Finite Group Databasebeta

← C83×#11727 #35239551

Subgroups — C83×#11727 #35239551

Order 21248.

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

normal subgroups only

This group has 456 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×#2395C211yesmaximal, characteristic
2C4×#99×C83C211yesmaximal, characteristic
3C83×#1460C211yesmaximal, characteristic
4C83×#1352C212yesmaximal, complemented
5C83×#1344C212yesmaximal, complemented
6C2×C4×D4×C83C2211yescharacteristic
7C2×#117×C83C411yescharacteristic, complemented
8#431×C83—21no—
9C2×#99×C83—21no—
10#400×C83C411yescharacteristic, complemented
11#399×C83C2211yescharacteristic
12#398×C83C411yescharacteristic, complemented
13#397×C83—21no—
14#397×C83C2211yescharacteristic
15#394×C83—21no—
16C4×C4⋊C4×C83—21no—
17C4×C22⋊C4×C83—21no—
18C4×C22⋊C4×C83—21no—
19C4×C22⋊C4×C83C411yescharacteristic, complemented
20#342×C83C2212yes—
21#323×C83—21no—
22#323×C83C2212yes—
23C22×D4×C83D411yescharacteristic
24C23×C4×C83C2×C411yescharacteristic
25C23×C4×C83D411yescharacteristic
26C4×D4×C83—41no—
27C4×D4×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
33C2×C4⋊C4×C83C2×C411yescharacteristic
34C2×C4⋊C4×C83C2×C411yescharacteristic
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×C83D411yescharacteristic
41C2×C22⋊C4×C83C2311yescharacteristic
42C2×C42×C83—21no—
43C2×C42×C83—21no—
44C2×C42×C83—41no—
45C2×C42×C83D411yescharacteristic
46C4⋊C8×C83—41no—
47#99×C83—24no—
48C22⋊C8×C83—22no—
49C4×C8×C83—41no—
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×D4×C83—22no—
60C2×D4×C83—22no—
61C2×D4×C83—41no—
62C22×C4×C83—21no—
63C22×C4×C83—21no—
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—42no—
71C22×C4×C83—42no—
72C22×C4×C83—42no—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C2×C8×C83—42no—
79C4⋊C4×C83—41no—
80C4⋊C4×C83—41no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—42no—
83C4⋊C4×C83—42no—
84C22⋊C4×C83—24no—
85C22⋊C4×C83—41no—
86C22⋊C4×C83—41no—
87C22⋊C4×C83—42no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—42no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C42×C83—21no—
93C42×C83—21no—
94C42×C83—42no—
95C42×C83—41no—
96C42×C83—41no—
97C42×C83—82no—
98C23×C83—21no—
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×C83C2×C22⋊C411yescharacteristic
108C23×C83#13512yes—
109D4×C83—44no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—44no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—81no—
136C2×C4×C83—82no—
137C2×C4×C83—82no—
138C2×C4×C83#9911yescharacteristic
139C2×C4×C83#9911yescharacteristic
140C2×C4×C83#10412yes—
141C8×C83—82no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—22no—
145C22×C83—42no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83C2×#9911yescharacteristic
155C22×C83#41611yescharacteristic
156C22×C83C2×#10411yescharacteristic
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—42no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—81no—
164C4×C83—84no—
165C4×C83—81no—
166C4×C83—81no—
167C4×C83#35212yes—
168#11727C8311yesmaximal, characteristic, direct product factor, Sylow
169C2×C83—21no—
170C2×C83—21no—
171C2×C83—42no—
172C2×C83—81no—
173C2×C83#150111yescharacteristic
174C2×C83C2×#35211yescharacteristic
175C2×C83#212611yescharacteristic
176#2395C2×C8311yescharacteristic
177C4×#99C2×C8311yescharacteristic
178#1460C2×C8311yescharacteristic
179#1352C2×C8312yescomplemented
180#1344C2×C8312yescomplemented
181C83#1172711yescharacteristic, direct product factor, Sylow
182C2×C4×D4C22×C8311yescharacteristic
183C2×#117C4×C8311yescharacteristic, complemented
184#431—21no—
185C2×#99—21no—
186#400C4×C8311yescharacteristic, complemented
187#399C22×C8311yescharacteristic
188#398C4×C8311yescharacteristic, complemented
189#397—21no—
190#397C22×C8311yescharacteristic
191#394—21no—
192C4×C4⋊C4—21no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4C4×C8311yescharacteristic, complemented
196#342C22×C8312yes—
197#323—21no—
198#323C22×C8312yes—
199C22×D4D4×C8311yescharacteristic
200C23×C4C2×C4×C8311yescharacteristic
201C23×C4D4×C8311yescharacteristic
202C4×D4—41no—
203C4×D4—41no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×C4×C8311yescharacteristic
209C2×C4⋊C4C2×C4×C8311yescharacteristic
210C2×C4⋊C4C2×C4×C8311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4D4×C8311yescharacteristic
217C2×C22⋊C4C23×C8311yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—41no—
221C2×C42D4×C8311yescharacteristic
222C4⋊C8—41no—
223#99—24no—
224C22⋊C8—22no—
225C4×C8—41no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—41no—
231#95C2×C4×C8311yescharacteristic
232#95C2×C4×C8311yescharacteristic
233C24C22⋊C4×C8311yescharacteristic
234C24C2×D4×C8311yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C22⋊C4×C8311yescharacteristic
250C22×C4C22⋊C4×C8311yescharacteristic
251C22×C4C22⋊C4×C8311yescharacteristic
252C22×C4C22×C4×C8311yescharacteristic
253C22×C4C2×D4×C8311yescharacteristic
254C2×C8—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—24no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—41no—
272C42—41no—
273C42—82no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×C22⋊C4×C8311yescharacteristic
284C23#135×C8312yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#99×C8311yescharacteristic
315C2×C4#99×C8311yescharacteristic
316C2×C4#104×C8312yes—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×#99×C8311yescharacteristic
331C22#416×C8311yescharacteristic
332C22C2×#104×C8311yescharacteristic
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—84no—
341C4—81no—
342C4—81no—
343C4#352×C8312yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2C83×#150111yescharacteristic
349C2C2×#352×C8311yescharacteristic
350C2C83×#212611yescharacteristic
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×#1352C2semidirect
5. C83×#1344C2semidirect
7. C2×#117×C83C4semidirect
10. #400×C83C4semidirect
12. #398×C83C4semidirect
19. C4×C22⋊C4×C83C4semidirect
168. #11727C83direct (N × H)
179. #1352C2×C83semidirect
180. #1344C2×C83semidirect
181. C83#11727direct (N × H)
183. C2×#117C4×C83semidirect
186. #400C4×C83semidirect
188. #398C4×C83semidirect
195. C4×C22⋊C4C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#23952
C4×#99×C832
C83×#14602
C83×#13522
C83×#13442
#1172783