Finite Group Databasebeta

← #35307396

Subgroups — #35307396

Order 21248.

640 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#24907397C212yesmaximal
2#24905733C211yesmaximal, characteristic
3#24905341C212yesmaximal
4#24905306C211yesmaximal, characteristic
5C83×#2646C211yesmaximal, characteristic
6C23×Dic166C2211yescharacteristic
7C2×#14581715—22no—
8C2×#14581715—22no—
9C2×C22⋊(C83⋊C8)C2212yes—
10C2×C83⋊#95—22no—
11C2×C83⋊#95C2211yescharacteristic
12#400×C83—22no—
13#398×C83—22no—
14#385×C83—22no—
15#380×C83C2212yes—
16C2×#95×C83C2211yescharacteristic
17C22×Dic166—22no—
18C22×Dic166—21no—
19C22×Dic166—22no—
20C22×Dic166—44no—
21C22×Dic166Q812yes—
22C23×Dic83—21no—
23C23×Dic83—21no—
24C22×C83⋊C8—22no—
25#14581715—44no—
26#14581715—44no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—22no—
29C83⋊#95—22no—
30C83⋊#95—44no—
31C23×C4×C83D412yes—
32C23×C4×C83C2311yescharacteristic
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83D412yes—
36C2×C4⋊C4×C83D412yes—
37C2×C22⋊C4×C83—22no—
38#95×C83—22no—
39#95×C83—22no—
40#95×C83—22no—
41#95×C83—22no—
42C2×Dic166—24no—
43C2×Dic166—24no—
44C2×Dic166—24no—
45C2×Dic166—44no—
46C2×Dic166—48no—
47C2×Dic166—44no—
48C2×Dic166—42no—
49C2×Dic166—42no—
50C2×Dic166—42no—
51C22×Dic83—24no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C22×Dic83—42no—
55C22×Dic83—41no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83—42no—
59C22×Dic83—41no—
60C22×Dic83—42no—
61C2×C83⋊C8—42no—
62C2×C83⋊C8—42no—
63C2×C83⋊C8—42no—
64C24×C83C2×D411yescharacteristic
65C22×C4×C83—22no—
66C22×C4×C83—22no—
67C22×C4×C83—22no—
68C22×C4×C83—22no—
69C22×C4×C83—22no—
70C22×C4×C83—42no—
71C22×C4×C83—42no—
72C22×C4×C83—42no—
73C22×C4×C83C2×D412yes—
74C22×C4×C83C2×Q811yescharacteristic
75C22×C4×C83#4112yes—
76C22×C4×C83#4111yescharacteristic
77C4⋊C4×C83—42no—
78C4⋊C4×C83—42no—
79C4⋊C4×C83—82no—
80C4⋊C4×C83—82no—
81C22⋊C4×C83—42no—
82C22⋊C4×C83—42no—
83C23×C83—21no—
84C23×C83—21no—
85C23×C83—22no—
86C23×C83#12011yescharacteristic
87C23×C83C22⋊Q811yescharacteristic
88C23×C83#12411yescharacteristic
89C23×C83#13512yes—
90C23×C83#13712yes—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—41no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83—44no—
97C2×C4×C83—42no—
98C2×C4×C83—42no—
99C2×C4×C83—42no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83—42no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—82no—
107C2×C4×C83C22⋊Q812yes—
108C2×C4×C83#12412yes—
109C2×Dic83—24no—
110C2×Dic83—42no—
111C2×Dic83—44no—
112C2×Dic83—42no—
113C2×Dic83—44no—
114C2×Dic83—44no—
115C2×Dic83—42no—
116C2×Dic83—82no—
117C2×Dic83—81no—
118C2×Dic83—81no—
119Dic166—28no—
120Dic166—44no—
121Dic166—44no—
122Dic166—44no—
123C83⋊C8—82no—
124C22×C83—22no—
125C22×C83—22no—
126C22×C83—22no—
127C22×C83—22no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83#39311yescharacteristic
133C22×C83#45212yes—
134C22×C83#45511yescharacteristic
135C22×C83#45611yescharacteristic
136C22×C83B(4,𝔽2)12yes—
137C22×C83#47912yes—
138C22×C83#48811yescharacteristic
139C22×C83#49111yescharacteristic
140C4×C83—22no—
141C4×C83—22no—
142C4×C83—82no—
143C4×C83—82no—
144C4×C83—82no—
145Dic83—44no—
146Dic83—82no—
147#10345—831nomaximal, Sylow
148C2×C83—21no—
149C2×C83—22no—
150C2×C83—21no—
151C2×C83#160312yes—
152C2×C83#160612yes—
153C2×C83#202611yescharacteristic
154C2×C83#202811yescharacteristic
155C2×C83#202911yescharacteristic
156#2646D8311yescharacteristic
157#1787—832no—
158#1446—831no—
159#1294—832no—
160#1280—831no—
161C83#1034511yescharacteristic, complemented, Sylow
162C23×Q8—831no—
163C2×#103—1662no—
164C2×#103—1662no—
165C2×C22⋊C8—832no—
166#400—22no—
167#398—22no—
168#385—22no—
169#380C2×D8312yes—
170C2×#95—831no—
171C2×#95—1662no—
172C2×#95C2×D8311yescharacteristic
173C22×Q8—3324no—
174C22×Q8—832no—
175C22×Q8—1662no—
176C22×Q8—1662no—
177C22×Q8—1661no—
178C23×C4—1661no—
179C23×C4—1661no—
180C23×C4C22⋊D8312yes—
181C23×C4C22×D8311yescharacteristic
182C22×C8—1662no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊D8312yes—
186C2×C4⋊C4C22⋊D8312yes—
187C2×C22⋊C4—22no—
188#103—3324no—
189#103—3324no—
190C22⋊C8—1662no—
191C22⋊C8—1662no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—3324no—
197#95—1662no—
198C24C2×C22⋊D8311yescharacteristic
199C2×Q8—3322no—
200C2×Q8—3322no—
201C2×Q8—3328no—
202C2×Q8—3324no—
203C2×Q8—3324no—
204C2×Q8—3322no—
205C2×Q8—1664no—
206C2×Q8—1664no—
207C2×Q8—1664no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—3322no—
217C22×C4—3321no—
218C22×C4—3321no—
219C22×C4—3322no—
220C22×C4—3321no—
221C22×C4—3321no—
222C22×C4—3322no—
223C22×C4—1662no—
224C22×C4—1664no—
225C22×C4—1662no—
226C22×C4C2×C22⋊D8312yes—
227C22×C4Q8×D8311yescharacteristic
228C22×C4#1305059112yes—
229C22×C4#1305059111yescharacteristic
230C2×C8—3322no—
231C2×C8—3322no—
232C2×C8—3322no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1458176111yescharacteristic
243C23#1458176811yescharacteristic
244C23#1458178111yescharacteristic
245C23#1458181212yes—
246C23#1458182212yes—
247Q8—3324no—
248Q8—3324no—
249Q8—3324no—
250Q8—1668no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—3324no—
268C2×C4—3322no—
269C2×C4—3322no—
270C2×C4—3324no—
271C2×C4—3322no—
272C2×C4—3324no—
273C2×C4—6641no—
274C2×C4—6642no—
275C2×C4—6641no—
276C2×C4—1664no—
277C2×C4#1458176812yes—
278C2×C4#1458178112yes—
279C8—6642no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22#1912761211yescharacteristic
289C22#1912786312yes—
290C22#1912787911yescharacteristic
291C22#1912788411yescharacteristic
292C22#1912788812yes—
293C22#1912800512yes—
294C22#1912805811yescharacteristic
295C22#1912807111yescharacteristic
296C4—22no—
297C4—22no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—3324no—
302C4—6642no—
303C2—21no—
304C2—21no—
305C2—22no—
306C2#2490642612yes—
307C2#2490644312yes—
308C2#2490861111yescharacteristic
309C2#2490862111yescharacteristic
310C2#2490862611yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC23×C4×C83C23
FrattiniC23×C4C22×D83
FittingC83×#2646C2
socleC23×C83#120
prime radicalC24×C83C2×D4
hypercenter#2646D83
hypocenterC83#10345
LCSG ▷ C23×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
161. C83#10345semidirect

Maximal subgroups

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

subgroupindex
#249073972
#249057332
#249053412
#249053062
C83×#26462
#1034583