Finite Group Databasebeta

← #35299728

Subgroups — #35299728

Order 21248.

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

normal subgroups only

This group has 426 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#24917788C211yesmaximal, characteristic
2#24916565C211yesmaximal, characteristic, complemented
3#24905881C212yesmaximal
4#24905301C212yesmaximal, complemented
5C83×#1526C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19128058—22no—
8#19128053C412yes—
9#19128049C412yes—
10#19127832—22no—
11Q8×C83⋊C8—22no—
12#19127762—22no—
13#19127754C2212yes—
14#19127695—22no—
15C2×C4×C83⋊C8C2211yescharacteristic
16#19127430—22no—
17#19127430—22no—
18#533×C83C2211yescharacteristic
19C4×M16×C83C2212yes—
20#330×C83—22no—
21C2×#13050590—21no—
22C22×C83⋊C8—21no—
23#14581792C2×C411yescharacteristic
24#14581789C2×C411yescharacteristic
25#14581780—21no—
26#14581766—22no—
27C4×D332—22no—
28C4⋊(C83⋊C8)—24no—
29C4⋊(C83⋊C8)D412yes—
30C4⋊(C83⋊C8)D412yes—
31#14581715—24no—
32#14581712—24no—
33C22⋊(C83⋊C8)—22no—
34#14581703—22no—
35C4×C83⋊C8—22no—
36C4×C83⋊C8—22no—
37C4×C83⋊C8—21no—
38C4×C83⋊C8C2×C412yes—
39C2×M16×C83—22no—
40C4⋊Q8×C83C2×C412yes—
41#125×C83D412yes—
42C22⋊Q8×C83—22no—
43C4×Q8×C83—22no—
44#117×C83D412yes—
45C2×C42×C83C2311yescharacteristic
46#97×C83—22no—
47C4×C8×C83—22no—
48#13050590—44no—
49C2×Dic166—21no—
50C2×D332—21no—
51C2×D332—22no—
52C2×C22⋊D83—41no—
53C2×C4×D83—41no—
54C2×C83⋊C8—22no—
55C2×C83⋊C8—22no—
56C2×C83⋊C8—22no—
57C2×C83⋊C8—42no—
58C4⋊Dic83—22no—
59(C2×D83)⋊C4—42no—
60C2×Q8×C83—22no—
61C22×C4×C83C2×D411yescharacteristic
62C22×C4×C83#4112yes—
63M16×C83—24no—
64M16×C83—42no—
65C2×C8×C83—42no—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—24no—
68C4⋊C4×C83C22⋊C414yes—
69C22⋊C4×C83—22no—
70C42×C83—22no—
71C42×C83C22×C411yescharacteristic
72C42×C83C2×D412yes—
73C42×C83C2×D411yescharacteristic
74C23×C83#12311yescharacteristic
75C22×D83—41no—
76Q8×C83—42no—
77Q8×C83—42no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—24no—
82C2×C4×C83C2×C22⋊C411yescharacteristic
83C2×C4×C83C4×D412yes—
84C2×C4×C83#12011yescharacteristic
85C2×C4×C83#12112yes—
86C22⋊D83—82no—
87C2×Dic83—41no—
88D332—42no—
89D332—42no—
90D332—42no—
91C4×D83—82no—
92Dic166—42no—
93C8×C83—44no—
94C83⋊C8—42no—
95C83⋊C8—42no—
96C83⋊C8—42no—
97C22×C83—22no—
98C22×C83—21no—
99C22×C83#38611yescharacteristic
100C2×D83—82no—
101C2×D83—81no—
102C4×C83—22no—
103C4×C83—21no—
104C4×C83—44no—
105C4×C83#44312yes—
106C4×C83#44412yes—
107C4×C83#45412yes—
108Dic83—81no—
109#9070—831nomaximal, Sylow
110C2×C83—41no—
111C2×C83#187612yes—
112C2×C83#188811yescharacteristic
113D83—161no—
114#3131—831no—
115#2986—831no—
116#1526D8311yescharacteristic, complemented
117#1484—832no—
118#1278—832no—
119C83#907011yescharacteristic, complemented, Sylow
120#533C2×D8311yescharacteristic
121#532—831no—
122#488—1662no—
123#487—832no—
124#486—832no—
125#446—1662no—
126C8×Q8—1662no—
127#433—1662no—
128#431—832no—
129#416—1662no—
130C4×M16C2×D8312yes—
131C2×C4×C8—831no—
132#330—22no—
133D4⋊C8—1662no—
134D4⋊C8—1662no—
135C2×#41—1661no—
136C2×M16—22no—
137C22×C8—1661no—
138C4⋊Q8—831no—
139C4⋊Q8C4×D8312yes—
140#127—831no—
141#125C22⋊D8312yes—
142#124—1661no—
143C22⋊Q8—22no—
144#121—1662no—
145C4×Q8—22no—
146C4×D4—1662no—
147#117D33212yes—
148C2×C42C22×D8311yescharacteristic
149C4⋊C8—832no—
150C4⋊C8—832no—
151C4⋊C8—1664no—
152#103—1664no—
153#102—1664no—
154C22⋊C8—1662no—
155#97—22no—
156#97—1662no—
157C4×C8—22no—
158C4×C8—832no—
159C4×C8—1662no—
160C4×C8—1662no—
161C4×C8—1661no—
162#41—3324no—
163C2×Q8—22no—
164C2×Q8—1661no—
165C2×D4—3321no—
166C2×D4—1661no—
167C2×D4—1662no—
168C22×C4—3321no—
169C22×C4C2×D33211yescharacteristic
170C22×C4#1305058912yes—
171M16—24no—
172M16—42no—
173C2×C8—42no—
174C2×C8—3322no—
175C2×C8—1662no—
176C2×C8—1662no—
177C2×C8—1662no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—1662no—
181C4⋊C4(C2×D83)⋊C414yes—
182C22⋊C4—22no—
183C22⋊C4—3322no—
184C42—22no—
185C42C2×C4×D8311yescharacteristic
186C42D4×D8312yes—
187C42C2×C22⋊D8311yescharacteristic
188C23—3321no—
189C23#1458177611yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—3322no—
193D4—3322no—
194D4—3322no—
195D4—3322no—
196D4—6642no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—3321no—
202C2×C4—6642no—
203C2×C4C2×(C2×D83)⋊C411yescharacteristic
204C2×C4(C22⋊D83)⋊C412yes—
205C2×C4#1458176211yescharacteristic
206C2×C4#1458176412yes—
207C8—44no—
208C8—3322no—
209C8—3322no—
210C8—3322no—
211C22—22no—
212C22—21no—
213C22—6642no—
214C22—6641no—
215C22#1912757811yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—6641no—
220C4#1912781512yes—
221C4#1912782012yes—
222C4#1912787612yes—
223C2—41no—
224C2—13281no—
225C2#2490778812yes—
226C2#2490786411yescharacteristic
subgroupG/H
centerC2×C4#14581762
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1526C2
socleC22×C83#386
prime radical#19128251C22
cosocleC2×C42×C83C23
hypercenter#1526D83
hypocenterC83#9070
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #24916565C2semidirect
4. #24905301C2semidirect
5. C83×#1526C2semidirect
116. #1526D83semidirect
119. C83#9070semidirect

Maximal subgroups

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

subgroupindex
#249177882
#249165652
#249058812
#249053012
C83×#15262
#907083