Finite Group Databasebeta

← C83×#11557 #35239381

Subgroups — C83×#11557 #35239381

Order 21248.

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

normal subgroups only

This group has 750 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×#2408C211yesmaximal, characteristic
2C83×#1466C211yesmaximal, characteristic, complemented
3C83×#1459C211yesmaximal, characteristic, complemented
4C83×#1316C214yesmaximal, complemented
5C23×D4×C83C411yescharacteristic
6C2×#121×C83—21no—
7C2×#120×C83C412yes—
8C2×C4×D4×C83C2211yescharacteristic
9#432×C83—21no—
10C2×C22⋊C8×C83—21no—
11#394×C83—21no—
12#392×C83—21no—
13#390×C83—21no—
14#380×C83C411yescharacteristic
15#379×C83C2212yes—
16#324×C83—21no—
17#323×C83—21no—
18#323×C83C2212yes—
19#323×C83C2212yes—
20C25×C83C2×C412yes—
21C22×D4×C83—22no—
22C22×D4×C83—21no—
23C22×D4×C83—21no—
24C22×D4×C83—41no—
25C22×D4×C83—41no—
26C22×D4×C83—44no—
27C22×D4×C83C2×C411yescharacteristic
28C23×C4×C83C2×C411yescharacteristic
29C23×C4×C83D411yescharacteristic
30C22×C8×C83—41no—
31#121×C83—44no—
32#120×C83—44no—
33C4×D4×C83—41no—
34C4×D4×C83—41no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83D411yescharacteristic
37C2×C22⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—22no—
40C2×C22⋊C4×C83C2×C412yes—
41C2×C22⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83C2311yescharacteristic
43C2×C42×C83D411yescharacteristic
44C4⋊C8×C83—41no—
45C22⋊C8×C83—22no—
46C22⋊C8×C83—22no—
47C22⋊C8×C83—41no—
48#97×C83—41no—
49#95×C83—21no—
50C24×C83—22no—
51C24×C83—24no—
52C24×C83—42no—
53C24×C83—44no—
54C24×C83—42no—
55C24×C83—82no—
56C24×C83C22⋊C411yescharacteristic
57C24×C83C22×C411yescharacteristic
58C2×D4×C83—24no—
59C2×D4×C83—22no—
60C2×D4×C83—22no—
61C2×D4×C83—41no—
62C2×D4×C83—42no—
63C2×D4×C83—42no—
64C2×D4×C83—44no—
65C2×D4×C83—48no—
66C2×D4×C83—44no—
67C2×D4×C83—44no—
68C2×D4×C83—44no—
69C2×D4×C83—44no—
70C2×D4×C83—41no—
71C2×D4×C83—41no—
72C2×D4×C83—82no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—22no—
78C22×C4×C83—42no—
79C22×C4×C83—42no—
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C2×C8×C83—42no—
86C2×C8×C83—42no—
87C2×C8×C83—82no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—42no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—44no—
93C22⋊C4×C83—44no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—82no—
96C42×C83—21no—
97C42×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—24no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—42no—
108C23×C83—42no—
109C23×C83—42no—
110C23×C83—44no—
111C23×C83—44no—
112C23×C83—42no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83—84no—
116C23×C83—88no—
117C23×C83—88no—
118C23×C83—84no—
119C23×C83#10012yes—
120C23×C83C2×C22⋊C411yescharacteristic
121C23×C83#13112yes—
122D4×C83—44no—
123D4×C83—44no—
124D4×C83—48no—
125D4×C83—44no—
126D4×C83—88no—
127C2×C4×C83—22no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—42no—
135C2×C4×C83—42no—
136C2×C4×C83—42no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—42no—
141C2×C4×C83—42no—
142C2×C4×C83—42no—
143C2×C4×C83—81no—
144C2×C4×C83#9911yescharacteristic
145C2×C4×C83#9911yescharacteristic
146C8×C83—82no—
147C8×C83—82no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—42no—
153C22×C83—42no—
154C22×C83—42no—
155C22×C83—44no—
156C22×C83—44no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—44no—
162C22×C83—81no—
163C22×C83—82no—
164C22×C83—82no—
165C22×C83—88no—
166C22×C83—88no—
167C22×C83—82no—
168C22×C83—82no—
169C22×C83#40811yescharacteristic
170C22×C83C2×#9911yescharacteristic
171C22×C83C2×#10011yescharacteristic
172C4×C83—21no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83—41no—
177C4×C83—82no—
178C4×C83AΣL(1,𝔽16)12yes—
179#11557C8311yesmaximal, characteristic, direct product factor, Sylow
180C2×C83—21no—
181C2×C83—21no—
182C2×C83—42no—
183C2×C83—42no—
184C2×C83—84no—
185C2×C83#147111yescharacteristic
186C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
187C2×C83#212211yescharacteristic
188#2408C2×C8311yescharacteristic
189#1466C2×C8311yescharacteristic, complemented
190#1459C2×C8311yescharacteristic, complemented
191#1316C2×C8314yescomplemented
192C83#1155711yescharacteristic, direct product factor, Sylow
193C23×D4C4×C8311yescharacteristic
194C2×#121—21no—
195C2×#120C4×C8312yes—
196C2×C4×D4C22×C8311yescharacteristic
197#432—21no—
198C2×C22⋊C8—21no—
199#394—21no—
200#392—21no—
201#390—21no—
202#380C4×C8311yescharacteristic
203#379C22×C8312yes—
204#324—21no—
205#323—21no—
206#323C22×C8312yes—
207#323C22×C8312yes—
208C25C2×C4×C8312yes—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—44no—
215C22×D4C2×C4×C8311yescharacteristic
216C23×C4C2×C4×C8311yescharacteristic
217C23×C4D4×C8311yescharacteristic
218C22×C8—41no—
219#121—44no—
220#120—44no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4D4×C8311yescharacteristic
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4C2×C4×C8312yes—
229C2×C22⋊C4D4×C8311yescharacteristic
230C2×C22⋊C4C23×C8311yescharacteristic
231C2×C42D4×C8311yescharacteristic
232C4⋊C8—41no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—41no—
236#97—41no—
237#95—21no—
238C24—22no—
239C24—24no—
240C24—42no—
241C24—44no—
242C24—42no—
243C24—82no—
244C24C22⋊C4×C8311yescharacteristic
245C24C22×C4×C8311yescharacteristic
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4C22⋊C4×C8311yescharacteristic
269C22×C4C22⋊C4×C8311yescharacteristic
270C22×C4C22⋊C4×C8311yescharacteristic
271C22×C4C2×D4×C8311yescharacteristic
272C22×C4C2×D4×C8311yescharacteristic
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—82no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—21no—
285C42—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—24no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—84no—
304C23—88no—
305C23—88no—
306C23—84no—
307C23#100×C8312yes—
308C23C2×C22⋊C4×C8311yescharacteristic
309C23#131×C8312yes—
310D4—44no—
311D4—44no—
312D4—48no—
313D4—44no—
314D4—88no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4#99×C8311yescharacteristic
333C2×C4#99×C8311yescharacteristic
334C8—82no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—44no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—44no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22—88no—
354C22—88no—
355C22—82no—
356C22—82no—
357C22#408×C8311yescharacteristic
358C22C2×#99×C8311yescharacteristic
359C22C2×#100×C8311yescharacteristic
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4AΣL(1,𝔽16)×C8312yes—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—42no—
371C2—84no—
372C2C83×#147111yescharacteristic
373C2C2×AΣL(1,𝔽16)×C8311yescharacteristic
374C2C83×#212211yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC24C22×C4×C83
Frattini, cosocleC2×C22⋊C4C23×C83
prime radicalC83×#2408C2
LCSG ▷ C24 ▷ C22 ▷ 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
2. C83×#1466C2semidirect
3. C83×#1459C2semidirect
4. C83×#1316C2semidirect
179. #11557C83direct (N × H)
189. #1466C2×C83semidirect
190. #1459C2×C83semidirect
191. #1316C2×C83semidirect
192. C83#11557direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#24082
C83×#14662
C83×#14592
C83×#13162
#1155783