Finite Group Databasebeta

← #35314928

Subgroups — #35314928

Order 21248.

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

normal subgroups only

This group has 420 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#24913161C211yesmaximal, characteristic
2#24905797C211yesmaximal, characteristic
3#24905489C212yesmaximal, complemented
4#24905468C212yesmaximal, complemented
5C83×#1742C211yesmaximal, characteristic
6C2×#14581784C411yescharacteristic
7#19127758—21no—
8#19127640C2211yescharacteristic
9#19127637C411yescharacteristic
10#19127631C2211yescharacteristic
11#19127626C411yescharacteristic
12#19127596—21no—
13#19127595—21no—
14#19127585—21no—
15#19127574—21no—
16#19127547C411yescharacteristic
17#19127427C2212yes—
18#19127425—21no—
19C22×M16×C83—21no—
20C2×#117×C83C2211yescharacteristic
21#343×C83—21no—
22#343×C83C2212yes—
23#14581784—41no—
24#14581784—41no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—21no—
34C4⋊(C83⋊C8)—41no—
35C22⋊(C83⋊C8)—22no—
36#14581703—41no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C2×M16×C83—24no—
46C2×M16×C83—22no—
47C2×M16×C83—41no—
48C2×M16×C83—42no—
49C22×C8×C83—41no—
50#117×C83—41no—
51#117×C83—41no—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C4⋊C4×C83C2311yescharacteristic
54C2×C22⋊C4×C83C2×C411yescharacteristic
55C2×C22⋊C4×C83D411yescharacteristic
56C2×C42×C83C2×C411yescharacteristic
57C2×C42×C83D411yescharacteristic
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65Dic83⋊C4—42no—
66Dic83⋊C4—42no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71C22⋊Dic83—42no—
72C22⋊Dic83—42no—
73C4×Dic83—42no—
74C24×C83C22⋊C411yescharacteristic
75C22×C4×C83—22no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86M16×C83—24no—
87M16×C83—44no—
88M16×C83—42no—
89C2×C8×C83—44no—
90C2×C8×C83—42no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—41no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83#10412yes—
104C23×C83C2×C22⋊C411yescharacteristic
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
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—21no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83#9911yescharacteristic
122C2×C4×C83#9911yescharacteristic
123C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C83#13711yescharacteristic
125C2×Dic83—22no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C8×C83—44no—
138C83⋊C8—82no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—22no—
142C22×C83—42no—
143C22×C83—41no—
144C22×C83—41no—
145C22×C83—81no—
146C22×C83C2×#9911yescharacteristic
147C22×C83#41711yescharacteristic
148C22×C83C2×#10411yescharacteristic
149C4×C83—21no—
150C4×C83—21no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—41no—
154C4×C83—41no—
155C4×C83—41no—
156C4×C83—41no—
157Dic83—42no—
158Dic83—81no—
159Dic83—81no—
160Dic83—81no—
161#11767—831nomaximal, Sylow
162C2×C83—21no—
163C2×C83—21no—
164C2×C83—42no—
165C2×C83#150411yescharacteristic
166C2×C83#212311yescharacteristic
167C2×C83#213211yescharacteristic
168#2611—831no—
169#1742D8311yescharacteristic
170#1464—831no—
171#1353—832no—
172#1346—832no—
173C83#1176711yescharacteristic, complemented, Sylow
174C22×M16—21no—
175C2×#125—831no—
176C2×#117C2×D8311yescharacteristic
177#432—1661no—
178#400—831no—
179#399—831no—
180#398—831no—
181#397—831no—
182#389—1661no—
183#388—1661no—
184#387—1661no—
185#385—1661no—
186#376—831no—
187#343—21no—
188#343C2×D8312yes—
189#324—832no—
190#323—1661no—
191C23×C4C22⋊D8311yescharacteristic
192C2×M16—24no—
193C2×M16—22no—
194C2×M16—41no—
195C2×M16—42no—
196C22×C8—41no—
197#125—3321no—
198#125—3321no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4C22⋊D8311yescharacteristic
208C2×C4⋊C4C22×D8311yescharacteristic
209C2×C22⋊C4—1661no—
210C2×C22⋊C4—1661no—
211C2×C22⋊C4C4×D8311yescharacteristic
212C2×C22⋊C4D33211yescharacteristic
213C2×C42—1661no—
214C2×C42C4×D8311yescharacteristic
215C2×C42D33211yescharacteristic
216C4⋊C8—3321no—
217C22⋊C8—1662no—
218#97—3321no—
219#95—831no—
220#95—831no—
221#95—1661no—
222#95—1661no—
223#95—1661no—
224#95—1661no—
225#95—1661no—
226C24(C2×D83)⋊C411yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1661no—
233C22×C4—1661no—
234C22×C4—1661no—
235C22×C4—1661no—
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—44no—
244M16—42no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—3322no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—3321no—
251C4⋊C4—3321no—
252C4⋊C4—3322no—
253C4⋊C4—3322no—
254C4⋊C4—3321no—
255C4⋊C4—3321no—
256C4⋊C4—3321no—
257C4⋊C4—3321no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—3322no—
261C22⋊C4—3322no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—3322no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1458172012yes—
272C23C2×(C2×D83)⋊C411yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
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—3321no—
299C2×C4—3321no—
300C2×C4—1662no—
301C2×C4#1458170611yescharacteristic
302C2×C4#1458170611yescharacteristic
303C2×C4#1458181811yescharacteristic
304C2×C4#1458182311yescharacteristic
305C8—44no—
306C8—6642no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1458170611yescharacteristic
315C22#1912770011yescharacteristic
316C22C2×#1458172011yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—3322no—
326C4—6641no—
327C4—6641no—
328C4—6641no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#2490596511yescharacteristic
333C2#2490911811yescharacteristic
334C2#2490914611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1742C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1742D83
hypocenterC83#11767
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #24905489C2semidirect
4. #24905468C2semidirect
173. C83#11767semidirect

Maximal subgroups

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

subgroupindex
#249131612
#249057972
#249054892
#249054682
C83×#17422
#1176783