Finite Group Databasebeta

← #35302006

Subgroups — #35302006

Order 21248.

400 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 103 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#24917845C211yesmaximal, characteristic
2#24916565C211yesmaximal, characteristic
3#24906149C212yesmaximal
4#24905352C212yesmaximal
5C83×#1561C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7#19128071C414yes—
8#19128053—22no—
9#19127832—22no—
10Q8×C83⋊C8—22no—
11#19127762—22no—
12#19127754C2212yes—
13#19127709—22no—
14C4×#13050574C2211yescharacteristic
15#19127446—22no—
16#19127446—22no—
17#533×C83C2211yescharacteristic
18#423×C83C2212yes—
19#332×C83—22no—
20C22×Dic166—21no—
21C2×#13050574—21no—
22#14581792—22no—
23#14581792—21no—
24#14581792C2×C412yes—
25C2×C4⋊Dic83—22no—
26C4⋊(C83⋊C8)—24no—
27C4⋊(C83⋊C8)D412yes—
28C4⋊(C83⋊C8)D412yes—
29#14581715—28no—
30C22⋊(C83⋊C8)—22no—
31#14581703—22no—
32#14581703C2×C412yes—
33C4×C83⋊C8—22no—
34C4×C83⋊C8—22no—
35C4×C83⋊C8—21no—
36C2×M16×C83—22no—
37C4⋊Q8×C83D412yes—
38#125×C83C2×C412yes—
39C22⋊Q8×C83—22no—
40C4×Q8×C83—22no—
41#117×C83D412yes—
42C2×C42×C83C2311yescharacteristic
43C4⋊C8×C83—22no—
44C4⋊C8×C83—22no—
45C2×Dic166—22no—
46C2×Dic166—44no—
47C2×Dic166—41no—
48C22×Dic83—41no—
49#13050574—42no—
50C2×C83⋊C8—22no—
51C2×C83⋊C8—22no—
52C2×C83⋊C8—22no—
53C4⋊Dic83—24no—
54C4⋊Dic83—42no—
55C2×Q8×C83—22no—
56C22×C4×C83C2×D411yescharacteristic
57C22×C4×C83#4112yes—
58M16×C83—82no—
59C2×C8×C83—42no—
60C4⋊C4×C83—24no—
61C4⋊C4×C83—22no—
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—42no—
70Q8×C83—42no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83C2×C22⋊C411yescharacteristic
76C2×C4×C83C4×D412yes—
77C2×C4×C83#12011yescharacteristic
78C2×C4×C83#12112yes—
79C2×Dic83—42no—
80C2×Dic83—82no—
81Dic166—44no—
82Dic166—82no—
83C8×C83—82no—
84C83⋊C8—42no—
85C83⋊C8—42no—
86C83⋊C8—42no—
87C22×C83—21no—
88C22×C83—22no—
89C22×C83#38611yescharacteristic
90C4×C83—21no—
91C4×C83—22no—
92C4×C83—44no—
93C4×C83#45412yes—
94C4×C83#47112yes—
95C4×C83#47312yes—
96Dic83—82no—
97#9436—831nomaximal, Sylow
98C2×C83—41no—
99C2×C83#188811yescharacteristic
100C2×C83#190812yes—
101#3139—831no—
102#2986—831no—
103#1561D8311yescharacteristic
104#1549—832no—
105#1299—832no—
106C83#943611yescharacteristic, complemented, Sylow
107#533C2×D8311yescharacteristic
108C2×C4⋊Q8—831no—
109#491—834no—
110#487—1662no—
111#446—1662no—
112C8×Q8—1662no—
113#433—1662no—
114#431—832no—
115#423C2×D8312yes—
116#419—1662no—
117C4×M16—831no—
118#332—22no—
119#332—1662no—
120#332—1662no—
121C22×Q8—1661no—
122C2×M16—22no—
123C2×M16—1661no—
124C4⋊Q8—832no—
125C4⋊Q8—1662no—
126C4⋊Q8—1661no—
127C4⋊Q8C22⋊D8312yes—
128#125C4×D8312yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117D33212yes—
132C2×C4⋊C4—1662no—
133C2×C42C22×D8311yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—832no—
137C4⋊C8—832no—
138C4⋊C8—1664no—
139#103—1668no—
140C22⋊C8—1662no—
141#97—832no—
142#97—1662no—
143C4×C8—1662no—
144C4×C8—1662no—
145C4×C8—1661no—
146C2×Q8—22no—
147C2×Q8—3324no—
148C2×Q8—3321no—
149C2×Q8—1662no—
150C22×C4—3321no—
151C22×C4C2×D33211yescharacteristic
152C22×C4#1305058912yes—
153M16—82no—
154M16—3322no—
155C2×C8—42no—
156C2×C8—1662no—
157C2×C8—1662no—
158C2×C8—1662no—
159C4⋊C4—22no—
160C4⋊C4—24no—
161C4⋊C4—3322no—
162C4⋊C4—1664no—
163C4⋊C4(C2×D83)⋊C414yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D8311yescharacteristic
167C42D4×D8312yes—
168C42C2×C22⋊D8311yescharacteristic
169C23#1458177611yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—3324no—
173Q8—6642no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—3322no—
179C2×C4—6642no—
180C2×C4C2×(C2×D83)⋊C411yescharacteristic
181C2×C4(C22⋊D83)⋊C412yes—
182C2×C4#1458176211yescharacteristic
183C2×C4#1458176412yes—
184C8—82no—
185C8—3322no—
186C8—3322no—
187C8—3322no—
188C22—21no—
189C22—22no—
190C22#1912757811yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—6642no—
195C4#1912787612yes—
196C4#1912796212yes—
197C4#1912797412yes—
198C2—41no—
199C2#2490786411yescharacteristic
200C2#2490797612yes—
subgroupG/H
centerC22#19127578
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1561C2
socleC22×C83#386
prime radicalC23×C83#123
cosocleC2×C42×C83C23
hypercenter#1561D83
hypocenterC83#9436
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

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
#249178452
#249165652
#249061492
#249053522
C83×#15612
#943683