Finite Group Databasebeta

← #35315078

Subgroups — #35315078

Order 21248.

502 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 87 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#24907540C211yesmaximal, characteristic, complemented
2#24905797C211yesmaximal, characteristic, complemented
3#24905460C214yesmaximal, complemented
4C83×#2619C211yesmaximal, characteristic
5#19127758—21no—
6#19127564—21no—
7#19127468—21no—
8#19127468C2212yes—
9#19127427—21no—
10#19127425C2212yes—
11C2×#127×C83C411yescharacteristic, complemented
12C2×#121×C83—21no—
13C2×#117×C83C2211yescharacteristic
14#394×C83—21no—
15#394×C83C412yescomplemented
16#392×C83—21no—
17#392×C83C2212yes—
18#390×C83—21no—
19#382×C83C411yescharacteristic, complemented
20C2×C22⋊Dic83—22no—
21C4⋊(C83⋊C8)—41no—
22C22⋊(C83⋊C8)—22no—
23#14581703—41no—
24C83⋊#95—41no—
25C83⋊#95—41no—
26C22×D4×C83—22no—
27C22×D4×C83—21no—
28C22×D4×C83—21no—
29C22×D4×C83C2×C412yes—
30C23×C4×C83D411yescharacteristic
31#127×C83—41no—
32#127×C83—41no—
33#121×C83—44no—
34#117×C83—41no—
35#117×C83—41no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83C2×C411yescharacteristic
38C2×C4⋊C4×C83D411yescharacteristic
39C2×C22⋊C4×C83—22no—
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83D411yescharacteristic
43C2×C22⋊C4×C83C2311yescharacteristic
44C2×C42×C83C2×C411yescharacteristic
45C2×C42×C83D411yescharacteristic
46#95×C83—21no—
47#95×C83C2×C412yes—
48C22×Dic83—42no—
49C2×C83⋊C8—42no—
50C22⋊Dic83—42no—
51C22⋊Dic83—42no—
52C24×C83—22no—
53C24×C83C2×D411yescharacteristic
54C2×D4×C83—42no—
55C2×D4×C83—42no—
56C2×D4×C83—42no—
57C2×D4×C83—42no—
58C2×D4×C83—44no—
59C2×D4×C83—41no—
60C2×D4×C83—41no—
61C2×D4×C83—82no—
62C2×D4×C83—82no—
63C2×D4×C83—82no—
64C2×D4×C83—82no—
65C22×C4×C83—22no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—22no—
69C22×C4×C83—41no—
70C22×C4×C83C22⋊C411yescharacteristic
71C22×C4×C83C22⋊C411yescharacteristic
72C22×C4×C83C22⋊C411yescharacteristic
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22×C411yescharacteristic
75C22×C4×C83C2×D411yescharacteristic
76C4⋊C4×C83—41no—
77C4⋊C4×C83—41no—
78C4⋊C4×C83—42no—
79C22⋊C4×C83—41no—
80C22⋊C4×C83—41no—
81C22⋊C4×C83—44no—
82C22⋊C4×C83—82no—
83C22⋊C4×C83—82no—
84C42×C83—21no—
85C42×C83—21no—
86C42×C83—21no—
87C42×C83—21no—
88C23×C83—21no—
89C23×C83—21no—
90C23×C83—41no—
91C23×C83—41no—
92C23×C83—42no—
93C23×C83—42no—
94C23×C83—42no—
95C23×C83—42no—
96C23×C83—42no—
97C23×C83—42no—
98C23×C83—42no—
99C23×C83C2×C22⋊C411yescharacteristic
100C23×C83AGL(1,ℤ/8)12yes—
101D4×C83—82no—
102D4×C83—82no—
103D4×C83—84no—
104D4×C83—84no—
105D4×C83—88no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83#9911yescharacteristic
125C2×C4×C83#9911yescharacteristic
126C2×Dic83—82no—
127C2×Dic83—82no—
128C2×Dic83—82no—
129C83⋊C8—82no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83—22no—
133C22×C83—42no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—81no—
137C22×C83—82no—
138C22×C83—82no—
139C22×C83—82no—
140C22×C83—82no—
141C22×C83—82no—
142C22×C83—82no—
143C22×C83—82no—
144C22×C83C2×#9911yescharacteristic
145C22×C83#41811yescharacteristic
146C22×C83#42111yescharacteristic
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—41no—
150C4×C83—41no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—82no—
155Dic83—162no—
156#11800—831nomaximal, Sylow
157C2×C83—21no—
158C2×C83—21no—
159C2×C83—42no—
160C2×C83—162no—
161C2×C83#151011yescharacteristic
162C2×C83#212211yescharacteristic
163C2×C83#212811yescharacteristic
164#2619D8311yescharacteristic
165#1820—831no—
166#1464—831no—
167#1343—834no—
168C83#1180011yescharacteristic, complemented, Sylow
169C2×#127Dic8311yescharacteristic, complemented
170C2×#121—21no—
171C2×#117C2×D8311yescharacteristic
172#432—1661no—
173#394—21no—
174#394Dic8312yescomplemented
175#392—21no—
176#392C2×D8312yes—
177#390—21no—
178#382Dic8311yescharacteristic, complemented
179#381—1661no—
180#342—832no—
181#342—1661no—
182#324—1661no—
183#323—832no—
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C2×Dic8312yes—
188C23×C4C22⋊D8311yescharacteristic
189#127—41no—
190#127—41no—
191#121—44no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic8311yescharacteristic
196C2×C4⋊C4C22⋊D8311yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1662no—
201C2×C22⋊C4C22⋊D8311yescharacteristic
202C2×C22⋊C4C22×D8311yescharacteristic
203C2×C42C2×Dic8311yescharacteristic
204C2×C42C22⋊D8311yescharacteristic
205C4⋊C8—3321no—
206C22⋊C8—1662no—
207#97—3321no—
208#95—21no—
209#95—3321no—
210#95—3321no—
211#95C2×Dic8312yes—
212C24—22no—
213C24C2×C22⋊D8311yescharacteristic
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—3322no—
231C22×C4C22⋊Dic8311yescharacteristic
232C22×C4C22⋊Dic8311yescharacteristic
233C22×C4C22⋊Dic8311yescharacteristic
234C22×C4C22⋊Dic8311yescharacteristic
235C22×C4C22×Dic8311yescharacteristic
236C22×C4C2×C22⋊D8311yescharacteristic
237C2×C8—3322no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—3322no—
247C22⋊C4—3322no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23C2×C22⋊Dic8311yescharacteristic
264C23#1458181812yes—
265D4—82no—
266D4—82no—
267D4—84no—
268D4—84no—
269D4—88no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—6642no—
289C2×C4—6642no—
290C2×C4—6642no—
291C2×C4#1458170711yescharacteristic
292C2×C4#1458170711yescharacteristic
293C8—6642no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C2×#1458170711yescharacteristic
309C22#1912770611yescharacteristic
310C22#1912771811yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—13282no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—162no—
324C2#2490599211yescharacteristic
325C2#2490911311yescharacteristic
326C2#2490913511yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
Fitting, prime radicalC83×#2619C2
socleC22×C83C2×#99
cosocleC2×C22⋊C4×C83C23
hypercenter#2619D83
hypocenterC83#11800
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #24907540C2semidirect
2. #24905797C2semidirect
3. #24905460C2semidirect
11. C2×#127×C83C4semidirect
15. #394×C83C4semidirect
19. #382×C83C4semidirect
168. C83#11800semidirect
169. C2×#127Dic83semidirect
174. #394Dic83semidirect
178. #382Dic83semidirect

Maximal subgroups

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

subgroupindex
#249075402
#249057972
#249054602
C83×#26192
#1180083