Finite Group Databasebeta

← #35308032

Subgroups — #35308032

Order 21248.

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

normal subgroups only

This group has 408 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
1C2×#19127978C211yesmaximal, characteristic
2#24907452C211yesmaximal, characteristic
3#24906238C211yesmaximal, characteristic, complemented
4#24906144C211yesmaximal, characteristic, complemented
5#24905853C211yesmaximal, characteristic
6#24905358C211yesmaximal, characteristic, complemented
7C83×#1549C211yesmaximal, characteristic, complemented
8C2×C4×D332C2211yescharacteristic
9#19127978—22no—
10#19127978—22no—
11#19127754C2211yescharacteristic
12C2×#14581725—21no—
13#19127725C2211yescharacteristic
14C2×C4⋊(C83⋊C8)C2211yescharacteristic
15C2×#14581712—21no—
16C2×#14581708—21no—
17#19127469—21no—
18#19127454—22no—
19#19127450—22no—
20#19127428—22no—
21C2×C4⋊Q8×C83C2211yescharacteristic
22#431×C83C2211yescharacteristic
23#423×C83C2211yescharacteristic
24#332×C83—22no—
25C22×D332D411yescharacteristic
26C22×C4×D83—21no—
27C2×#13050574—21no—
28C22×C83⋊C8—21no—
29C4×D332—21no—
30C4×D332—21no—
31C4×D332—21no—
32C4×D332—21no—
33C2×C4⋊Dic83D411yescharacteristic
34C2×(C2×D83)⋊C4—21no—
35#14581725—24no—
36C4⋊(C83⋊C8)—21no—
37C4⋊(C83⋊C8)—21no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)D412yes—
41#14581712—24no—
42#14581708—42no—
43C22⋊(C83⋊C8)—21no—
44C4×C83⋊C8—21no—
45C22×Q8×C83—21no—
46C2×M16×C83—21no—
47C4⋊Q8×C83—21no—
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83D412yes—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C42×C83C2311yescharacteristic
54C4⋊C8×C83—21no—
55C4⋊C8×C83—21no—
56C4⋊C8×C83D412yes—
57C22⋊C8×C83—21no—
58C4×C8×C83—21no—
59C23×D83—21no—
60C2×D332—21no—
61C2×D332—21no—
62C2×D332—21no—
63C2×D332—21no—
64C2×D332—21no—
65C2×D332Q1612yes—
66C22×Dic83—21no—
67C2×C4×D83—41no—
68C2×C4×D83—41no—
69C2×C4×D83—82no—
70#13050574—41no—
71#13050574—41no—
72C2×C83⋊C8—22no—
73C2×C83⋊C8—41no—
74C2×C83⋊C8—41no—
75C2×C83⋊C8—41no—
76C2×C83⋊C8—41no—
77C4⋊Dic83—21no—
78C4⋊Dic83—21no—
79(C2×D83)⋊C4—41no—
80(C2×D83)⋊C4—41no—
81C2×Q8×C83—22no—
82C2×Q8×C83—41no—
83C2×Q8×C83—82no—
84C22×C4×C83—41no—
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83#4111yescharacteristic
87C22×C4×C83#4111yescharacteristic
88M16×C83—41no—
89M16×C83—41no—
90C2×C8×C83—22no—
91C2×C8×C83—41no—
92C4⋊C4×C83—22no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—41no—
96C42×C83C2×D411yescharacteristic
97C42×C83C2×D411yescharacteristic
98C42×C83C2×D411yescharacteristic
99C42×C83#4111yescharacteristic
100C23×C83#12311yescharacteristic
101C22×D83—22no—
102C22×D83—41no—
103C22×D83—41no—
104C22×D83—81no—
105C22×D83—81no—
106Q8×C83—82no—
107Q8×C83—82no—
108C2×C4×C83—22no—
109C2×C4×C83—41no—
110C2×C4×C83—42no—
111C2×C4×C83—82no—
112C2×C4×C83#12011yescharacteristic
113C2×C4×C83#12111yescharacteristic
114C2×C4×C83#12111yescharacteristic
115C2×C4×C83#12111yescharacteristic
116C2×C4×C83#12311yescharacteristic
117C2×C4×C83#12411yescharacteristic
118C2×C4×C83C2×Q1611yescharacteristic
119C2×C4×C83#13511yescharacteristic
120C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C83#13711yescharacteristic
122C2×Dic83—41no—
123C2×Dic83—81no—
124D332—22no—
125D332—22no—
126D332—41no—
127D332—41no—
128C4×D83—82no—
129C4×D83—82no—
130C8×C83—42no—
131C8×C83—81no—
132C83⋊C8—42no—
133C83⋊C8—81no—
134C83⋊C8—81no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83#39411yescharacteristic
138C22×C83#48411yescharacteristic
139C22×C83#48511yescharacteristic
140C2×D83—42no—
141C2×D83—82no—
142C2×D83—81no—
143C2×D83—81no—
144C2×D83—82no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—82no—
149C4×C83#44911yescharacteristic
150C4×C83#45111yescharacteristic
151C4×C83#46711yescharacteristic
152C4×C83#46811yescharacteristic
153C4×C83#47212yes—
154Dic83—81no—
155#10449—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83#203611yescharacteristic
159C2×C83#204511yescharacteristic
160C2×C83#204911yescharacteristic
161D83—82no—
162C2×#474—831no—
163#1800—831no—
164#1571—831no—
165#1549D8311yescharacteristic, complemented
166#1548—831no—
167#1478—831no—
168#1301—831no—
169C83#1044911yescharacteristic, complemented, Sylow
170C2×C4⋊Q8C2×D8311yescharacteristic
171C2×C4×D4—831no—
172#474—1662no—
173#474—1662no—
174#431—831no—
175#431C2×D8311yescharacteristic
176C2×#107—1661no—
177#423—831no—
178#423C2×D8311yescharacteristic
179C2×C4⋊C8—831no—
180C2×#102—1661no—
181C2×#100—1661no—
182#343—1661no—
183#335—1662no—
184#333—1662no—
185#332—22no—
186D4⋊C8—1662no—
187C22×Q8—21no—
188C22×D4—831no—
189C23×C4—1661no—
190C2×M16—21no—
191C2×M16—1661no—
192C22×C8—1661no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8D33212yes—
197C4×D4—1661no—
198C4×D4—1661no—
199C4×D4—1661no—
200C4×D4—1661no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—831no—
204C2×C22⋊C4—1661no—
205C2×C42C22×D8311yescharacteristic
206#107—1664no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—832no—
210C4⋊C8—1661no—
211C4⋊C8—1661no—
212C4⋊C8—1661no—
213C4⋊C8—1661no—
214C4⋊C8C22⋊D8312yes—
215#102—1664no—
216#100—3322no—
217C22⋊C8—21no—
218C22⋊C8—1661no—
219C4×C8—21no—
220C4×C8—1661no—
221C24—1661no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—832no—
226C2×D4—1661no—
227C2×D4—1661no—
228C2×D4—1661no—
229C2×D4—1661no—
230C2×D4—1661no—
231C22×C4—41no—
232C22×C4—3321no—
233C22×C4—3321no—
234C22×C4—6642no—
235C22×C4—1661no—
236C22×C4D4×D8311yescharacteristic
237C22×C4#1305058911yescharacteristic
238C22×C4#1305059011yescharacteristic
239M16—41no—
240M16—41no—
241M16—3321no—
242M16—3321no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—3321no—
246C2×C8—3321no—
247C2×C8—3321no—
248C2×C8—3321no—
249C2×C8—1662no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1661no—
255C4⋊C4—1661no—
256C22⋊C4—3321no—
257C22⋊C4—3321no—
258C42D4×D8311yescharacteristic
259C42C2×C22⋊D8311yescharacteristic
260C42C2×D33211yescharacteristic
261C42#1305058911yescharacteristic
262C23—3321no—
263C23—3321no—
264C23—6641no—
265C23—6641no—
266C23—1662no—
267C23#1458177411yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—3321no—
271D4—3321no—
272D4—1662no—
273D4—1662no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—3321no—
279C2×C4—6642no—
280C2×C4—6642no—
281C2×C4—6641no—
282C2×C4#1458176211yescharacteristic
283C2×C4#1458176311yescharacteristic
284C2×C4#1458176411yescharacteristic
285C2×C4#1458176611yescharacteristic
286C2×C4#1458177611yescharacteristic
287C2×C4#1458177811yescharacteristic
288C2×C4Q16×D8311yescharacteristic
289C2×C4#1458181511yescharacteristic
290C2×C4#1458181911yescharacteristic
291C2×C4#1458182111yescharacteristic
292C8—42no—
293C8—81no—
294C8—3322no—
295C8—6641no—
296C8—6641no—
297C22—21no—
298C22—21no—
299C22—3322no—
300C22—6642no—
301C22—6642no—
302C22—6641no—
303C22—6641no—
304C22#1912761711yescharacteristic
305C22#1912803511yescharacteristic
306C22#1912804111yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—6641no—
312C4#1912784911yescharacteristic
313C4#1912785811yescharacteristic
314C4#1912793911yescharacteristic
315C4#1912794411yescharacteristic
316C4#1912796912yes—
317C2—21no—
318C2—21no—
319C2—6642no—
320C2#2490866511yescharacteristic
321C2#2490871811yescharacteristic
322C2#2490874211yescharacteristic
subgroupG/H
centerC22#19127617
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1549C2
socleC22×C83#394
prime radicalC22×D332D4
hypercenter#1549D83
hypocenterC83#10449
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. #24906238C2semidirect
4. #24906144C2semidirect
6. #24905358C2semidirect
7. C83×#1549C2semidirect
165. #1549D83semidirect
169. C83#10449semidirect

Maximal subgroups

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

subgroupindex
C2×#191279782
#249074522
#249062382
#249061442
#249058532
#249053582
C83×#15492
#1044983