Finite Group Databasebeta

← #35301883

Subgroups — #35301883

Order 21248.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 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#24918204C211yesmaximal, characteristic
2#24916568C211yesmaximal, characteristic
3#24906101C211yesmaximal, characteristic
4#24906013C211yesmaximal, characteristic
5#24905336C211yesmaximal, characteristic
6#24905327C211yesmaximal, characteristic
7C83×#1515C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19128097C411yescharacteristic, complemented
10#19128080C411yescharacteristic, complemented
11#19128053C411yescharacteristic, complemented
12#19128051C411yescharacteristic, complemented
13#19127957—22no—
14#19127830—22no—
15#19127829—22no—
16#19127806—22no—
17#19127759—22no—
18#19127758C2211yescharacteristic
19#19127758C2211yescharacteristic
20C4×#13050574C2211yescharacteristic
21#19127443—22no—
22#19127437—22no—
23#533×C83C2211yescharacteristic
24#432×C83C2211yescharacteristic
25#383×C83C2211yescharacteristic
26#328×C83—22no—
27C2×#13050580—22no—
28C2×#13050574—21no—
29#14581792C2×C411yescharacteristic
30#14581792C2×C411yescharacteristic
31#14581784—21no—
32#14581771—21no—
33#14581771D412yes—
34C4×Dic166—21no—
35C4×Dic166D412yes—
36#14581751—21no—
37#14581723—22no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40Dic83⋊C8—24no—
41#14581716—22no—
42#14581715—22no—
43C22⋊(C83⋊C8)—21no—
44C22⋊(C83⋊C8)—21no—
45#14581703—21no—
46#14581703—21no—
47#14581703D412yes—
48#14581702—22no—
49C4×C83⋊C8C2×C411yescharacteristic
50C4×C83⋊C8C2×C411yescharacteristic
51C8×Dic83—22no—
52C4⋊Q8×C83C2×C411yescharacteristic
53C4⋊Q8×C83C2×C411yescharacteristic
54#125×C83—21no—
55C22⋊Q8×C83—21no—
56C22⋊Q8×C83—21no—
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59#117×C83—21no—
60C2×C42×C83C2311yescharacteristic
61C4⋊C8×C83—21no—
62C22⋊C8×C83D412yes—
63#97×C83—21no—
64#95×C83—21no—
65#95×C83—21no—
66#95×C83—21no—
67C2×Dic166—21no—
68C2×Dic166C22⋊C412yes—
69#13050580—44no—
70#13050574—42no—
71C2×C83⋊C8—21no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—22no—
74C2×C83⋊C8—41no—
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83C22⋊C412yes—
79Dic83⋊C4—22no—
80Dic83⋊C4—22no—
81C22⋊Dic83—22no—
82C4×Dic83—22no—
83C2×Q8×C83—21no—
84C2×Q8×C83—21no—
85C22×C4×C83—41no—
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83#4111yescharacteristic
89C2×C8×C83—22no—
90C4⋊C4×C83—21no—
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—41no—
96C22⋊C4×C83—41no—
97C42×C83—41no—
98C42×C83C22×C411yescharacteristic
99C42×C83C2×D411yescharacteristic
100C42×C83C2×D411yescharacteristic
101C42×C83#4111yescharacteristic
102C23×C83#12111yescharacteristic
103Q8×C83—42no—
104Q8×C83—81no—
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—81no—
111C2×C4×C83—81no—
112C2×C4×C83—81no—
113C2×C4×C83C2×C22⋊C411yescharacteristic
114C2×C4×C83C4×D411yescharacteristic
115C2×C4×C83C4×D411yescharacteristic
116C2×C4×C83#12011yescharacteristic
117C2×C4×C83#12111yescharacteristic
118C2×C4×C83#12311yescharacteristic
119C2×Dic83—22no—
120C2×Dic83—22no—
121Dic166—42no—
122Dic166—81no—
123C8×C83—42no—
124C83⋊C8—22no—
125C83⋊C8—42no—
126C83⋊C8—41no—
127C83⋊C8—81no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83—21no—
131C22×C83#38611yescharacteristic
132C4×C83—21no—
133C4×C83—21no—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—21no—
137C4×C83—21no—
138C4×C83—81no—
139C4×C83—81no—
140C4×C83—161no—
141Dic83—42no—
142Dic83—42no—
143#9416—831nomaximal, Sylow
144C2×C83—41no—
145C2×C83#188411yescharacteristic
146C2×C83#190211yescharacteristic
147C2×C83#190511yescharacteristic
148#3185—831no—
149#2987—831no—
150#1537—831no—
151#1515—831no—
152#1515D8311yescharacteristic
153#1291—831no—
154#1288—831no—
155C83#941611yescharacteristic, complemented, Sylow
156#533—831no—
157#533C2×D8311yescharacteristic
158#498—831no—
159#494—831no—
160#487—831no—
161#487—831no—
162#470—1662no—
163#446—1662no—
164#446—1662no—
165#441—1662no—
166#433—1662no—
167#432—831no—
168#432—831no—
169#432C2×D8311yescharacteristic
170C4×M16—831no—
171#383C2×D8311yescharacteristic
172#330—1662no—
173#328—22no—
174#328—1662no—
175C2×Q16—1662no—
176C2×M16—1661no—
177C4⋊Q8—831no—
178C4⋊Q8—831no—
179C4⋊Q8C4×D8311yescharacteristic
180C4⋊Q8C4×D8311yescharacteristic
181#125—21no—
182#125—1661no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—832no—
186C22⋊Q8—1661no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—832no—
190C4×Q8—1661no—
191#117—21no—
192#117—1661no—
193C2×C42C22×D8311yescharacteristic
194#106—1662no—
195C4⋊C8—21no—
196C4⋊C8—1664no—
197C4⋊C8—1661no—
198C4⋊C8—1661no—
199#103—1662no—
200#103—1662no—
201C22⋊C8—1661no—
202C22⋊C8—1661no—
203C22⋊C8C22⋊D8312yes—
204#97—21no—
205#97—832no—
206#97—1662no—
207#97—1661no—
208#97—1661no—
209C4×C8—831no—
210C4×C8—831no—
211C4×C8—1662no—
212#95—21no—
213#95—21no—
214#95—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—832no—
218C2×Q8—1661no—
219C22×C4—41no—
220C22×C4D4×D8311yescharacteristic
221C22×C4C2×C22⋊D8311yescharacteristic
222C22×C4#1305058911yescharacteristic
223Q16—3324no—
224M16—3322no—
225C2×C8—22no—
226C2×C8—3321no—
227C2×C8—1662no—
228C2×C8—1661no—
229C2×C8—1661no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—832no—
237C4⋊C4—1662no—
238C4⋊C4—1662no—
239C4⋊C4—1661no—
240C4⋊C4—1661no—
241C4⋊C4—1661no—
242C22⋊C4—41no—
243C22⋊C4—1662no—
244C42—41no—
245C42—1662no—
246C42C2×C4×D8311yescharacteristic
247C42D4×D8311yescharacteristic
248C42D4×D8311yescharacteristic
249C42#1305059011yescharacteristic
250C23#1458176311yescharacteristic
251Q8—42no—
252Q8—81no—
253Q8—3322no—
254Q8—6641no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—1662no—
264C2×C4—1662no—
265C2×C4C22⋊C4×D8311yescharacteristic
266C2×C4(C22⋊D83)⋊C411yescharacteristic
267C2×C4C4×C22⋊D8311yescharacteristic
268C2×C4#1458176011yescharacteristic
269C2×C4#1458176511yescharacteristic
270C2×C4#1458177411yescharacteristic
271C8—42no—
272C8—3321no—
273C8—3322no—
274C8—6641no—
275C8—1662no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1912757711yescharacteristic
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—81no—
287C4—81no—
288C4—161no—
289C4—3322no—
290C4—3322no—
291C2—41no—
292C2#2490784311yescharacteristic
293C2#2490793311yescharacteristic
294C2#2490796011yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1515C2
socleC22×C83#386
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1515D83
hypocenterC83#9416
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
9. #19128097C4semidirect
10. #19128080C4semidirect
11. #19128053C4semidirect
12. #19128051C4semidirect
155. C83#9416semidirect

Maximal subgroups

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

subgroupindex
#249182042
#249165682
#249061012
#249060132
#249053362
#249053272
C83×#15152
#941683