Finite Group Databasebeta

← #35315313

Subgroups — #35315313

Order 21248.

376 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 79 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#24913040C211yesmaximal, characteristic
2#24907546C211yesmaximal, characteristic
3#24905494C212yesmaximal, complemented
4#24905470C212yesmaximal, complemented
5C83×#1458C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic
7#19127665—21no—
8#19127631C2211yescharacteristic
9#19127626C411yescharacteristic
10#19127621C2211yescharacteristic
11#19127613C411yescharacteristic
12#19127589—21no—
13#19127585—21no—
14#19127582—21no—
15#19127574—21no—
16#19127569C411yescharacteristic
17#19127470—21no—
18#19127470C2212yes—
19C2×#117×C83C2211yescharacteristic
20#431×C83—21no—
21#324×C83C2212yes—
22#323×C83—21no—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—22no—
25C4×Dic166—41no—
26C4×Dic166—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—21no—
34C22⋊(C83⋊C8)—41no—
35C22⋊(C83⋊C8)—41no—
36C83⋊#95—21no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C23×C4×C83D411yescharacteristic
45#117×C83—41no—
46#117×C83—41no—
47C2×C4⋊C4×C83D411yescharacteristic
48C2×C4⋊C4×C83C2311yescharacteristic
49C2×C22⋊C4×C83C2×C411yescharacteristic
50C2×C22⋊C4×C83D411yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C4⋊C8×C83—41no—
54C22⋊C8×C83—22no—
55C4×C8×C83—41no—
56C2×Dic166—41no—
57C2×Dic166—42no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62#13050574—42no—
63#13050574—42no—
64C2×C83⋊C8—42no—
65C4⋊Dic83—41no—
66Dic83⋊C4—42no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69Dic83⋊C4—42no—
70C22⋊Dic83—42no—
71C22⋊Dic83—42no—
72C4×Dic83—42no—
73C24×C83C22⋊C411yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—22no—
77C22×C4×C83—41no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C2×C8×C83—42no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C22⋊C4×C83—41no—
88C22⋊C4×C83—41no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C42×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83C2×C22⋊C411yescharacteristic
99C2×C4×C83—21no—
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—42no—
108C2×C4×C83—42no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83#9911yescharacteristic
114C2×C4×C83#9911yescharacteristic
115C2×C4×C83#13512yes—
116C2×Dic83—22no—
117C2×Dic83—41no—
118C2×Dic83—41no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128Dic166—42no—
129Dic166—42no—
130C8×C83—82no—
131C83⋊C8—82no—
132C22×C83—21no—
133C22×C83—21no—
134C22×C83—42no—
135C22×C83—41no—
136C22×C83—41no—
137C22×C83—41no—
138C22×C83—81no—
139C22×C83C2×#9911yescharacteristic
140C22×C83#41611yescharacteristic
141C22×C83#42111yescharacteristic
142C4×C83—22no—
143C4×C83—21no—
144C4×C83—21no—
145C4×C83—41no—
146C4×C83—41no—
147C4×C83—41no—
148C4×C83—41no—
149Dic83—42no—
150Dic83—81no—
151Dic83—81no—
152Dic83—81no—
153#11849—831nomaximal, Sylow
154C2×C83—21no—
155C2×C83—21no—
156C2×C83—81no—
157C2×C83#151611yescharacteristic
158C2×C83#212011yescharacteristic
159C2×C83#213511yescharacteristic
160#2599—831no—
161#1822—831no—
162#1458D8311yescharacteristic
163#1354—832no—
164#1346—832no—
165C83#1184911yescharacteristic, complemented, Sylow
166C2×C4×Q8—831no—
167C2×#117C2×D8311yescharacteristic
168#431—21no—
169#407—1661no—
170#398—831no—
171#397—831no—
172#395—831no—
173#393—831no—
174#388—1661no—
175#387—1661no—
176#387—1661no—
177#385—1661no—
178#383—831no—
179#343—832no—
180#343—1661no—
181#324C2×D8312yes—
182#323—21no—
183C22×Q8—831no—
184C23×C4D33211yescharacteristic
185C2×M16—1662no—
186C4×Q8—3321no—
187C4×Q8—3321no—
188#117—41no—
189#117—41no—
190C2×C4⋊C4—831no—
191C2×C4⋊C4—1661no—
192C2×C4⋊C4—1661no—
193C2×C4⋊C4—1661no—
194C2×C4⋊C4D33211yescharacteristic
195C2×C4⋊C4C22×D8311yescharacteristic
196C2×C22⋊C4—1661no—
197C2×C22⋊C4—1661no—
198C2×C22⋊C4C4×D8311yescharacteristic
199C2×C22⋊C4C22⋊D8311yescharacteristic
200C2×C42—1661no—
201C2×C42C4×D8311yescharacteristic
202C2×C42C22⋊D8311yescharacteristic
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—3321no—
206C22⋊C8—3321no—
207C4×C8—41no—
208#95—831no—
209#95—831no—
210#95—1661no—
211#95—1661no—
212#95—1661no—
213#95—1661no—
214#95—1661no—
215#95—1661no—
216C24(C2×D83)⋊C411yescharacteristic
217C2×Q8—3321no—
218C2×Q8—3322no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—1661no—
224C22×C4—1661no—
225C22×C4—1661no—
226C22×C4—1661no—
227C22×C4(C2×D83)⋊C411yescharacteristic
228C22×C4(C2×D83)⋊C411yescharacteristic
229C22×C4(C2×D83)⋊C411yescharacteristic
230C22×C4C2×C4×D8311yescharacteristic
231C22×C4C2×C22⋊D8311yescharacteristic
232C22×C4C2×D33211yescharacteristic
233M16—3322no—
234M16—3322no—
235C2×C8—42no—
236C2×C8—3322no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—3321no—
240C4⋊C4—3322no—
241C4⋊C4—3322no—
242C4⋊C4—3321no—
243C4⋊C4—3321no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—3322no—
247C22⋊C4—3322no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—3322no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×(C2×D83)⋊C411yescharacteristic
259Q8—3322no—
260Q8—3322no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—3321no—
276C2×C4—3321no—
277C2×C4—3321no—
278C2×C4—3321no—
279C2×C4—3321no—
280C2×C4—3321no—
281C2×C4—3321no—
282C2×C4—3321no—
283C2×C4—3321no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—1662no—
287C2×C4#1458170611yescharacteristic
288C2×C4#1458170611yescharacteristic
289C2×C4#1458181412yes—
290C8—82no—
291C8—6642no—
292C22—21no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22C2×#1458170611yescharacteristic
300C22#1912769811yescharacteristic
301C22#1912771711yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—3322no—
310C4—6641no—
311C4—6641no—
312C4—6641no—
313C2—21no—
314C2—21no—
315C2—81no—
316C2#2490602011yescharacteristic
317C2#2490910211yescharacteristic
318C2#2490915711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1458C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1458D83
hypocenterC83#11849
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #24905494C2semidirect
4. #24905470C2semidirect
165. C83#11849semidirect

Maximal subgroups

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

subgroupindex
#249130402
#249075462
#249054942
#249054702
C83×#14582
#1184983