Finite Group Databasebeta

← #35302253

Subgroups — #35302253

Order 21248.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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#24912360C211yesmaximal, characteristic
2#24907455C211yesmaximal, characteristic, complemented
3#24905889C212yesmaximal, complemented
4#24905322C212yesmaximal, complemented
5C83×#2972C211yesmaximal, characteristic, complemented
6C2×#14581763—21no—
7C2×C4×D332C2211yescharacteristic
8#19127725C2212yes—
9C2×#14581709—21no—
10#19127626C412yes—
11#19127610C412yes—
12#19127580—22no—
13#19127567C2211yescharacteristic
14#19127558—21no—
15#19127470—21no—
16#19127430—24no—
17C2×C4×D4×C83C2211yescharacteristic
18C8×D4×C83—24no—
19#431×C83C2212yes—
20C2×C22⋊C8×C83—22no—
21C22×D332C2×C411yescharacteristic
22C22×C22⋊D83—22no—
23C22×C4×D83—21no—
24C2×#13050574—22no—
25#14581763—42no—
26#14581763—42no—
27C4×D332—22no—
28C4×D332—22no—
29C2×C4⋊Dic83—21no—
30C2×C4⋊Dic83C2×C411yescharacteristic
31C2×C22⋊Dic83—22no—
32C2×(C2×D83)⋊C4—22no—
33C2×(C2×D83)⋊C4—21no—
34C4⋊(C83⋊C8)—22no—
35C4⋊(C83⋊C8)—22no—
36#14581709—42no—
37C83⋊#95—22no—
38C83⋊#95—21no—
39C83⋊#95C2×C412yes—
40C22×D4×C83D411yescharacteristic
41C23×C4×C83D412yes—
42C22×C8×C83—24no—
43C4×D4×C83—22no—
44C4×D4×C83—22no—
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C22⋊C4×C83C2×C412yes—
47C2×C42×C83C2311yescharacteristic
48C4⋊C8×C83—22no—
49C22⋊C8×C83—24no—
50C22⋊C8×C83—22no—
51C4×C8×C83D414yescomplemented
52C23×D83—21no—
53C2×D332—22no—
54C2×D332—42no—
55C2×C22⋊D83—44no—
56C2×C22⋊D83—44no—
57C2×C22⋊D83—82no—
58C22×Dic83—21no—
59C22×Dic83—22no—
60C2×C4×D83—42no—
61C2×C4×D83—41no—
62C2×C4×D83—41no—
63C2×C4×D83—42no—
64#13050574—42no—
65#13050574—42no—
66C2×C83⋊C8—42no—
67C4⋊Dic83—22no—
68C4⋊Dic83—41no—
69C4⋊Dic83—41no—
70C22⋊Dic83—44no—
71(C2×D83)⋊C4—41no—
72(C2×D83)⋊C4—41no—
73(C2×D83)⋊C4—82no—
74C24×C83C22⋊C412yes—
75C2×D4×C83—22no—
76C2×D4×C83—22no—
77C2×D4×C83D812yes—
78C2×D4×C83SD1612yes—
79C22×C4×C83—22no—
80C22×C4×C83—22no—
81C22×C4×C83—82no—
82C22×C4×C83C22⋊C412yes—
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C2×C8×C83—24no—
87C2×C8×C83—84no—
88C4⋊C4×C83—22no—
89C22⋊C4×C83—24no—
90C42×C83C2×D412yes—
91C42×C83#4112yes—
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—42no—
95C23×C83—42no—
96C23×C83#10214yes—
97C23×C83C2×C22⋊C411yescharacteristic
98C22×D83—22no—
99C22×D83—41no—
100C22×D83—41no—
101C22×D83—41no—
102C22×D83—41no—
103C22×D83—41no—
104C22×D83—41no—
105D4×C83—24no—
106D4×C83—24no—
107C2×C4×C83—22no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—44no—
111C2×C4×C83—42no—
112C2×C4×C83—82no—
113C2×C4×C83C4×D412yes—
114C2×C4×C83#12011yescharacteristic
115C2×C4×C83#12112yes—
116C2×C4×C83#12311yescharacteristic
117C2×C4×C83C2×D811yescharacteristic
118C2×C4×C83C2×SD1611yescharacteristic
119C2×C4×C83#13512yes—
120C22⋊D83—88no—
121C2×Dic83—42no—
122C2×Dic83—42no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—42no—
126C2×Dic83—41no—
127D332—24no—
128D332—42no—
129C4×D83—82no—
130C4×D83—82no—
131C8×C83—44no—
132C83⋊C8—82no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—24no—
136C22×C83—44no—
137C22×C83—42no—
138C22×C83—42no—
139C22×C83—82no—
140C22×C83#38611yescharacteristic
141C22×C83C2×#10211yescharacteristic
142C22×C83#41611yescharacteristic
143C2×D83—42no—
144C2×D83—42no—
145C2×D83—42no—
146C2×D83—81no—
147C2×D83—81no—
148C2×D83—81no—
149C2×D83—81no—
150C4×C83—22no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—42no—
154C4×C83#46512yes—
155C4×C83#46612yes—
156Dic83—82no—
157Dic83—81no—
158#9485—831nomaximal, Sylow
159C2×C83—21no—
160C2×C83—21no—
161C2×C83—44no—
162C2×C83#189111yescharacteristic
163C2×C83#189311yescharacteristic
164C2×C83#191111yescharacteristic
165D83—82no—
166#2972D8311yescharacteristic, complemented
167#2535—831no—
168#1800—831no—
169#1486—832no—
170#1286—832no—
171C83#948511yescharacteristic, complemented, Sylow
172C2×#121—1661no—
173C2×C4×D4—831no—
174C2×C4×D4C2×D8311yescharacteristic
175C8×D4—24no—
176#431C2×D8312yes—
177#423—832no—
178C2×#100—1661no—
179C2×C22⋊C8—22no—
180#397—832no—
181#392—832no—
182#386—1662no—
183#382—831no—
184#380—1661no—
185#343—1661no—
186D4⋊C8—1664no—
187C22×D4—831no—
188C22×D4—1662no—
189C22×D4D33211yescharacteristic
190C23×C4—1661no—
191C23×C4C22⋊D8312yes—
192C2×M16—1662no—
193C22×C8—24no—
194#121—3322no—
195#121—3322no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—1662no—
199C4×D4—1662no—
200C2×C4⋊C4—831no—
201C2×C4⋊C4—1661no—
202C2×C4⋊C4D33211yescharacteristic
203C2×C22⋊C4—1662no—
204C2×C22⋊C4—1661no—
205C2×C22⋊C4—1662no—
206C2×C22⋊C4C4×D8312yes—
207C2×C42C22×D8311yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—1662no—
210C4⋊C8—1662no—
211#100—3322no—
212C22⋊C8—24no—
213C22⋊C8—22no—
214C4×C8C22⋊D8314yescomplemented
215#95—832no—
216#95—1662no—
217#95—1661no—
218C24—1661no—
219C24(C2×D83)⋊C412yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—3322no—
223C2×D4—3324no—
224C2×D4—3324no—
225C2×D4—6642no—
226C2×D4—1662no—
227C2×D4D66412yes—
228C2×D4#1305057912yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—3322no—
233C22×C4—3321no—
234C22×C4—3321no—
235C22×C4—3322no—
236C22×C4—1662no—
237C22×C4—1661no—
238C22×C4(C2×D83)⋊C412yes—
239C22×C4C2×C4×D8311yescharacteristic
240C22×C4C2×C22⋊D8311yescharacteristic
241C22×C4C2×D33211yescharacteristic
242M16—3322no—
243M16—3322no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—3322no—
247C4⋊C4—22no—
248C4⋊C4—3321no—
249C4⋊C4—3321no—
250C4⋊C4—1662no—
251C22⋊C4—24no—
252C22⋊C4—3324no—
253C22⋊C4—3321no—
254C22⋊C4—3321no—
255C22⋊C4—6642no—
256C42C2×C22⋊D8312yes—
257C42#1305059012yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—3321no—
263C23—3321no—
264C23—3321no—
265C23—3321no—
266C23—3321no—
267C23—3321no—
268C23—1662no—
269C23#1458171414yes—
270C23C2×(C2×D83)⋊C411yescharacteristic
271D4—24no—
272D4—24no—
273D4—3322no—
274D4—6648no—
275D4—1664no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—3322no—
283C2×C4—3322no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—3322no—
287C2×C4—3321no—
288C2×C4—6642no—
289C2×C4—6642no—
290C2×C4C4×C22⋊D8312yes—
291C2×C4#1458176111yescharacteristic
292C2×C4#1458176612yes—
293C2×C4#1458177511yescharacteristic
294C2×C4C2×D66411yescharacteristic
295C2×C4C2×#1305057911yescharacteristic
296C2×C4#1458181412yes—
297C8—44no—
298C8—6642no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—3322no—
307C22—3322no—
308C22—3322no—
309C22—6641no—
310C22—6641no—
311C22—6641no—
312C22—6641no—
313C22#1912758011yescharacteristic
314C22C2×#1458171411yescharacteristic
315C22#1912769811yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—6642no—
321C4—6641no—
322C4#1912793012yes—
323C4#1912793712yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—6642no—
328C2#2490788111yescharacteristic
329C2#2490789111yescharacteristic
330C2#2490799511yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2972C2
socleC22×C83#386
prime radical#24912360C2
cosocleC2×C42×C83C23
hypercenter#2972D83
hypocenterC83#9485
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #24907455C2semidirect
3. #24905889C2semidirect
4. #24905322C2semidirect
5. C83×#2972C2semidirect
51. C4×C8×C83D4semidirect
166. #2972D83semidirect
171. C83#9485semidirect
214. C4×C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#249123602
#249074552
#249058892
#249053222
C83×#29722
#948583