Finite Group Databasebeta

← #35302246

Subgroups — #35302246

Order 21248.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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#24916544C211yesmaximal, characteristic
2#24912825C211yesmaximal, characteristic
3#24905891C212yesmaximal
4#24905323C212yesmaximal
5C83×#1801C211yesmaximal, characteristic
6C2×C4×Dic166C2211yescharacteristic
7Q8×C83⋊C8—24no—
8#19127754—22no—
9#19127754C2212yes—
10#19127640C412yes—
11#19127621C412yes—
12#19127608—21no—
13#19127597—21no—
14#19127585—22no—
15#19127567C2211yescharacteristic
16#19127431—24no—
17C2×C4×Q8×C83C2211yescharacteristic
18#423×C83C2212yes—
19C2×#101×C83—21no—
20#343×C83—21no—
21C22×Dic166C2×C411yescharacteristic
22C4×Dic166—22no—
23C4×Dic166—22no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×Dic83⋊C4—22no—
27C2×Dic83⋊C4—21no—
28C2×C4×Dic83—21no—
29C4⋊(C83⋊C8)—24no—
30C4⋊(C83⋊C8)—22no—
31C22⋊(C83⋊C8)—22no—
32C4×C83⋊C8—24no—
33C4×C83⋊C8D414yes—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C83⋊#95—21no—
38C83⋊#95C2×C412yes—
39C22×Q8×C83D411yescharacteristic
40C2×M16×C83—22no—
41C4×Q8×C83—22no—
42C4×Q8×C83—22no—
43C2×C4⋊C4×C83C2×C412yes—
44C2×C4⋊C4×C83D411yescharacteristic
45C2×C42×C83D412yes—
46C2×C42×C83C2311yescharacteristic
47C4⋊C8×C83—22no—
48C4⋊C8×C83—22no—
49#101×C83—42no—
50C2×Dic166—22no—
51C2×Dic166—42no—
52C22×Dic83—21no—
53C22×Dic83—22no—
54C22×Dic83—21no—
55C2×C83⋊C8—24no—
56C4⋊Dic83—22no—
57C4⋊Dic83—42no—
58Dic83⋊C4—44no—
59Dic83⋊C4—41no—
60Dic83⋊C4—41no—
61C4×Dic83—42no—
62C2×Q8×C83—22no—
63C2×Q8×C83—22no—
64C2×Q8×C83SD1612yes—
65C2×Q8×C83Q1612yes—
66C22×C4×C83C22⋊C412yes—
67C22×C4×C83C22⋊C412yes—
68C22×C4×C83C22×C411yescharacteristic
69C22×C4×C83C2×D411yescharacteristic
70C22×C4×C83C2×D411yescharacteristic
71M16×C83—42no—
72M16×C83—42no—
73C2×C8×C83—42no—
74C4⋊C4×C83—24no—
75C4⋊C4×C83—22no—
76C42×C83—22no—
77C42×C83—22no—
78C42×C83C2×D412yes—
79C42×C83#4112yes—
80C23×C83C2×C22⋊C411yescharacteristic
81Q8×C83—24no—
82Q8×C83—24no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83—42no—
89C2×C4×C83#10314yes—
90C2×C4×C83C4×D412yes—
91C2×C4×C83#12011yescharacteristic
92C2×C4×C83#12112yes—
93C2×C4×C83#12311yescharacteristic
94C2×C4×C83C2×SD1611yescharacteristic
95C2×C4×C83C2×Q1611yescharacteristic
96C2×C4×C83#13512yes—
97C2×Dic83—22no—
98C2×Dic83—41no—
99C2×Dic83—41no—
100C2×Dic83—42no—
101C2×Dic83—42no—
102C2×Dic83—42no—
103C2×Dic83—41no—
104C2×Dic83—42no—
105Dic166—24no—
106Dic166—42no—
107C8×C83—82no—
108C83⋊C8—44no—
109C22×C83—21no—
110C22×C83—21no—
111C22×C83#38611yescharacteristic
112C22×C83C2×#10311yescharacteristic
113C22×C83#41611yescharacteristic
114C4×C83—22no—
115C4×C83—24no—
116C4×C83—21no—
117C4×C83—21no—
118C4×C83—42no—
119C4×C83#46512yes—
120C4×C83#46712yes—
121Dic83—42no—
122Dic83—82no—
123Dic83—81no—
124#9484—831nomaximal, Sylow
125C2×C83—21no—
126C2×C83—21no—
127C2×C83#189111yescharacteristic
128C2×C83#189211yescharacteristic
129C2×C83#191211yescharacteristic
130#2982—831no—
131#2581—831no—
132#1801D8311yescharacteristic
133#1487—832no—
134#1286—832no—
135C83#948411yescharacteristic, complemented, Sylow
136C2×C4×Q8—831no—
137C2×C4×Q8C2×D8311yescharacteristic
138C8×Q8—1664no—
139#431—832no—
140#431—1662no—
141#423C2×D8312yes—
142C2×#101—21no—
143#400—832no—
144#395—832no—
145#391—1661no—
146#389—1661no—
147#387—1662no—
148#382—831no—
149#343—21no—
150Q8⋊C8—1664no—
151C22×Q8—831no—
152C22×Q8C22⋊D8311yescharacteristic
153C2×M16—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1662no—
157C4×Q8—1662no—
158C2×C4⋊C4—831no—
159C2×C4⋊C4—1662no—
160C2×C4⋊C4—1661no—
161C2×C4⋊C4—1661no—
162C2×C4⋊C4C4×D8312yes—
163C2×C4⋊C4C22⋊D8311yescharacteristic
164C2×C42—1661no—
165C2×C42D33212yes—
166C2×C42C22×D8311yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—1664no—
170C4⋊C8—1662no—
171#101—42no—
172C22⋊C8—1662no—
173C4×C8—834no—
174C4×C8—1664no—
175#95—832no—
176#95—1662no—
177#95—1662no—
178#95—1662no—
179#95—1661no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—3322no—
183C2×Q8—1662no—
184C2×Q8#1305057812yes—
185C2×Q8#1305058012yes—
186C22×C4—1662no—
187C22×C4—1661no—
188C22×C4—1661no—
189C22×C4(C2×D83)⋊C412yes—
190C22×C4(C2×D83)⋊C412yes—
191C22×C4C2×C4×D8311yescharacteristic
192C22×C4C2×C22⋊D8311yescharacteristic
193C22×C4C2×D33211yescharacteristic
194M16—42no—
195M16—42no—
196C2×C8—42no—
197C2×C8—1664no—
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—3324no—
201C4⋊C4—3321no—
202C4⋊C4—3321no—
203C4⋊C4—3322no—
204C4⋊C4—1662no—
205C42—22no—
206C42—22no—
207C42—3322no—
208C42D4×D8312yes—
209C42#1305058912yes—
210C23C2×(C2×D83)⋊C411yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—3322no—
214Q8—1664no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—3321no—
222C2×C4—3321no—
223C2×C4—3322no—
224C2×C4—3322no—
225C2×C4—3322no—
226C2×C4—3321no—
227C2×C4—3322no—
228C2×C4—1662no—
229C2×C4#1458171514yes—
230C2×C4(C22⋊D83)⋊C412yes—
231C2×C4#1458176211yescharacteristic
232C2×C4#1458176412yes—
233C2×C4#1458177611yescharacteristic
234C2×C4C2×#1305057811yescharacteristic
235C2×C4C2×#1305058011yescharacteristic
236C2×C4#1458181212yes—
237C8—82no—
238C8—3324no—
239C22—21no—
240C22—21no—
241C22#1912757811yescharacteristic
242C22C2×#1458171511yescharacteristic
243C22#1912769511yescharacteristic
244C4—22no—
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—3322no—
250C4—6642no—
251C4—6641no—
252C4#1912792812yes—
253C4#1912794012yes—
254C2—21no—
255C2—21no—
256C2#2490787911yescharacteristic
257C2#2490788411yescharacteristic
258C2#2490799811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1801C2
socleC22×C83#386
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1801D83
hypocenterC83#9484
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
135. C83#9484semidirect

Maximal subgroups

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

subgroupindex
#249165442
#249128252
#249058912
#249053232
C83×#18012
#948483