Finite Group Databasebeta

← C83×#11732 #35239556

Subgroups — C83×#11732 #35239556

Order 21248.

442 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2633C211yesmaximal, characteristic
2C4×#99×C83C211yesmaximal, characteristic
3C83×#1459C211yesmaximal, characteristic
4C83×#1352C212yesmaximal, complemented
5C83×#1344C212yesmaximal, complemented
6C2×#125×C83C411yescharacteristic, complemented
7C2×C4×D4×C83C2211yescharacteristic
8C2×#99×C83—21no—
9C2×C22⋊C8×C83—21no—
10#399×C83C411yescharacteristic, complemented
11#399×C83C2211yescharacteristic
12#398×C83—21no—
13#397×C83C411yescharacteristic, complemented
14#397×C83C2211yescharacteristic
15#394×C83—21no—
16#385×C83—21no—
17#385×C83—21no—
18#383×C83C411yescharacteristic, complemented
19C4×C22⋊C4×C83—21no—
20#342×C83C2212yes—
21#324×C83—21no—
22#323×C83C2212yes—
23C22×D4×C83D411yescharacteristic
24C23×C4×C83D411yescharacteristic
25C23×C4×C83D411yescharacteristic
26C22×C8×C83—41no—
27#125×C83—41no—
28#125×C83—41no—
29C4×D4×C83—41no—
30C4×D4×C83—41no—
31C2×C4⋊C4×C83—21no—
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83—21no—
34C2×C4⋊C4×C83—21no—
35C2×C4⋊C4×C83C2×C411yescharacteristic
36C2×C4⋊C4×C83C2×C411yescharacteristic
37C2×C4⋊C4×C83D411yescharacteristic
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83C2×C411yescharacteristic
44C2×C22⋊C4×C83C2311yescharacteristic
45C2×C42×C83—41no—
46C2×C42×C83C2×C411yescharacteristic
47#99×C83—24no—
48C22⋊C8×C83—22no—
49C22⋊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—81no—
72C22×C4×C83—81no—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C2×C8×C83—42no—
79C2×C8×C83—82no—
80C4⋊C4×C83—41no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—42no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—81no—
87C4⋊C4×C83—81no—
88C22⋊C4×C83—24no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—81no—
95C22⋊C4×C83—81no—
96C42×C83—21no—
97C42×C83—21no—
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—
110D4×C83—44no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—41no—
118C2×C4×C83—42no—
119C2×C4×C83—42no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—44no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
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—41no—
136C2×C4×C83—81no—
137C2×C4×C83—81no—
138C2×C4×C83—82no—
139C2×C4×C83—82no—
140C2×C4×C83#9911yescharacteristic
141C2×C4×C83#9911yescharacteristic
142C8×C83—82no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83C2×#9911yescharacteristic
156C22×C83#41611yescharacteristic
157C22×C83C2×#10411yescharacteristic
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—81no—
165C4×C83—81no—
166C4×C83—84no—
167C4×C83—81no—
168C4×C83—81no—
169#11732C8311yesmaximal, characteristic, direct product factor, Sylow
170C2×C83—21no—
171C2×C83—21no—
172C2×C83—42no—
173C2×C83—42no—
174C2×C83#150111yescharacteristic
175C2×C83#212311yescharacteristic
176C2×C83#212711yescharacteristic
177#2633C2×C8311yescharacteristic
178C4×#99C2×C8311yescharacteristic
179#1459C2×C8311yescharacteristic
180#1352C2×C8312yescomplemented
181#1344C2×C8312yescomplemented
182C83#1173211yescharacteristic, direct product factor, Sylow
183C2×#125C4×C8311yescharacteristic, complemented
184C2×C4×D4C22×C8311yescharacteristic
185C2×#99—21no—
186C2×C22⋊C8—21no—
187#399C4×C8311yescharacteristic, complemented
188#399C22×C8311yescharacteristic
189#398—21no—
190#397C4×C8311yescharacteristic, complemented
191#397C22×C8311yescharacteristic
192#394—21no—
193#385—21no—
194#385—21no—
195#383C4×C8311yescharacteristic, complemented
196C4×C22⋊C4—21no—
197#342C22×C8312yes—
198#324—21no—
199#323C22×C8312yes—
200C22×D4D4×C8311yescharacteristic
201C23×C4D4×C8311yescharacteristic
202C23×C4D4×C8311yescharacteristic
203C22×C8—41no—
204#125—41no—
205#125—41no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×C4×C8311yescharacteristic
213C2×C4⋊C4C2×C4×C8311yescharacteristic
214C2×C4⋊C4D4×C8311yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4C2×C4×C8311yescharacteristic
221C2×C22⋊C4C23×C8311yescharacteristic
222C2×C42—41no—
223C2×C42C2×C4×C8311yescharacteristic
224#99—24no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—41no—
232#95C2×C4×C8311yescharacteristic
233#95C2×C4×C8311yescharacteristic
234C24C22⋊C4×C8311yescharacteristic
235C24C2×D4×C8311yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4C22⋊C4×C8311yescharacteristic
251C22×C4C22⋊C4×C8311yescharacteristic
252C22×C4C22⋊C4×C8311yescharacteristic
253C22×C4C22×C4×C8311yescharacteristic
254C22×C4C2×D4×C8311yescharacteristic
255C2×C8—42no—
256C2×C8—82no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—21no—
274C42—21no—
275C42—82no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#104×C8312yes—
285C23C2×C22⋊C4×C8311yescharacteristic
286C23#135×C8312yes—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—41no—
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—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4#99×C8311yescharacteristic
318C2×C4#99×C8311yescharacteristic
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22C2×#99×C8311yescharacteristic
333C22#416×C8311yescharacteristic
334C22C2×#104×C8311yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—84no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—42no—
350C2C83×#150111yescharacteristic
351C2C83×#212311yescharacteristic
352C2C83×#212711yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C22⋊C4C23×C83
prime radicalC22×D4×C83D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×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
6. C2×#125×C83C4semidirect
10. #399×C83C4semidirect
13. #397×C83C4semidirect
18. #383×C83C4semidirect
169. #11732C83direct (N × H)
180. #1352C2×C83semidirect
181. #1344C2×C83semidirect
182. C83#11732direct (N × H)
183. C2×#125C4×C83semidirect
187. #399C4×C83semidirect
190. #397C4×C83semidirect
195. #383C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#26332
C4×#99×C832
C83×#14592
C83×#13522
C83×#13442
#1173283