Finite Group Databasebeta

← C83×#9287 #35237111

Subgroups — C83×#9287 #35237111

Order 21248.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3227C211yesmaximal, characteristic
2C83×#2932C211yesmaximal, characteristic
3C83×#1993C211yesmaximal, characteristic, complemented
4C83×#1548C211yesmaximal, 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#498×C83C412yescomplemented
12#486×C83C412yescomplemented
13#481×C83—22no—
14#476×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#333×C83—22no—
26#327×C83—22no—
27C2×#41×C83—21no—
28C22×Q8×C83—21no—
29#131×C83—42no—
30#131×C83—42no—
31C2×M16×C83—22no—
32C2×M16×C83—21no—
33C22×C8×C83—22no—
34C22×C8×C83D411yescharacteristic
35C22×C8×C83D411yescharacteristic
36C4⋊Q8×C83—21no—
37C4⋊Q8×C83—21no—
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83C2×C412yes—
40C4⋊Q8×C83C2×C411yescharacteristic
41#127×C83C2×C411yescharacteristic
42#124×C83—21no—
43#121×C83—21no—
44#121×C83D412yes—
45C4×D4×C83—21no—
46C4×D4×C83D412yes—
47C2×C4⋊C4×C83—21no—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C42×C83C2311yescharacteristic
51#106×C83—24no—
52C4⋊C8×C83—21no—
53C4⋊C8×C83—21no—
54C4⋊C8×C83—21no—
55C4⋊C8×C83D412yes—
56#102×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
84M16×C83—44no—
85M16×C83—41no—
86M16×C83—41no—
87C2×C8×C83—21no—
88C2×C8×C83—21no—
89C2×C8×C83—22no—
90C2×C8×C83—21no—
91C2×C8×C83—22no—
92C2×C8×C83—21no—
93C2×C8×C83—21no—
94C2×C8×C83—44no—
95C2×C8×C83—41no—
96C2×C8×C83SD1612yes—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—22no—
99C4⋊C4×C83—22no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83C22⋊C412yes—
105C22⋊C4×C83—22no—
106C22⋊C4×C83—22no—
107C42×C83C22×C411yescharacteristic
108C42×C83C2×D411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83#4111yescharacteristic
111C23×C83—22no—
112C23×C83#12111yescharacteristic
113Q8×C83—41no—
114Q8×C83—41no—
115Q8×C83—82no—
116Q8×C83—82no—
117D4×C83—42no—
118D4×C83—42no—
119D4×C83—42no—
120D4×C83—41no—
121D4×C83—41no—
122D4×C83—81no—
123C2×C4×C83—22no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—42no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—82no—
130C2×C4×C83—81no—
131C2×C4×C83—81no—
132C2×C4×C83—81no—
133C2×C4×C83C2×C22⋊C411yescharacteristic
134C2×C4×C83C4×D411yescharacteristic
135C2×C4×C83C4×D411yescharacteristic
136C2×C4×C83#12011yescharacteristic
137C2×C4×C83#12111yescharacteristic
138C2×C4×C83#12311yescharacteristic
139C2×C4×C83C2×SD1611yescharacteristic
140C2×C4×C83#13511yescharacteristic
141C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C83#13711yescharacteristic
143C8×C83—22no—
144C8×C83—42no—
145C8×C83—42no—
146C8×C83—41no—
147C8×C83—81no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—42no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83#38611yescharacteristic
154C22×C83#46111yescharacteristic
155C22×C83#46311yescharacteristic
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—42no—
161C4×C83—82no—
162C4×C83—161no—
163C4×C83C4×SD1611yescharacteristic
164C4×C83#44011yescharacteristic
165C4×C83#45011yescharacteristic
166C4×C83#45211yescharacteristic
167#9287C8311yesmaximal, characteristic, direct product factor, Sylow
168C2×C83—21no—
169C2×C83—21no—
170C2×C83—82no—
171C2×C83#187711yescharacteristic
172C2×C83#190311yescharacteristic
173C2×C83#190911yescharacteristic
174#3227C2×C8311yescharacteristic
175#2932C2×C8311yescharacteristic
176#1993C2×C8311yescharacteristic, complemented
177#1548C2×C8311yescharacteristic, complemented
178#1517C2×C8311yescharacteristic
179#1286C2×C8311yescharacteristic, complemented
180#1275C2×C8311yescharacteristic, complemented
181C83#928711yescharacteristic, direct product factor, Sylow
182C2×#131—21no—
183#532C22×C8311yescharacteristic
184C2×C4⋊Q8C22×C8311yescharacteristic
185#498C4×C8312yescomplemented
186#486C4×C8312yescomplemented
187#481—22no—
188#476—22no—
189#435—22no—
190#435—22no—
191#431C22×C8311yescharacteristic
192#431C22×C8311yescharacteristic
193#423C22×C8311yescharacteristic
194#408—21no—
195C2×#97C22×C8311yescharacteristic
196#382C22×C8311yescharacteristic
197#340—21no—
198#340—21no—
199#333—22no—
200#327—22no—
201C2×#41—21no—
202C22×Q8—21no—
203#131—42no—
204#131—42no—
205C2×M16—22no—
206C2×M16—21no—
207C22×C8—22no—
208C22×C8D4×C8311yescharacteristic
209C22×C8D4×C8311yescharacteristic
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C2×C4×C8312yes—
214C4⋊Q8C2×C4×C8311yescharacteristic
215#127C2×C4×C8311yescharacteristic
216#124—21no—
217#121—21no—
218#121D4×C8312yes—
219C4×D4—21no—
220C4×D4D4×C8312yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C42C23×C8311yescharacteristic
225#106—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8D4×C8312yes—
230#102—24no—
231C22⋊C8—24no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234#97—21no—
235#97—21no—
236C4×C8—21no—
237C4×C8C2×C4×C8312yes—
238#95—21no—
239#95—21no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊C4×C8312yes—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×D4×C8311yescharacteristic
256C22×C4C2×D4×C8311yescharacteristic
257C22×C4#41×C8311yescharacteristic
258M16—44no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—21no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—44no—
269C2×C8—41no—
270C2×C8SD16×C8312yes—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4C22⋊C4×C8312yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C22×C4×C8311yescharacteristic
282C42C2×D4×C8311yescharacteristic
283C42C2×D4×C8311yescharacteristic
284C42#41×C8311yescharacteristic
285C23—22no—
286C23#121×C8311yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—82no—
290Q8—82no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4C2×C22⋊C4×C8311yescharacteristic
308C2×C4C4×D4×C8311yescharacteristic
309C2×C4C4×D4×C8311yescharacteristic
310C2×C4#120×C8311yescharacteristic
311C2×C4#121×C8311yescharacteristic
312C2×C4#123×C8311yescharacteristic
313C2×C4C2×SD16×C8311yescharacteristic
314C2×C4#135×C8311yescharacteristic
315C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
316C2×C4#137×C8311yescharacteristic
317C8—22no—
318C8—42no—
319C8—42no—
320C8—41no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22#386×C8311yescharacteristic
328C22#461×C8311yescharacteristic
329C22#463×C8311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—82no—
336C4—161no—
337C4C4×SD16×C8311yescharacteristic
338C4#440×C8311yescharacteristic
339C4#450×C8311yescharacteristic
340C4#452×C8311yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2C83×#187711yescharacteristic
345C2C83×#190311yescharacteristic
346C2C83×#190911yescharacteristic
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×#1548C2semidirect
6. C83×#1286C2semidirect
7. C83×#1275C2semidirect
11. #498×C83C4semidirect
12. #486×C83C4semidirect
167. #9287C83direct (N × H)
176. #1993C2×C83semidirect
177. #1548C2×C83semidirect
179. #1286C2×C83semidirect
180. #1275C2×C83semidirect
181. C83#9287direct (N × H)
185. #498C4×C83semidirect
186. #486C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32272
C83×#29322
C83×#19932
C83×#15482
C83×#15172
C83×#12862
C83×#12752
#928783