Finite Group Databasebeta

← #35321773

Subgroups — #35321773

Order 21248.

558 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581731C211yesmaximal, characteristic, complemented
2C2×#19127934C211yesmaximal, characteristic
3C2×#19127502C211yesmaximal, characteristic, complemented
4#24907489C211yesmaximal, characteristic, complemented
5#24905555C211yesmaximal, characteristic, complemented
6#24905554C211yesmaximal, characteristic, complemented
7C2×#357×C83C211yesmaximal, characteristic, complemented
8C22×C8×D83C2211yescharacteristic
9C2×#14581763—21no—
10C2×#14581731—22no—
11C2×#14581731—22no—
12C2×#14581731—22no—
13C22×C83⋊C16C2211yescharacteristic
14#19127934—22no—
15#19127934C414yescomplemented
16C2×#14581726C2211yescharacteristic
17C2×#14581726C2211yescharacteristic
18C2×#14581713—21no—
19#19127558—21no—
20#19127502—22no—
21#19127463—21no—
22C22×D8×C83C2211yescharacteristic
23C22×C16×C83C2211yescharacteristic
24C2×#107×C83C2211yescharacteristic
25#357×C83—22no—
26C22×C22⋊D83—41no—
27C22×C4×D83D411yescharacteristic
28C22×C83⋊C8D411yescharacteristic
29C2×C8×D83—21no—
30C2×C8×D83—21no—
31C2×C8×D83—21no—
32C2×C8×D83—21no—
33C2×C8×D83C2×C412yes—
34C2×C8×D83D412yes—
35#14581763—24no—
36#14581763—42no—
37C2×C4⋊Dic83—21no—
38C2×C4⋊Dic83—21no—
39C2×C22⋊Dic83—41no—
40C2×(C2×D83)⋊C4—41no—
41#14581731—28no—
42C2×C83⋊C16—21no—
43C2×C83⋊C16—21no—
44C2×C83⋊C16D412yescomplemented
45#14581726—21no—
46#14581726—21no—
47#14581726—21no—
48#14581726C2×C412yes—
49#14581713—24no—
50C83⋊#95—41no—
51C22×D4×C83—21no—
52C2×D8×C83—21no—
53C2×D8×C83—21no—
54C2×D8×C83—21no—
55C2×D8×C83—21no—
56C2×D8×C83—21no—
57C2×D8×C83C2×C412yes—
58C22×C8×C83C2311yescharacteristic
59C2×C4⋊C4×C83—21no—
60C2×C16×C83—21no—
61C2×C16×C83—21no—
62C2×C16×C83D412yescomplemented
63#107×C83—21no—
64#107×C83—21no—
65C23×D83D811yescharacteristic
66C2×C22⋊D83—44no—
67C2×C22⋊D83—82no—
68C2×C22⋊D83—82no—
69C22×Dic83—41no—
70C22×Dic83—41no—
71C22×Dic83D811yescharacteristic
72C2×C4×D83—21no—
73C2×C4×D83—21no—
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76C2×C4×D83C22⋊C412yes—
77C2×C4×D83SD1612yes—
78C2×C83⋊C8—21no—
79C2×C83⋊C8—21no—
80C2×C83⋊C8C22⋊C412yes—
81C8×D83—22no—
82C8×D83—22no—
83C8×D83—22no—
84C8×D83—22no—
85C4⋊Dic83—22no—
86C4⋊Dic83—41no—
87C4⋊Dic83—41no—
88C4⋊Dic83—41no—
89C22⋊Dic83—44no—
90(C2×D83)⋊C4—82no—
91C83⋊C16—22no—
92C24×C83—41no—
93C2×D4×C83—22no—
94C2×D4×C83—41no—
95C2×D4×C83—41no—
96C2×D4×C83—41no—
97C2×D4×C83—41no—
98C2×D4×C83—41no—
99C22×C4×C83—41no—
100C22×C4×C83C2×D411yescharacteristic
101D8×C83—22no—
102D8×C83—22no—
103D8×C83—41no—
104D8×C83—41no—
105C2×C8×C83C22×C411yescharacteristic
106C2×C8×C83C2×D411yescharacteristic
107C2×C8×C83C2×D411yescharacteristic
108C2×C8×C83C2×D411yescharacteristic
109C2×C8×C83#4111yescharacteristic
110C2×C8×C83#4111yescharacteristic
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C16×C83—22no—
114C23×C83—42no—
115C23×C83—81no—
116C23×C83—81no—
117C23×C83—81no—
118C23×C83—81no—
119C23×C83—81no—
120C23×C83—81no—
121C23×C83C2×D811yescharacteristic
122C22×D83—21no—
123C22×D83—21no—
124C22×D83—21no—
125C22×D83—21no—
126C22×D83—21no—
127C22×D83—21no—
128C22×D83#10212yes—
129D4×C83—42no—
130D4×C83—42no—
131D4×C83—81no—
132D4×C83—81no—
133C2×C4×C83—81no—
134C2×C4×C83—81no—
135C2×C4×C83—81no—
136C2×C4×C83C2×C22⋊C411yescharacteristic
137C2×C4×C83#12111yescharacteristic
138C2×C4×C83#12111yescharacteristic
139C2×C4×C83C2×SD1611yescharacteristic
140C2×C4×C83#13511yescharacteristic
141C2×C4×C83#13511yescharacteristic
142C22⋊D83—88no—
143C2×Dic83—21no—
144C2×Dic83—21no—
145C2×Dic83—42no—
146C2×Dic83—81no—
147C2×Dic83—81no—
148C2×Dic83—81no—
149C2×Dic83—81no—
150C2×Dic83—81no—
151C2×Dic83#10212yes—
152C4×D83—22no—
153C4×D83—22no—
154C4×D83—22no—
155C4×D83—22no—
156C8×C83C4×D411yescharacteristic
157C8×C83C4×D411yescharacteristic
158C8×C83#12011yescharacteristic
159C8×C83#12311yescharacteristic
160C83⋊C8—22no—
161C22×C83—82no—
162C22×C83—82no—
163C22×C83—82no—
164C22×C83—161no—
165C22×C83—161no—
166C22×C83—161no—
167C22×C83—161no—
168C22×C83C2×#10211yescharacteristic
169C22×C83#46611yescharacteristic
170C22×C83#46611yescharacteristic
171C22×C83#50911yescharacteristic
172C22×C83#50911yescharacteristic
173C22×C83#50911yescharacteristic
174C22×C83#51011yescharacteristic
175C2×D83—22no—
176C2×D83—22no—
177C2×D83—22no—
178C2×D83—41no—
179C2×D83—41no—
180C2×D83—41no—
181C2×D83—41no—
182C4×C83—161no—
183C4×C83#38611yescharacteristic
184C4×C83#41611yescharacteristic
185C4×C83#46511yescharacteristic
186C4×C83#46511yescharacteristic
187Dic83—22no—
188Dic83—82no—
189Dic83—161no—
190#13061—831nomaximal, Sylow
191C2×C83—162no—
192C2×C83#189311yescharacteristic
193C2×C83#213911yescharacteristic
194C2×C83#214011yescharacteristic
195C2×C83#221811yescharacteristic
196C2×C83#221811yescharacteristic
197C2×C83#222011yescharacteristic
198C2×C83#222011yescharacteristic
199D83—42no—
200C22×M32—831no—
201C2×#466—831no—
202C2×#357—831no—
203C2×#357D8311yescharacteristic, complemented
204#1808—831no—
205#1380—831no—
206#1380—831no—
207C83#1306111yescharacteristic, complemented, Sylow
208C22×D8C2×D8311yescharacteristic
209C23×C8—831no—
210C2×#121—1661no—
211C2×M32—1662no—
212C2×M32—1662no—
213C2×M32—1662no—
214C22×C16—831no—
215C22×C16C2×D8311yescharacteristic
216#466—834no—
217#466—1662no—
218C2×#107—831no—
219C2×#107—831no—
220C2×#107C2×D8311yescharacteristic
221C2×#102—1661no—
222#380—1661no—
223#357—22no—
224#357—1662no—
225#340—1661no—
226C22×D4—21no—
227C22×D4—3321no—
228C23×C4—831no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8C4×D8312yes—
235C22×C8—831no—
236C22×C8—832no—
237C22×C8—832no—
238C22×C8—1661no—
239C22×C8—1661no—
240C22×C8—1661no—
241C22×C8—1661no—
242C22×C8C22×D8311yescharacteristic
243#121—3322no—
244#121—1664no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1661no—
247C2×C4⋊C4—1661no—
248C2×C22⋊C4—3321no—
249C2×C22⋊C4—3321no—
250M32—1668no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—832no—
254C2×C16—1661no—
255C2×C16—1661no—
256C2×C16C22⋊D8312yescomplemented
257#107—21no—
258#107—21no—
259#107—832no—
260#107—1661no—
261#107—1661no—
262#107—1661no—
263#102—1664no—
264#95—3321no—
265C24—41no—
266C24—831no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—3324no—
274C2×D4—6642no—
275C2×D4—6642no—
276C22×C4—41no—
277C22×C4—3321no—
278C22×C4—3321no—
279C22×C4—831no—
280C22×C4—832no—
281C22×C4—832no—
282C22×C4—1661no—
283C22×C4—1661no—
284C22×C4—1661no—
285C22×C4—1661no—
286C22×C4D4×D8311yescharacteristic
287D8—22no—
288D8—22no—
289D8—41no—
290D8—41no—
291C2×C8—832no—
292C2×C8—1661no—
293C2×C8—1661no—
294C2×C8—1662no—
295C2×C8—1662no—
296C2×C8—1662no—
297C2×C8—1662no—
298C2×C8C2×C4×D8311yescharacteristic
299C2×C8D4×D8311yescharacteristic
300C2×C8D4×D8311yescharacteristic
301C2×C8C2×C22⋊D8311yescharacteristic
302C2×C8#1305058911yescharacteristic
303C2×C8#1305059011yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—3321no—
307C4⋊C4—3321no—
308C4⋊C4—3321no—
309C4⋊C4—1662no—
310C22⋊C4—3324no—
311C22⋊C4—6642no—
312C16—22no—
313C16—1662no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—832no—
322C23—1661no—
323C23—1661no—
324C23—1661no—
325C23—1661no—
326C23—1661no—
327C23—1661no—
328C23D8×D8311yescharacteristic
329D4—42no—
330D4—42no—
331D4—81no—
332D4—81no—
333D4—6648no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—3322no—
338C2×C4—832no—
339C2×C4—6641no—
340C2×C4—6641no—
341C2×C4—6641no—
342C2×C4—6641no—
343C2×C4—6641no—
344C2×C4—1661no—
345C2×C4—1661no—
346C2×C4—1662no—
347C2×C4—1662no—
348C2×C4—1662no—
349C2×C4—1662no—
350C2×C4C22⋊C4×D8311yescharacteristic
351C2×C4#1458176311yescharacteristic
352C2×C4#1458176511yescharacteristic
353C2×C4SD16×D8311yescharacteristic
354C2×C4#1458181111yescharacteristic
355C2×C4#1458181311yescharacteristic
356C8—1662no—
357C8(C22⋊D83)⋊C411yescharacteristic
358C8C4×C22⋊D8311yescharacteristic
359C8#1458176011yescharacteristic
360C8#1458177411yescharacteristic
361C22—82no—
362C22—82no—
363C22—82no—
364C22—161no—
365C22—161no—
366C22—161no—
367C22—161no—
368C22—3321no—
369C22—3321no—
370C22—3321no—
371C22—3321no—
372C22—1662no—
373C22—1662no—
374C22—1662no—
375C22#102×D8311yescharacteristic
376C22#1912793411yescharacteristic
377C22#1912793611yescharacteristic
378C22#1912813511yescharacteristic
379C22#1912813511yescharacteristic
380C22#1912813811yescharacteristic
381C22#1912814011yescharacteristic
382C4—161no—
383C4—13281no—
384C4—6642no—
385C4—1662no—
386C4#1912757711yescharacteristic
387C4#1912769411yescharacteristic
388C4#1912792711yescharacteristic
389C4#1912792911yescharacteristic
390C2—162no—
391C2—3322no—
392C2#2490788811yescharacteristic
393C2#2490917211yescharacteristic
394C2#2490917711yescharacteristic
395C2#2490956311yescharacteristic
396C2#2490956511yescharacteristic
397C2#2490957311yescharacteristic
398C2#2490957511yescharacteristic
subgroupG/H
centerC23D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC2×#357×C83C2
socleC23×C83C2×D8
prime radicalC2×#19127934C2
cosocleC22×C8×C83C23
hypercenterC2×#357D83
hypocenterC83#13061
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. C22×#14581731C2semidirect
3. C2×#19127502C2semidirect
4. #24907489C2semidirect
5. #24905555C2semidirect
6. #24905554C2semidirect
7. C2×#357×C83C2semidirect
15. #19127934C4semidirect
44. C2×C83⋊C16D4semidirect
62. C2×C16×C83D4semidirect
203. C2×#357D83semidirect
207. C83#13061semidirect
256. C2×C16C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C22×#145817312
C2×#191279342
C2×#191275022
#249074892
#249055552
#249055542
C2×#357×C832
#1306183