Finite Group Databasebeta

← C83×#11731 #35239555

Subgroups — C83×#11731 #35239555

Order 21248.

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

normal subgroups only

This group has 436 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×#2638C211yesmaximal, characteristic
2C4×#99×C83C211yesmaximal, characteristic
3C83×#1460C211yesmaximal, characteristic
4C83×#1351C212yesmaximal, complemented
5C83×#1344C212yesmaximal, complemented
6C2×C22⋊Q8×C83C411yescharacteristic, complemented
7C2×C4×D4×C83C2211yescharacteristic
8#431×C83—21no—
9C2×#99×C83—21no—
10#398×C83C411yescharacteristic, complemented
11#397×C83—21no—
12#397×C83C2211yescharacteristic
13#395×C83C411yescharacteristic, complemented
14#394×C83—21no—
15#393×C83C2211yescharacteristic
16#387×C83—21no—
17#385×C83—21no—
18#381×C83C411yescharacteristic, complemented
19C4×C22⋊C4×C83—21no—
20#342×C83C2212yes—
21#323×C83—21no—
22#323×C83C2212yes—
23C22×Q8×C83C2×C411yescharacteristic
24C22×D4×C83D411yescharacteristic
25C23×C4×C83C2×C411yescharacteristic
26C23×C4×C83D411yescharacteristic
27C22⋊Q8×C83—41no—
28C22⋊Q8×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×C83C2×C411yescharacteristic
35C2×C4⋊C4×C83C2×C411yescharacteristic
36C2×C22⋊C4×C83—22no—
37C2×C22⋊C4×C83—21no—
38C2×C22⋊C4×C83—21no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83C2311yescharacteristic
43C2×C42×C83—41no—
44C2×C42×C83D411yescharacteristic
45C4⋊C8×C83—41no—
46#99×C83—24no—
47C22⋊C8×C83—22no—
48C4×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—41no—
60C2×Q8×C83—81no—
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—81no—
74C22×C4×C83—81no—
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C2×C8×C83—42no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—42no—
86C4⋊C4×C83—81no—
87C22⋊C4×C83—24no—
88C22⋊C4×C83—41no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—81no—
94C22⋊C4×C83—81no—
95C42×C83—21no—
96C42×C83—21no—
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—
109Q8×C83—82no—
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—41no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—44no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
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—81no—
135C2×C4×C83—81no—
136C2×C4×C83—81no—
137C2×C4×C83—82no—
138C2×C4×C83—82no—
139C2×C4×C83#9911yescharacteristic
140C2×C4×C83#9911yescharacteristic
141C2×C4×C83#10412yes—
142C8×C83—82no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—22no—
146C22×C83—42no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83C2×#9911yescharacteristic
156C22×C83#41611yescharacteristic
157C22×C83C2×#10411yescharacteristic
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—22no—
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#11731C8311yesmaximal, characteristic, direct product factor, Sylow
170C2×C83—21no—
171C2×C83—21no—
172C2×C83—42no—
173C2×C83—81no—
174C2×C83#150111yescharacteristic
175C2×C83#212111yescharacteristic
176C2×C83#212911yescharacteristic
177#2638C2×C8311yescharacteristic
178C4×#99C2×C8311yescharacteristic
179#1460C2×C8311yescharacteristic
180#1351C2×C8312yescomplemented
181#1344C2×C8312yescomplemented
182C83#1173111yescharacteristic, direct product factor, Sylow
183C2×C22⋊Q8C4×C8311yescharacteristic, complemented
184C2×C4×D4C22×C8311yescharacteristic
185#431—21no—
186C2×#99—21no—
187#398C4×C8311yescharacteristic, complemented
188#397—21no—
189#397C22×C8311yescharacteristic
190#395C4×C8311yescharacteristic, complemented
191#394—21no—
192#393C22×C8311yescharacteristic
193#387—21no—
194#385—21no—
195#381C4×C8311yescharacteristic, complemented
196C4×C22⋊C4—21no—
197#342C22×C8312yes—
198#323—21no—
199#323C22×C8312yes—
200C22×Q8C2×C4×C8311yescharacteristic
201C22×D4D4×C8311yescharacteristic
202C23×C4C2×C4×C8311yescharacteristic
203C23×C4D4×C8311yescharacteristic
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×C4×C8311yescharacteristic
212C2×C4⋊C4C2×C4×C8311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4D4×C8311yescharacteristic
219C2×C22⋊C4C23×C8311yescharacteristic
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—21no—
231#95—41no—
232#95C2×C4×C8311yescharacteristic
233#95C2×C4×C8311yescharacteristic
234C24C22⋊C4×C8311yescharacteristic
235C24C2×D4×C8311yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4C22⋊C4×C8311yescharacteristic
253C22×C4C22⋊C4×C8311yescharacteristic
254C22×C4C22⋊C4×C8311yescharacteristic
255C22×C4C22×C4×C8311yescharacteristic
256C22×C4C2×D4×C8311yescharacteristic
257C2×C8—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C42—82no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊C4×C8311yescharacteristic
285C23#135×C8312yes—
286Q8—82no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—41no—
300C2×C4—41no—
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—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4#99×C8311yescharacteristic
317C2×C4#99×C8311yescharacteristic
318C2×C4#104×C8312yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#99×C8311yescharacteristic
333C22#416×C8311yescharacteristic
334C22C2×#104×C8311yescharacteristic
335C4—22no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—84no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2C83×#150111yescharacteristic
351C2C83×#212111yescharacteristic
352C2C83×#212911yescharacteristic
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×#1351C2semidirect
5. C83×#1344C2semidirect
6. C2×C22⋊Q8×C83C4semidirect
10. #398×C83C4semidirect
13. #395×C83C4semidirect
18. #381×C83C4semidirect
169. #11731C83direct (N × H)
180. #1351C2×C83semidirect
181. #1344C2×C83semidirect
182. C83#11731direct (N × H)
183. C2×C22⋊Q8C4×C83semidirect
187. #398C4×C83semidirect
190. #395C4×C83semidirect
195. #381C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#26382
C4×#99×C832
C83×#14602
C83×#13512
C83×#13442
#1173183