Finite Group Databasebeta

← #35305399

Subgroups — #35305399

Order 21248.

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

normal subgroups only

This group has 476 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#24917987C211yesmaximal, characteristic
2#24917502C212yesmaximal
3#24905865C212yesmaximal
4#24905294C211yesmaximal, characteristic
5C83×#1985C211yesmaximal, characteristic
6#19128258C2212yes—
7#19128110—22no—
8#19128110—22no—
9#19128036—22no—
10#19127996—22no—
11#19127911—22no—
12#19127857—22no—
13C4×#13050580—24no—
14#19127754C2212yes—
15#19127739—22no—
16#19127431—24no—
17C2×C4×Q8×C83C2211yescharacteristic
18C2×C4⋊C8×C83C2211yescharacteristic
19C2×#97×C83C2211yescharacteristic
20#336×C83—22no—
21C2×#13050580—24no—
22#14581792D412yes—
23#14581792D412yes—
24#14581784—22no—
25#14581771—22no—
26#14581771—22no—
27C4×Dic166—22no—
28C4×Dic166—22no—
29#14581751—22no—
30#14581726—22no—
31#14581726—22no—
32#14581725—24no—
33#14581724—24no—
34C4⋊(C83⋊C8)—22no—
35#14581716—24no—
36#14581715—24no—
37C22⋊(C83⋊C8)—22no—
38C4×C83⋊C8D414yes—
39C22×Q8×C83D411yescharacteristic
40C22×C8×C83—21no—
41C22×C8×C83D412yes—
42C4×Q8×C83—22no—
43C4×Q8×C83—22no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C42×C83—21no—
47C2×C42×C83C2311yescharacteristic
48C4⋊C8×C83—22no—
49#97×C83—22no—
50C2×Dic166—22no—
51C2×Dic166—22no—
52#13050580—48no—
53C2×C83⋊C8—24no—
54C4⋊Dic83—22no—
55C4⋊Dic83—22no—
56C4⋊Dic83—22no—
57C4⋊Dic83—22no—
58Dic83⋊C4—42no—
59Dic83⋊C4—42no—
60C22⋊Dic83—42no—
61C4×Dic83—42no—
62C2×Q8×C83—22no—
63C2×Q8×C83—42no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83C2×D411yescharacteristic
67C22×C4×C83C2×D411yescharacteristic
68C22×C4×C83#4111yescharacteristic
69C2×C8×C83—22no—
70C2×C8×C83—22no—
71C2×C8×C83—41no—
72C2×C8×C83—42no—
73C2×C8×C83Q1614yescomplemented
74C4⋊C4×C83—22no—
75C4⋊C4×C83—42no—
76C42×C83—41no—
77C42×C83—41no—
78C42×C83C2×D412yes—
79C42×C83C2×D412yes—
80C23×C83#12111yescharacteristic
81Q8×C83—24no—
82Q8×C83—42no—
83C2×C4×C83—22no—
84C2×C4×C83—41no—
85C2×C4×C83—41no—
86C2×C4×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83—42no—
89C2×C4×C83—41no—
90C2×C4×C83#12011yescharacteristic
91C2×C4×C83#12012yes—
92C2×C4×C83#12112yes—
93C2×C4×C83#12711yescharacteristic
94C2×C4×C83C2×Q1612yes—
95C2×C4×C83#13712yes—
96C2×Dic83—42no—
97C2×Dic83—42no—
98Dic166—44no—
99Dic166—44no—
100C8×C83—24no—
101C8×C83—81no—
102C83⋊C8—44no—
103C22×C83—21no—
104C22×C83—21no—
105C22×C83#39211yescharacteristic
106C22×C83#46212yes—
107C4×C83—22no—
108C4×C83—22no—
109C4×C83—42no—
110C4×C83—81no—
111C4×C83#45112yes—
112C4×C83#49411yescharacteristic
113C4×C83#49711yescharacteristic
114Dic83—82no—
115Dic83—82no—
116#10037—831nomaximal, Sylow
117C2×C83—22no—
118C2×C83#201011yescharacteristic
119C2×C83#201611yescharacteristic
120C2×C83#201711yescharacteristic
121#3162—831no—
122#3104—832no—
123#1985D8311yescharacteristic
124#1481—832no—
125#1275—831no—
126C83#1003711yescharacteristic, complemented, Sylow
127#533—832no—
128C2×C4×Q8C2×D8311yescharacteristic
129#501—1662no—
130#501—1662no—
131#484—1662no—
132#477—1662no—
133#462—1662no—
134#451—1662no—
135C4×Q16—1664no—
136#431—832no—
137#427—1662no—
138C2×C4⋊C8C2×D8311yescharacteristic
139C2×#97C2×D8311yescharacteristic
140#336—22no—
141Q8⋊C8—1664no—
142C22×Q8D33211yescharacteristic
143C2×Q16—1664no—
144C22×C8—21no—
145C22×C8C22⋊D8312yes—
146C4⋊Q8—832no—
147C4⋊Q8—832no—
148#125—1662no—
149C22⋊Q8—1662no—
150C22⋊Q8—1662no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1662no—
154C4×Q8—1662no—
155#117—1662no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D33211yescharacteristic
158C2×C42—21no—
159C2×C42C22×D8311yescharacteristic
160#107—1664no—
161#107—1662no—
162#107—1662no—
163#106—1664no—
164C4⋊C8—22no—
165C4⋊C8—1662no—
166#103—1664no—
167#103—1664no—
168C22⋊C8—1662no—
169#97—22no—
170C4×C8—834no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×Q8—1662no—
174C2×Q8—1662no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4C2×C22⋊D8311yescharacteristic
178C22×C4C2×D33211yescharacteristic
179C22×C4#1305059011yescharacteristic
180Q16—3328no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—1664no—
186C2×C8#1305058014yescomplemented
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—3322no—
190C4⋊C4—3322no—
191C4⋊C4—1662no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C22⋊C4—3322no—
196C42—41no—
197C42—41no—
198C42—3322no—
199C42D4×D8312yes—
200C42D4×D8312yes—
201C23#1458176611yescharacteristic
202Q8—24no—
203Q8—42no—
204Q8—3324no—
205Q8—3324no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—3322no—
214C2×C4—3322no—
215C2×C4#1458176011yescharacteristic
216C2×C4#1458176212yes—
217C2×C4#1458176512yes—
218C2×C4#1458178811yescharacteristic
219C2×C4C2×#1305058012yes—
220C2×C4#1458182212yes—
221C8—24no—
222C8—81no—
223C8—3324no—
224C22—21no—
225C22—21no—
226C22#1912761011yescharacteristic
227C22#1912791112yes—
228C4—22no—
229C4—22no—
230C4—42no—
231C4—81no—
232C4—6642no—
233C4—6642no—
234C4#1912785612yes—
235C4#1912808011yescharacteristic
236C4#1912809611yescharacteristic
237C2—22no—
238C2#2490853011yescharacteristic
239C2#2490856211yescharacteristic
240C2#2490856711yescharacteristic
subgroupG/H
centerC2×C4#14581788
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1985C2
socleC22×C83#392
prime radicalC23×C83#121
hypercenter#1985D83
hypocenterC83#10037
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
73. C2×C8×C83Q16semidirect
126. C83#10037semidirect
186. C2×C8#13050580semidirect

Maximal subgroups

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

subgroupindex
#249179872
#249175022
#249058652
#249052942
C83×#19852
#1003783