Finite Group Databasebeta

← C83×#9289 #35237113

Subgroups — C83×#9289 #35237113

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3206C211yesmaximal, characteristic
2C83×#2932C211yesmaximal, characteristic
3C83×#1993C211yesmaximal, characteristic, complemented
4C83×#1549C211yesmaximal, characteristic, complemented
5C83×#1517C211yesmaximal, characteristic
6C83×#1286C211yesmaximal, characteristic, complemented
7C83×#1275C211yesmaximal, characteristic, complemented
8C2×#131×C83—21no—
9#532×C83C2211yescharacteristic
10C2×C4⋊Q8×C83C2211yescharacteristic
11#492×C83C412yescomplemented
12#487×C83C412yescomplemented
13#461×C83—22no—
14#448×C83—22no—
15#435×C83—22no—
16#435×C83—22no—
17#431×C83C2211yescharacteristic
18#431×C83C2211yescharacteristic
19#423×C83C2211yescharacteristic
20#408×C83—21no—
21C2×#97×C83C2211yescharacteristic
22#382×C83C2211yescharacteristic
23#340×C83—21no—
24#340×C83—21no—
25#332×C83—22no—
26#327×C83—22no—
27C2×#41×C83—21no—
28C22×Q8×C83—21no—
29C2×SD16×C83—24no—
30#131×C83—42no—
31#131×C83—42no—
32C2×M16×C83—22no—
33C2×M16×C83—21no—
34C22×C8×C83—22no—
35C22×C8×C83D411yescharacteristic
36C22×C8×C83D411yescharacteristic
37C4⋊Q8×C83—21no—
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—21no—
40C4⋊Q8×C83C2×C412yes—
41C4⋊Q8×C83C2×C411yescharacteristic
42#127×C83C2×C411yescharacteristic
43#124×C83—21no—
44#121×C83—21no—
45#121×C83D412yes—
46C4×D4×C83—21no—
47C4×D4×C83D412yes—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C42×C83C2311yescharacteristic
52C4⋊C8×C83—21no—
53C4⋊C8×C83—21no—
54C4⋊C8×C83—21no—
55C4⋊C8×C83D412yes—
56#103×C83—24no—
57C22⋊C8×C83—24no—
58C22⋊C8×C83—21no—
59C22⋊C8×C83—21no—
60#97×C83—21no—
61#97×C83—21no—
62C4×C8×C83—21no—
63C4×C8×C83C2×C412yes—
64#95×C83—21no—
65#95×C83—21no—
66#41×C83—41no—
67#41×C83—41no—
68#41×C83—41no—
69#41×C83—41no—
70C2×Q8×C83—21no—
71C2×Q8×C83—22no—
72C2×Q8×C83—41no—
73C2×Q8×C83—82no—
74C2×D4×C83—22no—
75C2×D4×C83—21no—
76C2×D4×C83—21no—
77C2×D4×C83C22⋊C412yes—
78C22×C4×C83—22no—
79C22×C4×C83—41no—
80C22×C4×C83—41no—
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83#4111yescharacteristic
84SD16×C83—84no—
85M16×C83—44no—
86M16×C83—41no—
87M16×C83—41no—
88C2×C8×C83—21no—
89C2×C8×C83—21no—
90C2×C8×C83—22no—
91C2×C8×C83—21no—
92C2×C8×C83—22no—
93C2×C8×C83—21no—
94C2×C8×C83—21no—
95C2×C8×C83—44no—
96C2×C8×C83—41no—
97C2×C8×C83SD1612yescomplemented
98C4⋊C4×C83—21no—
99C4⋊C4×C83—22no—
100C4⋊C4×C83—22no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83C22⋊C412yes—
106C22⋊C4×C83—22no—
107C22⋊C4×C83—22no—
108C42×C83C22×C411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83C2×D411yescharacteristic
111C42×C83#4111yescharacteristic
112C23×C83—22no—
113C23×C83#12111yescharacteristic
114Q8×C83—44no—
115Q8×C83—41no—
116Q8×C83—41no—
117Q8×C83—82no—
118D4×C83—42no—
119D4×C83—42no—
120D4×C83—42no—
121D4×C83—41no—
122D4×C83—41no—
123D4×C83—81no—
124C2×C4×C83—22no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—82no—
131C2×C4×C83—81no—
132C2×C4×C83—81no—
133C2×C4×C83—81no—
134C2×C4×C83C2×C22⋊C411yescharacteristic
135C2×C4×C83C4×D411yescharacteristic
136C2×C4×C83C4×D411yescharacteristic
137C2×C4×C83#12011yescharacteristic
138C2×C4×C83#12111yescharacteristic
139C2×C4×C83#12311yescharacteristic
140C2×C4×C83C2×SD1611yescharacteristic
141C2×C4×C83#13511yescharacteristic
142C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C83#13711yescharacteristic
144C8×C83—22no—
145C8×C83—42no—
146C8×C83—42no—
147C8×C83—41no—
148C8×C83—81no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83—42no—
154C22×C83#38611yescharacteristic
155C22×C83#46111yescharacteristic
156C22×C83#46311yescharacteristic
157C4×C83—21no—
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—42no—
162C4×C83—82no—
163C4×C83—161no—
164C4×C83C4×SD1611yescharacteristic
165C4×C83#44011yescharacteristic
166C4×C83#45011yescharacteristic
167C4×C83#45211yescharacteristic
168#9289C8311yesmaximal, characteristic, direct product factor, Sylow
169C2×C83—21no—
170C2×C83—21no—
171C2×C83—82no—
172C2×C83#187711yescharacteristic
173C2×C83#190211yescharacteristic
174C2×C83#191011yescharacteristic
175#3206C2×C8311yescharacteristic
176#2932C2×C8311yescharacteristic
177#1993C2×C8311yescharacteristic, complemented
178#1549C2×C8311yescharacteristic, complemented
179#1517C2×C8311yescharacteristic
180#1286C2×C8311yescharacteristic, complemented
181#1275C2×C8311yescharacteristic, complemented
182C83#928911yescharacteristic, direct product factor, Sylow
183C2×#131—21no—
184#532C22×C8311yescharacteristic
185C2×C4⋊Q8C22×C8311yescharacteristic
186#492C4×C8312yescomplemented
187#487C4×C8312yescomplemented
188#461—22no—
189#448—22no—
190#435—22no—
191#435—22no—
192#431C22×C8311yescharacteristic
193#431C22×C8311yescharacteristic
194#423C22×C8311yescharacteristic
195#408—21no—
196C2×#97C22×C8311yescharacteristic
197#382C22×C8311yescharacteristic
198#340—21no—
199#340—21no—
200#332—22no—
201#327—22no—
202C2×#41—21no—
203C22×Q8—21no—
204C2×SD16—24no—
205#131—42no—
206#131—42no—
207C2×M16—22no—
208C2×M16—21no—
209C22×C8—22no—
210C22×C8D4×C8311yescharacteristic
211C22×C8D4×C8311yescharacteristic
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C2×C4×C8312yes—
216C4⋊Q8C2×C4×C8311yescharacteristic
217#127C2×C4×C8311yescharacteristic
218#124—21no—
219#121—21no—
220#121D4×C8312yes—
221C4×D4—21no—
222C4×D4D4×C8312yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C42C23×C8311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8D4×C8312yes—
231#103—24no—
232C22⋊C8—24no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#97—21no—
236#97—21no—
237C4×C8—21no—
238C4×C8C2×C4×C8312yes—
239#95—21no—
240#95—21no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—82no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C22⋊C4×C8312yes—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×D4×C8311yescharacteristic
257C22×C4C2×D4×C8311yescharacteristic
258C22×C4#41×C8311yescharacteristic
259SD16—84no—
260M16—44no—
261M16—41no—
262M16—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—44no—
271C2×C8—41no—
272C2×C8SD16×C8312yescomplemented
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4C22⋊C4×C8312yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C42C22×C4×C8311yescharacteristic
284C42C2×D4×C8311yescharacteristic
285C42C2×D4×C8311yescharacteristic
286C42#41×C8311yescharacteristic
287C23—22no—
288C23#121×C8311yescharacteristic
289Q8—44no—
290Q8—41no—
291Q8—41no—
292Q8—82no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—41no—
297D4—41no—
298D4—81no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C2×C22⋊C4×C8311yescharacteristic
310C2×C4C4×D4×C8311yescharacteristic
311C2×C4C4×D4×C8311yescharacteristic
312C2×C4#120×C8311yescharacteristic
313C2×C4#121×C8311yescharacteristic
314C2×C4#123×C8311yescharacteristic
315C2×C4C2×SD16×C8311yescharacteristic
316C2×C4#135×C8311yescharacteristic
317C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
318C2×C4#137×C8311yescharacteristic
319C8—22no—
320C8—42no—
321C8—42no—
322C8—41no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#386×C8311yescharacteristic
330C22#461×C8311yescharacteristic
331C22#463×C8311yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—161no—
339C4C4×SD16×C8311yescharacteristic
340C4#440×C8311yescharacteristic
341C4#450×C8311yescharacteristic
342C4#452×C8311yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2C83×#187711yescharacteristic
347C2C83×#190211yescharacteristic
348C2C83×#191011yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radical#532×C83C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1993C2semidirect
4. C83×#1549C2semidirect
6. C83×#1286C2semidirect
7. C83×#1275C2semidirect
11. #492×C83C4semidirect
12. #487×C83C4semidirect
97. C2×C8×C83SD16semidirect
168. #9289C83direct (N × H)
177. #1993C2×C83semidirect
178. #1549C2×C83semidirect
180. #1286C2×C83semidirect
181. #1275C2×C83semidirect
182. C83#9289direct (N × H)
186. #492C4×C83semidirect
187. #487C4×C83semidirect
272. C2×C8SD16×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32062
C83×#29322
C83×#19932
C83×#15492
C83×#15172
C83×#12862
C83×#12752
#928983