Finite Group Databasebeta

← #35304127

Subgroups — #35304127

Order 21248.

484 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 107 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#24907239C211yesmaximal, characteristic
3#24905984C212yesmaximal
4#24905828C212yesmaximal
5C83×#1810C211yesmaximal, characteristic
6C22×Dic332C411yescharacteristic, complemented
7C2×#14581771C2212yes—
8#19127942—21no—
9#19127942—21no—
10#19127942—21no—
11#19127942—21no—
12C2×#14581726C411yescharacteristic, complemented
13C2×#14581717C412yescomplemented
14C2×C22⋊(C83⋊C8)C2211yescharacteristic
15C2×#14581702—21no—
16C4×C22⋊Dic83C2211yescharacteristic
17#19127455—21no—
18#19127439—24no—
19#19127426—24no—
20C23×C8×C83C2211yescharacteristic
21C2×#108×C83—21no—
22#407×C83C2212yes—
23#341×C83—21no—
24C22×Dic166C2×C412yes—
25C2×Dic332—21no—
26C2×Dic332—21no—
27C2×Dic332—42no—
28C22×C83⋊C8—21no—
29#14581771—22no—
30#14581771—22no—
31#14581771—22no—
32#14581771—22no—
33C2×C4⋊Dic83C2×C412yes—
34C2×Dic83⋊C4—22no—
35C2×C22⋊Dic83—22no—
36C2×C22⋊Dic83—21no—
37C2×C4×Dic83—21no—
38#14581726—21no—
39#14581726—21no—
40#14581717—22no—
41#14581717—22no—
42C22⋊(C83⋊C8)—21no—
43C22⋊(C83⋊C8)D412yes—
44#14581702—42no—
45C83⋊#95D412yes—
46C23×C4×C83C2311yescharacteristic
47C2×M16×C83—22no—
48C22×C8×C83—42no—
49C22×C8×C83C2×C411yescharacteristic
50C22×C8×C83C2×C411yescharacteristic
51C22×C8×C83D411yescharacteristic
52C22×C8×C83D411yescharacteristic
53C22×C8×C83D411yescharacteristic
54C22×C8×C83D411yescharacteristic
55#108×C83—42no—
56C22⋊C8×C83—22no—
57C22⋊C8×C83—22no—
58C2×Dic166—22no—
59C2×Dic166—22no—
60C2×Dic166—42no—
61C2×Dic166—42no—
62C22×Dic83—22no—
63C22×Dic83—22no—
64C22×Dic83—22no—
65Dic332—42no—
66Dic332—42no—
67C2×C83⋊C8—22no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8—41no—
70C4⋊Dic83—22no—
71C4⋊Dic83—22no—
72Dic83⋊C4—42no—
73Dic83⋊C4—42no—
74C22⋊Dic83—42no—
75C22⋊Dic83—42no—
76C22⋊Dic83—42no—
77C4×Dic83—41no—
78C4×Dic83—41no—
79C24×C83#4111yescharacteristic
80C22×C4×C83—41no—
81C22×C4×C83—41no—
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83#4111yescharacteristic
88M16×C83—42no—
89M16×C83—42no—
90C2×C8×C83—22no—
91C2×C8×C83—22no—
92C2×C8×C83—22no—
93C2×C8×C83—41no—
94C2×C8×C83—41no—
95C2×C8×C83—41no—
96C2×C8×C83—42no—
97C2×C8×C83C22⋊C411yescharacteristic
98C2×C8×C83C22⋊C411yescharacteristic
99C2×C8×C83C22⋊C411yescharacteristic
100C2×C8×C83C22⋊C411yescharacteristic
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×C83C2×C22⋊C411yescharacteristic
117C2×C4×C83C4×D411yescharacteristic
118C2×C4×C83#12111yescharacteristic
119C2×C4×C83#12711yescharacteristic
120C2×Dic83—42no—
121C2×Dic83—42no—
122C2×Dic83—42no—
123C2×Dic83—42no—
124C2×Dic83—44no—
125C2×Dic83—42no—
126C2×Dic83—82no—
127Dic166—42no—
128Dic166—42no—
129Dic166—82no—
130C8×C83—21no—
131C8×C83—21no—
132C8×C83—21no—
133C8×C83—82no—
134C8×C83#9912yes—
135C83⋊C8—42no—
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×C83C2×#9911yescharacteristic
151C4×C83#41011yescharacteristic
152Dic83—82no—
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#1810D8311yescharacteristic
164#1752—831no—
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—21no—
176C2×#107—831no—
177C2×#103—832no—
178#407C2×D8312yes—
179C2×C22⋊C8—831no—
180C2×#97—1661no—
181C4×C22⋊C4—831no—
182#341—21no—
183#336—1661no—
184#328—1664no—
185#324—1664no—
186C22×Q8—832no—
187C23×C4C22×D8311yescharacteristic
188C2×Q16—3322no—
189C2×Q16—1661no—
190C2×Q16—1661no—
191C2×M16—22no—
192C22×C8—42no—
193C22×C8—1661no—
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—1662no—
207C2×C22⋊C4—1661no—
208C2×C42—1661no—
209#108—42no—
210#107—1661no—
211#107—1661no—
212#103—1662no—
213#103—1662no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—832no—
217C22⋊C8—1661no—
218#97—3322no—
219#95—832no—
220C24#1305059111yescharacteristic
221C2×Q8—3322no—
222C2×Q8—3322no—
223C2×Q8—1662no—
224C2×Q8—1662no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—1662no—
228C22×C4—1662no—
229C22×C4—1662no—
230C22×C4C2×C4×D8311yescharacteristic
231C22×C4D4×D8311yescharacteristic
232C22×C4D4×D8311yescharacteristic
233C22×C4C2×C22⋊D8311yescharacteristic
234C22×C4C2×D33211yescharacteristic
235C22×C4#1305059111yescharacteristic
236Q16—3322no—
237Q16—3322no—
238M16—42no—
239M16—42no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—42no—
247C2×C8—3321no—
248C2×C8—3321no—
249C2×C8—1662no—
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—3322no—
259C22⋊C4—3322no—
260C22⋊C4—3322no—
261C42—3321no—
262C42—3321no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23#1458175411yescharacteristic
267C23#1458176711yescharacteristic
268C23#1458178111yescharacteristic
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—3322no—
282C2×C4—3322no—
283C2×C4—3324no—
284C2×C4—3322no—
285C2×C4—3322no—
286C2×C4—3322no—
287C2×C4—6642no—
288C2×C4C2×(C2×D83)⋊C411yescharacteristic
289C2×C4#1458175411yescharacteristic
290C2×C4#1458176711yescharacteristic
291C2×C4#1458178811yescharacteristic
292C8—21no—
293C8—21no—
294C8—21no—
295C8—82no—
296C8—3322no—
297C8#1458170612yes—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—44no—
306C22#1912760211yescharacteristic
307C22#1912782212yes—
308C22#1912797712yes—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—6642no—
313C4—6642no—
314C4—6642no—
315C4C2×#1458170611yescharacteristic
316C4#1912767411yescharacteristic
317C2—22no—
318C2—22no—
319C2—41no—
320C2#2490829811yescharacteristic
321C2#2490835411yescharacteristic
322C2#2490837611yescharacteristic
subgroupG/H
centerC22#19127602
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1810C2
socleC22×C83#390
prime radicalC24×C83#41
cosocleC23×C4×C83C23
hypercenter#1810D83
hypocenterC83#9817
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
6. C22×Dic332C4semidirect
12. C2×#14581726C4semidirect
13. 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
#249072392
#249059842
#249058282
C83×#18102
#981783