Finite Group Databasebeta

← #35297421

Subgroups — #35297421

Order 21248.

404 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 101 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#24916174C211yesmaximal, characteristic
2#24906218C212yesmaximal, complemented
3#24906196C212yesmaximal
4C83⋊#1309C211yesmaximal, characteristic, complemented
5C83×#1309C211yesmaximal, characteristic, complemented
6C2×#14581800C412yes—
7#19128251C2211yescharacteristic
8#19127778—22no—
9#19127766—22no—
10#19127725C2212yes—
11#19127669C412yes—
12C2×C4×C83⋊C8C2211yescharacteristic
13#19127451—22no—
14#19127451—22no—
15#19127449—22no—
16#19127444—22no—
17#423×C83C2212yes—
18#423×C83C2211yescharacteristic
19C2×#13050590C2×C412yes—
20#14581800—44no—
21C2×#13050574—22no—
22C2×#13050573—22no—
23C22×C83⋊C8C2×C411yescharacteristic
24C22×C83⋊C8C2×C411yescharacteristic
25C2×C8×D83—22no—
26#14581792D411yescharacteristic
27#14581789D411yescharacteristic
28#14581780Q812yes—
29#14581766—22no—
30C4×D332—22no—
31C4⋊(C83⋊C8)—22no—
32C4⋊(C83⋊C8)—22no—
33(C2×D83)⋊C8—24no—
34C4×C83⋊C8D412yes—
35C4×C83⋊C8D412yes—
36C2×M16×C83—22no—
37C2×M16×C83C2×C412yes—
38C2×C42×C83C2311yescharacteristic
39C4⋊C8×C83—22no—
40C4⋊C8×C83—22no—
41C4⋊C8×C83—22no—
42#13050590—42no—
43#13050590—42no—
44C2×Dic166C4⋊C412yes—
45C2×D332—22no—
46C2×D332C4⋊C412yes—
47C2×C22⋊D83—22no—
48C2×C4×D83—22no—
49#13050574—82no—
50#13050573—44no—
51C2×C83⋊C8—22no—
52C2×C83⋊C8—41no—
53C2×C83⋊C8—42no—
54C2×C83⋊C8C22⋊C411yescharacteristic
55C2×C83⋊C8C22⋊C412yes—
56C2×C83⋊C8C22⋊C411yescharacteristic
57C8×D83—44no—
58C4⋊Dic83—22no—
59(C2×D83)⋊C4—24no—
60C22×C4×C83C22×C411yescharacteristic
61C22×C4×C83#4112yes—
62M16×C83—42no—
63M16×C83—82no—
64C2×C8×C83—22no—
65C2×C8×C83—42no—
66C42×C83C2×D412yes—
67C42×C83C2×D411yescharacteristic
68C42×C83C2×Q811yescharacteristic
69C23×C83#11711yescharacteristic
70C22×D83—22no—
71C2×C4×C83—21no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83C2×C22⋊C411yescharacteristic
76C2×C4×C83C2×C4⋊C411yescharacteristic
77C2×C4×C83#12112yes—
78C2×C4×C83C22⋊Q812yes—
79C22⋊D83—44no—
80C2×Dic83—22no—
81D332—42no—
82D332—42no—
83D332—42no—
84C4×D83—42no—
85C4×D83—42no—
86Dic166—42no—
87C8×C83—42no—
88C8×C83—82no—
89C83⋊C8—21no—
90C83⋊C8—22no—
91C83⋊C8—82no—
92C83⋊C8#10012yes—
93C22×C83—21no—
94C22×C83—22no—
95C22×C83#38011yescharacteristic
96C2×D83—42no—
97C2×D83—44no—
98C4×C83—21no—
99C4×C83—22no—
100C4×C83C2×#10011yescharacteristic
101C4×C83#41311yescharacteristic
102C4×C83#47112yes—
103C4×C83#47112yes—
104Dic83—42no—
105#8636—831nomaximal, Sylow
106C2×C83—41no—
107C2×C83#180011yescharacteristic
108C2×C83#181512yes—
109D83—82no—
110#2931—831no—
111#1566—832no—
112#1560—832no—
113#1309—831no—
114#1309D8311yescharacteristic, complemented
115C83#863611yescharacteristic, complemented, Sylow
116C2×#131—832no—
117#532—831no—
118#436—1662no—
119C8×D4—1662no—
120#423—832no—
121#423C2×D8312yes—
122#423C2×D8311yescharacteristic
123#408—832no—
124C2×C4×C8—831no—
125#334—1662no—
126#334—1662no—
127#333—1662no—
128#331—1662no—
129C2×#41—832no—
130#131—3324no—
131C2×M16—22no—
132C2×M16—1662no—
133C2×M16—1662no—
134C2×M16C4×D8312yes—
135C22×C8—831no—
136C22×C8—831no—
137C22×C8—1662no—
138C4⋊Q8—831no—
139#127—831no—
140#124—832no—
141#121—1662no—
142C4×D4—1662no—
143C2×C42C22×D8311yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—1662no—
148C4⋊C8—1662no—
149C22⋊C8—1664no—
150C4×C8—832no—
151C4×C8—832no—
152#41—3322no—
153#41—3322no—
154C2×Q8—832no—
155C2×D4—832no—
156C2×D4—1662no—
157C2×D4—1662no—
158C22×C4—1662no—
159C22×C4C2×C4×D8311yescharacteristic
160C22×C4#1305059012yes—
161M16—42no—
162M16—82no—
163M16—3324no—
164M16—6642no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—3324no—
168C2×C8—3322no—
169C2×C8—3321no—
170C2×C8—831no—
171C2×C8—832no—
172C2×C8—831no—
173C2×C8—1662no—
174C4⋊C4—1662no—
175C22⋊C4—1664no—
176C42D4×D8311yescharacteristic
177C42D4×D8312yes—
178C42Q8×D8311yescharacteristic
179C23—1662no—
180C23#1458174911yescharacteristic
181Q8—3322no—
182D4—3322no—
183D4—3324no—
184D4—3322no—
185D4—3322no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—3322no—
191C2×C4—3322no—
192C2×C4—1662no—
193C2×C4C22⋊C4×D8311yescharacteristic
194C2×C4C4⋊C4×D8311yescharacteristic
195C2×C4#1458176512yes—
196C2×C4#1458177012yes—
197C8—42no—
198C8—82no—
199C8—832no—
200C8—6642no—
201C8—1661no—
202C8—1662no—
203C22—22no—
204C22—21no—
205C22—3324no—
206C22—3322no—
207C22#1912756011yescharacteristic
208C4—22no—
209C4—21no—
210C4—3322no—
211C4#100×D8311yescharacteristic
212C4#1912768411yescharacteristic
213C4#1912796312yes—
214C4#1912796712yes—
215C2—41no—
216C2—6642no—
217C2#2490745411yescharacteristic
218C2#2490752112yes—
subgroupG/H
centerC22#19127560
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1309C2
socleC22×C83#380
prime radical#19128251C22
cosocleC2×C42×C83C23
hypercenter#1309D83
hypocenterC83#8636
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #24906218C2semidirect
4. C83⋊#1309C2semidirect
5. C83×#1309C2semidirect
114. #1309D83semidirect
115. C83#8636semidirect

Maximal subgroups

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

subgroupindex
#249161742
#249062182
#249061962
C83⋊#13092
C83×#13092
#863683