Finite Group Databasebeta

← #35302196

Subgroups — #35302196

Order 21248.

576 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 115 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#24916492C211yesmaximal, characteristic
2#24911771C211yesmaximal, characteristic
3#24907547C212yesmaximal, complemented
4#24905330C212yesmaximal, complemented
5C83×#1799C211yesmaximal, characteristic
6C2×#14581773—21no—
7C22×C4⋊Dic83C2211yescharacteristic
8#19127773—24no—
9#19127758—21no—
10#19127665—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—
24C2×#13050574—22no—
25C22×C83⋊C8D414yes—
26#14581773—44no—
27C2×C4⋊Dic83—22no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83—22no—
31C2×C4⋊Dic83C2×C412yes—
32C2×Dic83⋊C4—22no—
33C2×C22⋊Dic83—22no—
34C2×C22⋊Dic83—21no—
35C4⋊(C83⋊C8)—22no—
36C22⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—22no—
38C22⋊(C83⋊C8)—22no—
39#14581703—22no—
40C83⋊#95—22no—
41C83⋊#95—22no—
42C83⋊#95C2×C412yes—
43C22×D4×C83D411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C23×C4×C83C2311yescharacteristic
46C4×D4×C83—22no—
47C4×D4×C83—22no—
48C2×C4⋊C4×C83—22no—
49C2×C4⋊C4×C83C2×C411yescharacteristic
50C2×C22⋊C4×C83—22no—
51C2×C22⋊C4×C83C2×C411yescharacteristic
52C2×C22⋊C4×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54#99×C83—42no—
55#95×C83—22no—
56C22×Dic83—22no—
57C22×Dic83—22no—
58C22×Dic83—42no—
59C22×Dic83—42no—
60C22×Dic83—41no—
61C22×Dic83—41no—
62#13050574—82no—
63C2×C83⋊C8—24no—
64C2×C83⋊C8—44no—
65C4⋊Dic83—24no—
66C4⋊Dic83—42no—
67C4⋊Dic83—41no—
68C4⋊Dic83—41no—
69Dic83⋊C4—44no—
70C22⋊Dic83—44no—
71C22⋊Dic83—42no—
72C24×C83C22⋊C411yescharacteristic
73C24×C83C2×D411yescharacteristic
74C2×D4×C83—22no—
75C2×D4×C83—22no—
76C2×D4×C83—21no—
77C2×D4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83—42no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D412yes—
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83#4112yes—
90C4⋊C4×C83—22no—
91C4⋊C4×C83—44no—
92C22⋊C4×C83—22no—
93C22⋊C4×C83—22no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C42×C83—21no—
97C42×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83C2×C22⋊C411yescharacteristic
106C23×C83#12112yes—
107C23×C83AGL(1,ℤ/8)12yes—
108C23×C83AGL(1,ℤ/8)11yescharacteristic
109C23×C83#13711yescharacteristic
110D4×C83—42no—
111D4×C83—42no—
112C2×C4×C83—24no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—81no—
125C2×C4×C83—82no—
126C2×C4×C83—82no—
127C2×C4×C83—82no—
128C2×C4×C83C4×D412yes—
129C2×C4×C83#12011yescharacteristic
130C2×C4×C83#12311yescharacteristic
131C2×Dic83—42no—
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—42no—
137C2×Dic83—82no—
138C2×Dic83—82no—
139C83⋊C8—44no—
140C22×C83—21no—
141C22×C83—22no—
142C22×C83—21no—
143C22×C83—22no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—42no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—81no—
152C22×C83#38611yescharacteristic
153C22×C83#41711yescharacteristic
154C22×C83#41811yescharacteristic
155C22×C83#46812yes—
156C22×C83#46912yes—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—162no—
163C4×C83#45812yes—
164Dic83—82no—
165Dic83—82no—
166#9474—831nomaximal, Sylow
167C2×C83—21no—
168C2×C83—22no—
169C2×C83—21no—
170C2×C83—22no—
171C2×C83—81no—
172C2×C83#189011yescharacteristic
173C2×C83#189511yescharacteristic
174C2×C83#190911yescharacteristic
175#2973—831no—
176#2486—831no—
177#1823—832no—
178#1799D8311yescharacteristic
179#1289—832no—
180C83#947411yescharacteristic, complemented, Sylow
181C2×#123—1661no—
182C2×C4×D4C2×D8311yescharacteristic
183C22×C4⋊C4—831no—
184#435—1664no—
185#432—1661no—
186C2×#99—21no—
187#407—1661no—
188C2×C22⋊C8—832no—
189#398—832no—
190#397—832no—
191#389—1661no—
192#381—1661no—
193#380—831no—
194#380—1661no—
195#380C2×D8312yes—
196#342—21no—
197#340—1664no—
198C22×D4C22⋊D8311yescharacteristic
199C23×C4—1661no—
200C23×C4D33211yescharacteristic
201C23×C4C22×D8311yescharacteristic
202C2×M16—1662no—
203C22×C8—834no—
204#123—3324no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—832no—
209C2×C4⋊C4—1662no—
210C2×C4⋊C4—1661no—
211C2×C4⋊C4—1661no—
212C2×C4⋊C4—1662no—
213C2×C4⋊C4—1662no—
214C2×C4⋊C4C4×D8311yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1662no—
217C2×C22⋊C4—1661no—
218C2×C22⋊C4C4×D8311yescharacteristic
219C2×C22⋊C4C22⋊D8311yescharacteristic
220C2×C42D33211yescharacteristic
221C4⋊C8—1662no—
222#99—42no—
223C22⋊C8—1662no—
224C22⋊C8—1662no—
225C22⋊C8—1662no—
226#97—1662no—
227#95—22no—
228#95—832no—
229#95—1662no—
230#95—1662no—
231C24(C2×D83)⋊C411yescharacteristic
232C24C2×C22⋊D8311yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—3322no—
243C22×C4—3322no—
244C22×C4—3321no—
245C22×C4—3321no—
246C22×C4—1662no—
247C22×C4—1662no—
248C22×C4(C2×D83)⋊C411yescharacteristic
249C22×C4(C2×D83)⋊C411yescharacteristic
250C22×C4(C2×D83)⋊C411yescharacteristic
251C22×C4C2×C4×D8311yescharacteristic
252C22×C4D4×D8312yes—
253C22×C4C2×D33211yescharacteristic
254C22×C4#1305058912yes—
255M16—6642no—
256C2×C8—3324no—
257C2×C8—1664no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—3324no—
261C4⋊C4—3322no—
262C4⋊C4—3321no—
263C4⋊C4—3321no—
264C4⋊C4—1664no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—3324no—
270C22⋊C4—3322no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D83)⋊C411yescharacteristic
281C23#1458176412yes—
282C23#1458181712yes—
283C23#1458181811yescharacteristic
284C23#1458182311yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—3322no—
304C2×C4—3322no—
305C2×C4—3322no—
306C2×C4—3322no—
307C2×C4—3322no—
308C2×C4—3322no—
309C2×C4—6642no—
310C2×C4—6642no—
311C2×C4(C22⋊D83)⋊C412yes—
312C2×C4#1458176211yescharacteristic
313C2×C4#1458177611yescharacteristic
314C8—3324no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1912757811yescharacteristic
328C22#1912770011yescharacteristic
329C22#1912770511yescharacteristic
330C22#1912794912yes—
331C22#1912795212yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—6642no—
339C4—6642no—
340C4#1912789212yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2490787411yescharacteristic
347C2#2490789911yescharacteristic
348C2#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#9474
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
180. C83#9474semidirect

Maximal subgroups

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

subgroupindex
#249164922
#249117712
#249075472
#249053302
C83×#17992
#947483