharacteristic
17#390×C83C2211yescharacteristic
18#336×C83C2212yes—
19C22×Dic166D412yes—
20C23×Dic83—21no—
21C22×C83⋊C8—21no—
22C2×C4⋊Dic83—21no—
23C2×C4⋊Dic83—21no—
24C2×C4⋊Dic83—22no—
25C2×C4⋊Dic83D412yes—
26C2×C22⋊Dic83—22no—
27C2×C22⋊Dic83—21no—
28#14581726—44no—
29#14581723—42no—
30#14581717—44no—
31#14581713—42no—
32C83⋊#95—22no—
33C83⋊#95—21no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C22×D4×C83D411yescharacteristic
37C22×C8×C83—22no—
38C2×C4⋊C4×C83D411yescharacteristic
39C2×C22⋊C4×C83—22no—
40C2×C42×C83C2311yescharacteristic
41C2×Dic166—22no—
42C2×Dic166—22no—
43C2×Dic166—42no—
44C2×Dic166—42no—
45C22×Dic83—22no—
46C22×Dic83—22no—
47C22×Dic83—41no—
48C22×Dic83—82no—
49C22×Dic83—81no—
50C2×C83⋊C8—22no—
51C2×C83⋊C8—41no—
52C2×C83⋊C8—41no—
53C4⋊Dic83—42no—
54C4⋊Dic83—41no—
55C4⋊Dic83—42no—
56C4⋊Dic83—41no—
57C4⋊Dic83—42no—
58C4⋊Dic83—81no—
59C22⋊Dic83—82no—
60C22⋊Dic83—81no—
61C24×C83—21no—
62C2×D4×C83—21no—
63C2×D4×C83—41no—
64C2×D4×C83—41no—
65C2×D4×C83Q1612yes—
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—
73C22⋊C4×C83—42no—
74C22⋊C4×C83—42no—
75C42×C83—41no—
76C23×C83—22no—
77C23×C83—42no—
78C23×C83—41no—
79C23×C83—42no—
80C23×C83—41no—
81C23×C83#12011yescharacteristic
82D4×C83—42no—
83D4×C83—81no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—42no—
89C2×C4×C83—41no—
90C2×C4×C83C2×Q1611yescharacteristic
91C2×C4×C83#13512yes—
92C2×C4×C83#13712yes—
93C2×C4×C83#13711yescharacteristic
94C2×Dic83—42no—
95C2×Dic83—42no—
96C2×Dic83—42no—
97C2×Dic83—42no—
98C2×Dic83—42no—
99C2×Dic83—42no—
100C2×Dic83—84no—
101C2×Dic83—82no—
102C2×Dic83—81no—
103Dic166—44no—
104Dic166—82no—
105C8×C83—44no—
106C83⋊C8—42no—
107C22×C83—42no—
108C22×C83—44no—
109C22×C83—82no—
110C22×C83—81no—
111C22×C83—81no—
112C22×C83#45111yescharacteristic
113C22×C83#45212yes—
114C22×C83#45312yes—
115C22×C83#45311yescharacteristic
116C22×C83#45811yescharacteristic
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—22no—
120C4×C83—22no—
121C4×C83—81no—
122Dic83—82no—
123Dic83—82no—
124Dic83—161no—
125#7833—831nomaximal, Sylow
126C2×C83—82no—
127C2×C83#161411yescharacteristic
128C2×C83#161812yes—
129C2×C83#163011yescharacteristic
130C2×C83#163212yes—
131C2×C83#165511yescharacteristic
132#2804—831no—
133#2095—832no—
134#2094—831no—
135#1272—832no—
136#1270D8311yescharacteristic
137C83#783311yescharacteristic, complemented, Sylow
138C2×#107—1662no—
139C2×#106—1661no—
140C2×#103—1662no—
141C2×#102—1661no—
142#400—1661no—
143#393—1662no—
144#391—832no—
145#390C2×D8311yescharacteristic
146#386—1661no—
147#385—1662no—
148#384—831no—
149#336—831no—
150#336C2×D8312yes—
151C22×Q8—832no—
152C22×D4D33211yescharacteristic
153C23×C4—1661no—
154C22×C8—22no—
155C22×C8—1661no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—1662no—
158C2×C4⋊C4—1661no—
159C2×C4⋊C4—1661no—
160C2×C4⋊C4D33211yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1662no—
163C2×C22⋊C4—1661no—
164C2×C42C22×D8311yescharacteristic
165#107—3324no—
166#106—3322no—
167#103—3324no—
168#102—3322no—
169#95—1662no—
170#95—1661no—
171#95—1662no—
172#95—1662no—
173C24—21no—
174C2×Q8—3322no—
175C2×Q8—3322no—
176C2×Q8—1662no—
177C2×Q8—1662no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4Dic33212yes—
182C22×C4—21no—
183C22×C4—3321no—
184C22×C4—6642no—
185C22×C4—6641no—
186C22×C4—1662no—
187C22×C4—1662no—
188C22×C4D4×D8312yes—
189C22×C4C2×D33211yescharacteristic
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—3321no—
194C2×C8—3321no—
195C2×C8—1662no—
196C4⋊C4—41no—
197C4⋊C4—3322no—
198C4⋊C4—3321no—
199C4⋊C4—3322no—
200C4⋊C4—3321no—
201C4⋊C4—3322no—
202C4⋊C4—6641no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—6642no—
206C22⋊C4—6641no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1458176211yescharacteristic
214Q8—3324no—
215Q8—6642no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—3322no—
225C2×C4—3322no—
226C2×C4—3322no—
227C2×C4—3322no—
228C2×C4—3322no—
229C2×C4—3322no—
230C2×C4—6642no—
231C2×C4—6641no—
232C2×C4—6644no—
233C2×C4C2×Dic33211yescharacteristic
234C2×C4#1458181112yes—
235C2×C4#1458182112yes—
236C2×C4#1458182511yescharacteristic
237C8—44no—
238C8—3322no—
239C22—44no—
240C22—42no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1912785911yescharacteristic
245C22#1912786412yes—
246C22#1912787012yes—
247C22#1912787111yescharacteristic
248C22#1912789211yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—13281no—
255C4—6642no—
256C4—6642no—
257C2—82no—
258C2#2490649011yescharacteristic
259C2#2490651212yes—
260C2#2490658511yescharacteristic
261C2#2490659912yes—
262C2#2490675311yescharacteristic
subgroupG/H
centerC23#14581762
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1270C2
socleC23×C83#120
prime radicalC22×D4×C83D4
hypercenter#1270D83
hypocenterC83#7833
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #24909000C2semidirect
4. #24905288C2semidirect
137. C83#7833semidirect

Maximal subgroups

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

subgroupindex
#249154092
#249090002
#249089962
#249052882
C83×#12702
#783383