Finite Group Databasebeta

← #35302009

Subgroups — #35302009

Order 21248.

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

normal subgroups only

This group has 400 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#24916565C211yesmaximal, characteristic
2#24906201C211yesmaximal, characteristic
3#24906148C212yesmaximal
4#24905353C212yesmaximal
5C83×#3139C211yesmaximal, characteristic
6#19127832—22no—
7Q8×C83⋊C8—22no—
8#19127762—22no—
9#19127754C2212yes—
10#19127725C2212yes—
11#19127447—22no—
12#19127447—22no—
13#19127447—22no—
14#533×C83C2211yescharacteristic
15C2×C4⋊Q8×C83C2211yescharacteristic
16#491×C83C414yes—
17#487×C83—22no—
18#419×C83—22no—
19C4×M16×C83C2211yescharacteristic
20C2×#13050574—22no—
21C4⋊(C83⋊C8)—24no—
22C4⋊(C83⋊C8)—22no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)D412yes—
25C4⋊(C83⋊C8)D412yes—
26C22⋊(C83⋊C8)—22no—
27#14581703—22no—
28C4×C83⋊C8—22no—
29C4×C83⋊C8—22no—
30C22×Q8×C83—21no—
31C2×M16×C83—21no—
32C4⋊Q8×C83—22no—
33C4⋊Q8×C83—21no—
34C4⋊Q8×C83C2×C412yes—
35C4⋊Q8×C83D412yes—
36#125×C83C2×C412yes—
37C22⋊Q8×C83—22no—
38C4×Q8×C83—22no—
39#117×C83D412yes—
40C2×C4⋊C4×C83—22no—
41C2×C42×C83C2311yescharacteristic
42#103×C83—28no—
43#97×C83C2×C412yes—
44C4×C8×C83—21no—
45#13050574—82no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8—22no—
48C2×C83⋊C8—42no—
49C2×Q8×C83—22no—
50C2×Q8×C83—22no—
51C2×Q8×C83—44no—
52C2×Q8×C83—41no—
53C22×C4×C83—41no—
54C22×C4×C83C2×D411yescharacteristic
55C22×C4×C83#4112yes—
56M16×C83—42no—
57C2×C8×C83—22no—
58C4⋊C4×C83—24no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83—24no—
61C4⋊C4×C83—42no—
62C4⋊C4×C83C22⋊C414yes—
63C22⋊C4×C83—22no—
64C42×C83—22no—
65C42×C83C22×C411yescharacteristic
66C42×C83C2×D412yes—
67C42×C83C2×D411yescharacteristic
68C23×C83#12311yescharacteristic
69Q8×C83—44no—
70Q8×C83—42no—
71Q8×C83—42no—
72Q8×C83—82no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—24no—
77C2×C4×C83—42no—
78C2×C4×C83—82no—
79C2×C4×C83C2×C22⋊C411yescharacteristic
80C2×C4×C83C4×D412yes—
81C2×C4×C83#12011yescharacteristic
82C2×C4×C83#12112yes—
83C8×C83—42no—
84C83⋊C8—42no—
85C83⋊C8—42no—
86C83⋊C8—82no—
87C22×C83—21no—
88C22×C83—22no—
89C22×C83#38611yescharacteristic
90C4×C83—21no—
91C4×C83—22no—
92C4×C83—44no—
93C4×C83—82no—
94C4×C83#45412yes—
95C4×C83#47112yes—
96C4×C83#47312yes—
97#9436—831nomaximal, Sylow
98C2×C83—41no—
99C2×C83#188811yescharacteristic
100C2×C83#190812yes—
101#3139D8311yescharacteristic
102#2986—831no—
103#1561—831no—
104#1549—832no—
105#1299—832no—
106C83#943611yescharacteristic, complemented, Sylow
107#533C2×D8311yescharacteristic
108C2×C4⋊Q8C2×D8311yescharacteristic
109#491Dic8314yes—
110#487—22no—
111#446—1662no—
112C8×Q8—1662no—
113#433—1662no—
114#431—832no—
115#423—832no—
116#419—22no—
117C4×M16C2×D8311yescharacteristic
118#332—1662no—
119#332—1662no—
120#332—1662no—
121C22×Q8—21no—
122C2×M16—21no—
123C2×M16—1662no—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8C2×Dic8312yes—
127C4⋊Q8C22⋊D8312yes—
128#125C2×Dic8312yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D8312yes—
132C2×C4⋊C4—22no—
133C2×C42C22×D8311yescharacteristic
134C4⋊C8—832no—
135C4⋊C8—832no—
136C4⋊C8—1664no—
137C4⋊C8—1662no—
138C4⋊C8—1662no—
139#103—28no—
140C22⋊C8—1662no—
141#97—1662no—
142#97C2×Dic8312yes—
143C4×C8—21no—
144C4×C8—1662no—
145C4×C8—1662no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—44no—
149C2×Q8—41no—
150C22×C4—41no—
151C22×C4C2×C22⋊D8311yescharacteristic
152C22×C4#1305058912yes—
153M16—42no—
154M16—6642no—
155C2×C8—22no—
156C2×C8—3322no—
157C2×C8—1662no—
158C2×C8—1662no—
159C4⋊C4—22no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—42no—
163C4⋊C4C22⋊Dic8314yes—
164C22⋊C4—22no—
165C42—22no—
166C42C22×Dic8311yescharacteristic
167C42D4×D8312yes—
168C42C2×C22⋊D8311yescharacteristic
169C23#1458177311yescharacteristic
170Q8—44no—
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4C2×C22⋊Dic8311yescharacteristic
181C2×C4D4×Dic8312yes—
182C2×C4#1458176011yescharacteristic
183C2×C4#1458176412yes—
184C8—42no—
185C8—3322no—
186C8—3322no—
187C8—6642no—
188C22—21no—
189C22—22no—
190C22#1912758111yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—82no—
195C4#1912787312yes—
196C4#1912796612yes—
197C4#1912797412yes—
198C2—41no—
199C2#2490786711yescharacteristic
200C2#2490798012yes—
subgroupG/H
centerC22#19127581
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3139C2
socleC22×C83#386
prime radicalC23×C83#123
cosocleC2×C42×C83C23
hypercenter#3139D83
hypocenterC83#9436
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
106. C83#9436semidirect

Maximal subgroups

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

subgroupindex
#249165652
#249062012
#249061482
#249053532
C83×#31392
#943683