Finite Group Databasebeta

← #35314791

Subgroups — #35314791

Order 21248.

400 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 83 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#24913323C211yesmaximal, characteristic
2#24905787C211yesmaximal, characteristic, complemented
3#24905492C212yesmaximal, complemented
4#24905465C212yesmaximal
5C4×#101×C83C211yesmaximal, characteristic, complemented
6C2×#14581780C411yescharacteristic
7#19127754—21no—
8#19127636—21no—
9#19127621C411yescharacteristic
10#19127621C2211yescharacteristic
11#19127618C411yescharacteristic
12#19127618C2211yescharacteristic
13#19127612—21no—
14#19127592—21no—
15#19127592—21no—
16#19127569C411yescharacteristic
17#19127427—21no—
18#19127427C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20C2×#101×C83—21no—
21C4×M16×C83—21no—
22#343×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C22×D332C2×C411yescharacteristic
25#14581780—41no—
26#14581780—41no—
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×(C2×D83)⋊C4—21no—
31C2×(C2×D83)⋊C4—21no—
32C2×(C2×D83)⋊C4—21no—
33C2×(C2×D83)⋊C4—21no—
34C2×(C2×D83)⋊C4—21no—
35C4⋊(C83⋊C8)—41no—
36C22⋊(C83⋊C8)—22no—
37C4×C83⋊C8—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C22×Q8×C83D411yescharacteristic
45C2×M16×C83—22no—
46C4×Q8×C83—41no—
47C4×Q8×C83—41no—
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83C2311yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54#101×C83—24no—
55#97×C83—41no—
56C4×C8×C83—41no—
57C23×D83—21no—
58C2×Dic166—41no—
59C2×Dic166—81no—
60C2×D332—41no—
61C2×D332—81no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C22×Dic83—21no—
65C2×C83⋊C8—42no—
66Dic83⋊C4—81no—
67Dic83⋊C4—81no—
68(C2×D83)⋊C4—42no—
69(C2×D83)⋊C4—41no—
70(C2×D83)⋊C4—41no—
71(C2×D83)⋊C4—42no—
72(C2×D83)⋊C4—42no—
73(C2×D83)⋊C4—81no—
74(C2×D83)⋊C4—81no—
75C2×Q8×C83—22no—
76C2×Q8×C83—22no—
77C2×Q8×C83—41no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85M16×C83—24no—
86M16×C83—42no—
87C2×C8×C83—42no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C42×C83—21no—
92C42×C83—21no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—21no—
97C23×C83C2×C22⋊C411yescharacteristic
98C22×D83—41no—
99C22×D83—41no—
100C22×D83—41no—
101C22×D83—41no—
102C22×D83—41no—
103C22×D83—41no—
104C22×D83—41no—
105Q8×C83—44no—
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—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83#9911yescharacteristic
121C2×C4×C83#9911yescharacteristic
122C2×C4×C83#10412yes—
123C2×C4×C83#13512yes—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133D332—82no—
134Dic166—82no—
135C8×C83—44no—
136C83⋊C8—82no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83C2×#9911yescharacteristic
140C22×C83#41611yescharacteristic
141C22×C83C2×#10411yescharacteristic
142C2×D83—81no—
143C2×D83—81no—
144C2×D83—81no—
145C2×D83—81no—
146C2×D83—81no—
147C2×D83—81no—
148C2×D83—81no—
149C4×C83—21no—
150C4×C83—21no—
151C4×C83—22no—
152C4×C83—22no—
153C4×C83—41no—
154C4×C83—41no—
155C4×C83—41no—
156C4×C83—41no—
157Dic83—81no—
158Dic83—81no—
159Dic83—81no—
160#11738—831nomaximal, Sylow
161C2×C83—21no—
162C2×C83—21no—
163C2×C83#150111yescharacteristic
164C2×C83#212211yescharacteristic
165C2×C83#213311yescharacteristic
166D83—161no—
167#2632—831no—
168C4×#101D8311yescharacteristic, complemented
169#1461—831no—
170#1354—832no—
171#1345—832no—
172C83#1173811yescharacteristic, complemented, Sylow
173C2×#124—831no—
174C2×C4×Q8C2×D8311yescharacteristic
175#431—1661no—
176C2×#101—21no—
177C4×M16—21no—
178#399—1661no—
179#395—831no—
180#395—831no—
181#394—831no—
182#394—831no—
183#393—1661no—
184#388—1661no—
185#388—1661no—
186#383—831no—
187#343C2×D8312yes—
188#324—832no—
189#324—1661no—
190C22×Q8—831no—
191C22×Q8C22⋊D8311yescharacteristic
192C22×D4—831no—
193C2×M16—22no—
194#124—3321no—
195#124—3321no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—1661no—
199C2×C4⋊C4—1661no—
200C2×C4⋊C4C4×D8311yescharacteristic
201C2×C4⋊C4D33211yescharacteristic
202C2×C4⋊C4C22×D8311yescharacteristic
203C2×C22⋊C4—1661no—
204C2×C22⋊C4—1661no—
205C2×C22⋊C4—1661no—
206C2×C22⋊C4—1661no—
207C2×C22⋊C4—1661no—
208C2×C22⋊C4—1661no—
209C2×C42C4×D8311yescharacteristic
210C2×C42D33211yescharacteristic
211C2×C42C22⋊D8311yescharacteristic
212C4⋊C8—3321no—
213#101—24no—
214C22⋊C8—1662no—
215#97—41no—
216C4×C8—41no—
217C4×C8—3321no—
218#95—831no—
219#95—831no—
220#95—1661no—
221#95—1661no—
222#95—1661no—
223#95—1661no—
224C24—1661no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—3321no—
229C2×Q8—6641no—
230C2×D4—3321no—
231C2×D4—6641no—
232C22×C4—1661no—
233C22×C4—1661no—
234C22×C4—1661no—
235C22×C4(C2×D83)⋊C411yescharacteristic
236C22×C4(C2×D83)⋊C411yescharacteristic
237C22×C4(C2×D83)⋊C411yescharacteristic
238C22×C4(C2×D83)⋊C411yescharacteristic
239C22×C4C2×C4×D8311yescharacteristic
240C22×C4C2×C22⋊D8311yescharacteristic
241C22×C4C2×D33211yescharacteristic
242M16—24no—
243M16—42no—
244C2×C8—42no—
245C2×C8—3322no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—6641no—
250C4⋊C4—6641no—
251C22⋊C4—3322no—
252C22⋊C4—3322no—
253C22⋊C4—3321no—
254C22⋊C4—3321no—
255C22⋊C4—3322no—
256C22⋊C4—6641no—
257C22⋊C4—6641no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—3321no—
265C23—3321no—
266C23—3321no—
267C23—3321no—
268C23—3321no—
269C23—3321no—
270C23—3321no—
271C23C2×(C2×D83)⋊C411yescharacteristic
272Q8—44no—
273Q8—6642no—
274D4—6642no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—3321no—
290C2×C4—3321no—
291C2×C4—3321no—
292C2×C4—3321no—
293C2×C4—3321no—
294C2×C4—3321no—
295C2×C4—3321no—
296C2×C4—3321no—
297C2×C4—3321no—
298C2×C4#1458170611yescharacteristic
299C2×C4#1458170611yescharacteristic
300C2×C4#1458172012yes—
301C2×C4#1458181212yes—
302C8—44no—
303C8—6642no—
304C22—21no—
305C22—21no—
306C22—6641no—
307C22—6641no—
308C22—6641no—
309C22—6641no—
310C22—6641no—
311C22—6641no—
312C22—6641no—
313C22C2×#1458170611yescharacteristic
314C22#1912769511yescharacteristic
315C22C2×#1458172011yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—6641no—
325C4—6641no—
326C4—6641no—
327C2—21no—
328C2—21no—
329C2—13281no—
330C2#2490595211yescharacteristic
331C2#2490911411yescharacteristic
332C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC4×#101×C83C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenterC4×#101D83
hypocenterC83#11738
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #24905787C2semidirect
3. #24905492C2semidirect
5. C4×#101×C83C2semidirect
168. C4×#101D83semidirect
172. C83#11738semidirect

Maximal subgroups

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

subgroupindex
#249133232
#249057872
#249054922
#249054652
C4×#101×C832
#1173883