Finite Group Databasebeta

← #35305397

Subgroups — #35305397

Order 21248.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#24917498C212yesmaximal
2#24908403C211yesmaximal, characteristic
3#24905866C212yesmaximal
4#24905294C211yesmaximal, characteristic
5C83×#3162C211yesmaximal, characteristic
6C2×C4×Dic166C2211yescharacteristic
7#19128034—22no—
8#19127993—22no—
9#19127908—22no—
10#19127856—22no—
11C4×#13050580—24no—
12#19127754C2212yes—
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14#19127455—22no—
15Q8⋊(C83⋊C8)—24no—
16#533×C83C2212yes—
17#501×C83—22no—
18#501×C83—22no—
19#427×C83—22no—
20C2×#97×C83C2211yescharacteristic
21C22×Dic166D411yescharacteristic
22C2×#13050580—24no—
23C22×C83⋊C8—21no—
24C4×Dic166—22no—
25C4×Dic166—22no—
26C2×C4⋊Dic83D411yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×C4×Dic83—21no—
29#14581725—24no—
30C4⋊(C83⋊C8)—22no—
31C4⋊(C83⋊C8)—22no—
32#14581716—24no—
33#14581715—24no—
34C22⋊(C83⋊C8)—22no—
35C4×C83⋊C8D414yes—
36C22×C8×C83D412yes—
37C4⋊Q8×C83D412yes—
38C4⋊Q8×C83D412yes—
39#125×C83—22no—
40C22⋊Q8×C83—22no—
41C22⋊Q8×C83—22no—
42C4×Q8×C83—22no—
43C4×Q8×C83—22no—
44#117×C83—22no—
45C2×C42×C83C2311yescharacteristic
46#107×C83—22no—
47#107×C83—22no—
48#106×C83—24no—
49#97×C83—22no—
50C2×Dic166—22no—
51C2×Dic166—42no—
52C22×Dic83—21no—
53C22×Dic83—21no—
54#13050580—48no—
55C2×C83⋊C8—24no—
56C2×C83⋊C8—41no—
57C2×C83⋊C8—42no—
58C4⋊Dic83—22no—
59Dic83⋊C4—42no—
60C4×Dic83—41no—
61C4×Dic83—41no—
62C2×Q8×C83—22no—
63C2×Q8×C83—22no—
64C22×C4×C83C2×D411yescharacteristic
65C22×C4×C83C2×D411yescharacteristic
66C22×C4×C83#4111yescharacteristic
67C2×C8×C83—22no—
68C2×C8×C83—22no—
69C2×C8×C83Q1614yescomplemented
70C4⋊C4×C83—22no—
71C4⋊C4×C83—22no—
72C4⋊C4×C83—22no—
73C4⋊C4×C83—22no—
74C4⋊C4×C83—42no—
75C4⋊C4×C83—42no—
76C22⋊C4×C83—42no—
77C42×C83—42no—
78C42×C83C2×D412yes—
79C42×C83C2×D412yes—
80C23×C83#12111yescharacteristic
81Q8×C83—44no—
82Q8×C83—44no—
83C2×C4×C83—41no—
84C2×C4×C83—41no—
85C2×C4×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83#12012yes—
88C2×C4×C83#12011yescharacteristic
89C2×C4×C83#12112yes—
90C2×C4×C83#12711yescharacteristic
91C2×C4×C83C2×Q1612yes—
92C2×C4×C83#13712yes—
93C2×Dic83—22no—
94C2×Dic83—41no—
95C2×Dic83—41no—
96C2×Dic83—42no—
97C2×Dic83—41no—
98Dic166—24no—
99Dic166—42no—
100C8×C83—24no—
101C83⋊C8—44no—
102C83⋊C8—81no—
103C22×C83—21no—
104C22×C83—21no—
105C22×C83#39211yescharacteristic
106C22×C83#46212yes—
107C4×C83—22no—
108C4×C83—22no—
109C4×C83—82no—
110C4×C83—82no—
111C4×C83#45112yes—
112C4×C83#49411yescharacteristic
113C4×C83#49711yescharacteristic
114Dic83—42no—
115Dic83—81no—
116#10037—831nomaximal, Sylow
117C2×C83—22no—
118C2×C83#201011yescharacteristic
119C2×C83#201611yescharacteristic
120C2×C83#201711yescharacteristic
121#3162D8311yescharacteristic
122#3104—832no—
123#1985—831no—
124#1481—832no—
125#1275—831no—
126C83#1003711yescharacteristic, complemented, Sylow
127#533C2×D8312yes—
128C2×C4×Q8—831no—
129#501—22no—
130#501—22no—
131#484—1662no—
132#477—1662no—
133#462—1662no—
134#451—1662no—
135C4×Q16—1664no—
136#431—832no—
137#427—22no—
138C2×C4⋊C8—831no—
139C2×#97C2×D8311yescharacteristic
140#336—1662no—
141Q8⋊C8—1664no—
142C22×Q8—831no—
143C2×Q16—1664no—
144C22×C8—1661no—
145C22×C8C22⋊D8312yes—
146C4⋊Q8C22⋊D8312yes—
147C4⋊Q8C22⋊D8312yes—
148#125—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1662no—
154C4×Q8—1662no—
155#117—22no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—1661no—
158C2×C42—1661no—
159C2×C42C22×D8311yescharacteristic
160#107—22no—
161#107—22no—
162#107—1664no—
163#106—24no—
164C4⋊C8—1662no—
165C4⋊C8—1662no—
166#103—1664no—
167#103—1664no—
168C22⋊C8—1662no—
169#97—22no—
170C4×C8—834no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—3322no—
174C2×Q8—1662no—
175C22×C4—1661no—
176C22×C4—1661no—
177C22×C4D4×D8311yescharacteristic
178C22×C4C2×C22⋊D8311yescharacteristic
179C22×C4#1305058911yescharacteristic
180Q16—3328no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—3321no—
184C2×C8—3322no—
185C2×C8—1664no—
186C2×C8#1305058014yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—3322no—
194C4⋊C4—1662no—
195C22⋊C4—42no—
196C42—42no—
197C42—3321no—
198C42—3321no—
199C42D4×D8312yes—
200C42C2×C22⋊D8312yes—
201C23#1458176311yescharacteristic
202Q8—44no—
203Q8—44no—
204Q8—3322no—
205Q8—1664no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—3321no—
211C2×C4—3321no—
212C2×C4—3322no—
213C2×C4—3321no—
214C2×C4—1662no—
215C2×C4#1458176012yes—
216C2×C4#1458176111yescharacteristic
217C2×C4#1458176412yes—
218C2×C4#1458178811yescharacteristic
219C2×C4C2×#1305058012yes—
220C2×C4#1458182312yes—
221C8—24no—
222C8—3324no—
223C8—6641no—
224C22—21no—
225C22—21no—
226C22#1912760911yescharacteristic
227C22#1912790812yes—
228C4—22no—
229C4—22no—
230C4—82no—
231C4—82no—
232C4—3322no—
233C4—6641no—
234C4#1912785712yes—
235C4#1912808011yescharacteristic
236C4#1912809311yescharacteristic
237C2—22no—
238C2#2490852711yescharacteristic
239C2#2490856311yescharacteristic
240C2#2490856611yescharacteristic
subgroupG/H
centerC2×C4#14581788
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3162C2
socleC22×C83#392
prime radicalC23×C83#121
hypercenter#3162D83
hypocenterC83#10037
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
69. 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
#249174982
#249084032
#249058662
#249052942
C83×#31622
#1003783