Finite Group Databasebeta

← #35305274

Subgroups — #35305274

Order 21248.

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

normal subgroups only

This group has 528 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#24917502C212yesmaximal
2#24908444C211yesmaximal, characteristic
3#24905892C212yesmaximal
4#24905292C211yesmaximal, characteristic
5C83×#2399C211yesmaximal, characteristic
6#19128258C2211yescharacteristic
7#19128036—22no—
8#19127996—22no—
9#19127911—22no—
10#19127857—22no—
11C4×#13050580—24no—
12#19127754C2212yes—
13#19127725C2211yescharacteristic
14#19127461—22no—
15Q8⋊(C83⋊C8)—24no—
16C2×C4×Q8×C83C2212yes—
17#398×C83—22no—
18#395×C83—22no—
19C4×C4⋊C4×C83—22no—
20#376×C83C2211yescharacteristic
21C2×#13050580—24no—
22C2×#13050574—21no—
23#14581792D412yes—
24#14581784—21no—
25#14581771—22no—
26C4×Dic166—22no—
27#14581751—21no—
28#14581725—24no—
29C4⋊(C83⋊C8)—22no—
30C4⋊(C83⋊C8)—22no—
31#14581716—24no—
32#14581715—24no—
33C22⋊(C83⋊C8)—22no—
34C4×C83⋊C8D414yes—
35C22×Q8×C83D412yes—
36C4×Q8×C83—22no—
37C4×Q8×C83—22no—
38C4×Q8×C83—22no—
39C4×Q8×C83—22no—
40C2×C4⋊C4×C83—24no—
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83D412yes—
43C2×C42×C83—22no—
44C2×C42×C83D412yes—
45C2×C42×C83C2311yescharacteristic
46#95×C83—22no—
47#95×C83—22no—
48#95×C83—22no—
49C2×Dic166—22no—
50#13050580—48no—
51#13050574—42no—
52C2×C83⋊C8—24no—
53C2×C83⋊C8—41no—
54C4⋊Dic83—22no—
55C4⋊Dic83—22no—
56Dic83⋊C4—42no—
57C22⋊Dic83—41no—
58C4×Dic83—41no—
59C2×Q8×C83—22no—
60C2×Q8×C83—22no—
61C2×Q8×C83—44no—
62C22×C4×C83—22no—
63C22×C4×C83—22no—
64C22×C4×C83—22no—
65C22×C4×C83C2×D412yes—
66C22×C4×C83C2×D411yescharacteristic
67C4⋊C4×C83—22no—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—48no—
70C4⋊C4×C83—42no—
71C4⋊C4×C83—42no—
72C42×C83—22no—
73C42×C83—42no—
74C42×C83—42no—
75C42×C83Q1614yescomplemented
76C42×C83C2×D411yescharacteristic
77C42×C83C2×D412yes—
78C42×C83#4111yescharacteristic
79C23×C83#12011yescharacteristic
80Q8×C83—24no—
81Q8×C83—24no—
82Q8×C83—42no—
83Q8×C83—42no—
84C2×C4×C83—24no—
85C2×C4×C83—24no—
86C2×C4×C83—42no—
87C2×C4×C83—44no—
88C2×C4×C83—42no—
89C2×C4×C83—42no—
90C2×C4×C83—44no—
91C2×C4×C83—42no—
92C2×C4×C83#12012yes—
93C2×C4×C83#12111yescharacteristic
94C2×C4×C83#12112yes—
95C2×C4×C83#12711yescharacteristic
96C2×C4×C83C2×Q1612yes—
97C2×C4×C83#13712yes—
98C2×Dic83—42no—
99Dic166—44no—
100C83⋊C8—44no—
101C83⋊C8—81no—
102C22×C83—22no—
103C22×C83#39211yescharacteristic
104C22×C83#45112yes—
105C4×C83—22no—
106C4×C83—22no—
107C4×C83—44no—
108C4×C83—44no—
109C4×C83—82no—
110C4×C83#46212yes—
111C4×C83#49411yescharacteristic
112C4×C83#49711yescharacteristic
113Dic83—82no—
114#10017—831nomaximal, Sylow
115C2×C83—22no—
116C2×C83#201012yes—
117C2×C83#201611yescharacteristic
118#3104—832no—
119#2399D8311yescharacteristic
120#1995—831no—
121#1487—832no—
122#1274—831no—
123C83#1001711yescharacteristic, complemented, Sylow
124#533—831no—
125C2×C4×Q8C2×D8312yes—
126#484—1662no—
127#477—1662no—
128#462—1662no—
129#451—1662no—
130C4×Q16—1664no—
131#431—832no—
132#423—831no—
133#398—22no—
134#395—22no—
135C4×C4⋊C4—22no—
136#376C2×D8311yescharacteristic
137#339—1662no—
138Q8⋊C8—1664no—
139C22×Q8C22⋊D8312yes—
140C2×Q16—1664no—
141C2×M16—1661no—
142C4⋊Q8—832no—
143#125—1661no—
144C22⋊Q8—1662no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—1662no—
150#117—1661no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D8312yes—
154C2×C42—22no—
155C2×C42C22⋊D8312yes—
156C2×C42C22×D8311yescharacteristic
157#107—1664no—
158C4⋊C8—1662no—
159C4⋊C8—1662no—
160#103—1664no—
161#103—1664no—
162C22⋊C8—1662no—
163C4×C8—834no—
164#95—22no—
165#95—22no—
166#95—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—1662no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×C22⋊D8312yes—
175C22×C4C2×C22⋊D8311yescharacteristic
176Q16—3328no—
177M16—3322no—
178C2×C8—3321no—
179C2×C8—1664no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—3322no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C22⋊C4—3321no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—3321no—
193C42#1305058014yescomplemented
194C42D4×D8311yescharacteristic
195C42D4×D8312yes—
196C42#1305058911yescharacteristic
197C23#1458176111yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—3324no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—3322no—
212C2×C4#1458176012yes—
213C2×C4#1458176311yescharacteristic
214C2×C4#1458176412yes—
215C2×C4#1458178811yescharacteristic
216C2×C4C2×#1305058012yes—
217C2×C4#1458182312yes—
218C8—3324no—
219C8—6641no—
220C22—22no—
221C22#1912760911yescharacteristic
222C22#1912785712yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—6642no—
229C4#1912790812yes—
230C4#1912808011yescharacteristic
231C4#1912809311yescharacteristic
232C2—22no—
233C2#2490852812yes—
234C2#2490856311yescharacteristic
subgroupG/H
centerC2×C4#14581788
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2399C2
socleC22×C83#392
prime radicalC23×C83#120
hypercenter#2399D83
hypocenterC83#10017
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
75. C42×C83Q16semidirect
123. C83#10017semidirect
193. C42#13050580semidirect

Maximal subgroups

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

subgroupindex
#249175022
#249084442
#249058922
#249052922
C83×#23992
#1001783