Finite Group Databasebeta

← #35315079

Subgroups — #35315079

Order 21248.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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#24913213C211yesmaximal, characteristic
2#24907540C211yesmaximal, characteristic, complemented
3#24905461C214yesmaximal, complemented
4C83×#1464C211yesmaximal, characteristic, complemented
5C2×#14581789C411yescharacteristic, complemented
6C2×#14581766—21no—
7#19127618—21no—
8#19127618C412yescomplemented
9#19127610—21no—
10#19127610C2212yes—
11#19127603—21no—
12#19127567C411yescharacteristic, complemented
13#19127564—21no—
14#19127468—21no—
15#19127468C2212yes—
16C2×#117×C83C2211yescharacteristic
17#432×C83—21no—
18#324×C83—21no—
19#323×C83C2212yes—
20C22×D332—21no—
21C22×D332—21no—
22C22×D332C2×C412yes—
23C22×C22⋊D83—22no—
24#14581789—41no—
25#14581789—41no—
26#14581766—44no—
27C2×C4⋊Dic83—21no—
28C2×C22⋊Dic83—22no—
29C2×(C2×D83)⋊C4—22no—
30C2×(C2×D83)⋊C4—22no—
31C2×(C2×D83)⋊C4—22no—
32C83⋊#95—21no—
33C83⋊#95—41no—
34C83⋊#95—41no—
35C83⋊#95C2×C412yes—
36C23×C4×C83D411yescharacteristic
37#117×C83—41no—
38#117×C83—41no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C22⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83C2311yescharacteristic
43C2×C42×C83C2×C411yescharacteristic
44C2×C42×C83D411yescharacteristic
45C4⋊C8×C83—41no—
46C22⋊C8×C83—22no—
47#97×C83—41no—
48C23×D83—22no—
49C2×D332—42no—
50C2×D332—42no—
51C2×D332—42no—
52C2×D332—42no—
53C2×D332—41no—
54C2×D332—41no—
55C2×D332—82no—
56C2×D332—82no—
57C2×C22⋊D83—44no—
58C2×C22⋊D83—82no—
59C2×C22⋊D83—82no—
60C22×Dic83—22no—
61C22×Dic83—42no—
62C4⋊Dic83—42no—
63C22⋊Dic83—42no—
64C22⋊Dic83—42no—
65(C2×D83)⋊C4—44no—
66(C2×D83)⋊C4—82no—
67(C2×D83)⋊C4—82no—
68C24×C83C2×D411yescharacteristic
69C22×C4×C83—22no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—41no—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C2×C8×C83—42no—
80C4⋊C4×C83—41no—
81C4⋊C4×C83—41no—
82C22⋊C4×C83—41no—
83C22⋊C4×C83—41no—
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×C83C2×C22⋊C411yescharacteristic
93C23×C83AGL(1,ℤ/8)12yes—
94C22×D83—42no—
95C22×D83—42no—
96C22×D83—42no—
97C22×D83—42no—
98C22×D83—42no—
99C22×D83—42no—
100C22×D83—42no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83#9911yescharacteristic
117C2×C4×C83#9911yescharacteristic
118C22⋊D83—88no—
119C2×Dic83—42no—
120C2×Dic83—42no—
121C2×Dic83—42no—
122C2×Dic83—82no—
123C2×Dic83—82no—
124C2×Dic83—82no—
125D332—82no—
126D332—82no—
127D332—84no—
128D332—84no—
129C8×C83—82no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83—22no—
133C22×C83—42no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—81no—
137C22×C83C2×#9911yescharacteristic
138C22×C83#41811yescharacteristic
139C22×C83#42111yescharacteristic
140C2×D83—82no—
141C2×D83—82no—
142C2×D83—82no—
143C2×D83—82no—
144C2×D83—82no—
145C2×D83—82no—
146C2×D83—82no—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—41no—
150C4×C83—41no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83—41no—
154Dic83—82no—
155Dic83—162no—
156#11800—831nomaximal, Sylow
157C2×C83—21no—
158C2×C83—21no—
159C2×C83—42no—
160C2×C83#151011yescharacteristic
161C2×C83#212211yescharacteristic
162C2×C83#212811yescharacteristic
163D83—162no—
164#2619—831no—
165#1820—831no—
166#1464D8311yescharacteristic, complemented
167#1343—834no—
168C83#1180011yescharacteristic, complemented, Sylow
169C2×#127—831no—
170C2×#121—1661no—
171C2×#117C2×D8311yescharacteristic
172#432—21no—
173#394—832no—
174#394—1661no—
175#392—832no—
176#392—1661no—
177#390—1661no—
178#382—831no—
179#381—1661no—
180#342—832no—
181#342—1661no—
182#324—21no—
183#323C2×D8312yes—
184C22×D4—832no—
185C22×D4—1662no—
186C22×D4—1661no—
187C22×D4—1661no—
188C23×C4D33211yescharacteristic
189#127—3321no—
190#127—3321no—
191#121—3324no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—1661no—
195C2×C4⋊C4C4×D8311yescharacteristic
196C2×C4⋊C4C22⋊D8311yescharacteristic
197C2×C22⋊C4—1662no—
198C2×C22⋊C4—1662no—
199C2×C22⋊C4—1662no—
200C2×C22⋊C4—1662no—
201C2×C22⋊C4D33211yescharacteristic
202C2×C22⋊C4C22×D8311yescharacteristic
203C2×C42C4×D8311yescharacteristic
204C2×C42C22⋊D8311yescharacteristic
205C4⋊C8—41no—
206C22⋊C8—22no—
207#97—41no—
208#95—3321no—
209#95—3321no—
210#95—832no—
211#95—1661no—
212C24—1662no—
213C24C2×D33211yescharacteristic
214C2×D4—3322no—
215C2×D4—3322no—
216C2×D4—3322no—
217C2×D4—3322no—
218C2×D4—3324no—
219C2×D4—3321no—
220C2×D4—3321no—
221C2×D4—6642no—
222C2×D4—6642no—
223C2×D4—6642no—
224C2×D4—6642no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—3322no—
230C22×C4—1662no—
231C22×C4(C2×D83)⋊C411yescharacteristic
232C22×C4(C2×D83)⋊C411yescharacteristic
233C22×C4(C2×D83)⋊C411yescharacteristic
234C22×C4(C2×D83)⋊C411yescharacteristic
235C22×C4C2×C4×D8311yescharacteristic
236C22×C4C2×C22⋊D8311yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—3322no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—3324no—
244C22⋊C4—3322no—
245C22⋊C4—3322no—
246C22⋊C4—6642no—
247C22⋊C4—6642no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—3322no—
257C23—3322no—
258C23—3322no—
259C23—3322no—
260C23—3322no—
261C23—3322no—
262C23—3322no—
263C23C2×(C2×D83)⋊C411yescharacteristic
264C23#1458182012yes—
265D4—6642no—
266D4—6642no—
267D4—6644no—
268D4—6644no—
269D4—6648no—
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—3322no—
286C2×C4—3322no—
287C2×C4—3322no—
288C2×C4—6642no—
289C2×C4—6642no—
290C2×C4—6642no—
291C2×C4#1458170611yescharacteristic
292C2×C4#1458170611yescharacteristic
293C8—82no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—6642no—
302C22—6642no—
303C22—6642no—
304C22—6642no—
305C22—6642no—
306C22—6642no—
307C22—6642no—
308C22C2×#1458170611yescharacteristic
309C22#1912770711yescharacteristic
310C22#1912771711yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—13282no—
319C4—6642no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—13282no—
324C2#2490599311yescharacteristic
325C2#2490911411yescharacteristic
326C2#2490913411yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1464C2
socleC22×C83C2×#99
prime radical#24913213C2
cosocleC2×C22⋊C4×C83C23
hypercenter#1464D83
hypocenterC83#11800
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #24907540C2semidirect
3. #24905461C2semidirect
4. C83×#1464C2semidirect
5. C2×#14581789C4semidirect
8. #19127618C4semidirect
12. #19127567C4semidirect
166. #1464D83semidirect
168. C83#11800semidirect

Maximal subgroups

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

subgroupindex
#249132132
#249075402
#249054612
C83×#14642
#1180083