Finite Group Databasebeta

← #35307077

Subgroups — #35307077

Order 21248.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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#24918385C212yesmaximal
2#24915841C211yesmaximal, characteristic, complemented
3#24906145C212yesmaximal, complemented
4#24905296C211yesmaximal, characteristic, complemented
5C83×#1962C211yesmaximal, characteristic
6C2×#14581792C2212yes—
7#19128073—22no—
8#19128053—22no—
9#19127906—24no—
10#19127839—24no—
11#19127754C2212yes—
12#19127607—24no—
13#19127567—22no—
14#19127449—24no—
15#532×C83C2211yescharacteristic
16#423×C83C2211yescharacteristic
17C2×#104×C83—22no—
18C43×C83C2211yescharacteristic
19C22×Dic166D414yes—
20C2×#13050578—28no—
21#14581792—22no—
22#14581792—22no—
23#14581792—22no—
24#14581792—22no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83—22no—
27C4⋊(C83⋊C8)D414yes—
28#14581715—28no—
29C22⋊(C83⋊C8)—22no—
30C4×C83⋊C8—22no—
31C83⋊#95—28no—
32C2×#41×C83—21no—
33C2×M16×C83—21no—
34C4⋊Q8×C83D411yescharacteristic
35#127×C83D411yescharacteristic
36#124×C83—21no—
37#121×C83—22no—
38C4×D4×C83—22no—
39C2×C42×C83—22no—
40C2×C42×C83D412yes—
41C2×C42×C83C2311yescharacteristic
42C4⋊C8×C83—22no—
43#104×C83—44no—
44C2×Dic166—24no—
45C2×Dic166—24no—
46C2×Dic166—48no—
47C22×Dic83—24no—
48#13050578—88no—
49C2×C83⋊C8—24no—
50C4⋊Dic83—42no—
51C4⋊Dic83—42no—
52C4⋊Dic83—42no—
53C4⋊Dic83—42no—
54#41×C83—42no—
55#41×C83—42no—
56C2×Q8×C83—21no—
57C2×D4×C83—22no—
58C2×D4×C83—21no—
59C2×D4×C83—41no—
60C22×C4×C83—22no—
61C22×C4×C83—41no—
62C22×C4×C83C2×D412yes—
63C22×C4×C83C2×D411yescharacteristic
64M16×C83—42no—
65C2×C8×C83—41no—
66C4⋊C4×C83—22no—
67C22⋊C4×C83—42no—
68C42×C83—22no—
69C42×C83—44no—
70C42×C83SD1614yescomplemented
71C42×C83C2×D412yes—
72C42×C83C2×D411yescharacteristic
73C42×C83#4111yescharacteristic
74C23×C83—41no—
75C23×C83#12011yescharacteristic
76Q8×C83—42no—
77D4×C83—44no—
78D4×C83—42no—
79D4×C83—82no—
80C2×C4×C83—24no—
81C2×C4×C83—24no—
82C2×C4×C83—44no—
83C2×C4×C83—41no—
84C2×C4×C83—81no—
85C2×C4×C83—81no—
86C2×C4×C83#12012yes—
87C2×C4×C83#12112yes—
88C2×C4×C83#12111yescharacteristic
89C2×C4×C83#12711yescharacteristic
90C2×C4×C83C2×SD1612yes—
91C2×C4×C83#13712yes—
92C2×Dic83—48no—
93C2×Dic83—44no—
94Dic166—48no—
95Dic166—84no—
96C8×C83—81no—
97C83⋊C8—44no—
98C22×C83—22no—
99C22×C83—82no—
100C22×C83—81no—
101C22×C83#39211yescharacteristic
102C22×C83#44812yes—
103C4×C83—22no—
104C4×C83—44no—
105C4×C83—81no—
106C4×C83#45514yes—
107C4×C83#46112yes—
108C4×C83#49211yescharacteristic
109C4×C83#49711yescharacteristic
110Dic83—84no—
111#10287—831nomaximal, Sylow
112C2×C83—22no—
113C2×C83—161no—
114C2×C83#200412yes—
115C2×C83#201311yescharacteristic
116#3206—832no—
117#2862—831no—
118#1962D8311yescharacteristic
119#1548—832no—
120#1276—831no—
121C83#1028711yescharacteristic, complemented, Sylow
122#532C2×D8311yescharacteristic
123C2×C4⋊Q8—832no—
124#492—1662no—
125#487—1662no—
126#461—1664no—
127#448—1664no—
128#431—832no—
129#423C2×D8311yescharacteristic
130C2×#104—22no—
131#391—1664no—
132#382—1662no—
133C43C2×D8311yescharacteristic
134#333—1664no—
135C2×#41—21no—
136C22×Q8—834no—
137C2×SD16—1668no—
138C2×M16—21no—
139C4⋊Q8—1662no—
140C4⋊Q8—1662no—
141C4⋊Q8—1662no—
142C4⋊Q8—1662no—
143C4⋊Q8D33211yescharacteristic
144#127D33211yescharacteristic
145#124—21no—
146#121—22no—
147C4×D4—22no—
148C2×C4⋊C4—1662no—
149C2×C4⋊C4—1662no—
150C2×C42—22no—
151C2×C42C22⋊D8312yes—
152C2×C42C22×D8311yescharacteristic
153C4⋊C8—22no—
154C4⋊C8—834no—
155#104—44no—
156#103—1668no—
157C22⋊C8—1662no—
158C4×C8—1662no—
159#95—1668no—
160#41—42no—
161#41—42no—
162C2×Q8—21no—
163C2×Q8—3328no—
164C2×Q8—1664no—
165C2×Q8—1664no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—41no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—1664no—
172C22×C4D4×D8312yes—
173C22×C4C2×C22⋊D8311yescharacteristic
174SD16—6648no—
175M16—42no—
176C2×C8—41no—
177C2×C8—1664no—
178C4⋊C4—22no—
179C4⋊C4—3322no—
180C4⋊C4—3322no—
181C4⋊C4—3322no—
182C4⋊C4—3322no—
183C22⋊C4—42no—
184C42—22no—
185C42—44no—
186C42#1305057814yescomplemented
187C42D4×D8312yes—
188C42C2×D33211yescharacteristic
189C42#1305059011yescharacteristic
190C23—41no—
191C23#1458176011yescharacteristic
192Q8—42no—
193Q8—3328no—
194Q8—6644no—
195D4—44no—
196D4—42no—
197D4—82no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—41no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—3328no—
205C2×C4—3324no—
206C2×C4#1458176212yes—
207C2×C4#1458176512yes—
208C2×C4#1458176611yescharacteristic
209C2×C4#1458178811yescharacteristic
210C2×C4C2×#1305057812yes—
211C2×C4#1458182212yes—
212C8—81no—
213C8—3324no—
214C22—22no—
215C22—81no—
216C22—82no—
217C22#1912761011yescharacteristic
218C22#1912783912yes—
219C4—22no—
220C4—44no—
221C4—81no—
222C4—6644no—
223C4#1912788014yes—
224C4#1912790612yes—
225C4#1912807311yescharacteristic
226C4#1912809611yescharacteristic
227C2—22no—
228C2—161no—
229C2#2490849112yes—
230C2#2490854511yescharacteristic
subgroupG/H
centerC22#19127610
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1962C2
socleC22×C83#392
prime radical#532×C83C22
hypercenter#1962D83
hypocenterC83#10287
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #24915841C2semidirect
3. #24906145C2semidirect
4. #24905296C2semidirect
70. C42×C83SD16semidirect
121. C83#10287semidirect
186. C42#13050578semidirect

Maximal subgroups

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

subgroupindex
#249183852
#249158412
#249061452
#249052962
C83×#19622
#1028783