Finite Group Databasebeta

← #35304129

Subgroups — #35304129

Order 21248.

484 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#19127942C211yesmaximal, characteristic
2#24907501C211yesmaximal, characteristic
3#24905982C212yesmaximal
4#24905826C212yesmaximal
5C83×#1752C211yesmaximal, characteristic
6C22×Dic332C411yescharacteristic, complemented
7C2×#14581771C2212yes—
8#19127942—21no—
9#19127942—21no—
10#19127942—21no—
11#19127942—21no—
12C2×#14581728—21no—
13C2×#14581726C411yescharacteristic, complemented
14C2×#14581717C412yescomplemented
15#19127665C2212yes—
16#19127466—21no—
17#19127437—24no—
18#19127426—24no—
19C23×C8×C83C2211yescharacteristic
20C2×C22⋊C8×C83C2211yescharacteristic
21C2×#97×C83—21no—
22C4×C22⋊C4×C83C2211yescharacteristic
23#336×C83—21no—
24C22×Dic166C2×C412yes—
25C2×Dic332—21no—
26C2×Dic332—21no—
27C2×Dic332—42no—
28C2×#13050574—22no—
29#14581771—22no—
30#14581771—22no—
31#14581771—22no—
32#14581771—22no—
33C2×C4⋊Dic83C2×C412yes—
34C2×Dic83⋊C4—22no—
35C2×C22⋊Dic83—22no—
36#14581728—42no—
37#14581726—21no—
38#14581726—21no—
39#14581717—22no—
40#14581717—22no—
41C22⋊(C83⋊C8)—22no—
42C22⋊(C83⋊C8)—22no—
43C23×C4×C83C2311yescharacteristic
44C22×C8×C83—21no—
45C22×C8×C83—42no—
46C22×C8×C83C2×C411yescharacteristic
47C22×C8×C83C2×C411yescharacteristic
48C22×C8×C83D411yescharacteristic
49C22×C8×C83D411yescharacteristic
50C22×C8×C83D411yescharacteristic
51C22×C8×C83D411yescharacteristic
52C2×C22⋊C4×C83—21no—
53C2×C42×C83—21no—
54C22⋊C8×C83—21no—
55C22⋊C8×C83D412yes—
56#97×C83—42no—
57#95×C83D412yes—
58C2×Dic166—22no—
59C2×Dic166—22no—
60C2×Dic166—42no—
61C2×Dic166—42no—
62C22×Dic83—22no—
63C22×Dic83—22no—
64Dic332—42no—
65Dic332—42no—
66#13050574—42no—
67#13050574—42no—
68C2×C83⋊C8—42no—
69C4⋊Dic83—22no—
70C4⋊Dic83—22no—
71Dic83⋊C4—42no—
72Dic83⋊C4—42no—
73C22⋊Dic83—42no—
74C22⋊Dic83—42no—
75C24×C83#4111yescharacteristic
76C22×C4×C83—22no—
77C22×C4×C83—41no—
78C22×C4×C83—41no—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83#4111yescharacteristic
85C2×C8×C83—22no—
86C2×C8×C83—22no—
87C2×C8×C83—22no—
88C2×C8×C83—22no—
89C2×C8×C83—41no—
90C2×C8×C83—41no—
91C2×C8×C83—41no—
92C2×C8×C83—41no—
93C2×C8×C83—41no—
94C2×C8×C83C22⋊C411yescharacteristic
95C2×C8×C83C22⋊C411yescharacteristic
96C2×C8×C83C22⋊C411yescharacteristic
97C2×C8×C83C22⋊C411yescharacteristic
98C22⋊C4×C83—42no—
99C42×C83—41no—
100C42×C83—41no—
101C23×C83—22no—
102C23×C83—22no—
103C23×C83—41no—
104C23×C83C4×D411yescharacteristic
105C23×C83#12111yescharacteristic
106C23×C83#12411yescharacteristic
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—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83C2×C22⋊C411yescharacteristic
120C2×C4×C83C4×D411yescharacteristic
121C2×C4×C83#12111yescharacteristic
122C2×C4×C83#12711yescharacteristic
123C2×Dic83—42no—
124C2×Dic83—42no—
125C2×Dic83—44no—
126C2×Dic83—82no—
127Dic166—42no—
128Dic166—42no—
129Dic166—82no—
130C8×C83—21no—
131C8×C83—21no—
132C8×C83—21no—
133C8×C83—42no—
134C8×C83#9912yes—
135C83⋊C8—82no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—44no—
144C22×C83#39011yescharacteristic
145C22×C83#44412yes—
146C22×C83#47312yes—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—21no—
150C4×C83—82no—
151C4×C83C2×#9911yescharacteristic
152C4×C83#41011yescharacteristic
153Dic83—82no—
154Dic83—82no—
155#9817—831nomaximal, Sylow
156C2×C83—22no—
157C2×C83—22no—
158C2×C83—41no—
159C2×C83#196411yescharacteristic
160C2×C83#197511yescharacteristic
161C2×C83#197911yescharacteristic
162C2×#467—831no—
163#1810—831no—
164#1752D8311yescharacteristic
165#1508—832no—
166#1472—832no—
167C83#981711yescharacteristic, complemented, Sylow
168C22×Q16—831no—
169C23×C8C2×D8311yescharacteristic
170C2×C22⋊Q8—832no—
171#467—1661no—
172#467—1661no—
173#467—1661no—
174#467—1661no—
175C2×#108—1661no—
176C2×#107—831no—
177C2×#103—832no—
178#407—832no—
179C2×C22⋊C8C2×D8311yescharacteristic
180C2×#97—21no—
181C4×C22⋊C4C2×D8311yescharacteristic
182#341—1661no—
183#336—21no—
184#328—1664no—
185#324—1664no—
186C22×Q8—832no—
187C23×C4C22×D8311yescharacteristic
188C2×Q16—3322no—
189C2×Q16—1661no—
190C2×Q16—1661no—
191C2×M16—1662no—
192C22×C8—21no—
193C22×C8—42no—
194C22×C8C4×D8311yescharacteristic
195C22×C8C4×D8311yescharacteristic
196C22×C8D33211yescharacteristic
197C22×C8D33211yescharacteristic
198C22×C8C22⋊D8311yescharacteristic
199C22×C8C22⋊D8311yescharacteristic
200C22⋊Q8—1662no—
201C22⋊Q8—1662no—
202C22⋊Q8—1662no—
203C22⋊Q8—1662no—
204C2×C4⋊C4—832no—
205C2×C4⋊C4—1662no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1662no—
208C2×C42—21no—
209#108—3322no—
210#107—1661no—
211#107—1661no—
212#103—1662no—
213#103—1662no—
214C22⋊C8—21no—
215C22⋊C8—1662no—
216C22⋊C8—1662no—
217C22⋊C8D33212yes—
218#97—42no—
219#95D33212yes—
220C24#1305059011yescharacteristic
221C2×Q8—3322no—
222C2×Q8—3322no—
223C2×Q8—1662no—
224C2×Q8—1662no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—1662no—
229C22×C4—1662no—
230C22×C4C2×C4×D8311yescharacteristic
231C22×C4C2×C22⋊D8311yescharacteristic
232C22×C4C2×D33211yescharacteristic
233C22×C4C2×D33211yescharacteristic
234C22×C4C2×D33211yescharacteristic
235C22×C4#1305059011yescharacteristic
236Q16—3322no—
237Q16—3322no—
238M16—3322no—
239M16—3322no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—3322no—
250C2×C8(C2×D83)⋊C411yescharacteristic
251C2×C8(C2×D83)⋊C411yescharacteristic
252C2×C8(C2×D83)⋊C411yescharacteristic
253C2×C8(C2×D83)⋊C411yescharacteristic
254C4⋊C4—3322no—
255C4⋊C4—3322no—
256C4⋊C4—1662no—
257C4⋊C4—1662no—
258C22⋊C4—42no—
259C22⋊C4—3322no—
260C22⋊C4—3322no—
261C42—41no—
262C42—41no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23C4×D33211yescharacteristic
267C23#1458176611yescharacteristic
268C23#1458178011yescharacteristic
269Q8—3322no—
270Q8—3322no—
271Q8—6642no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—3324no—
285C2×C4—3322no—
286C2×C4—3322no—
287C2×C4—6642no—
288C2×C4C2×(C2×D83)⋊C411yescharacteristic
289C2×C4C4×D33211yescharacteristic
290C2×C4#1458176611yescharacteristic
291C2×C4#1458178911yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—42no—
296C8—6642no—
297C8#1458170612yes—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—44no—
306C22#1912760311yescharacteristic
307C22#1912782412yes—
308C22#1912797612yes—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—6642no—
314C4—6642no—
315C4C2×#1458170611yescharacteristic
316C4#1912767411yescharacteristic
317C2—22no—
318C2—22no—
319C2—41no—
320C2#2490829911yescharacteristic
321C2#2490835611yescharacteristic
322C2#2490837811yescharacteristic
subgroupG/H
centerC22#19127603
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1752C2
socleC22×C83#390
prime radicalC24×C83#41
cosocleC23×C4×C83C23
hypercenter#1752D83
hypocenterC83#9817
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
6. C22×Dic332C4semidirect
13. C2×#14581726C4semidirect
14. C2×#14581717C4semidirect
167. C83#9817semidirect

Maximal subgroups

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

subgroupindex
C2×#191279422
#249075012
#249059822
#249058262
C83×#17522
#981783