Finite Group Databasebeta

← #35302018

Subgroups — #35302018

Order 21248.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#24917839C211yesmaximal, characteristic
2#24906201C211yesmaximal, characteristic, complemented
3#24906130C212yesmaximal
4#24905354C212yesmaximal, complemented
5C83×#2986C211yesmaximal, characteristic, complemented
6C2×#14581789C2211yescharacteristic
7#19128067C414yes—
8#19128049—22no—
9#19127725C2212yes—
10#19127705—22no—
11C4×#13050574C2211yescharacteristic
12#19127447—22no—
13#19127444—22no—
14#19127444—22no—
15#533×C83C2211yescharacteristic
16#446×C83—22no—
17C8×Q8×C83—22no—
18#433×C83—22no—
19#431×C83C2212yes—
20C22×D332—21no—
21C22×D332—22no—
22C2×#13050574—21no—
23C2×#13050574—22no—
24#14581789—22no—
25#14581789—21no—
26#14581789C2×C412yes—
27C4⋊(C83⋊C8)—22no—
28C4⋊(C83⋊C8)—22no—
29#14581712—28no—
30#14581703C2×C412yes—
31C4×C83⋊C8—21no—
32C4⋊Q8×C83D412yes—
33#125×C83C2×C412yes—
34C22⋊Q8×C83—22no—
35C4×Q8×C83—22no—
36#117×C83D412yes—
37C2×C42×C83C2311yescharacteristic
38C4⋊C8×C83—24no—
39C4⋊C8×C83D412yes—
40C4⋊C8×C83D412yes—
41C22⋊C8×C83—22no—
42#97×C83—22no—
43C4×C8×C83—22no—
44C4×C8×C83—22no—
45C23×D83—41no—
46C2×D332—22no—
47C2×D332—24no—
48C2×D332—44no—
49C2×D332—41no—
50C2×D332—42no—
51C2×D332—82no—
52C2×D332—82no—
53#13050574—42no—
54#13050574—82no—
55C2×C83⋊C8—22no—
56C2×C83⋊C8—42no—
57C2×Q8×C83—22no—
58C22×C4×C83C2×D411yescharacteristic
59C22×C4×C83#4112yes—
60C2×C8×C83—22no—
61C2×C8×C83—22no—
62C4⋊C4×C83—24no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83C22⋊C414yes—
65C22⋊C4×C83—22no—
66C42×C83—22no—
67C42×C83C22×C411yescharacteristic
68C42×C83C2×D412yes—
69C42×C83C2×D411yescharacteristic
70C23×C83#12311yescharacteristic
71C22×D83—42no—
72C22×D83—82no—
73C22×D83—84no—
74Q8×C83—42no—
75Q8×C83—42no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—24no—
80C2×C4×C83C2×C22⋊C411yescharacteristic
81C2×C4×C83C4×D412yes—
82C2×C4×C83#12011yescharacteristic
83C2×C4×C83#12112yes—
84D332—44no—
85D332—84no—
86D332—82no—
87D332—82no—
88D332—82no—
89C8×C83—42no—
90C8×C83—42no—
91C83⋊C8—42no—
92C83⋊C8—82no—
93C22×C83—21no—
94C22×C83—22no—
95C22×C83#38611yescharacteristic
96C2×D83—82no—
97C2×D83—84no—
98C2×D83—164no—
99C4×C83—21no—
100C4×C83—22no—
101C4×C83—44no—
102C4×C83#45412yes—
103C4×C83#47112yes—
104C4×C83#47212yes—
105#9438—831nomaximal, Sylow
106C2×C83—41no—
107C2×C83#188811yescharacteristic
108C2×C83#190712yes—
109D83—162no—
110#3138—831no—
111#2986D8311yescharacteristic, complemented
112#1561—831no—
113#1545—832no—
114#1299—832no—
115C83#943811yescharacteristic, complemented, Sylow
116#533C2×D8311yescharacteristic
117C2×#127—831no—
118#490—834no—
119#486—1662no—
120#446—22no—
121C8×Q8—22no—
122#433—22no—
123#431C2×D8312yes—
124#423—832no—
125#418—1662no—
126C4×M16—831no—
127#332—1662no—
128#331—1662no—
129#331—1662no—
130C22×D4—1661no—
131C22×D4—1662no—
132C2×M16—1662no—
133C2×M16—1661no—
134C4⋊Q8D33212yes—
135#127—832no—
136#127—1662no—
137#127—1661no—
138#125C4×D8312yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D8312yes—
142C2×C42C22×D8311yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—1662no—
145C4⋊C8—1662no—
146C4⋊C8C22⋊D8312yes—
147C4⋊C8C22⋊D8312yes—
148#102—1668no—
149C22⋊C8—22no—
150#97—22no—
151#97—832no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—1661no—
155C24—3321no—
156C2×Q8—22no—
157C2×D4—3324no—
158C2×D4—3321no—
159C2×D4—3322no—
160C2×D4—6642no—
161C2×D4—6642no—
162C2×D4—1662no—
163C2×D4—1664no—
164C22×C4C2×C22⋊D8311yescharacteristic
165C22×C4#1305059012yes—
166M16—3322no—
167M16—6642no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—3322no—
171C2×C8—1662no—
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4(C2×D83)⋊C414yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D8311yescharacteristic
178C42C2×C22⋊D8312yes—
179C42C2×D33211yescharacteristic
180C23—3322no—
181C23—6642no—
182C23—6644no—
183C23#1458177511yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—3324no—
187D4—6644no—
188D4—6642no—
189D4—6642no—
190D4—6642no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×(C2×D83)⋊C411yescharacteristic
196C2×C4C4×C22⋊D8312yes—
197C2×C4#1458176111yescharacteristic
198C2×C4#1458176612yes—
199C8—42no—
200C8—42no—
201C8—3322no—
202C8—6642no—
203C22—21no—
204C22—22no—
205C22—13284no—
206C22—6642no—
207C22—6644no—
208C22#1912758011yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1912787412yes—
213C4#1912796412yes—
214C4#1912797112yes—
215C2—41no—
216C2—13282no—
217C2#2490786611yescharacteristic
218C2#2490797112yes—
subgroupG/H
centerC22#19127580
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2986C2
socleC22×C83#386
prime radicalC2×#14581789C22
cosocleC2×C42×C83C23
hypercenter#2986D83
hypocenterC83#9438
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #24906201C2semidirect
4. #24905354C2semidirect
5. C83×#2986C2semidirect
111. #2986D83semidirect
115. C83#9438semidirect

Maximal subgroups

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

subgroupindex
#249178392
#249062012
#249061302
#249053542
C83×#29862
#943883