Finite Group Databasebeta

← #35291989

Subgroups — #35291989

Order 21248.

226 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 43 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#24905588C211yesmaximal, characteristic
2#24905395C211yesmaximal, characteristic
3C83×#2025C211yesmaximal, characteristic
4C2×#14581707—21no—
5#19127427—21no—
6C2×#125×C83C411yescharacteristic, complemented
7C2×#102×C83—21no—
8#390×C83C2211yescharacteristic
9#340×C83—21no—
10#336×C83C411yescharacteristic, complemented
11C2×C22⋊Dic83—41no—
12#14581707—22no—
13#14581707—41no—
14C22⋊(C83⋊C8)—41no—
15C22×D4×C83D411yescharacteristic
16C22×C8×C83—21no—
17#125×C83—42no—
18C2×C4⋊C4×C83—21no—
19C2×C4⋊C4×C83—41no—
20C2×C4⋊C4×C83D411yescharacteristic
21C2×C22⋊C4×C83—41no—
22C2×C42×C83C2×C411yescharacteristic
23#102×C83—42no—
24C22×Dic83—81no—
25C2×C83⋊C8—81no—
26C22⋊Dic83—42no—
27C22⋊Dic83—81no—
28C24×C83—21no—
29C2×D4×C83—21no—
30C2×D4×C83—41no—
31C2×D4×C83—41no—
32C2×D4×C83SD1611yescharacteristic
33C2×D4×C83Q1611yescharacteristic
34C22×C4×C83—21no—
35C22×C4×C83—21no—
36C22×C4×C83—41no—
37C22×C4×C83C22⋊C411yescharacteristic
38C2×C8×C83—41no—
39C2×C8×C83—41no—
40C2×C8×C83—41no—
41C4⋊C4×C83—22no—
42C4⋊C4×C83—41no—
43C4⋊C4×C83—41no—
44C4⋊C4×C83—42no—
45C4⋊C4×C83—42no—
46C22⋊C4×C83—82no—
47C42×C83—41no—
48C23×C83—21no—
49C23×C83—21no—
50C23×C83—41no—
51C23×C83—41no—
52C23×C83—81no—
53C23×C83—81no—
54C23×C83#9911yescharacteristic
55D4×C83—41no—
56D4×C83—41no—
57D4×C83—81no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—41no—
61C2×C4×C83—41no—
62C2×C4×C83—41no—
63C2×C4×C83—42no—
64C2×C4×C83—41no—
65C2×C4×C83—81no—
66C2×C4×C83—81no—
67C2×C4×C83—81no—
68C2×C4×C83#10311yescharacteristic
69C2×C4×C83#10411yescharacteristic
70C2×Dic83—82no—
71C2×Dic83—161no—
72C2×Dic83—161no—
73C8×C83—81no—
74C83⋊C8—161no—
75C22×C83—21no—
76C22×C83—21no—
77C22×C83—41no—
78C22×C83—41no—
79C22×C83—81no—
80C22×C83—81no—
81C22×C83—81no—
82C22×C83—81no—
83C22×C83—161no—
84C22×C83#32811yescharacteristic
85C22×C83AΣL(1,𝔽16)11yescharacteristic
86C22×C83#35411yescharacteristic
87C4×C83—21no—
88C4×C83—21no—
89C4×C83—81no—
90C4×C83—82no—
91C4×C83—81no—
92Dic83—162no—
93#7428—831nomaximal, Sylow
94C2×C83—21no—
95C2×C83—21no—
96C2×C83—81no—
97C2×C83—81no—
98C2×C83#135111yescharacteristic
99C2×C83#141211yescharacteristic
100C2×C83#141411yescharacteristic
101#2025D8311yescharacteristic
102#1393—831no—
103#1318—831no—
104C83#742811yescharacteristic, complemented, Sylow
105C2×#125Dic8311yescharacteristic, complemented
106C2×#102—21no—
107C2×#99—1661no—
108#390C2×D8311yescharacteristic
109#340—21no—
110#336Dic8311yescharacteristic, complemented
111#324—1661no—
112C22×D4C22⋊D8311yescharacteristic
113C22×C8—21no—
114#125—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊D8311yescharacteristic
118C2×C22⋊C4—41no—
119C2×C22⋊C4—3321no—
120C2×C42C2×Dic8311yescharacteristic
121#102—42no—
122#99—3321no—
123#99—1662no—
124C22⋊C8—3321no—
125C24—21no—
126C2×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4#1305057711yescharacteristic
130C2×D4#1305058011yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—6641no—
135C22×C4C22⋊Dic8311yescharacteristic
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—6641no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—3322no—
147C22⋊C4—6641no—
148C42—41no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23#1458170711yescharacteristic
156D4—41no—
157D4—41no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—13281no—
170C2×C4—13281no—
171C2×C4—6642no—
172C2×C4#1458171611yescharacteristic
173C2×C4#1458171911yescharacteristic
174C8—81no—
175C8—13281no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—161no—
185C22#1912743811yescharacteristic
186C22#1912748811yescharacteristic
187C22#1912749611yescharacteristic
188C4—21no—
189C4—21no—
190C4—81no—
191C4—82no—
192C4—81no—
193C4—13282no—
194C2—21no—
195C2—21no—
196C2—81no—
197C2—81no—
198C2#2490548411yescharacteristic
199C2#2490563411yescharacteristic
200C2#2490563811yescharacteristic
subgroupG/H
centerC22#19127496
commutatorC2×C42×C83C2×C4
Frattini#390C2×D83
FittingC83×#2025C2
socleC22×C83#354
prime radicalC22×D4×C83D4
cosocle#390×C83C22
hypercenter#2025D83
hypocenterC83#7428
LCSG ▷ C2×C42×C83 ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2025

Complements

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

NH ≅ G/Ntype
6. C2×#125×C83C4semidirect
10. #336×C83C4semidirect
104. C83#7428semidirect
105. C2×#125Dic83semidirect
110. #336Dic83semidirect

Maximal subgroups

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

subgroupindex
#249055882
#249053952
C83×#20252
#742883