Finite Group Databasebeta

← #35314979

Subgroups — #35314979

Order 21248.

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

normal subgroups only

This group has 480 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#24912455C211yesmaximal, characteristic
2C4×#14581707C211yesmaximal, characteristic
3#24905464C214yesmaximal, complemented
4C83×#1465C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6C2×#14581779—21no—
7C2×#14581707—21no—
8#19127626C412yescomplemented
9#19127626C2212yes—
10#19127606—21no—
11#19127561—21no—
12#19127561—21no—
13C4×C4⋊Dic83C411yescharacteristic, complemented
14C4×C22⋊Dic83—21no—
15#19127468C2212yes—
16C2×C4×D4×C83C2211yescharacteristic
17#407×C83—21no—
18#324×C83—21no—
19#323×C83C2212yes—
20C22×Dic166—21no—
21#14581792—41no—
22#14581792—41no—
23#14581779—44no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2×C412yes—
27C2×C22⋊Dic83—22no—
28C2×C22⋊Dic83—22no—
29C2×C22⋊Dic83—22no—
30C2×C4×Dic83—21no—
31C2×C4×Dic83—41no—
32#14581707—24no—
33C83⋊#95—22no—
34C83⋊#95—22no—
35C83⋊#95—41no—
36C83⋊#95C2×C412yes—
37C22×D4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C23×C4×C83D411yescharacteristic
40C2×M16×C83—41no—
41C4×D4×C83—41no—
42C4×D4×C83—41no—
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83C2×C411yescharacteristic
45C2×C22⋊C4×C83C2311yescharacteristic
46C2×C42×C83C2×C411yescharacteristic
47C22⋊C8×C83—22no—
48C22⋊C8×C83—41no—
49C2×Dic166—42no—
50C2×Dic166—42no—
51C2×Dic166—42no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C22×Dic83—42no—
55C4⋊Dic83—42no—
56C4⋊Dic83—41no—
57C4⋊Dic83—41no—
58C4⋊Dic83—42no—
59C22⋊Dic83—24no—
60C22⋊Dic83—44no—
61C22⋊Dic83—44no—
62C22⋊Dic83—42no—
63C4×Dic83—42no—
64C4×Dic83—82no—
65C24×C83C22⋊C411yescharacteristic
66C24×C83C2×D411yescharacteristic
67C2×D4×C83—22no—
68C2×D4×C83—22no—
69C2×D4×C83—41no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—42no—
75C22×C4×C83—42no—
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81M16×C83—82no—
82C2×C8×C83—42no—
83C4⋊C4×C83—41no—
84C22⋊C4×C83—41no—
85C22⋊C4×C83—41no—
86C42×C83—21no—
87C42×C83—21no—
88C23×C83—21no—
89C23×C83—21no—
90C23×C83—21no—
91C23×C83—21no—
92C23×C83—21no—
93C23×C83—41no—
94C23×C83—41no—
95C23×C83—41no—
96C23×C83—41no—
97C23×C83C2×C22⋊C411yescharacteristic
98C23×C83AGL(1,ℤ/8)12yes—
99D4×C83—44no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—22no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—81no—
112C2×C4×C83—81no—
113C2×C4×C83#9911yescharacteristic
114C2×C4×C83#9911yescharacteristic
115C2×C4×C83#10412yes—
116C2×Dic83—44no—
117C2×Dic83—42no—
118C2×Dic83—42no—
119C2×Dic83—42no—
120C2×Dic83—42no—
121C2×Dic83—42no—
122C2×Dic83—42no—
123C2×Dic83—82no—
124C2×Dic83—82no—
125Dic166—84no—
126C8×C83—82no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83—22no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83—41no—
133C22×C83—41no—
134C22×C83—41no—
135C22×C83—41no—
136C22×C83—41no—
137C22×C83—81no—
138C22×C83—81no—
139C22×C83C2×#9911yescharacteristic
140C22×C83#41811yescharacteristic
141C22×C83C2×#10411yescharacteristic
142C4×C83—21no—
143C4×C83—22no—
144C4×C83—41no—
145C4×C83—41no—
146C4×C83—41no—
147C4×C83#35412yes—
148Dic83—82no—
149Dic83—84no—
150Dic83—82no—
151#11777—831nomaximal, Sylow
152C2×C83—21no—
153C2×C83—21no—
154C2×C83—42no—
155C2×C83—81no—
156C2×C83#150711yescharacteristic
157C2×C83#212311yescharacteristic
158C2×C83C2×#35411yescharacteristic
159#2547—831no—
160C4×#99—831no—
161#1465D8311yescharacteristic
162#1344—834no—
163C83#1177711yescharacteristic, complemented, Sylow
164C2×C4⋊Q8—831no—
165C2×#124—1661no—
166C2×C4×D4C2×D8311yescharacteristic
167C2×#99—1661no—
168#407—21no—
169#397—832no—
170#397—832no—
171#391—1661no—
172#380—1661no—
173#380—1661no—
174C4×C4⋊C4—831no—
175C4×C22⋊C4—1661no—
176#342—832no—
177#324—21no—
178#323C2×D8312yes—
179C22×Q8—1661no—
180C22×D4D33211yescharacteristic
181C23×C4D33211yescharacteristic
182C23×C4C22⋊D8311yescharacteristic
183C2×M16—41no—
184C4⋊Q8—3321no—
185C4⋊Q8—3321no—
186#124—3324no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—832no—
190C2×C4⋊C4—1661no—
191C2×C4⋊C4—1661no—
192C2×C4⋊C4C22⋊D8311yescharacteristic
193C2×C22⋊C4—1662no—
194C2×C22⋊C4—1662no—
195C2×C22⋊C4—1662no—
196C2×C22⋊C4C4×D8311yescharacteristic
197C2×C22⋊C4C22×D8311yescharacteristic
198C2×C42—3321no—
199C2×C42—1661no—
200C2×C42C4×D8311yescharacteristic
201#99—1664no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204#95—3321no—
205#95—832no—
206#95—1662no—
207#95—1662no—
208C24(C2×D83)⋊C411yescharacteristic
209C24C2×D33211yescharacteristic
210C2×Q8—3322no—
211C2×Q8—3322no—
212C2×Q8—3322no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—3322no—
223C22×C4—1662no—
224C22×C4—1662no—
225C22×C4(C2×D83)⋊C411yescharacteristic
226C22×C4(C2×D83)⋊C411yescharacteristic
227C22×C4(C2×D83)⋊C411yescharacteristic
228C22×C4C2×C4×D8311yescharacteristic
229C22×C4C2×C22⋊D8311yescharacteristic
230M16—82no—
231C2×C8—42no—
232C4⋊C4—41no—
233C4⋊C4—3322no—
234C4⋊C4—3322no—
235C4⋊C4—3321no—
236C4⋊C4—3321no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—3324no—
240C22⋊C4—3324no—
241C22⋊C4—3322no—
242C22⋊C4—1664no—
243C42—21no—
244C42—21no—
245C42—3322no—
246C42—6642no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D83)⋊C411yescharacteristic
257C23#1458182012yes—
258Q8—6644no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—3324no—
274C2×C4—3322no—
275C2×C4—3322no—
276C2×C4—3322no—
277C2×C4—3322no—
278C2×C4—3322no—
279C2×C4—3322no—
280C2×C4—6642no—
281C2×C4—6642no—
282C2×C4#1458170611yescharacteristic
283C2×C4#1458170611yescharacteristic
284C2×C4#1458171812yes—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×#1458170611yescharacteristic
299C22#1912770711yescharacteristic
300C22C2×#1458171811yescharacteristic
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—6642no—
307C4—6644no—
308C4—6642no—
309C4#1912749512yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2490598011yescharacteristic
315C2#2490911811yescharacteristic
316C2C2×#1912749511yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1465C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1465D83
hypocenterC83#11777
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #24905464C2semidirect
5. C2×#14581792C4semidirect
8. #19127626C4semidirect
13. C4×C4⋊Dic83C4semidirect
163. C83#11777semidirect

Maximal subgroups

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

subgroupindex
#249124552
C4×#145817072
#249054642
C83×#14652
#1177783