Finite Group Databasebeta

← C83×#11755 #35239579

Subgroups — C83×#11755 #35239579

Order 21248.

586 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C83×#2482C211yesmaximal, characteristic
2C2×#342×C83C211yesmaximal, characteristic
3C83×#1462C211yesmaximal, characteristic
4C83×#1352C212yesmaximal, complemented
5C83×#1344C212yesmaximal, complemented
6C2×#117×C83C411yescharacteristic, complemented
7C22×C22⋊C4×C83—21no—
8C22×C22⋊C4×C83C2211yescharacteristic
9#407×C83—21no—
10#399×C83C411yescharacteristic, complemented
11#399×C83C2211yescharacteristic
12#397×C83C411yescharacteristic, complemented
13#397×C83C2211yescharacteristic
14#388×C83—21no—
15#388×C83—21no—
16#385×C83—21no—
17#381×C83C411yescharacteristic, complemented
18#379×C83—21no—
19#342×C83—21no—
20#342×C83C2212yes—
21#323×C83—21no—
22#323×C83C2212yes—
23C25×C83D411yescharacteristic
24C23×C4×C83—41no—
25C23×C4×C83C2×C411yescharacteristic
26C23×C4×C83D411yescharacteristic
27C2×M16×C83—41no—
28#117×C83—41no—
29#117×C83—41no—
30C2×C4⋊C4×C83—21no—
31C2×C4⋊C4×C83—21no—
32C2×C4⋊C4×C83C2×C411yescharacteristic, complemented
33C2×C4⋊C4×C83C2×C411yescharacteristic, complemented
34C2×C22⋊C4×C83—24no—
35C2×C22⋊C4×C83—22no—
36C2×C22⋊C4×C83—21no—
37C2×C22⋊C4×C83—21no—
38C2×C22⋊C4×C83—21no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—41no—
43C2×C22⋊C4×C83—41no—
44C2×C22⋊C4×C83—41no—
45C2×C22⋊C4×C83—42no—
46C2×C22⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83D411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83C2311yescharacteristic
50C2×C42×C83—21no—
51C22⋊C8×C83—22no—
52C22⋊C8×C83—41no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56#95×C83C2×C411yescharacteristic, complemented
57#95×C83C2×C411yescharacteristic, complemented
58C24×C83—21no—
59C24×C83—21no—
60C24×C83—22no—
61C24×C83—21no—
62C24×C83—21no—
63C24×C83—42no—
64C24×C83—81no—
65C24×C83C22⋊C411yescharacteristic
66C24×C83C22⋊C411yescharacteristic
67C24×C83C2×D411yescharacteristic
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—42no—
77C22×C4×C83—44no—
78C22×C4×C83—42no—
79C22×C4×C83—81no—
80C22×C4×C83—82no—
81C22×C4×C83—82no—
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86M16×C83—82no—
87C2×C8×C83—42no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—81no—
91C4⋊C4×C83—81no—
92C22⋊C4×C83—24no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—41no—
97C22⋊C4×C83—42no—
98C22⋊C4×C83—44no—
99C22⋊C4×C83—41no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—42no—
102C22⋊C4×C83—41no—
103C22⋊C4×C83—42no—
104C22⋊C4×C83—41no—
105C22⋊C4×C83—81no—
106C22⋊C4×C83—81no—
107C42×C83—42no—
108C23×C83—22no—
109C23×C83—22no—
110C23×C83—22no—
111C23×C83—22no—
112C23×C83—22no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83—21no—
119C23×C83—41no—
120C23×C83—42no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—42no—
126C23×C83—81no—
127C23×C83—81no—
128C23×C83—84no—
129C23×C83—82no—
130C23×C83—82no—
131C23×C83#9911yescharacteristic
132C23×C83#9911yescharacteristic
133C23×C83#10412yes—
134C23×C83C2×C22⋊C411yescharacteristic
135C23×C83AGL(1,ℤ/8)11yescharacteristic
136C23×C83#13711yescharacteristic
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—42no—
142C2×C4×C83—42no—
143C2×C4×C83—42no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—44no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83—41no—
156C2×C4×C83—41no—
157C2×C4×C83—41no—
158C2×C4×C83—41no—
159C2×C4×C83—41no—
160C2×C4×C83—41no—
161C2×C4×C83—81no—
162C2×C4×C83—88no—
163C2×C4×C83—82no—
164C2×C4×C83—82no—
165C8×C83—82no—
166C22×C83—21no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—22no—
175C22×C83—21no—
176C22×C83—42no—
177C22×C83—42no—
178C22×C83—42no—
179C22×C83—41no—
180C22×C83—41no—
181C22×C83—44no—
182C22×C83—44no—
183C22×C83—41no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—81no—
188C22×C83—81no—
189C22×C83—82no—
190C22×C83—82no—
191C22×C83#35212yes—
192C22×C83C2×#9911yescharacteristic
193C22×C83#41711yescharacteristic
194C22×C83C2×#10411yescharacteristic
195C4×C83—41no—
196C4×C83—41no—
197C4×C83—41no—
198C4×C83—41no—
199C4×C83—81no—
200C4×C83—81no—
201C4×C83—84no—
202C4×C83—81no—
203C4×C83—81no—
204#11755C8311yesmaximal, characteristic, direct product factor, Sylow
205C2×C83—21no—
206C2×C83—21no—
207C2×C83—22no—
208C2×C83—41no—
209C2×C83—41no—
210C2×C83—42no—
211C2×C83—41no—
212C2×C83#150411yescharacteristic
213C2×C83C2×#35211yescharacteristic
214C2×C83#212711yescharacteristic
215#2482C2×C8311yescharacteristic
216C2×#342C2×C8311yescharacteristic
217#1462C2×C8311yescharacteristic
218#1352C2×C8312yescomplemented
219#1344C2×C8312yescomplemented
220C83#1175511yescharacteristic, direct product factor, Sylow
221C2×#117C4×C8311yescharacteristic, complemented
222C22×C22⋊C4—21no—
223C22×C22⋊C4C22×C8311yescharacteristic
224#407—21no—
225#399C4×C8311yescharacteristic, complemented
226#399C22×C8311yescharacteristic
227#397C4×C8311yescharacteristic, complemented
228#397C22×C8311yescharacteristic
229#388—21no—
230#388—21no—
231#385—21no—
232#381C4×C8311yescharacteristic, complemented
233#379—21no—
234#342—21no—
235#342C22×C8312yes—
236#323—21no—
237#323C22×C8312yes—
238C25D4×C8311yescharacteristic
239C23×C4—41no—
240C23×C4C2×C4×C8311yescharacteristic
241C23×C4D4×C8311yescharacteristic
242C2×M16—41no—
243#117—41no—
244#117—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C2×C4×C8311yescharacteristic, complemented
248C2×C4⋊C4C2×C4×C8311yescharacteristic, complemented
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4C2×C4×C8311yescharacteristic
262C2×C22⋊C4D4×C8311yescharacteristic
263C2×C22⋊C4D4×C8311yescharacteristic
264C2×C22⋊C4C23×C8311yescharacteristic
265C2×C42—21no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95C2×C4×C8311yescharacteristic, complemented
272#95C2×C4×C8311yescharacteristic, complemented
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24C22⋊C4×C8311yescharacteristic
281C24C22⋊C4×C8311yescharacteristic
282C24C2×D4×C8311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C22⋊C4×C8311yescharacteristic
298C22×C4C22⋊C4×C8311yescharacteristic
299C22×C4C22×C4×C8311yescharacteristic
300C22×C4C2×D4×C8311yescharacteristic
301M16—82no—
302C2×C8—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C42—42no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#99×C8311yescharacteristic
347C23#99×C8311yescharacteristic
348C23#104×C8312yes—
349C23C2×C22⋊C4×C8311yescharacteristic
350C23AGL(1,ℤ/8)×C8311yescharacteristic
351C23#137×C8311yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—88no—
378C2×C4—82no—
379C2×C4—82no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22#352×C8312yes—
407C22C2×#99×C8311yescharacteristic
408C22#417×C8311yescharacteristic
409C22C2×#104×C8311yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—81no—
415C4—81no—
416C4—84no—
417C4—81no—
418C4—81no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2C83×#150411yescharacteristic
427C2C2×#352×C8311yescharacteristic
428C2C83×#212711yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C22⋊C4C23×C83
prime radicalC25×C83D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C24×C83 ◁ C22×C22⋊C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
4. C83×#1352C2semidirect
5. C83×#1344C2semidirect
6. C2×#117×C83C4semidirect
10. #399×C83C4semidirect
12. #397×C83C4semidirect
17. #381×C83C4semidirect
32. C2×C4⋊C4×C83C2×C4semidirect
33. C2×C4⋊C4×C83C2×C4semidirect
56. #95×C83C2×C4semidirect
57. #95×C83C2×C4semidirect
204. #11755C83direct (N × H)
218. #1352C2×C83semidirect
219. #1344C2×C83semidirect
220. C83#11755direct (N × H)
221. C2×#117C4×C83semidirect
225. #399C4×C83semidirect
227. #397C4×C83semidirect
232. #381C4×C83semidirect
247. C2×C4⋊C4C2×C4×C83semidirect
248. C2×C4⋊C4C2×C4×C83semidirect
271. #95C2×C4×C83semidirect
272. #95C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#24822
C2×#342×C832
C83×#14622
C83×#13522
C83×#13442
#1175583