Finite Group Databasebeta

← #35299448

Subgroups — #35299448

Order 21248.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19127637C211yesmaximal, characteristic
2#24907551C212yesmaximal, complemented
3C2×#19127425C211yesmaximal, characteristic
4#24905341C212yesmaximal, complemented
5C83×#2312C211yesmaximal, characteristic
6C22×C4⋊Dic83C2211yescharacteristic
7C22×Dic83⋊C4—21no—
8C22×C22⋊Dic83—21no—
9C2×C22⋊(C83⋊C8)C2212yes—
10#19127637—22no—
11#19127637C414yescomplemented
12C2×C83⋊#95C2211yescharacteristic
13#19127463—24no—
14#19127425—22no—
15C22×C22⋊C4×C83C2211yescharacteristic
16#388×C83—24no—
17#380×C83C2212yes—
18#379×C83—21no—
19C4×C22⋊C4×C83—21no—
20C23×Dic83—21no—
21C23×Dic83—21no—
22C22×C83⋊C8—22no—
23C2×C4⋊Dic83—22no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83C2×C412yes—
27C2×Dic83⋊C4—24no—
28C2×Dic83⋊C4—44no—
29C2×C22⋊Dic83—24no—
30C2×C22⋊Dic83—44no—
31C22⋊(C83⋊C8)—22no—
32C22⋊(C83⋊C8)—22no—
33C83⋊#95—21no—
34C83⋊#95C2×C412yes—
35C25×C83D411yescharacteristic
36C23×C4×C83D411yescharacteristic
37C23×C4×C83C2311yescharacteristic
38C2×C4⋊C4×C83D414yes—
39C2×C22⋊C4×C83—22no—
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83C2×C412yes—
45C2×C22⋊C4×C83D412yes—
46C2×C42×C83—22no—
47#95×C83—22no—
48#95×C83—22no—
49C22×Dic83—24no—
50C22×Dic83—22no—
51C22×Dic83—22no—
52C22×Dic83—42no—
53C22×Dic83—42no—
54C22×Dic83—42no—
55C22×Dic83—42no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C2×C83⋊C8—44no—
59C2×C83⋊C8—42no—
60C4⋊Dic83—24no—
61C4⋊Dic83—42no—
62Dic83⋊C4—48no—
63C22⋊Dic83—48no—
64C24×C83—21no—
65C24×C83—22no—
66C24×C83—21no—
67C24×C83—22no—
68C24×C83—42no—
69C24×C83—42no—
70C24×C83C22⋊C412yes—
71C24×C83C2×D411yescharacteristic
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—24no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83C22⋊C412yes—
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D412yes—
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83#4112yes—
82C4⋊C4×C83—44no—
83C22⋊C4×C83—24no—
84C22⋊C4×C83—24no—
85C22⋊C4×C83—42no—
86C22⋊C4×C83—42no—
87C22⋊C4×C83—82no—
88C22⋊C4×C83—82no—
89C42×C83—82no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83—42no—
96C23×C83—42no—
97C23×C83—48no—
98C23×C83—42no—
99C23×C83—42no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—81no—
103C23×C83—81no—
104C23×C83—84no—
105C23×C83C2×C22⋊C411yescharacteristic
106C23×C83#12011yescharacteristic
107C23×C83#12311yescharacteristic
108C23×C83#13512yes—
109C23×C83#13712yes—
110C2×C4×C83—24no—
111C2×C4×C83—22no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—44no—
118C2×C4×C83—44no—
119C2×C4×C83—44no—
120C2×C4×C83—81no—
121C2×C4×C83C4×D412yes—
122C2×C4×C83#12112yes—
123C2×Dic83—24no—
124C2×Dic83—48no—
125C2×Dic83—42no—
126C2×Dic83—42no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129C2×Dic83—42no—
130C2×Dic83—82no—
131C2×Dic83—82no—
132C83⋊C8—82no—
133C22×C83—24no—
134C22×C83—24no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—44no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83—82no—
144C22×C83—84no—
145C22×C83—84no—
146C22×C83#38611yescharacteristic
147C22×C83#41611yescharacteristic
148C22×C83#41911yescharacteristic
149C22×C83#45212yes—
150C22×C83#45312yes—
151C22×C83#45812yes—
152C22×C83#48512yes—
153C4×C83—22no—
154C4×C83—22no—
155C4×C83—42no—
156C4×C83—84no—
157Dic83—44no—
158Dic83—82no—
159#9024—831nomaximal, Sylow
160C2×C83—21no—
161C2×C83—21no—
162C2×C83—22no—
163C2×C83—82no—
164C2×C83#161814yes—
165C2×C83#187311yescharacteristic
166C2×C83#188711yescharacteristic
167C2×C83#189011yescharacteristic
168C2×#399—831no—
169#2312D8311yescharacteristic
170#1824—832no—
171C2×#323—831no—
172#1294—832no—
173C83#902411yescharacteristic, complemented, Sylow
174C22×C4⋊C4—831no—
175C22×C4⋊C4—1661no—
176C22×C22⋊C4—1661no—
177C22×C22⋊C4C2×D8311yescharacteristic
178C2×C22⋊C8—832no—
179#399—834no—
180#399—1662no—
181#388—24no—
182#380C2×D8312yes—
183#379—21no—
184C4×C22⋊C4—21no—
185C2×#95—831no—
186#340—1664no—
187#323—1662no—
188C25C22⋊D8311yescharacteristic
189C23×C4—1661no—
190C23×C4—1661no—
191C23×C4C22⋊D8311yescharacteristic
192C23×C4C22×D8311yescharacteristic
193C22×C8—1662no—
194C2×C4⋊C4—3324no—
195C2×C4⋊C4—832no—
196C2×C4⋊C4—1664no—
197C2×C4⋊C4—1661no—
198C2×C4⋊C4—1662no—
199C2×C4⋊C4—1662no—
200C2×C4⋊C4C22⋊D8314yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—3324no—
207C2×C22⋊C4—1664no—
208C2×C22⋊C4C4×D8312yes—
209C2×C22⋊C4D33212yes—
210C2×C42—22no—
211C22⋊C8—1662no—
212C22⋊C8—1662no—
213#95—22no—
214#95—22no—
215#95—832no—
216#95—1661no—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24(C2×D83)⋊C412yes—
224C24C2×C22⋊D8311yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—3322no—
231C22×C4—3322no—
232C22×C4—3322no—
233C22×C4—3322no—
234C22×C4—3321no—
235C22×C4—3321no—
236C22×C4—1662no—
237C22×C4—1664no—
238C22×C4—1662no—
239C22×C4(C2×D83)⋊C412yes—
240C22×C4C2×C4×D8311yescharacteristic
241C22×C4C2×C22⋊D8312yes—
242C22×C4C2×D33211yescharacteristic
243C22×C4#1305059012yes—
244C2×C8—3324no—
245C2×C8—3322no—
246C4⋊C4—44no—
247C4⋊C4—3328no—
248C4⋊C4—3322no—
249C4⋊C4—1664no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—3328no—
257C42—82no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23C2×(C2×D83)⋊C411yescharacteristic
274C23#1458176111yescharacteristic
275C23#1458177511yescharacteristic
276C23#1458181212yes—
277C23#1458182212yes—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—3328no—
290C2×C4—3322no—
291C2×C4—3322no—
292C2×C4—3322no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4—6642no—
296C2×C4—6642no—
297C2×C4—1664no—
298C2×C4C4×C22⋊D8312yes—
299C2×C4#1458176612yes—
300C8—6642no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#1912758011yescharacteristic
315C22#1912769511yescharacteristic
316C22#1912770911yescharacteristic
317C22#1912786312yes—
318C22#1912786912yes—
319C22#1912789112yes—
320C22#1912804212yes—
321C4—22no—
322C4—22no—
323C4—42no—
324C4—84no—
325C4—3324no—
326C4—6642no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#2490651114yes—
332C2#2490777511yescharacteristic
333C2#2490786111yescharacteristic
334C2#2490787611yescharacteristic
subgroupG/H
centerC23#14581761
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2312C2
socleC23×C83#120
prime radicalC25×C83D4
cosocleC23×C4×C83C23
hypercenter#2312D83
hypocenterC83#9024
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #24907551C2semidirect
4. #24905341C2semidirect
11. #19127637C4semidirect
173. C83#9024semidirect

Maximal subgroups

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

subgroupindex
C2×#191276372
#249075512
C2×#191274252
#249053412
C83×#23122
#902483