Finite Group Databasebeta

← #35302176

Subgroups — #35302176

Order 21248.

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

normal subgroups only

This group has 598 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
1#24916487C211yesmaximal, characteristic
2#24911771C211yesmaximal, characteristic
3#24907547C212yesmaximal, complemented
4#24905330C212yesmaximal, complemented
5C83×#1799C211yesmaximal, characteristic
6C2×#14581773—21no—
7C22×C4⋊Dic83C2211yescharacteristic
8D4×C83⋊C8—24no—
9#19127754—21no—
10C2×C22⋊(C83⋊C8)—21no—
11C2×C22⋊(C83⋊C8)C2212yes—
12#19127631C412yescomplemented
13#19127626C412yescomplemented
14#19127599—21no—
15#19127562—21no—
16#19127561—21no—
17#19127561C2211yescharacteristic
18#19127463—24no—
19C2×C4×D4×C83C2211yescharacteristic
20C2×#99×C83—21no—
21#380×C83C2212yes—
22#342×C83—21no—
23C23×Dic83—21no—
24C22×C83⋊C8—22no—
25C22×C83⋊C8D414yes—
26#14581773—42no—
27#14581773—42no—
28C2×C4⋊Dic83—22no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83—21no—
31C2×C4⋊Dic83—22no—
32C2×C4⋊Dic83C2×C412yes—
33C2×Dic83⋊C4—22no—
34C2×C22⋊Dic83—22no—
35C2×C22⋊Dic83—21no—
36C4⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—22no—
38C22⋊(C83⋊C8)—22no—
39C22⋊(C83⋊C8)—22no—
40C4×C83⋊C8—22no—
41C83⋊#95—22no—
42C83⋊#95—22no—
43C83⋊#95C2×C412yes—
44C22×D4×C83D411yescharacteristic
45C23×C4×C83D411yescharacteristic
46C23×C4×C83C2311yescharacteristic
47C4×D4×C83—22no—
48C4×D4×C83—22no—
49C2×C4⋊C4×C83—22no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83—22no—
52C2×C22⋊C4×C83C2×C411yescharacteristic
53C2×C22⋊C4×C83D411yescharacteristic
54C2×C42×C83D411yescharacteristic
55#99×C83—42no—
56#95×C83—22no—
57C22×Dic83—22no—
58C22×Dic83—22no—
59C22×Dic83—42no—
60C22×Dic83—41no—
61C22×Dic83—41no—
62C22×Dic83—42no—
63C2×C83⋊C8—24no—
64C2×C83⋊C8—44no—
65C2×C83⋊C8—82no—
66C4⋊Dic83—24no—
67C4⋊Dic83—42no—
68C4⋊Dic83—41no—
69C4⋊Dic83—41no—
70Dic83⋊C4—44no—
71C22⋊Dic83—44no—
72C22⋊Dic83—41no—
73C22⋊Dic83—41no—
74C24×C83C22⋊C411yescharacteristic
75C24×C83C2×D411yescharacteristic
76C2×D4×C83—22no—
77C2×D4×C83—22no—
78C2×D4×C83D812yes—
79C2×D4×C83SD1612yes—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—42no—
83C22×C4×C83—42no—
84C22×C4×C83—42no—
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D412yes—
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83#4112yes—
92C4⋊C4×C83—22no—
93C4⋊C4×C83—44no—
94C22⋊C4×C83—22no—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—42no—
97C22⋊C4×C83—42no—
98C42×C83—21no—
99C42×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83#10212yes—
107C23×C83C2×C22⋊C411yescharacteristic
108C23×C83#12112yes—
109C23×C83C2×D811yescharacteristic
110C23×C83C2×SD1611yescharacteristic
111C23×C83#13512yes—
112D4×C83—24no—
113D4×C83—24no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—24no—
117C2×C4×C83—22no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—41no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—81no—
126C2×C4×C83—82no—
127C2×C4×C83—82no—
128C2×C4×C83—82no—
129C2×C4×C83#10212yes—
130C2×C4×C83C4×D412yes—
131C2×C4×C83#12011yescharacteristic
132C2×C4×C83#12311yescharacteristic
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—42no—
138C2×Dic83—42no—
139C2×Dic83—82no—
140C2×Dic83—82no—
141C83⋊C8—44no—
142C22×C83—21no—
143C22×C83—22no—
144C22×C83—21no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—42no—
150C22×C83—42no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—81no—
154C22×C83#38611yescharacteristic
155C22×C83C2×#10211yescharacteristic
156C22×C83#41611yescharacteristic
157C22×C83#46512yes—
158C22×C83#46612yes—
159C4×C83—21no—
160C4×C83—22no—
161C4×C83—22no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—162no—
165C4×C83#45812yes—
166Dic83—82no—
167Dic83—82no—
168#9470—831nomaximal, Sylow
169C2×C83—21no—
170C2×C83—22no—
171C2×C83—21no—
172C2×C83—22no—
173C2×C83—42no—
174C2×C83#189011yescharacteristic
175C2×C83#189311yescharacteristic
176C2×C83#190911yescharacteristic
177#2972—831no—
178#2486—831no—
179#1823—832no—
180#1799D8311yescharacteristic
181#1289—832no—
182C83#947011yescharacteristic, complemented, Sylow
183C2×#123—1661no—
184C2×C4×D4C2×D8311yescharacteristic
185C22×C4⋊C4—831no—
186C8×D4—1664no—
187#431—1661no—
188C2×#99—21no—
189C2×C22⋊C8—832no—
190C2×C22⋊C8—1661no—
191#398—832no—
192#397—832no—
193#389—1661no—
194#381—1661no—
195#380—831no—
196#380—1661no—
197#380C2×D8312yes—
198#342—21no—
199#340—1664no—
200C22×D4C22⋊D8311yescharacteristic
201C23×C4—1661no—
202C23×C4D33211yescharacteristic
203C23×C4C22×D8311yescharacteristic
204C22×C8—834no—
205C22×C8—1662no—
206#123—3322no—
207#123—3322no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—832no—
212C2×C4⋊C4—1662no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4—1661no—
215C2×C4⋊C4—1662no—
216C2×C4⋊C4—1662no—
217C2×C4⋊C4C4×D8311yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—1662no—
220C2×C22⋊C4—1661no—
221C2×C22⋊C4C4×D8311yescharacteristic
222C2×C22⋊C4C22⋊D8311yescharacteristic
223C2×C42D33211yescharacteristic
224C4⋊C8—1662no—
225#99—42no—
226C22⋊C8—1662no—
227C22⋊C8—1662no—
228C22⋊C8—1662no—
229C4×C8—1662no—
230#95—22no—
231#95—832no—
232#95—1662no—
233#95—1662no—
234C24(C2×D83)⋊C411yescharacteristic
235C24C2×C22⋊D8311yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1305057512yes—
239C2×D4#1305057712yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—3322no—
246C22×C4—3321no—
247C22×C4—3321no—
248C22×C4—3322no—
249C22×C4—1662no—
250C22×C4—1662no—
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4(C2×D83)⋊C411yescharacteristic
253C22×C4(C2×D83)⋊C411yescharacteristic
254C22×C4C2×C4×D8311yescharacteristic
255C22×C4D4×D8312yes—
256C22×C4C2×D33211yescharacteristic
257C22×C4#1305058912yes—
258C2×C8—3324no—
259C2×C8—6642no—
260C2×C8—1664no—
261C4⋊C4—22no—
262C4⋊C4—44no—
263C4⋊C4—3322no—
264C4⋊C4—3321no—
265C4⋊C4—3321no—
266C4⋊C4—3324no—
267C4⋊C4—1664no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—3321no—
273C22⋊C4—3321no—
274C22⋊C4—3324no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1458171212yes—
284C23C2×(C2×D83)⋊C411yescharacteristic
285C23#1458176412yes—
286C23C2×#1305057511yescharacteristic
287C23C2×#1305057711yescharacteristic
288C23#1458181212yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—3322no—
307C2×C4—3322no—
308C2×C4—3322no—
309C2×C4—3322no—
310C2×C4—3322no—
311C2×C4—3322no—
312C2×C4—6642no—
313C2×C4—6642no—
314C2×C4#1458171212yes—
315C2×C4(C22⋊D83)⋊C412yes—
316C2×C4#1458176211yescharacteristic
317C2×C4#1458177611yescharacteristic
318C8—3324no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1912757811yescharacteristic
332C22C2×#1458171211yescharacteristic
333C22#1912769511yescharacteristic
334C22#1912793212yes—
335C22#1912793512yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—162no—
342C4—6642no—
343C4—6642no—
344C4#1912789212yes—
345C2—22no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2490787411yescharacteristic
351C2#2490788911yescharacteristic
352C2#2490798311yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1799C2
socleC22×C83#386
prime radicalC22×D4×C83D4
cosocleC23×C4×C83C23
hypercenter#1799D83
hypocenterC83#9470
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #24907547C2semidirect
4. #24905330C2semidirect
12. #19127631C4semidirect
13. #19127626C4semidirect
182. C83#9470semidirect

Maximal subgroups

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

subgroupindex
#249164872
#249117712
#249075472
#249053302
C83×#17992
#947083