Finite Group Databasebeta

← #35305000

Subgroups — #35305000

Order 21248.

632 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 121 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×#19128032C212yesmaximal
2#24913577C211yesmaximal, characteristic
3C2×#19127439C212yesmaximal
4#24905730C211yesmaximal, characteristic
5C83×#1280C211yesmaximal, characteristic
6C2×#14581771C2212yes—
7C22×C4⋊Dic83C2211yescharacteristic
8#19128032—24no—
9#19128032—24no—
10C2×#14581724—22no—
11C2×#14581717—22no—
12#19127633—22no—
13#19127622—22no—
14#19127561—22no—
15C2×C83⋊#95—22no—
16C2×C83⋊#95C2211yescharacteristic
17#19127439—24no—
18C2×C22⋊C8×C83C2212yes—
19C2×#95×C83C2211yescharacteristic
20C22×Dic166D412yes—
21C23×Dic83—21no—
22C23×Dic83—21no—
23#14581771—22no—
24#14581771—22no—
25#14581771—22no—
26#14581771—22no—
27C2×C4⋊Dic83—22no—
28C2×C4⋊Dic83—22no—
29C2×C4⋊Dic83—22no—
30C2×C4⋊Dic83—22no—
31C2×C4⋊Dic83—21no—
32C2×C4⋊Dic83D412yes—
33C2×C4⋊Dic83D412yes—
34C2×Dic83⋊C4—22no—
35C2×C22⋊Dic83—22no—
36#14581724—28no—
37#14581717—28no—
38C83⋊#95—24no—
39C83⋊#95—22no—
40C83⋊#95—44no—
41C23×C4×C83D412yes—
42C23×C4×C83C2311yescharacteristic
43C22×C8×C83—22no—
44C22⋊C8×C83—22no—
45C22⋊C8×C83D414yes—
46#95×C83—22no—
47C2×Dic166—22no—
48C2×Dic166—22no—
49C2×Dic166—44no—
50C22×Dic83—22no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C22×Dic83—41no—
55C22×Dic83—42no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83—41no—
59C22×Dic83—42no—
60C22×Dic83—42no—
61C22×Dic83—42no—
62C4⋊Dic83—22no—
63C4⋊Dic83—24no—
64C4⋊Dic83—22no—
65C4⋊Dic83—42no—
66C4⋊Dic83—82no—
67C4⋊Dic83—82no—
68Dic83⋊C4—42no—
69Dic83⋊C4—42no—
70C22⋊Dic83—42no—
71C22⋊Dic83—42no—
72C24×C83C2×D411yescharacteristic
73C22×C4×C83—22no—
74C22×C4×C83—22no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83SD1614yes—
79C22×C4×C83C2×D412yes—
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D412yes—
82C22×C4×C83#4111yescharacteristic
83C2×C8×C83—24no—
84C2×C8×C83—44no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—22no—
88C23×C83#12011yescharacteristic
89C23×C83#12111yescharacteristic
90C23×C83#12711yescharacteristic
91C23×C83C2×SD1612yes—
92C23×C83#13712yes—
93C2×C4×C83—24no—
94C2×C4×C83—24no—
95C2×C4×C83—41no—
96C2×C4×C83—42no—
97C2×C4×C83—41no—
98C2×C4×C83—42no—
99C2×C4×C83—42no—
100C2×C4×C83—44no—
101C2×C4×C83—82no—
102C2×C4×C83#12012yes—
103C2×C4×C83#12112yes—
104C2×Dic83—44no—
105C2×Dic83—42no—
106C2×Dic83—42no—
107C2×Dic83—42no—
108C2×Dic83—42no—
109C2×Dic83—42no—
110C2×Dic83—42no—
111C2×Dic83—44no—
112C2×Dic83—44no—
113C2×Dic83—82no—
114C2×Dic83—81no—
115C2×Dic83—82no—
116C2×Dic83—81no—
117C2×Dic83—81no—
118C2×Dic83—81no—
119Dic166—44no—
120Dic166—82no—
121C8×C83—44no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—22no—
125C22×C83—22no—
126C22×C83—21no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83#39211yescharacteristic
131C22×C83#44812yes—
132C22×C83#45511yescharacteristic
133C22×C83#45611yescharacteristic
134C22×C83#45812yes—
135C22×C83#46112yes—
136C22×C83#49211yescharacteristic
137C22×C83#49711yescharacteristic
138C4×C83—22no—
139C4×C83—22no—
140C4×C83—44no—
141Dic83—82no—
142Dic83—82no—
143Dic83—82no—
144Dic83—82no—
145#9973—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—22no—
148C2×C83—21no—
149C2×C83#160812yes—
150C2×C83#161112yes—
151C2×C83#200711yescharacteristic
152C2×C83#201311yescharacteristic
153C2×C83#202211yescharacteristic
154C2×#483—832no—
155#2659—831no—
156C2×#328—832no—
157#1445—831no—
158#1280D8311yescharacteristic
159C83#997311yescharacteristic, complemented, Sylow
160C2×C22⋊Q8—832no—
161C22×C4⋊C4—831no—
162#483—1664no—
163#483—1664no—
164C2×#106—1662no—
165C2×#103—1662no—
166C2×C22⋊C8C2×D8312yes—
167#398—1662no—
168#395—1662no—
169#380—1662no—
170C2×#95—831no—
171C2×#95—1662no—
172C2×#95C2×D8311yescharacteristic
173#328—1664no—
174C22×Q8—832no—
175C23×C4—1661no—
176C23×C4—1661no—
177C23×C4D33212yes—
178C23×C4C22×D8311yescharacteristic
179C22×C8—22no—
180C22⋊Q8—1662no—
181C22⋊Q8—1662no—
182C22⋊Q8—1662no—
183C22⋊Q8—1662no—
184C2×C4⋊C4—832no—
185C2×C4⋊C4—832no—
186C2×C4⋊C4—1662no—
187C2×C4⋊C4—1661no—
188C2×C4⋊C4—1662no—
189C2×C4⋊C4—1662no—
190C2×C4⋊C4—1662no—
191C2×C4⋊C4—1662no—
192C2×C22⋊C4—1662no—
193#106—1668no—
194#103—1668no—
195C22⋊C8—22no—
196C22⋊C8D33214yes—
197#95—22no—
198#95—3324no—
199#95—1664no—
200#95—1662no—
201C24C2×D33211yescharacteristic
202C2×Q8—3324no—
203C2×Q8—1662no—
204C2×Q8—1662no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—3321no—
211C22×C4—3322no—
212C22×C4—3321no—
213C22×C4—3321no—
214C22×C4—3321no—
215C22×C4—3322no—
216C22×C4—3322no—
217C22×C4—3322no—
218C22×C4—1662no—
219C22×C4—1662no—
220C22×C4—1662no—
221C22×C4—1662no—
222C22×C4#1305057914yes—
223C22×C4D4×D8312yes—
224C22×C4D4×D8311yescharacteristic
225C22×C4C2×D33212yes—
226C22×C4#1305059111yescharacteristic
227C2×C8—24no—
228C2×C8—44no—
229C4⋊C4—3322no—
230C4⋊C4—3322no—
231C4⋊C4—3322no—
232C4⋊C4—6642no—
233C4⋊C4—6642no—
234C4⋊C4—1664no—
235C4⋊C4—1662no—
236C4⋊C4—1662no—
237C22⋊C4—3322no—
238C22⋊C4—3322no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1458176211yescharacteristic
243C23#1458176711yescharacteristic
244C23#1458178911yescharacteristic
245C23C2×#1305057912yes—
246C23#1458182512yes—
247Q8—3324no—
248Q8—6642no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—82no—
258C2×C4—3322no—
259C2×C4—3324no—
260C2×C4—3322no—
261C2×C4—3322no—
262C2×C4—3322no—
263C2×C4—3322no—
264C2×C4—3322no—
265C2×C4—3324no—
266C2×C4—3324no—
267C2×C4—6641no—
268C2×C4—6642no—
269C2×C4—6641no—
270C2×C4—6641no—
271C2×C4—6641no—
272C2×C4—6642no—
273C2×C4#1458176212yes—
274C2×C4#1458176712yes—
275C8—44no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22#1912761111yescharacteristic
285C22#1912784212yes—
286C22#1912788111yescharacteristic
287C22#1912788611yescharacteristic
288C22#1912789212yes—
289C22#1912790712yes—
290C22#1912807411yescharacteristic
291C22#1912809511yescharacteristic
292C4—22no—
293C4—22no—
294C4—44no—
295C4—6642no—
296C4—6642no—
297C4—6642no—
298C4—6642no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2#2490645612yes—
303C2#2490647312yes—
304C2#2490851211yescharacteristic
305C2#2490854811yescharacteristic
306C2#2490859511yescharacteristic
subgroupG/H
centerC23#14581762
commutator, cosocleC23×C4×C83C23
FrattiniC23×C4C22×D83
FittingC83×#1280C2
socleC23×C83#120
prime radicalC24×C83C2×D4
hypercenter#1280D83
hypocenterC83#9973
LCSG ▷ C23×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
159. C83#9973semidirect

Maximal subgroups

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

subgroupindex
C2×#191280322
#249135772
C2×#191274392
#249057302
C83×#12802
#997383