Finite Group Databasebeta

← #35292993

Subgroups — #35292993

Order 21248.

570 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 111 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#24915764C211yesmaximal, characteristic
2#24907443C211yesmaximal, characteristic, complemented
3#24905288C212yesmaximal, complemented
4#24905283C212yesmaximal, complemented
5C83×#1728C211yesmaximal, characteristic, complemented
6C2×#14581780C411yescharacteristic
7C2×#14581771C411yescharacteristic
8C2×#14581766C411yescharacteristic
9C2×C4⋊(C83⋊C8)—21no—
10C2×C22⋊(C83⋊C8)—21no—
11#19127607C2211yescharacteristic
12#19127603C2211yescharacteristic
13#19127592—21no—
14#19127592—21no—
15#19127580—21no—
16#19127580—21no—
17#19127567C411yescharacteristic
18#19127456C2212yes—
19C22×M16×C83—21no—
20C22×C42×C83C2211yescharacteristic
21#336×C83—21no—
22#336×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C22×D332C2×C411yescharacteristic
25C22×C22⋊D83—21no—
26C22×C83⋊C8—22no—
27#14581780—42no—
28#14581771—42no—
29#14581766—42no—
30C2×C4⋊Dic83C2×C411yescharacteristic
31C2×C4⋊Dic83C2×C411yescharacteristic
32C2×Dic83⋊C4—21no—
33C2×C22⋊Dic83—21no—
34C2×(C2×D83)⋊C4—21no—
35C2×(C2×D83)⋊C4—21no—
36C2×(C2×D83)⋊C4—21no—
37C2×(C2×D83)⋊C4—21no—
38C2×(C2×D83)⋊C4—21no—
39C4⋊(C83⋊C8)—42no—
40C22⋊(C83⋊C8)—42no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C23×C4×C83C2×C411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C2×M16×C83—24no—
49C2×M16×C83—42no—
50C2×M16×C83—42no—
51C22×C8×C83—22no—
52C2×C42×C83—42no—
53C2×C42×C83C2×C411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C2×C42×C83D411yescharacteristic
56C2×C42×C83C2311yescharacteristic
57C23×D83—21no—
58C2×Dic166—22no—
59C2×Dic166—41no—
60C2×Dic166—41no—
61C2×D332—22no—
62C2×D332—41no—
63C2×D332—41no—
64C2×C22⋊D83—42no—
65C2×C22⋊D83—42no—
66C2×C22⋊D83—42no—
67C22×Dic83—21no—
68C22×Dic83—21no—
69C22×Dic83—21no—
70C2×C83⋊C8—42no—
71C2×C83⋊C8—42no—
72C2×C83⋊C8—42no—
73C4⋊Dic83—41no—
74C4⋊Dic83—41no—
75Dic83⋊C4—42no—
76C22⋊Dic83—42no—
77(C2×D83)⋊C4—42no—
78(C2×D83)⋊C4—42no—
79(C2×D83)⋊C4—42no—
80(C2×D83)⋊C4—41no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—41no—
83(C2×D83)⋊C4—41no—
84C24×C83C22⋊C411yescharacteristic
85C22×C4×C83—22no—
86C22×C4×C83—22no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—22no—
90C22×C4×C83—22no—
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83—41no—
94C22×C4×C83—41no—
95C22×C4×C83—41no—
96C22×C4×C83C22⋊C411yescharacteristic
97C22×C4×C83C22⋊C411yescharacteristic
98C22×C4×C83C22⋊C411yescharacteristic
99C22×C4×C83C22×C411yescharacteristic
100C22×C4×C83C2×D411yescharacteristic
101C22×C4×C83C2×D411yescharacteristic
102M16×C83—48no—
103C2×C8×C83—24no—
104C2×C8×C83—42no—
105C2×C8×C83—42no—
106C42×C83—22no—
107C42×C83—22no—
108C42×C83—41no—
109C42×C83—41no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83#10412yes—
117C23×C83C2×C22⋊C411yescharacteristic
118C22×D83—41no—
119C22×D83—41no—
120C22×D83—41no—
121C22×D83—41no—
122C22×D83—41no—
123C22×D83—41no—
124C22×D83—41no—
125C2×C4×C83—24no—
126C2×C4×C83—24no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—44no—
141C2×C4×C83—42no—
142C2×C4×C83—42no—
143C2×C4×C83#9912yes—
144C2×C4×C83#10012yes—
145C2×C4×C83#13112yes—
146C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C83#13711yescharacteristic
149C2×C4×C83#13711yescharacteristic
150C22⋊D83—84no—
151C2×Dic83—41no—
152C2×Dic83—41no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—41no—
159C2×Dic83—41no—
160D332—82no—
161Dic166—82no—
162C8×C83—44no—
163C83⋊C8—82no—
164C22×C83—22no—
165C22×C83—22no—
166C22×C83—22no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83—41no—
171C22×C83#40811yescharacteristic
172C22×C83C2×#9911yescharacteristic
173C22×C83C2×#10011yescharacteristic
174C22×C83#41711yescharacteristic
175C22×C83#41711yescharacteristic
176C22×C83C2×#10411yescharacteristic
177C22×C83#42111yescharacteristic
178C2×D83—81no—
179C2×D83—81no—
180C2×D83—81no—
181C2×D83—81no—
182C2×D83—81no—
183C2×D83—81no—
184C2×D83—81no—
185C4×C83—22no—
186C4×C83—22no—
187C4×C83—21no—
188C4×C83—22no—
189C4×C83—21no—
190C4×C83—21no—
191C4×C83—21no—
192C4×C83—41no—
193Dic83—81no—
194Dic83—81no—
195Dic83—81no—
196#7710—831nomaximal, Sylow
197C2×C83—42no—
198C2×C83#147111yescharacteristic
199C2×C83#148311yescharacteristic
200C2×C83#149511yescharacteristic
201C2×C83#150411yescharacteristic
202C2×C83#151311yescharacteristic
203C2×C83#152711yescharacteristic
204C2×C83#156011yescharacteristic
205D83—161no—
206#2850—831no—
207#1797—831no—
208#1728D8311yescharacteristic, complemented
209#1272—832no—
210#1270—832no—
211C83#771011yescharacteristic, complemented, Sylow
212C22×M16—21no—
213C2×#124—831no—
214C2×C22⋊Q8—831no—
215C2×#121—831no—
216C22×C42C2×D8311yescharacteristic
217C2×C4⋊C8—1661no—
218C2×C22⋊C8—1661no—
219#391—831no—
220#390—831no—
221#388—1661no—
222#388—1661no—
223#386—1661no—
224#386—1661no—
225#382—831no—
226#336—21no—
227#336—832no—
228#336C2×D8312yes—
229C22×Q8—831no—
230C22×D4—831no—
231C22×D4—1661no—
232C23×C4C4×D8311yescharacteristic
233C23×C4D33211yescharacteristic
234C23×C4C22⋊D8311yescharacteristic
235C2×M16—24no—
236C2×M16—42no—
237C2×M16—42no—
238C22×C8—22no—
239C22×C8—1662no—
240#124—3322no—
241C22⋊Q8—3322no—
242#121—3322no—
243C2×C4⋊C4—831no—
244C2×C4⋊C4—831no—
245C2×C4⋊C4—1661no—
246C2×C22⋊C4—1661no—
247C2×C22⋊C4—1661no—
248C2×C22⋊C4—1661no—
249C2×C22⋊C4—1661no—
250C2×C22⋊C4—1661no—
251C2×C22⋊C4—1661no—
252C2×C42—42no—
253C2×C42C4×D8311yescharacteristic
254C2×C42D33211yescharacteristic
255C2×C42C22⋊D8311yescharacteristic
256C2×C42C22×D8311yescharacteristic
257C4⋊C8—3322no—
258C22⋊C8—3322no—
259#95—1661no—
260#95—1661no—
261#95—1661no—
262#95—1661no—
263C24—1661no—
264C24(C2×D83)⋊C411yescharacteristic
265C2×Q8—3321no—
266C2×Q8—3321no—
267C2×Q8—1662no—
268C2×D4—3321no—
269C2×D4—3321no—
270C2×D4—3322no—
271C2×D4—3322no—
272C2×D4—3322no—
273C2×D4—1662no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—1661no—
286C22×C4—1661no—
287C22×C4—1661no—
288C22×C4(C2×D83)⋊C411yescharacteristic
289C22×C4(C2×D83)⋊C411yescharacteristic
290C22×C4(C2×D83)⋊C411yescharacteristic
291C22×C4C2×C4×D8311yescharacteristic
292C22×C4C2×C22⋊D8311yescharacteristic
293C22×C4C2×D33211yescharacteristic
294M16—48no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—3322no—
299C2×C8—3322no—
300C2×C8—3322no—
301C4⋊C4—3321no—
302C4⋊C4—3321no—
303C4⋊C4—3322no—
304C22⋊C4—3322no—
305C22⋊C4—3322no—
306C22⋊C4—3321no—
307C22⋊C4—3321no—
308C22⋊C4—3321no—
309C22⋊C4—3321no—
310C22⋊C4—3322no—
311C22⋊C4—3322no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—3321no—
323C23—3321no—
324C23—3321no—
325C23—3321no—
326C23—3321no—
327C23—3321no—
328C23—3321no—
329C23#1458172012yes—
330C23C2×(C2×D83)⋊C411yescharacteristic
331Q8—6642no—
332D4—6642no—
333D4—6644no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3321no—
353C2×C4—3321no—
354C2×C4—3321no—
355C2×C4—3321no—
356C2×C4—3321no—
357C2×C4—3321no—
358C2×C4—3321no—
359C2×C4—3321no—
360C2×C4—3321no—
361C2×C4#1458170612yes—
362C2×C4#1458170812yes—
363C2×C4#1458180012yes—
364C2×C4#1458181811yescharacteristic
365C2×C4#1458182011yescharacteristic
366C2×C4#1458182311yescharacteristic
367C2×C4#1458182511yescharacteristic
368C8—44no—
369C8—6642no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—6641no—
378C22—6641no—
379C22—6641no—
380C22—6641no—
381C22—6641no—
382C22—6641no—
383C22—6641no—
384C22#1912766911yescharacteristic
385C22C2×#1458170611yescharacteristic
386C22C2×#1458170811yescharacteristic
387C22#1912770011yescharacteristic
388C22#1912770311yescharacteristic
389C22C2×#1458172011yescharacteristic
390C22#1912771711yescharacteristic
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—41no—
399C4—6641no—
400C4—6641no—
401C4—6641no—
402C2—42no—
403C2—13281no—
404C2#2490582411yescharacteristic
405C2#2490587711yescharacteristic
406C2#2490592611yescharacteristic
407C2#2490596511yescharacteristic
408C2#2490600711yescharacteristic
409C2#2490606011yescharacteristic
410C2#2490619611yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1728C2
socleC23×C83C2×C22⋊C4
prime radicalC2×#14581766C4
cosocleC2×C42×C83C23
hypercenter#1728D83
hypocenterC83#7710
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
2. #24907443C2semidirect
3. #24905288C2semidirect
4. #24905283C2semidirect
5. C83×#1728C2semidirect
208. #1728D83semidirect
211. C83#7710semidirect

Maximal subgroups

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

subgroupindex
#249157642
#249074432
#249052882
#249052832
C83×#17282
#771083