Finite Group Databasebeta

← #35307139

Subgroups — #35307139

Order 21248.

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

normal subgroups only

This group has 528 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#24918408C212yesmaximal
2#24915841C211yesmaximal, characteristic
3#24906145C212yesmaximal
4#24905296C211yesmaximal, characteristic
5C83×#1994C211yesmaximal, characteristic
6C2×#14581792C2212yes—
7#19128080—22no—
8#19128053—22no—
9#19127911—24no—
10#19127856—24no—
11#19127754C2212yes—
12#19127607—24no—
13#19127567—22no—
14#19127449—24no—
15#533×C83C2211yescharacteristic
16#423×C83C2211yescharacteristic
17C43×C83C2211yescharacteristic
18#339×C83—22no—
19C22×Dic166D414yes—
20C2×#13050580—28no—
21#14581792—22no—
22#14581792—22no—
23#14581792—22no—
24#14581792—22no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83—22no—
27C4⋊(C83⋊C8)D414yes—
28#14581715—28no—
29C22⋊(C83⋊C8)—22no—
30C4×C83⋊C8—22no—
31C83⋊#95—28no—
32C2×M16×C83—21no—
33C4⋊Q8×C83D412yes—
34#125×C83—21no—
35C22⋊Q8×C83—22no—
36C4×Q8×C83—22no—
37#117×C83—21no—
38C2×C42×C83—22no—
39C2×C42×C83D412yes—
40C2×C42×C83C2311yescharacteristic
41C4⋊C8×C83—22no—
42C2×Dic166—24no—
43C2×Dic166—24no—
44C2×Dic166—48no—
45C22×Dic83—24no—
46#13050580—88no—
47C2×C83⋊C8—24no—
48C4⋊Dic83—42no—
49C4⋊Dic83—42no—
50C4⋊Dic83—42no—
51C4⋊Dic83—42no—
52C2×Q8×C83—22no—
53C22×C4×C83—22no—
54C22×C4×C83C2×D412yes—
55C22×C4×C83C2×D411yescharacteristic
56M16×C83—42no—
57C2×C8×C83—41no—
58C4⋊C4×C83—22no—
59C4⋊C4×C83—22no—
60C4⋊C4×C83—42no—
61C22⋊C4×C83—41no—
62C42×C83—22no—
63C42×C83—44no—
64C42×C83—41no—
65C42×C83Q1614yescomplemented
66C42×C83C2×D412yes—
67C42×C83C2×D411yescharacteristic
68C42×C83#4111yescharacteristic
69C23×C83#12011yescharacteristic
70Q8×C83—44no—
71C2×C4×C83—24no—
72C2×C4×C83—24no—
73C2×C4×C83—44no—
74C2×C4×C83—42no—
75C2×C4×C83#12012yes—
76C2×C4×C83#12112yes—
77C2×C4×C83#12111yescharacteristic
78C2×C4×C83#12711yescharacteristic
79C2×C4×C83C2×Q1612yes—
80C2×C4×C83#13712yes—
81C2×Dic83—48no—
82C2×Dic83—44no—
83Dic166—48no—
84Dic166—84no—
85C8×C83—81no—
86C83⋊C8—44no—
87C22×C83—22no—
88C22×C83#39211yescharacteristic
89C22×C83#45112yes—
90C4×C83—22no—
91C4×C83—44no—
92C4×C83—82no—
93C4×C83#45514yes—
94C4×C83#46212yes—
95C4×C83#49411yescharacteristic
96C4×C83#49711yescharacteristic
97Dic83—84no—
98#10297—831nomaximal, Sylow
99C2×C83—22no—
100C2×C83#200412yes—
101C2×C83#201611yescharacteristic
102#3209—832no—
103#2862—831no—
104#1994D8311yescharacteristic
105#1548—832no—
106#1276—831no—
107C83#1029711yescharacteristic, complemented, Sylow
108#533C2×D8311yescharacteristic
109C2×C4⋊Q8—832no—
110#494—1662no—
111#487—1662no—
112#462—1664no—
113#451—1664no—
114#431—832no—
115#423C2×D8311yescharacteristic
116#391—1664no—
117#382—1662no—
118C43C2×D8311yescharacteristic
119#339—22no—
120#333—1664no—
121C22×Q8—834no—
122C2×Q16—1668no—
123C2×M16—21no—
124C4⋊Q8—1662no—
125C4⋊Q8—1662no—
126C4⋊Q8—1662no—
127C4⋊Q8—1662no—
128C4⋊Q8D33212yes—
129#125—21no—
130C22⋊Q8—22no—
131C4×Q8—22no—
132#117—21no—
133C2×C4⋊C4—1662no—
134C2×C4⋊C4—1662no—
135C2×C42—22no—
136C2×C42C22⋊D8312yes—
137C2×C42C22×D8311yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—834no—
140#103—1668no—
141C22⋊C8—1662no—
142C4×C8—1662no—
143#95—1668no—
144C2×Q8—22no—
145C2×Q8—3328no—
146C2×Q8—1664no—
147C2×Q8—1664no—
148C22×C4—22no—
149C22×C4—1664no—
150C22×C4D4×D8312yes—
151C22×C4C2×C22⋊D8311yescharacteristic
152Q16—6648no—
153M16—42no—
154C2×C8—41no—
155C2×C8—1664no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—3322no—
160C4⋊C4—3322no—
161C4⋊C4—3322no—
162C4⋊C4—3322no—
163C22⋊C4—41no—
164C42—22no—
165C42—44no—
166C42—41no—
167C42#1305058014yescomplemented
168C42D4×D8312yes—
169C42C2×D33211yescharacteristic
170C42#1305059011yescharacteristic
171C23#1458176011yescharacteristic
172Q8—44no—
173Q8—3328no—
174Q8—6644no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—3328no—
180C2×C4—3324no—
181C2×C4#1458176212yes—
182C2×C4#1458176512yes—
183C2×C4#1458176611yescharacteristic
184C2×C4#1458178811yescharacteristic
185C2×C4C2×#1305058012yes—
186C2×C4#1458182212yes—
187C8—81no—
188C8—3324no—
189C22—22no—
190C22#1912761011yescharacteristic
191C22#1912785612yes—
192C4—22no—
193C4—44no—
194C4—82no—
195C4—6644no—
196C4#1912788014yes—
197C4#1912791112yes—
198C4#1912808011yescharacteristic
199C4#1912809611yescharacteristic
200C2—22no—
201C2#2490849112yes—
202C2#2490856211yescharacteristic
subgroupG/H
centerC22#19127610
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1994C2
socleC22×C83#392
prime radicalC23×C83#120
hypercenter#1994D83
hypocenterC83#10297
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
65. C42×C83Q16semidirect
107. C83#10297semidirect
167. C42#13050580semidirect

Maximal subgroups

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

subgroupindex
#249184082
#249158412
#249061452
#249052962
C83×#19942
#1029783