Finite Group Databasebeta

← C83×#9459 #35237283

Subgroups — C83×#9459 #35237283

Order 21248.

904 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2435C211yesmaximal, characteristic
2C83×#2308C211yesmaximal, characteristic
3C83×#1783C211yesmaximal, characteristic
4C83×#1511C212yesmaximal, complemented
5C83×#1284C212yesmaximal, complemented
6C23×D4×C83C2211yescharacteristic
7C2×#124×C83C412yescomplemented
8C2×C22⋊Q8×C83C2211yescharacteristic
9C22×C22⋊C4×C83—21no—
10C2×#100×C83—22no—
11#407×C83C2212yes—
12#394×C83—22no—
13#390×C83C412yescomplemented
14#386×C83—24no—
15#381×C83C2212yes—
16#379×C83—21no—
17#379×C83—21no—
18#379×C83—21no—
19C4×C22⋊C4×C83C2211yescharacteristic
20#328×C83—24no—
21C25×C83D411yescharacteristic
22C25×C83D411yescharacteristic
23C22×Q8×C83C2×C411yescharacteristic, complemented
24C22×D4×C83—22no—
25C22×D4×C83—22no—
26C22×D4×C83—44no—
27C22×D4×C83C2×C412yes—
28C22×D4×C83D412yes—
29C23×C4×C83—22no—
30C23×C4×C83C2311yescharacteristic
31C2×M16×C83—22no—
32#124×C83—44no—
33C22⋊Q8×C83—22no—
34C22⋊Q8×C83—22no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83C2×C411yescharacteristic, complemented
37C2×C22⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—22no—
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—44no—
48C2×C42×C83C2×C412yescomplemented
49#100×C83—44no—
50C22⋊C8×C83—22no—
51C22⋊C8×C83—22no—
52#95×C83—21no—
53#95×C83—22no—
54#95×C83D414yes—
55C24×C83—21no—
56C24×C83—24no—
57C24×C83—21no—
58C24×C83—21no—
59C24×C83—21no—
60C24×C83—42no—
61C24×C83—42no—
62C24×C83—42no—
63C24×C83—81no—
64C24×C83—81no—
65C24×C83C22⋊C412yes—
66C24×C83C22⋊C412yes—
67C24×C83C2×D411yescharacteristic
68C2×Q8×C83—22no—
69C2×Q8×C83—42no—
70C2×D4×C83—24no—
71C2×D4×C83—24no—
72C2×D4×C83—24no—
73C2×D4×C83—24no—
74C2×D4×C83—42no—
75C2×D4×C83—42no—
76C2×D4×C83—44no—
77C2×D4×C83—44no—
78C2×D4×C83—48no—
79C22×C4×C83—22no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—24no—
83C22×C4×C83—42no—
84C22×C4×C83—44no—
85C22×C4×C83—82no—
86C22×C4×C83—82no—
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D412yes—
90C22×C4×C83#4112yes—
91M16×C83—44no—
92C2×C8×C83—42no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C22⋊C4×C83—44no—
97C22⋊C4×C83—44no—
98C22⋊C4×C83—44no—
99C22⋊C4×C83—44no—
100C22⋊C4×C83—41no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—82no—
103C22⋊C4×C83—82no—
104C22⋊C4×C83—82no—
105C22⋊C4×C83—82no—
106C42×C83—42no—
107C23×C83—22no—
108C23×C83—22no—
109C23×C83—24no—
110C23×C83—22no—
111C23×C83—22no—
112C23×C83—22no—
113C23×C83—22no—
114C23×C83—22no—
115C23×C83—22no—
116C23×C83—22no—
117C23×C83—41no—
118C23×C83—42no—
119C23×C83—42no—
120C23×C83—42no—
121C23×C83—42no—
122C23×C83—42no—
123C23×C83—42no—
124C23×C83—42no—
125C23×C83—42no—
126C23×C83—42no—
127C23×C83—42no—
128C23×C83—81no—
129C23×C83—81no—
130C23×C83—81no—
131C23×C83—81no—
132C23×C83—88no—
133C23×C83—88no—
134C23×C83#9914yes—
135C23×C83C2×C22⋊C411yescharacteristic
136C23×C83#12011yescharacteristic
137C23×C83#12311yescharacteristic
138Q8×C83—82no—
139D4×C83—48no—
140D4×C83—48no—
141C2×C4×C83—22no—
142C2×C4×C83—22no—
143C2×C4×C83—42no—
144C2×C4×C83—42no—
145C2×C4×C83—44no—
146C2×C4×C83—42no—
147C2×C4×C83—42no—
148C2×C4×C83—42no—
149C2×C4×C83—44no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—42no—
153C2×C4×C83—42no—
154C2×C4×C83—44no—
155C2×C4×C83—88no—
156C2×C4×C83C4×D412yes—
157C2×C4×C83#12112yes—
158C8×C83—82no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—21no—
165C22×C83—24no—
166C22×C83—21no—
167C22×C83—44no—
168C22×C83—44no—
169C22×C83—44no—
170C22×C83—44no—
171C22×C83—44no—
172C22×C83—42no—
173C22×C83—42no—
174C22×C83—42no—
175C22×C83—42no—
176C22×C83—42no—
177C22×C83—82no—
178C22×C83—82no—
179C22×C83—84no—
180C22×C83—84no—
181C22×C83—84no—
182C22×C83#38611yescharacteristic
183C22×C83C2×#9912yes—
184C22×C83#45512yes—
185C4×C83—22no—
186C4×C83—22no—
187C4×C83—82no—
188C4×C83—84no—
189C4×C83—81no—
190C4×C83—81no—
191#9459C8311yesmaximal, characteristic, direct product factor, Sylow
192C2×C83—22no—
193C2×C83—22no—
194C2×C83—41no—
195C2×C83—44no—
196C2×C83—82no—
197C2×C83#188511yescharacteristic
198C2×C83#189811yescharacteristic
199C2×C83#191111yescharacteristic
200#2435C2×C8311yescharacteristic
201#2308C2×C8311yescharacteristic
202#1783C2×C8311yescharacteristic
203#1511C2×C8312yescomplemented
204#1284C2×C8312yescomplemented
205C83#945911yescharacteristic, direct product factor, Sylow
206C23×D4C22×C8311yescharacteristic
207C2×#124C4×C8312yescomplemented
208C2×C22⋊Q8C22×C8311yescharacteristic
209C22×C22⋊C4—21no—
210C2×#100—22no—
211#407C22×C8312yes—
212#394—22no—
213#390C4×C8312yescomplemented
214#386—24no—
215#381C22×C8312yes—
216#379—21no—
217#379—21no—
218#379—21no—
219C4×C22⋊C4C22×C8311yescharacteristic
220#328—24no—
221C25D4×C8311yescharacteristic
222C25D4×C8311yescharacteristic
223C22×Q8C2×C4×C8311yescharacteristic, complemented
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4C2×C4×C8312yes—
228C22×D4D4×C8312yes—
229C23×C4—22no—
230C23×C4C23×C8311yescharacteristic
231C2×M16—22no—
232#124—44no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×C4×C8311yescharacteristic, complemented
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—44no—
248C2×C42C2×C4×C8312yescomplemented
249#100—44no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252#95—21no—
253#95—22no—
254#95D4×C8314yes—
255C24—21no—
256C24—24no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—81no—
264C24—81no—
265C24C22⋊C4×C8312yes—
266C24C22⋊C4×C8312yes—
267C24C2×D4×C8311yescharacteristic
268C2×Q8—22no—
269C2×Q8—42no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—48no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C22×C4×C8311yescharacteristic
288C22×C4C2×D4×C8311yescharacteristic
289C22×C4C2×D4×C8312yes—
290C22×C4#41×C8312yes—
291M16—44no—
292C2×C8—42no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—42no—
307C23—22no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—88no—
333C23—88no—
334C23#99×C8314yes—
335C23C2×C22⋊C4×C8311yescharacteristic
336C23#120×C8311yescharacteristic
337C23#123×C8311yescharacteristic
338Q8—82no—
339D4—48no—
340D4—48no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—88no—
356C2×C4C4×D4×C8312yes—
357C2×C4#121×C8312yes—
358C8—82no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22#386×C8311yescharacteristic
383C22C2×#99×C8312yes—
384C22#455×C8312yes—
385C4—22no—
386C4—22no—
387C4—84no—
388C4—82no—
389C4—81no—
390C4—81no—
391C2—22no—
392C2—22no—
393C2—41no—
394C2—44no—
395C2—82no—
396C2C83×#188511yescharacteristic
397C2C83×#189811yescharacteristic
398C2C83×#191111yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC23×D4×C83C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
4. C83×#1511C2semidirect
5. C83×#1284C2semidirect
7. C2×#124×C83C4semidirect
13. #390×C83C4semidirect
23. C22×Q8×C83C2×C4semidirect
36. C2×C4⋊C4×C83C2×C4semidirect
48. C2×C42×C83C2×C4semidirect
191. #9459C83direct (N × H)
203. #1511C2×C83semidirect
204. #1284C2×C83semidirect
205. C83#9459direct (N × H)
207. C2×#124C4×C83semidirect
213. #390C4×C83semidirect
223. C22×Q8C2×C4×C83semidirect
236. C2×C4⋊C4C2×C4×C83semidirect
248. C2×C42C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#24352
C83×#23082
C83×#17832
C83×#15112
C83×#12842
#945983