Finite Group Databasebeta

← #35323567

Subgroups — #35323567

Order 21248.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127937C211yesmaximal, characteristic
2C2×#19127529C211yesmaximal, characteristic, complemented
3C2×#19127503C211yesmaximal, characteristic, complemented
4C2×#19127503C211yesmaximal, characteristic, complemented
5#24907492C211yesmaximal, characteristic, complemented
6#24905555C211yesmaximal, characteristic, complemented
7C2×C22⋊C16×C83C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C2×#14581766—21no—
10#19127937—22no—
11#19127937—22no—
12C2×#14581726C2211yescharacteristic
13C2×#14581726C2211yescharacteristic
14C2×#14581726C2211yescharacteristic
15C2×#14581714—21no—
16#19127561—21no—
17#19127529—22no—
18#19127503—22no—
19#19127503—22no—
20#19127464—21no—
21C23×C8×C83C2211yescharacteristic
22C22×C16×C83C2211yescharacteristic
23C22×C16×C83C2211yescharacteristic
24C22⋊C16×C83—22no—
25C22×D332—21no—
26C22×C22⋊D83—41no—
27C2×D664—21no—
28C2×D664—21no—
29C2×D664—21no—
30C2×D664—21no—
31C2×D664—21no—
32C2×D664D412yes—
33#14581766—24no—
34#14581766—42no—
35C2×C4⋊Dic83—21no—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83—21no—
38C2×C22⋊Dic83—41no—
39C2×(C2×D83)⋊C4—41no—
40#14581726—21no—
41#14581726—21no—
42#14581726—21no—
43#14581726—21no—
44#14581726—21no—
45#14581726D412yes—
46#14581714—24no—
47C83⋊#95—41no—
48C23×C4×C83D411yescharacteristic
49C22×C8×C83—21no—
50C22×C8×C83—21no—
51C22×C8×C83—21no—
52C22×C8×C83—21no—
53C22×C8×C83—21no—
54C22×C8×C83—21no—
55C22×C8×C83D411yescharacteristic
56C22×C8×C83C2311yescharacteristic
57C2×C16×C83—21no—
58C2×C16×C83—21no—
59C2×C16×C83—21no—
60C2×C16×C83—21no—
61C2×C16×C83—21no—
62C2×C16×C83D412yescomplemented
63C23×D83—41no—
64C2×D332—22no—
65C2×D332—41no—
66C2×D332—41no—
67C2×D332—41no—
68C2×D332—41no—
69C2×D332—41no—
70C2×C22⋊D83—44no—
71C2×C22⋊D83—82no—
72C2×C22⋊D83—82no—
73C22×Dic83—41no—
74C22×Dic83—41no—
75C22×Dic83—41no—
76D664—22no—
77D664—22no—
78D664—41no—
79D664—41no—
80C4⋊Dic83—22no—
81C4⋊Dic83—41no—
82C4⋊Dic83—41no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C4⋊Dic83—41no—
86C22⋊Dic83—82no—
87(C2×D83)⋊C4—44no—
88C24×C83D811yescharacteristic
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83D811yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C2×C8×C83—21no—
98C2×C8×C83—21no—
99C2×C8×C83—21no—
100C2×C8×C83—41no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83C2×D411yescharacteristic
105C2×C8×C83C2×D411yescharacteristic
106C2×C8×C83C2×D411yescharacteristic
107C2×C8×C83#4111yescharacteristic
108C2×C8×C83#4111yescharacteristic
109C2×C8×C83#4111yescharacteristic
110C16×C83—22no—
111C16×C83—41no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83SD3212yes—
119C23×C83C2×D811yescharacteristic
120C22×D83—42no—
121C22×D83—81no—
122C22×D83—81no—
123C22×D83—81no—
124C22×D83—81no—
125C22×D83—81no—
126C22×D83—81no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83#12011yescharacteristic
135C2×C4×C83#12111yescharacteristic
136C2×C4×C83#12311yescharacteristic
137C2×C4×C83#13511yescharacteristic
138C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C83#13711yescharacteristic
140C22⋊D83—88no—
141C2×Dic83—42no—
142C2×Dic83—81no—
143C2×Dic83—81no—
144C2×Dic83—81no—
145C2×Dic83—81no—
146C2×Dic83—81no—
147C2×Dic83—81no—
148C2×Dic83—81no—
149C2×Dic83—81no—
150D332—42no—
151D332—42no—
152D332—81no—
153D332—81no—
154C8×C83—41no—
155C8×C83#12111yescharacteristic
156C8×C83#12111yescharacteristic
157C8×C83#12311yescharacteristic
158C8×C83#12411yescharacteristic
159C22×C83—22no—
160C22×C83—22no—
161C22×C83—22no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83#44711yescharacteristic
167C22×C83#46611yescharacteristic
168C22×C83#48011yescharacteristic
169C22×C83C2×SD3211yescharacteristic
170C22×C83#50811yescharacteristic
171C22×C83#50911yescharacteristic
172C22×C83#50911yescharacteristic
173C2×D83—82no—
174C2×D83—82no—
175C2×D83—82no—
176C2×D83—161no—
177C2×D83—161no—
178C2×D83—161no—
179C2×D83—161no—
180C4×C83—41no—
181C4×C83#39411yescharacteristic
182C4×C83#45211yescharacteristic
183C4×C83#46811yescharacteristic
184C4×C83#48211yescharacteristic
185Dic83—82no—
186Dic83—161no—
187Dic83—161no—
188#13381—831nomaximal, Sylow
189C2×C83—42no—
190C2×C83#203311yescharacteristic
191C2×C83#218611yescharacteristic
192C2×C83#218711yescharacteristic
193C2×C83#221711yescharacteristic
194C2×C83#221811yescharacteristic
195C2×C83#223311yescharacteristic
196C2×C83#223411yescharacteristic
197D83—162no—
198C2×#466—831no—
199C2×#367—831no—
200C2×#357—831no—
201C2×#357—831no—
202C2×C22⋊C16D8311yescharacteristic, complemented
203#1808—831no—
204#1380—831no—
205C83#1338111yescharacteristic, complemented, Sylow
206C22×D8—831no—
207C23×C8C2×D8311yescharacteristic
208C2×#121—1661no—
209C22×C16C2×D8311yescharacteristic
210C22×C16C2×D8311yescharacteristic
211#466—1662no—
212#466—1662no—
213C2×#107—831no—
214C2×#107—831no—
215C2×#107—831no—
216C2×#102—1661no—
217#380—1661no—
218#367—1662no—
219#357—1662no—
220#357—1662no—
221C22⋊C16—22no—
222#340—1661no—
223C22×D4—3321no—
224C22×D4—1661no—
225C23×C4D33211yescharacteristic
226C2×D8—832no—
227C2×D8—1661no—
228C2×D8—1661no—
229C2×D8—1661no—
230C2×D8—1661no—
231C2×D8—1661no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8D33211yescharacteristic
239C22×C8C22×D8311yescharacteristic
240#121—3322no—
241#121—1664no—
242C2×C4⋊C4—1661no—
243C2×C4⋊C4—1661no—
244C2×C4⋊C4—1661no—
245C2×C22⋊C4—3321no—
246C2×C22⋊C4—3321no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16C22⋊D8312yescomplemented
253#107—832no—
254#107—1661no—
255#107—1661no—
256#107—1661no—
257#107—1661no—
258#107—1661no—
259#102—1664no—
260#95—3321no—
261C24—3321no—
262C24D66411yescharacteristic
263C2×D4—3324no—
264C2×D4—3321no—
265C2×D4—3321no—
266C2×D4—3321no—
267C2×D4—3321no—
268C2×D4—3321no—
269C2×D4—6642no—
270C2×D4—6642no—
271C2×D4—1662no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—3321no—
279C22×C4—3321no—
280C22×C4—3321no—
281C22×C4D66411yescharacteristic
282C22×C4C2×D33211yescharacteristic
283D8—3321no—
284D8—3321no—
285D8—1662no—
286D8—1662no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D8311yescharacteristic
295C2×C8D4×D8311yescharacteristic
296C2×C8C2×C22⋊D8311yescharacteristic
297C2×C8#1305058911yescharacteristic
298C2×C8#1305058911yescharacteristic
299C2×C8#1305059011yescharacteristic
300C4⋊C4—3321no—
301C4⋊C4—3321no—
302C4⋊C4—3321no—
303C4⋊C4—3321no—
304C4⋊C4—3321no—
305C4⋊C4—1662no—
306C22⋊C4—3324no—
307C22⋊C4—6642no—
308C16—22no—
309C16—41no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—3322no—
317C23—6641no—
318C23—6641no—
319C23—6641no—
320C23—6641no—
321C23—6641no—
322C23—6641no—
323C23#1458173712yes—
324C23C2×D66411yescharacteristic
325D4—3322no—
326D4—3322no—
327D4—6648no—
328D4—6641no—
329D4—6641no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—3322no—
338C2×C4—6641no—
339C2×C4—6641no—
340C2×C4—6641no—
341C2×C4—6641no—
342C2×C4—6641no—
343C2×C4—6641no—
344C2×C4—6641no—
345C2×C4—6641no—
346C2×C4#1458176211yescharacteristic
347C2×C4#1458176611yescharacteristic
348C2×C4#1458177611yescharacteristic
349C2×C4#1458181411yescharacteristic
350C2×C4#1458182011yescharacteristic
351C2×C4#1458182511yescharacteristic
352C8—41no—
353C8#1458176311yescharacteristic
354C8#1458176411yescharacteristic
355C8#1458177411yescharacteristic
356C8#1458177811yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—13281no—
365C22—13281no—
366C22—13281no—
367C22—13281no—
368C22—6642no—
369C22—6642no—
370C22—6642no—
371C22#1912783711yescharacteristic
372C22#1912793711yescharacteristic
373C22#1912801511yescharacteristic
374C22C2×#1458173711yescharacteristic
375C22#1912813311yescharacteristic
376C22#1912813911yescharacteristic
377C22#1912813911yescharacteristic
378C4—41no—
379C4—13281no—
380C4—13281no—
381C4—6642no—
382C4#1912761711yescharacteristic
383C4#1912786511yescharacteristic
384C4#1912794711yescharacteristic
385C4#1912802611yescharacteristic
386C2—42no—
387C2—13282no—
388C2#2490864611yescharacteristic
389C2#2490938811yescharacteristic
390C2#2490939411yescharacteristic
391C2#2490955911yescharacteristic
392C2#2490956611yescharacteristic
393C2#2490964911yescharacteristic
394C2#2490965511yescharacteristic
subgroupG/H
centerC23C2×D664
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC2×C22⋊C16×C83C2
socleC23×C83C2×D8
prime radicalC2×#19127937C2
hypercenterC2×C22⋊C16D83
hypocenterC83#13381
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#19127529C2semidirect
3. C2×#19127503C2semidirect
4. C2×#19127503C2semidirect
5. #24907492C2semidirect
6. #24905555C2semidirect
7. C2×C22⋊C16×C83C2semidirect
62. C2×C16×C83D4semidirect
202. C2×C22⋊C16D83semidirect
205. C83#13381semidirect
252. C2×C16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191279372
C2×#191275292
C2×#191275032
#249074922
#249055552
C2×C22⋊C16×C832
#1338183