Finite Group Databasebeta

← #35313783

Subgroups — #35313783

Order 21248.

598 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 95 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×#19127606C211yesmaximal, characteristic
2#24905793C211yesmaximal, characteristic
3#24905396C214yesmaximal, complemented
4C83×#1463C211yesmaximal, characteristic
5C23×Dic166C411yescharacteristic
6C22×C4×Dic83C411yescharacteristic
7#19127665—21no—
8#19127606—21no—
9#19127606—21no—
10#19127606C412yes—
11#19127606C2212yes—
12C2×C83⋊#95—21no—
13C2×C83⋊#95—21no—
14#19127427—21no—
15#19127427C2212yes—
16C22×C4⋊C4×C83C2211yescharacteristic
17#407×C83—21no—
18#324×C83—21no—
19#324×C83C2212yes—
20C22×Dic166—24no—
21C22×Dic166—21no—
22C22×Dic166—42no—
23C22×Dic166—41no—
24C22×Dic166—41no—
25C22×Dic166C2×C412yes—
26C23×Dic83—21no—
27C23×Dic83—21no—
28C2×#13050574—41no—
29C2×C4×Dic83—21no—
30C2×C4×Dic83—42no—
31C2×C4×Dic83C2×C412yes—
32C22⋊(C83⋊C8)—22no—
33C22⋊(C83⋊C8)—41no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C23×C4×C83C2×C411yescharacteristic
39C23×C4×C83D411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C2×M16×C83—41no—
42C2×C4⋊C4×C83—41no—
43C2×C4⋊C4×C83—41no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C4⋊C4×C83C2311yescharacteristic
48C22⋊C8×C83—22no—
49C22⋊C8×C83—41no—
50C2×Dic166—24no—
51C2×Dic166—22no—
52C2×Dic166—42no—
53C2×Dic166—48no—
54C2×Dic166—48no—
55C2×Dic166—42no—
56C2×Dic166—42no—
57C2×Dic166—42no—
58C2×Dic166M1614yescomplemented
59C22×Dic83—24no—
60C22×Dic83—24no—
61C22×Dic83—22no—
62C22×Dic83—22no—
63C22×Dic83—42no—
64C22×Dic83—42no—
65C22×Dic83—42no—
66C22×Dic83—41no—
67C22×Dic83—41no—
68#13050574—82no—
69C2×C83⋊C8—42no—
70C4×Dic83—42no—
71C4×Dic83—42no—
72C24×C83C22⋊C411yescharacteristic
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—22no—
76C22×C4×C83—22no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—42no—
82C22×C4×C83—42no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89M16×C83—82no—
90C2×C8×C83—42no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C23×C83—22no—
96C23×C83—22no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83#9911yescharacteristic
100C23×C83#9911yescharacteristic
101C23×C83C2×C22⋊C411yescharacteristic
102C2×C4×C83—24no—
103C2×C4×C83—24no—
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—21no—
111C2×C4×C83—21no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—81no—
119C2×C4×C83—81no—
120C2×C4×C83#10012yes—
121C2×C4×C83C2×M1612yes—
122C2×Dic83—24no—
123C2×Dic83—24no—
124C2×Dic83—48no—
125C2×Dic83—42no—
126C2×Dic83—44no—
127C2×Dic83—44no—
128C2×Dic83—42no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131Dic166—48no—
132Dic166—44no—
133Dic166—44no—
134C8×C83—82no—
135C83⋊C8—82no—
136C22×C83—24no—
137C22×C83—24no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83#35412yes—
145C22×C83#35612yes—
146C22×C83#40711yescharacteristic
147C22×C83C2×#9911yescharacteristic
148C22×C83C2×#10011yescharacteristic
149C4×C83—22no—
150C4×C83—22no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—41no—
155Dic83—44no—
156Dic83—44no—
157#11506—831nomaximal, Sylow
158C2×C83—21no—
159C2×C83—22no—
160C2×C83—22no—
161C2×C83—21no—
162C2×C83#146211yescharacteristic
163C2×C83C2×#35411yescharacteristic
164C2×C83C2×#35611yescharacteristic
165C2×#391—831no—
166#1463—831no—
167#1463D8311yescharacteristic
168#1319—834no—
169C83#1150611yescharacteristic, complemented, Sylow
170C23×Q8—831no—
171C22×C4⋊C4C2×D8311yescharacteristic
172C22×C42—831no—
173#407—21no—
174#407—1661no—
175#391—832no—
176#391—832no—
177#391—1661no—
178#391—1661no—
179C2×#95—1661no—
180C2×#95—1661no—
181#324—21no—
182#324—832no—
183#324—1661no—
184#324C2×D8312yes—
185C22×Q8—3322no—
186C22×Q8—3321no—
187C22×Q8—3321no—
188C22×Q8—832no—
189C22×Q8—1664no—
190C22×Q8—1661no—
191C23×C4—1661no—
192C23×C4—1661no—
193C23×C4C4×D8311yescharacteristic
194C23×C4D33211yescharacteristic
195C23×C4C22⋊D8311yescharacteristic
196C2×M16—41no—
197C2×M16—3321no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C4×D8311yescharacteristic
201C2×C4⋊C4D33211yescharacteristic
202C2×C4⋊C4C22⋊D8311yescharacteristic
203C2×C4⋊C4C22×D8311yescharacteristic
204C2×C42—3322no—
205C2×C42—832no—
206C2×C42—1661no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—3321no—
210C22⋊C8—1662no—
211#95—1662no—
212#95—1662no—
213#95—1662no—
214#95—1662no—
215C24(C2×D83)⋊C411yescharacteristic
216C2×Q8—3328no—
217C2×Q8—3328no—
218C2×Q8—3322no—
219C2×Q8—3322no—
220C2×Q8—3322no—
221C2×Q8—3322no—
222C2×Q8—834no—
223C2×Q8—1664no—
224C2×Q8—1662no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—3322no—
236C22×C4—3322no—
237C22×C4—3322no—
238C22×C4—3321no—
239C22×C4—3321no—
240C22×C4—1664no—
241C22×C4—1664no—
242C22×C4—1662no—
243C22×C4—1662no—
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4(C2×D83)⋊C411yescharacteristic
246C22×C4(C2×D83)⋊C411yescharacteristic
247C22×C4C2×C4×D8311yescharacteristic
248C22×C4C2×C22⋊D8311yescharacteristic
249C22×C4C2×D33211yescharacteristic
250M16—82no—
251M16—6642no—
252C2×C8—42no—
253C2×C8—3322no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C42—3322no—
259C42—3322no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23#1458170611yescharacteristic
265C23#1458170611yescharacteristic
266C23C2×(C2×D83)⋊C411yescharacteristic
267Q8—3324no—
268Q8—3324no—
269Q8—3328no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—3324no—
289C2×C4—3324no—
290C2×C4—3328no—
291C2×C4—3322no—
292C2×C4—3322no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4—1664no—
296C2×C4—1664no—
297C2×C4#1458170812yes—
298C2×C4M16×D8312yes—
299C8—82no—
300C8—6642no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22#1912749512yes—
310C22#1912749912yes—
311C22#1912766411yescharacteristic
312C22C2×#1458170611yescharacteristic
313C22C2×#1458170811yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3324no—
321C4—3324no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—22no—
326C2#2490578911yescharacteristic
327C2C2×#1912749511yescharacteristic
328C2C2×#1912749911yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1463C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1463D83
hypocenterC83#11506
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #24905396C2semidirect
58. C2×Dic166M16semidirect
169. C83#11506semidirect

Maximal subgroups

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

subgroupindex
C2×#191276062
#249057932
#249053962
C83×#14632
#1150683