Finite Group Databasebeta

← #35314663

Subgroups — #35314663

Order 21248.

444 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24909074C211yesmaximal, characteristic, complemented
2C2×#19127466C211yesmaximal, characteristic, complemented
3#24905481C212yesmaximal, complemented
4#24905478C212yesmaximal, complemented
5C83×#3050C211yesmaximal, characteristic
6C22×#13050574—21no—
7C2×#14581732—21no—
8C22⋊(C83⋊C16)—21no—
9C22⋊(C83⋊C16)C2212yes—
10#19127466—21no—
11#19127466C2212yes—
12C2×#135×C83C411yescharacteristic
13C23×C8×C83C2211yescharacteristic
14#525×C83—21no—
15#467×C83C411yescharacteristic
16#467×C83C2211yescharacteristic
17#466×C83C411yescharacteristic
18#466×C83C2211yescharacteristic
19#465×C83—21no—
20#465×C83—21no—
21#427×C83C411yescharacteristic
22#416×C83—21no—
23C2×#13050574—24no—
24C2×#13050574—22no—
25C2×#13050574—42no—
26C2×#13050574—41no—
27C22×C83⋊C8—41no—
28#14581732—42no—
29C2×C83⋊C16—22no—
30C2×#41×C83—21no—
31C23×C4×C83D411yescharacteristic
32#135×C83—42no—
33C2×Q16×C83C2×C411yescharacteristic
34C2×SD16×C83—21no—
35C2×D8×C83C2×C411yescharacteristic
36C22×C8×C83—42no—
37C22×C8×C83C2×C411yescharacteristic
38C22×C8×C83C2×C411yescharacteristic
39C22×C8×C83D411yescharacteristic
40C22×C8×C83D411yescharacteristic
41C22×C8×C83D411yescharacteristic
42C22×C8×C83C2311yescharacteristic
43#123×C83—41no—
44C22⋊Q8×C83—21no—
45C22⋊Q8×C83—21no—
46#121×C83—21no—
47#121×C83—21no—
48#120×C83—41no—
49C4×D4×C83—41no—
50C4×D4×C83—41no—
51#117×C83—21no—
52#107×C83C2×C411yescharacteristic
53#107×C83C2×C411yescharacteristic
54#106×C83—21no—
55#103×C83—21no—
56#103×C83—21no—
57#102×C83—21no—
58#102×C83—21no—
59#13050574—24no—
60#13050574—44no—
61#13050574—42no—
62C2×C83⋊C8—44no—
63C2×C83⋊C8—42no—
64C83⋊C16—42no—
65C24×C83SD1611yescharacteristic
66#41×C83—42no—
67#41×C83—42no—
68C2×Q8×C83—21no—
69C2×D4×C83—21no—
70C2×D4×C83—41no—
71C2×D4×C83—41no—
72C2×D4×C83—41no—
73C22×C4×C83—22no—
74C22×C4×C83—22no—
75C22×C4×C83—41no—
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83D811yescharacteristic
79C22×C4×C83D811yescharacteristic
80C22×C4×C83SD1611yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82Q16×C83—41no—
83SD16×C83—42no—
84D8×C83—41no—
85C2×C8×C83—22no—
86C2×C8×C83—22no—
87C2×C8×C83—41no—
88C2×C8×C83—41no—
89C2×C8×C83—41no—
90C2×C8×C83—41no—
91C2×C8×C83C22⋊C411yescharacteristic
92C2×C8×C83C22⋊C411yescharacteristic
93C2×C8×C83C22×C411yescharacteristic
94C2×C8×C83C2×D411yescharacteristic
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C22⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C22⋊C4×C83—41no—
104C42×C83—41no—
105C23×C83—22no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83#10211yescharacteristic
110C23×C83#10211yescharacteristic
111C23×C83C2×SD1611yescharacteristic
112Q8×C83—41no—
113Q8×C83—41no—
114D4×C83—41no—
115D4×C83—41no—
116D4×C83—82no—
117D4×C83—82no—
118D4×C83—82no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—22no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—82no—
130C2×C4×C83#9912yes—
131C2×C4×C83#10211yescharacteristic
132C2×C4×C83#10211yescharacteristic
133C2×C4×C83#10412yes—
134C2×C4×C83C2×C22⋊C411yescharacteristic
135C2×C4×C83C2×D811yescharacteristic
136C8×C83—21no—
137C8×C83—21no—
138C8×C83—21no—
139C8×C83—21no—
140C83⋊C8—44no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—44no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83#32712yes—
154C22×C83C2×#10211yescharacteristic
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—81no—
158C4×C83—81no—
159C4×C83—81no—
160C4×C83#32712yes—
161C4×C83C2×#9911yescharacteristic
162C4×C83C2×#10411yescharacteristic
163#11710—831nomaximal, Sylow
164C2×C83—22no—
165C2×C83—41no—
166C2×C83—41no—
167C2×C83—161no—
168C2×C83C2×#32711yescharacteristic
169C2×C83#214311yescharacteristic
170C2×C83#214811yescharacteristic
171#3050D8311yescharacteristic
172#2112—831no—
173C2×#341—831no—
174#1350—832no—
175#1349—832no—
176C83#1171011yescharacteristic, complemented, Sylow
177C2×#135Dic8311yescharacteristic
178C22×M16—1661no—
179C23×C8C2×D8311yescharacteristic
180#525—21no—
181C2×M32—1661no—
182#467Dic8311yescharacteristic
183#467C2×D8311yescharacteristic
184#466Dic8311yescharacteristic
185#466C2×D8311yescharacteristic
186#465—21no—
187#465—21no—
188#427Dic8311yescharacteristic
189#416—21no—
190C22⋊C16—832no—
191C22⋊C16—1661no—
192#341—832no—
193#341—1661no—
194C2×#41—21no—
195C23×C4C22⋊D8311yescharacteristic
196#135—42no—
197C2×Q16C2×Dic8311yescharacteristic
198C2×SD16—21no—
199C2×D8C2×Dic8311yescharacteristic
200C2×M16—3322no—
201C2×M16—3321no—
202C2×M16—1664no—
203C2×M16—1662no—
204C22×C8—42no—
205C22×C8—3321no—
206C22×C8C2×Dic8311yescharacteristic
207C22×C8C2×Dic8311yescharacteristic
208C22×C8C22⋊D8311yescharacteristic
209C22×C8C22⋊D8311yescharacteristic
210C22×C8C22⋊D8311yescharacteristic
211C22×C8C22×D8311yescharacteristic
212#123—41no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215#121—21no—
216#121—21no—
217#120—41no—
218C4×D4—41no—
219C4×D4—41no—
220#117—21no—
221M32—3322no—
222C2×C16—1662no—
223#107C2×Dic8311yescharacteristic
224#107C2×Dic8311yescharacteristic
225#106—21no—
226#103—21no—
227#103—21no—
228#102—21no—
229#102—21no—
230C24#1305057811yescharacteristic
231#41—42no—
232#41—42no—
233C2×Q8—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4C22⋊Dic8311yescharacteristic
242C22×C4C22⋊Dic8311yescharacteristic
243C22×C4#1305057511yescharacteristic
244C22×C4#1305057511yescharacteristic
245C22×C4#1305057811yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247Q16—41no—
248SD16—42no—
249D8—41no—
250M16—3322no—
251M16—3324no—
252M16—1664no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—3324no—
260C2×C8—3322no—
261C2×C8C22⋊Dic8311yescharacteristic
262C2×C8C22⋊Dic8311yescharacteristic
263C2×C8C22×Dic8311yescharacteristic
264C2×C8C2×C22⋊D8311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—41no—
275C16—3322no—
276C23—22no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1458171311yescharacteristic
281C23#1458171311yescharacteristic
282C23C2×#1305057811yescharacteristic
283Q8—41no—
284Q8—41no—
285D4—41no—
286D4—41no—
287D4—82no—
288D4—82no—
289D4—82no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4#1458170712yes—
302C2×C4#1458171311yescharacteristic
303C2×C4#1458171311yescharacteristic
304C2×C4#1458171912yes—
305C2×C4C2×C22⋊Dic8311yescharacteristic
306C2×C4C2×#1305057511yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—3324no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—44no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1912743512yes—
325C22C2×#1458171311yescharacteristic
326C4—21no—
327C4—21no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4#1912743512yes—
332C4C2×#1458170711yescharacteristic
333C4C2×#1458171911yescharacteristic
334C2—22no—
335C2—41no—
336C2—41no—
337C2—161no—
338C2C2×#1912743511yescharacteristic
339C2#2490919111yescharacteristic
340C2#2490921211yescharacteristic
subgroupG/H
centerC22C2×#14581713
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
Fitting, prime radicalC83×#3050C2
socleC22×C83C2×#102
cosocleC22×C8×C83C23
hypercenter#3050D83
hypocenterC83#11710
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. #24909074C2semidirect
2. C2×#19127466C2semidirect
3. #24905481C2semidirect
4. #24905478C2semidirect
176. C83#11710semidirect

Maximal subgroups

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

subgroupindex
#249090742
C2×#191274662
#249054812
#249054782
C83×#30502
#1171083