Finite Group Databasebeta

← #35299458

Subgroups — #35299458

Order 21248.

616 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#19127631C211yesmaximal, characteristic
2#24907547C212yesmaximal
3C2×#19127427C211yesmaximal, characteristic
4#24905341C212yesmaximal
5C83×#2313C211yesmaximal, characteristic
6C22×C4⋊Dic83C2211yescharacteristic
7C22×Dic83⋊C4—21no—
8C2×C22⋊(C83⋊C8)C2212yes—
9#19127631—22no—
10#19127631C414yescomplemented
11C2×C83⋊#95—21no—
12C2×C83⋊#95C2211yescharacteristic
13#19127463—24no—
14#19127427—22no—
15C22×C4⋊C4×C83C2211yescharacteristic
16#387×C83—24no—
17#381×C83—21no—
18#380×C83C2212yes—
19C4×C22⋊C4×C83—21no—
20C23×Dic83—21no—
21C23×Dic83—21no—
22C22×C83⋊C8—22no—
23C2×C4⋊Dic83—22no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83C2×C412yes—
27C2×Dic83⋊C4—24no—
28C2×Dic83⋊C4—44no—
29C22⋊(C83⋊C8)—22no—
30C22⋊(C83⋊C8)—22no—
31C83⋊#95—24no—
32C83⋊#95—21no—
33C83⋊#95C2×C412yes—
34C23×C4×C83D411yescharacteristic
35C23×C4×C83D411yescharacteristic
36C23×C4×C83C2311yescharacteristic
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83C2×C412yes—
40C2×C4⋊C4×C83D412yes—
41C2×C4⋊C4×C83D414yes—
42C2×C22⋊C4×C83—22no—
43C2×C42×C83—22no—
44#95×C83—22no—
45#95×C83—24no—
46#95×C83—22no—
47C22×Dic83—24no—
48C22×Dic83—22no—
49C22×Dic83—22no—
50C22×Dic83—42no—
51C22×Dic83—42no—
52C22×Dic83—41no—
53C22×Dic83—41no—
54C22×Dic83—42no—
55C22×Dic83—42no—
56C2×C83⋊C8—44no—
57C2×C83⋊C8—42no—
58C4⋊Dic83—24no—
59C4⋊Dic83—42no—
60Dic83⋊C4—48no—
61C24×C83C2×D411yescharacteristic
62C22×C4×C83—21no—
63C22×C4×C83—21no—
64C22×C4×C83—21no—
65C22×C4×C83—22no—
66C22×C4×C83—22no—
67C22×C4×C83—21no—
68C22×C4×C83—24no—
69C22×C4×C83—42no—
70C22×C4×C83—42no—
71C22×C4×C83C22⋊C412yes—
72C22×C4×C83C22⋊C412yes—
73C22×C4×C83C22×C411yescharacteristic
74C22×C4×C83C2×D412yes—
75C22×C4×C83C2×D411yescharacteristic
76C22×C4×C83#4112yes—
77C4⋊C4×C83—24no—
78C4⋊C4×C83—24no—
79C4⋊C4×C83—44no—
80C22⋊C4×C83—42no—
81C22⋊C4×C83—42no—
82C42×C83—82no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83C2×C22⋊C411yescharacteristic
86C23×C83#12011yescharacteristic
87C23×C83#12311yescharacteristic
88C23×C83#13512yes—
89C23×C83AGL(1,ℤ/8)12yes—
90C2×C4×C83—24no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—41no—
96C2×C4×C83—41no—
97C2×C4×C83—42no—
98C2×C4×C83—42no—
99C2×C4×C83—42no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—44no—
103C2×C4×C83—44no—
104C2×C4×C83—44no—
105C2×C4×C83—81no—
106C2×C4×C83C4×D412yes—
107C2×C4×C83#12112yes—
108C2×Dic83—24no—
109C2×Dic83—48no—
110C2×Dic83—42no—
111C2×Dic83—42no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—42no—
115C2×Dic83—82no—
116C2×Dic83—82no—
117C83⋊C8—82no—
118C22×C83—24no—
119C22×C83—24no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83#38611yescharacteristic
125C22×C83#41611yescharacteristic
126C22×C83#41811yescharacteristic
127C22×C83#44912yes—
128C22×C83#45612yes—
129C22×C83#45812yes—
130C22×C83#48212yes—
131C4×C83—22no—
132C4×C83—22no—
133C4×C83—42no—
134C4×C83—42no—
135C4×C83—84no—
136Dic83—44no—
137Dic83—82no—
138#9026—831nomaximal, Sylow
139C2×C83—21no—
140C2×C83—21no—
141C2×C83—22no—
142C2×C83#160914yes—
143C2×C83#187511yescharacteristic
144C2×C83#188611yescharacteristic
145C2×C83#189011yescharacteristic
146C2×#398—831no—
147#2313D8311yescharacteristic
148#1823—832no—
149C2×#324—831no—
150#1294—832no—
151C83#902611yescharacteristic, complemented, Sylow
152C22×C4⋊C4—831no—
153C22×C4⋊C4—1661no—
154C22×C4⋊C4C2×D8311yescharacteristic
155C2×C22⋊C8—832no—
156#398—834no—
157#398—1662no—
158#387—24no—
159#381—21no—
160#380C2×D8312yes—
161C4×C22⋊C4—21no—
162C2×#95—831no—
163C2×#95—1661no—
164#340—1664no—
165#324—1662no—
166C23×C4—1661no—
167C23×C4—1661no—
168C23×C4C22⋊D8311yescharacteristic
169C23×C4C22⋊D8311yescharacteristic
170C23×C4C22×D8311yescharacteristic
171C22×C8—1662no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—3324no—
175C2×C4⋊C4—832no—
176C2×C4⋊C4—1664no—
177C2×C4⋊C4—1661no—
178C2×C4⋊C4—1662no—
179C2×C4⋊C4—1662no—
180C2×C4⋊C4C4×D8312yes—
181C2×C4⋊C4D33212yes—
182C2×C4⋊C4C22⋊D8314yes—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C22⋊C8—1662no—
186C22⋊C8—1662no—
187#95—22no—
188#95—24no—
189#95—22no—
190#95—832no—
191#95—1664no—
192#95—1661no—
193C24C2×C22⋊D8311yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—3322no—
204C22×C4—3322no—
205C22×C4—3321no—
206C22×C4—3321no—
207C22×C4—3322no—
208C22×C4—3322no—
209C22×C4—1662no—
210C22×C4—1664no—
211C22×C4—1662no—
212C22×C4(C2×D83)⋊C412yes—
213C22×C4(C2×D83)⋊C412yes—
214C22×C4C2×C4×D8311yescharacteristic
215C22×C4C2×C22⋊D8312yes—
216C22×C4C2×D33211yescharacteristic
217C22×C4#1305059012yes—
218C2×C8—3324no—
219C2×C8—3322no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—44no—
223C4⋊C4—3328no—
224C4⋊C4—3322no—
225C4⋊C4—1664no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—82no—
229C23—22no—
230C23—22no—
231C23C2×(C2×D83)⋊C411yescharacteristic
232C23#1458176111yescharacteristic
233C23#1458177511yescharacteristic
234C23#1458181212yes—
235C23#1458181712yes—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—81no—
252C2×C4—3322no—
253C2×C4—3328no—
254C2×C4—3322no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—6642no—
259C2×C4—6642no—
260C2×C4—1664no—
261C2×C4C4×C22⋊D8312yes—
262C2×C4#1458176612yes—
263C8—6642no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—24no—
270C22#1912758011yescharacteristic
271C22#1912769511yescharacteristic
272C22#1912770511yescharacteristic
273C22#1912784612yes—
274C22#1912788412yes—
275C22#1912789112yes—
276C22#1912802512yes—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—84no—
282C4—3324no—
283C4—6642no—
284C2—21no—
285C2—21no—
286C2—22no—
287C2#2490646014yes—
288C2#2490778511yescharacteristic
289C2#2490785611yescharacteristic
290C2#2490787611yescharacteristic
subgroupG/H
centerC23#14581761
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2313C2
socleC23×C83#120
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#2313D83
hypocenterC83#9026
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
10. #19127631C4semidirect
151. C83#9026semidirect

Maximal subgroups

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

subgroupindex
C2×#191276312
#249075472
C2×#191274272
#249053412
C83×#23132
#902683