Finite Group Databasebeta

← #35308184

Subgroups — #35308184

Order 21248.

366 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 97 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#24907478C211yesmaximal, characteristic
2#24906239C211yesmaximal, characteristic
3#24906120C211yesmaximal, characteristic
4#24906027C211yesmaximal, characteristic
5#24905787C211yesmaximal, characteristic
6#24905323C211yesmaximal, characteristic
7C83×#3078C211yesmaximal, characteristic
8#19127754C2211yescharacteristic
9#19127754C2211yescharacteristic
10C2×C4⋊(C83⋊C8)C2211yescharacteristic
11C2×#14581716—21no—
12#19127567C2211yescharacteristic
13#19127463—21no—
14#19127453—22no—
15#19127450—21no—
16#19127447—21no—
17#19127438—22no—
18#19127427—22no—
19#533×C83C2211yescharacteristic
20C2×C4×Q8×C83C2211yescharacteristic
21#479×C83—22no—
22#477×C83—21no—
23#475×C83—21no—
24#428×C83—21no—
25#423×C83C2211yescharacteristic
26#419×C83—21no—
27C22×C83⋊C8—21no—
28C2×C4⋊Dic83—21no—
29C4⋊(C83⋊C8)—21no—
30C4⋊(C83⋊C8)—21no—
31C4⋊(C83⋊C8)—21no—
32C4⋊(C83⋊C8)—21no—
33#14581716—42no—
34C22⋊(C83⋊C8)—21no—
35C22⋊(C83⋊C8)D412yes—
36C4×C83⋊C8—21no—
37C4×C83⋊C8—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C22×Q8×C83D411yescharacteristic
41C2×M16×C83—21no—
42C4⋊Q8×C83D411yescharacteristic
43C4⋊Q8×C83D411yescharacteristic
44#125×C83—21no—
45C22⋊Q8×C83—21no—
46C22⋊Q8×C83—21no—
47C4×Q8×C83—21no—
48C4×Q8×C83—21no—
49C4×Q8×C83—21no—
50C4×Q8×C83—21no—
51C4×Q8×C83—21no—
52C4×Q8×C83—21no—
53#117×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83D411yescharacteristic
56C2×C42×C83—21no—
57C2×C42×C83C2311yescharacteristic
58#107×C83—22no—
59#106×C83—22no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83D412yes—
62#103×C83—22no—
63#103×C83—22no—
64C22×Dic83—41no—
65C2×C83⋊C8—22no—
66C2×C83⋊C8—41no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8—41no—
70C4⋊Dic83—41no—
71C4⋊Dic83—41no—
72C2×Q8×C83—21no—
73C2×Q8×C83—21no—
74C2×Q8×C83—21no—
75C2×Q8×C83—21no—
76C2×Q8×C83—21no—
77C2×Q8×C83—21no—
78C2×Q8×C83—21no—
79C2×Q8×C83Q1612yes—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83#4111yescharacteristic
85M16×C83—41no—
86M16×C83—41no—
87C2×C8×C83—22no—
88C4⋊C4×C83—21no—
89C4⋊C4×C83—21no—
90C4⋊C4×C83—21no—
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C22⋊C4×C83—41no—
99C42×C83—41no—
100C42×C83—41no—
101C42×C83—41no—
102C42×C83C2×D411yescharacteristic
103C42×C83C2×D411yescharacteristic
104C42×C83#4111yescharacteristic
105C42×C83#4111yescharacteristic
106C23×C83#12111yescharacteristic
107Q8×C83—22no—
108Q8×C83—22no—
109Q8×C83—41no—
110Q8×C83—41no—
111Q8×C83—81no—
112Q8×C83—81no—
113C2×C4×C83—22no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—81no—
122C2×C4×C83#12011yescharacteristic
123C2×C4×C83#12111yescharacteristic
124C2×C4×C83#12111yescharacteristic
125C2×C4×C83#12311yescharacteristic
126C2×C4×C83#12311yescharacteristic
127C2×C4×C83#12411yescharacteristic
128C2×C4×C83C2×Q1611yescharacteristic
129C2×C4×C83#13511yescharacteristic
130C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C83#13711yescharacteristic
132C2×Dic83—81no—
133C2×Dic83—81no—
134C2×Dic83—81no—
135C8×C83—42no—
136C83⋊C8—42no—
137C83⋊C8—81no—
138C83⋊C8—81no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83#39411yescharacteristic
142C22×C83#46711yescharacteristic
143C22×C83#46811yescharacteristic
144C4×C83—21no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—42no—
149C4×C83—81no—
150C4×C83—81no—
151C4×C83—81no—
152C4×C83#44911yescharacteristic
153C4×C83#45111yescharacteristic
154C4×C83#48411yescharacteristic
155C4×C83#48511yescharacteristic
156Dic83—161no—
157#10471—831nomaximal, Sylow
158C2×C83—21no—
159C2×C83—21no—
160C2×C83#203611yescharacteristic
161C2×C83#204911yescharacteristic
162C2×C83#205311yescharacteristic
163#3078D8311yescharacteristic
164#1805—831no—
165#1571—831no—
166#1542—831no—
167#1518—831no—
168#1461—831no—
169#1286—831no—
170C83#1047111yescharacteristic, complemented, Sylow
171#533C2×D8311yescharacteristic
172C2×C4×Q8C2×D8311yescharacteristic
173#479—22no—
174#477—21no—
175#475—21no—
176#431—831no—
177#431—831no—
178#428—21no—
179#423C2×D8311yescharacteristic
180C2×C4⋊C8—831no—
181#419—21no—
182C2×#103—1661no—
183#382—831no—
184#340—1661no—
185#335—1662no—
186#333—1661no—
187#332—1661no—
188#328—1662no—
189#324—1662no—
190C22×Q8C22⋊D8311yescharacteristic
191C2×M16—21no—
192C22×C8—1661no—
193C4⋊Q8C22⋊D8311yescharacteristic
194C4⋊Q8C22⋊D8311yescharacteristic
195#125—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204#117—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4C22⋊D8311yescharacteristic
208C2×C42—21no—
209C2×C42C22×D8311yescharacteristic
210#107—22no—
211#106—22no—
212C4⋊C8—21no—
213C4⋊C8—1661no—
214C4⋊C8—1661no—
215C4⋊C8—1661no—
216C4⋊C8—1661no—
217C4⋊C8C22⋊D8312yes—
218#103—22no—
219#103—22no—
220#103—3322no—
221C22⋊C8—832no—
222C22⋊C8—1661no—
223C4×C8—1661no—
224C4×C8—1661no—
225#95—1661no—
226#95—1661no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8#1305058012yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—3321no—
238C22×C4D4×D8311yescharacteristic
239C22×C4C2×C22⋊D8311yescharacteristic
240C22×C4#1305058911yescharacteristic
241M16—41no—
242M16—41no—
243C2×C8—22no—
244C2×C8—3321no—
245C2×C8—3321no—
246C2×C8—3321no—
247C2×C8—3321no—
248C2×C8—1662no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—3321no—
260C4⋊C4—3321no—
261C22⋊C4—41no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42C2×C22⋊D8311yescharacteristic
266C42C2×C22⋊D8311yescharacteristic
267C42#1305058911yescharacteristic
268C42#1305058911yescharacteristic
269C23#1458176411yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274Q8—81no—
275Q8—81no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—6641no—
286C2×C4—6641no—
287C2×C4—6641no—
288C2×C4#1458176111yescharacteristic
289C2×C4#1458176311yescharacteristic
290C2×C4#1458176411yescharacteristic
291C2×C4#1458177311yescharacteristic
292C2×C4#1458177311yescharacteristic
293C2×C4#1458177911yescharacteristic
294C2×C4C2×#1305058011yescharacteristic
295C2×C4#1458181211yescharacteristic
296C2×C4#1458181711yescharacteristic
297C2×C4#1458182311yescharacteristic
298C8—42no—
299C8—3322no—
300C8—6641no—
301C8—6641no—
302C22—21no—
303C22—21no—
304C22#1912761511yescharacteristic
305C22#1912794011yescharacteristic
306C22#1912794911yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—13281no—
316C4#1912784611yescharacteristic
317C4#1912785711yescharacteristic
318C4#1912803411yescharacteristic
319C4#1912803911yescharacteristic
320C2—21no—
321C2—21no—
322C2#2490866311yescharacteristic
323C2#2490874011yescharacteristic
324C2#2490876611yescharacteristic
subgroupG/H
centerC22#19127615
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3078C2
socleC22×C83#394
prime radicalC23×C83#121
hypercenter#3078D83
hypocenterC83#10471
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
170. C83#10471semidirect

Maximal subgroups

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

subgroupindex
#249074782
#249062392
#249061202
#249060272
#249057872
#249053232
C83×#30782
#1047183