Finite Group Databasebeta

← C83×#11728 #35239552

Subgroups — C83×#11728 #35239552

Order 21248.

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

normal subgroups only

This group has 470 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×#2398C211yesmaximal, characteristic
2C4×#99×C83C211yesmaximal, characteristic
3C83×#1459C211yesmaximal, characteristic
4C83×#1351C212yesmaximal, complemented
5C83×#1344C212yesmaximal, complemented
6C2×C4×Q8×C83C411yescharacteristic, complemented
7C2×C4×D4×C83C2211yescharacteristic
8C2×#99×C83—21no—
9C2×C22⋊C8×C83—21no—
10#400×C83—21no—
11#397×C83C411yescharacteristic, complemented
12#397×C83C2211yescharacteristic
13#394×C83—21no—
14#393×C83C411yescharacteristic, complemented
15#393×C83C2211yescharacteristic
16C4×C22⋊C4×C83—21no—
17C4×C22⋊C4×C83—21no—
18C4×C22⋊C4×C83—21no—
19#376×C83C411yescharacteristic, complemented
20#342×C83C2212yes—
21#324×C83—21no—
22#323×C83C2212yes—
23C22×Q8×C83C2×C411yescharacteristic
24C22×D4×C83D411yescharacteristic
25C23×C4×C83D411yescharacteristic
26C23×C4×C83D411yescharacteristic
27C22×C8×C83—41no—
28C4×Q8×C83—41no—
29C4×Q8×C83—41no—
30C4×D4×C83—41no—
31C4×D4×C83—41no—
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83C2×C411yescharacteristic
34C2×C4⋊C4×C83D411yescharacteristic
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×C83C2×C411yescharacteristic
41C2×C22⋊C4×C83C2311yescharacteristic
42C2×C42×C83—21no—
43C2×C42×C83—21no—
44C2×C42×C83—41no—
45C2×C42×C83C2×C411yescharacteristic
46#99×C83—24no—
47C22⋊C8×C83—22no—
48C22⋊C8×C83—41no—
49#95×C83—21no—
50#95×C83—21no—
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—41no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57C24×C83C22⋊C411yescharacteristic
58C24×C83C2×D411yescharacteristic
59C2×Q8×C83—42no—
60C2×Q8×C83—41no—
61C2×D4×C83—22no—
62C2×D4×C83—22no—
63C2×D4×C83—41no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—42no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C2×C8×C83—42no—
81C2×C8×C83—82no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—42no—
85C22⋊C4×C83—24no—
86C22⋊C4×C83—41no—
87C22⋊C4×C83—41no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—42no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—42no—
96C42×C83—41no—
97C42×C83—41no—
98C42×C83—82no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83#10412yes—
108C23×C83C2×C22⋊C411yescharacteristic
109C23×C83#13512yes—
110Q8×C83—42no—
111Q8×C83—42no—
112D4×C83—44no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—44no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83—82no—
141C2×C4×C83—82no—
142C2×C4×C83#9911yescharacteristic
143C2×C4×C83#9911yescharacteristic
144C8×C83—82no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—22no—
148C22×C83—22no—
149C22×C83—42no—
150C22×C83—42no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—81no—
156C22×C83—81no—
157C22×C83C2×#9911yescharacteristic
158C22×C83#41611yescharacteristic
159C22×C83C2×#10411yescharacteristic
160C4×C83—21no—
161C4×C83—42no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—81no—
167C4×C83—81no—
168C4×C83—84no—
169C4×C83—81no—
170C4×C83#35412yes—
171#11728C8311yesmaximal, characteristic, direct product factor, Sylow
172C2×C83—21no—
173C2×C83—21no—
174C2×C83—42no—
175C2×C83—42no—
176C2×C83#150111yescharacteristic
177C2×C83#212011yescharacteristic
178C2×C83C2×#35411yescharacteristic
179#2398C2×C8311yescharacteristic
180C4×#99C2×C8311yescharacteristic
181#1459C2×C8311yescharacteristic
182#1351C2×C8312yescomplemented
183#1344C2×C8312yescomplemented
184C83#1172811yescharacteristic, direct product factor, Sylow
185C2×C4×Q8C4×C8311yescharacteristic, complemented
186C2×C4×D4C22×C8311yescharacteristic
187C2×#99—21no—
188C2×C22⋊C8—21no—
189#400—21no—
190#397C4×C8311yescharacteristic, complemented
191#397C22×C8311yescharacteristic
192#394—21no—
193#393C4×C8311yescharacteristic, complemented
194#393C22×C8311yescharacteristic
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198#376C4×C8311yescharacteristic, complemented
199#342C22×C8312yes—
200#324—21no—
201#323C22×C8312yes—
202C22×Q8C2×C4×C8311yescharacteristic
203C22×D4D4×C8311yescharacteristic
204C23×C4D4×C8311yescharacteristic
205C23×C4D4×C8311yescharacteristic
206C22×C8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×C4×C8311yescharacteristic
213C2×C4⋊C4D4×C8311yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×C4×C8311yescharacteristic
220C2×C22⋊C4C23×C8311yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—41no—
224C2×C42C2×C4×C8311yescharacteristic
225#99—24no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—41no—
234#95C2×C4×C8311yescharacteristic
235#95C2×C4×C8311yescharacteristic
236C24C22⋊C4×C8311yescharacteristic
237C24C2×D4×C8311yescharacteristic
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C22⋊C4×C8311yescharacteristic
255C22×C4C22⋊C4×C8311yescharacteristic
256C22×C4C22⋊C4×C8311yescharacteristic
257C22×C4C22×C4×C8311yescharacteristic
258C22×C4C2×D4×C8311yescharacteristic
259C2×C8—42no—
260C2×C8—82no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—82no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#104×C8312yes—
287C23C2×C22⋊C4×C8311yescharacteristic
288C23#135×C8312yes—
289Q8—42no—
290Q8—42no—
291D4—44no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4#99×C8311yescharacteristic
322C2×C4#99×C8311yescharacteristic
323C8—82no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#99×C8311yescharacteristic
337C22#416×C8311yescharacteristic
338C22C2×#104×C8311yescharacteristic
339C4—21no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—84no—
348C4—81no—
349C4#354×C8312yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2C83×#150111yescharacteristic
355C2C83×#212011yescharacteristic
356C2C2×#354×C8311yescharacteristic
subgroupG/H
centerC2×C4×C83#99
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C22⋊C4C23×C83
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C83 ◁ C22×C4×C83 ◁ C2×C4×D4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
4. C83×#1351C2semidirect
5. C83×#1344C2semidirect
6. C2×C4×Q8×C83C4semidirect
11. #397×C83C4semidirect
14. #393×C83C4semidirect
19. #376×C83C4semidirect
171. #11728C83direct (N × H)
182. #1351C2×C83semidirect
183. #1344C2×C83semidirect
184. C83#11728direct (N × H)
185. C2×C4×Q8C4×C83semidirect
190. #397C4×C83semidirect
193. #393C4×C83semidirect
198. #376C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#23982
C4×#99×C832
C83×#14592
C83×#13512
C83×#13442
#1172883