Finite Group Databasebeta

← #35315143

Subgroups — #35315143

Order 21248.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 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×#19127621C211yesmaximal, characteristic
2#24907543C211yesmaximal, characteristic
3#24905494C214yesmaximal, complemented
4C83×#1463C211yesmaximal, characteristic
5C23×Dic166C411yescharacteristic
6C22×Dic83⋊C4—21no—
7#19127665—21no—
8#19127621—21no—
9#19127621—21no—
10#19127621C412yes—
11#19127621C2212yes—
12C2×C83⋊#95—21no—
13C2×C83⋊#95C411yescharacteristic
14#19127470—21no—
15#19127470C2212yes—
16C22×C4⋊C4×C83C2211yescharacteristic
17#407×C83—21no—
18#324×C83—21no—
19#324×C83C2212yes—
20C22×Dic166—21no—
21C22×Dic166—41no—
22C22×Dic166—41no—
23C22×Dic166—44no—
24C22×Dic166C2×C412yes—
25C23×Dic83—21no—
26C23×Dic83—21no—
27C2×#13050574—22no—
28C2×Dic83⋊C4—22no—
29C2×Dic83⋊C4—22no—
30C2×Dic83⋊C4—44no—
31C22⋊(C83⋊C8)—41no—
32C22⋊(C83⋊C8)—41no—
33C83⋊#95—21no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95C2×C412yes—
37C23×C4×C83C2×C411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C23×C4×C83D411yescharacteristic
40C2×M16×C83—41no—
41C2×C4⋊C4×C83—41no—
42C2×C4⋊C4×C83—41no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C4⋊C4×C83D411yescharacteristic
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C4⋊C4×C83C2311yescharacteristic
47C22⋊C8×C83—22no—
48C22⋊C8×C83—41no—
49C2×Dic166—44no—
50C2×Dic166—42no—
51C2×Dic166—44no—
52C2×Dic166—48no—
53C2×Dic166—42no—
54C2×Dic166—44no—
55C2×Dic166—44no—
56C22×Dic83—24no—
57C22×Dic83—24no—
58C22×Dic83—22no—
59C22×Dic83—22no—
60C22×Dic83—42no—
61C22×Dic83—42no—
62C22×Dic83—42no—
63C22×Dic83—41no—
64C22×Dic83—41no—
65#13050574—42no—
66#13050574—42no—
67C2×C83⋊C8—42no—
68Dic83⋊C4—44no—
69Dic83⋊C4—44no—
70C24×C83C22⋊C411yescharacteristic
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—22no—
78C22×C4×C83—41no—
79C22×C4×C83—42no—
80C22×C4×C83—42no—
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87M16×C83—82no—
88C2×C8×C83—42no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83#9911yescharacteristic
98C23×C83#9911yescharacteristic
99C23×C83C2×C22⋊C411yescharacteristic
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
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#13712yes—
120C2×Dic83—24no—
121C2×Dic83—24no—
122C2×Dic83—42no—
123C2×Dic83—48no—
124C2×Dic83—42no—
125C2×Dic83—44no—
126C2×Dic83—44no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129Dic166—48no—
130Dic166—44no—
131Dic166—44no—
132C8×C83—82no—
133C83⋊C8—82no—
134C22×C83—24no—
135C22×C83—24no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83AΣL(1,𝔽16)12yes—
143C22×C83#35612yes—
144C22×C83C2×#9911yescharacteristic
145C22×C83#41911yescharacteristic
146C22×C83#42111yescharacteristic
147C4×C83—22no—
148C4×C83—41no—
149C4×C83—41no—
150C4×C83—41no—
151C4×C83—41no—
152C4×C83—41no—
153Dic83—44no—
154Dic83—44no—
155#11815—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—22no—
158C2×C83—22no—
159C2×C83—21no—
160C2×C83#151111yescharacteristic
161C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
162C2×C83C2×#35611yescharacteristic
163C2×#395—831no—
164#1821—831no—
165#1463D8311yescharacteristic
166#1354—834no—
167C83#1181511yescharacteristic, complemented, Sylow
168C23×Q8—831no—
169C22×C4⋊C4—1661no—
170C22×C4⋊C4C2×D8311yescharacteristic
171#407—21no—
172#407—1661no—
173#395—832no—
174#395—832no—
175#395—1661no—
176#395—1661no—
177C2×#95—831no—
178C2×#95—1661no—
179#343—832no—
180#343—1661no—
181#324—21no—
182#324C2×D8312yes—
183C22×Q8—3321no—
184C22×Q8—3321no—
185C22×Q8—3324no—
186C22×Q8—832no—
187C22×Q8—1661no—
188C23×C4—1661no—
189C23×C4—1661no—
190C23×C4C4×D8311yescharacteristic
191C23×C4D33211yescharacteristic
192C23×C4C22⋊D8311yescharacteristic
193C2×M16—41no—
194C2×M16—1662no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—3324no—
198C2×C4⋊C4—1662no—
199C2×C4⋊C4—1662no—
200C2×C4⋊C4C4×D8311yescharacteristic
201C2×C4⋊C4D33211yescharacteristic
202C2×C4⋊C4C22⋊D8311yescharacteristic
203C2×C4⋊C4C22×D8311yescharacteristic
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—3321no—
207C22⋊C8—3321no—
208#95—832no—
209#95—1661no—
210#95—1662no—
211#95—1662no—
212C24(C2×D83)⋊C411yescharacteristic
213C2×Q8—3322no—
214C2×Q8—3322no—
215C2×Q8—3328no—
216C2×Q8—3324no—
217C2×Q8—3324no—
218C2×Q8—3324no—
219C2×Q8—3324no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—3322no—
231C22×C4—3322no—
232C22×C4—3322no—
233C22×C4—3321no—
234C22×C4—3321no—
235C22×C4—1664no—
236C22×C4—1664no—
237C22×C4—1662no—
238C22×C4—1662no—
239C22×C4(C2×D83)⋊C411yescharacteristic
240C22×C4(C2×D83)⋊C411yescharacteristic
241C22×C4(C2×D83)⋊C411yescharacteristic
242C22×C4C2×C4×D8311yescharacteristic
243C22×C4C2×C22⋊D8311yescharacteristic
244C22×C4C2×D33211yescharacteristic
245M16—82no—
246M16—3322no—
247M16—3322no—
248C2×C8—42no—
249C2×C8—3322no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—3324no—
255C4⋊C4—3324no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1458170611yescharacteristic
261C23#1458170611yescharacteristic
262C23C2×(C2×D83)⋊C411yescharacteristic
263Q8—3324no—
264Q8—3328no—
265Q8—3324no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—3322no—
286C2×C4—3322no—
287C2×C4—3324no—
288C2×C4—3328no—
289C2×C4—3322no—
290C2×C4—3322no—
291C2×C4—3324no—
292C2×C4—1664no—
293C2×C4—1664no—
294C2×C4#1458182512yes—
295C8—82no—
296C8—6642no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1912748712yes—
306C22#1912749912yes—
307C22C2×#1458170611yescharacteristic
308C22#1912771111yescharacteristic
309C22#1912771711yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—3324no—
317C4—3324no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2490599711yescharacteristic
323C2C2×#1912748711yescharacteristic
324C2C2×#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#11815
LCSG ▷ C22×C4×C83 ▷ C23×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. #24905494C2semidirect
167. C83#11815semidirect

Maximal subgroups

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

subgroupindex
C2×#191276212
#249075432
#249054942
C83×#14632
#1181583