Finite Group Databasebeta

← #35308345

Subgroups — #35308345

Order 21248.

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

normal subgroups only

This group has 344 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#24908453C211yesmaximal, characteristic
2#24906251C211yesmaximal, characteristic
3#24906201C211yesmaximal, characteristic
4#24906148C211yesmaximal, characteristic
5#24905884C211yesmaximal, characteristic
6#24905354C211yesmaximal, characteristic
7C83×#3229C211yesmaximal, characteristic
8#19127754C2211yescharacteristic
9#19127725C2211yescharacteristic
10#19127725C2211yescharacteristic
11C4×#13050574C2211yescharacteristic
12#19127470—22no—
13#19127453—21no—
14#19127451—21no—
15#19127447—22no—
16#19127447—21no—
17#19127447—21no—
18Q8⋊(C83⋊C8)—22no—
19#533×C83C2211yescharacteristic
20C2×C4⋊Q8×C83C2211yescharacteristic
21#498×C83—21no—
22#487×C83—21no—
23#483×C83—22no—
24#475×C83—21no—
25#475×C83—21no—
26#431×C83C2211yescharacteristic
27C2×#13050574—21no—
28C2×#13050574—21no—
29C2×#13050574—21no—
30C4⋊(C83⋊C8)—21no—
31C4⋊(C83⋊C8)—21no—
32C4⋊(C83⋊C8)—21no—
33C4⋊(C83⋊C8)—21no—
34C4⋊(C83⋊C8)D412yes—
35C22⋊(C83⋊C8)—21no—
36#14581703—21no—
37C4×C83⋊C8—21no—
38C4×C83⋊C8—21no—
39C22×Q8×C83—21no—
40C4⋊Q8×C83—21no—
41C4⋊Q8×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83D411yescharacteristic
45C4⋊Q8×C83D411yescharacteristic
46#125×C83—21no—
47C22⋊Q8×C83—21no—
48C22⋊Q8×C83—21no—
49C4×Q8×C83—21no—
50C4×Q8×C83—21no—
51#117×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83D412yes—
54C2×C42×C83C2311yescharacteristic
55#106×C83—22no—
56#106×C83—22no—
57C4⋊C8×C83D411yescharacteristic
58C4⋊C8×C83D411yescharacteristic
59#103×C83—22no—
60#103×C83—22no—
61C22⋊C8×C83—21no—
62C4×C8×C83—21no—
63#13050574—22no—
64#13050574—41no—
65#13050574—42no—
66#13050574—81no—
67C2×C83⋊C8—22no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8—41no—
70C2×C83⋊C8—41no—
71C2×Q8×C83—21no—
72C2×Q8×C83—21no—
73C2×Q8×C83—41no—
74C2×Q8×C83—41no—
75C2×Q8×C83—82no—
76C22×C4×C83—22no—
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83#4111yescharacteristic
79C22×C4×C83#4111yescharacteristic
80C2×C8×C83—21no—
81C2×C8×C83—21no—
82C4⋊C4×C83—22no—
83C4⋊C4×C83—21no—
84C4⋊C4×C83—22no—
85C4⋊C4×C83—21no—
86C4⋊C4×C83—21no—
87C4⋊C4×C83—21no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C22⋊C4×C83—41no—
93C42×C83—41no—
94C42×C83C2×D411yescharacteristic
95C42×C83C2×D411yescharacteristic
96C42×C83C2×D411yescharacteristic
97C42×C83#4111yescharacteristic
98C23×C83#12311yescharacteristic
99Q8×C83—41no—
100Q8×C83—41no—
101Q8×C83—41no—
102Q8×C83—41no—
103Q8×C83—82no—
104Q8×C83—82no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—42no—
113C2×C4×C83—82no—
114C2×C4×C83#12011yescharacteristic
115C2×C4×C83#12111yescharacteristic
116C2×C4×C83#12111yescharacteristic
117C2×C4×C83#12111yescharacteristic
118C2×C4×C83#12311yescharacteristic
119C2×C4×C83#12411yescharacteristic
120C8×C83—41no—
121C8×C83—41no—
122C83⋊C8—42no—
123C83⋊C8—42no—
124C83⋊C8—81no—
125C83⋊C8—81no—
126C22×C83—21no—
127C22×C83—21no—
128C22×C83—21no—
129C22×C83#39411yescharacteristic
130C4×C83—21no—
131C4×C83—21no—
132C4×C83—21no—
133C4×C83—81no—
134C4×C83—81no—
135C4×C83—82no—
136C4×C83#45411yescharacteristic
137C4×C83#45611yescharacteristic
138C4×C83#47112yes—
139C4×C83#47111yescharacteristic
140C4×C83#47311yescharacteristic
141#10494—831nomaximal, Sylow
142C2×C83—41no—
143C2×C83#203811yescharacteristic
144C2×C83#204611yescharacteristic
145C2×C83#205211yescharacteristic
146#3229D8311yescharacteristic
147#1997—831no—
148#1574—831no—
149#1561—831no—
150#1549—831no—
151#1485—831no—
152#1299—831no—
153C83#1049411yescharacteristic, complemented, Sylow
154#533C2×D8311yescharacteristic
155C2×C4⋊Q8C2×D8311yescharacteristic
156#498—21no—
157#487—21no—
158#483—22no—
159#475—21no—
160#475—21no—
161#431—831no—
162#431C2×D8311yescharacteristic
163#423—831no—
164#423—831no—
165C4×M16—831no—
166#343—1662no—
167#335—1661no—
168#334—1661no—
169#332—1662no—
170#332—1661no—
171#332—1661no—
172Q8⋊C8—1662no—
173C22×Q8—21no—
174C2×M16—1661no—
175C2×M16—1661no—
176C2×M16—1661no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22⋊D8311yescharacteristic
182C4⋊Q8C22⋊D8311yescharacteristic
183#125—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188#117—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22⋊D8312yes—
191C2×C42C22×D8311yescharacteristic
192#106—22no—
193#106—22no—
194C4⋊C8—832no—
195C4⋊C8—1661no—
196C4⋊C8—1661no—
197C4⋊C8—1661no—
198C4⋊C8—1661no—
199C4⋊C8C22⋊D8311yescharacteristic
200C4⋊C8C22⋊D8311yescharacteristic
201#103—22no—
202#103—22no—
203C22⋊C8—21no—
204C22⋊C8—1661no—
205#97—1661no—
206C4×C8—21no—
207C4×C8—1661no—
208C4×C8—1661no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—82no—
214C22×C4—22no—
215C22×C4C2×C22⋊D8311yescharacteristic
216C22×C4#1305058911yescharacteristic
217C22×C4#1305058911yescharacteristic
218M16—3321no—
219M16—3322no—
220M16—6641no—
221M16—1662no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—3321no—
225C2×C8—3321no—
226C2×C8—3321no—
227C2×C8—1662no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C22⋊C4—41no—
239C42—41no—
240C42D4×D8311yescharacteristic
241C42C2×C22⋊D8311yescharacteristic
242C42C2×C22⋊D8311yescharacteristic
243C42#1305058911yescharacteristic
244C23#1458177311yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—82no—
250Q8—82no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—82no—
260C2×C4#1458176111yescharacteristic
261C2×C4#1458176311yescharacteristic
262C2×C4#1458176411yescharacteristic
263C2×C4#1458176411yescharacteristic
264C2×C4#1458177311yescharacteristic
265C2×C4#1458177911yescharacteristic
266C8—41no—
267C8—41no—
268C8—3322no—
269C8—3322no—
270C8—6641no—
271C8—6641no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22#1912761511yescharacteristic
276C4—21no—
277C4—21no—
278C4—21no—
279C4—81no—
280C4—82no—
281C4—81no—
282C4#1912787411yescharacteristic
283C4#1912788411yescharacteristic
284C4#1912796112yes—
285C4#1912796611yescharacteristic
286C4#1912797411yescharacteristic
287C2—41no—
288C2#2490867711yescharacteristic
289C2#2490872111yescharacteristic
290C2#2490875911yescharacteristic
subgroupG/H
centerC22#19127615
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3229C2
socleC22×C83#394
prime radicalC23×C83#123
hypercenter#3229D83
hypocenterC83#10494
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
153. C83#10494semidirect

Maximal subgroups

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

subgroupindex
#249084532
#249062512
#249062012
#249061482
#249058842
#249053542
C83×#32292
#1049483