Finite Group Databasebeta

← C83×#9169 #35236993

Subgroups — C83×#9169 #35236993

Order 21248.

478 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3079C211yesmaximal, characteristic
2C83×#2906C211yesmaximal, characteristic
3C83×#1549C211yesmaximal, characteristic, complemented
4C2×#333×C83C211yesmaximal, characteristic, complemented
5C83×#1477C211yesmaximal, characteristic
6C83×#1301C211yesmaximal, characteristic, complemented
7C83×#1292C211yesmaximal, characteristic, complemented
8C2×#131×C83—21no—
9C2×C4⋊Q8×C83C2211yescharacteristic
10C4×#41×C83C2211yescharacteristic
11#477×C83—21no—
12#476×C83—21no—
13#475×C83C412yes—
14#474×C83C412yes—
15#431×C83C2211yescharacteristic
16#428×C83—21no—
17#424×C83—21no—
18#424×C83—22no—
19#423×C83—21no—
20#423×C83C2211yescharacteristic
21#423×C83C2211yescharacteristic
22C2×C4⋊C8×C83—21no—
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25#417×C83—21no—
26#333×C83—21no—
27#333×C83—21no—
28#332×C83—22no—
29Q8⋊C8×C83—21no—
30D4⋊C8×C83—21no—
31C2×#41×C83D411yescharacteristic
32C22×Q8×C83—21no—
33#131×C83—42no—
34#131×C83—42no—
35C2×M16×C83—22no—
36C2×M16×C83—21no—
37C2×M16×C83—21no—
38C22×C8×C83—22no—
39C22×C8×C83—21no—
40C22×C8×C83—21no—
41C4⋊Q8×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83C2×C412yes—
45C4×Q8×C83C2×C411yescharacteristic
46C4×Q8×C83D411yescharacteristic
47C4×D4×C83—21no—
48C4×D4×C83—21no—
49C4×D4×C83C2×C411yescharacteristic
50C4×D4×C83D411yescharacteristic
51#117×C83—21no—
52#117×C83D411yescharacteristic
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C42×C83—21no—
56C2×C42×C83C2311yescharacteristic
57#107×C83—22no—
58#106×C83—22no—
59C4⋊C8×C83—22no—
60C4⋊C8×C83—22no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83—21no—
66C4⋊C8×C83—21no—
67C4⋊C8×C83C2×C412yes—
68C4⋊C8×C83D412yes—
69C4⋊C8×C83D411yescharacteristic
70C4⋊C8×C83D411yescharacteristic
71#103×C83—22no—
72#102×C83—22no—
73C22⋊C8×C83—21no—
74C4×C8×C83—21no—
75#41×C83—21no—
76#41×C83—21no—
77#41×C83SD1612yes—
78#41×C83Q1612yes—
79C2×Q8×C83—22no—
80C2×Q8×C83—41no—
81C2×Q8×C83—82no—
82C2×Q8×C83C22⋊C411yescharacteristic
83C2×D4×C83—21no—
84C2×D4×C83C22⋊C411yescharacteristic
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—41no—
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83#4111yescharacteristic
91C22×C4×C83#4111yescharacteristic
92M16×C83—44no—
93M16×C83—41no—
94M16×C83—41no—
95M16×C83—41no—
96M16×C83—41no—
97C2×C8×C83—22no—
98C2×C8×C83—21no—
99C2×C8×C83—22no—
100C2×C8×C83—21no—
101C2×C8×C83—44no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83—81no—
107C4⋊C4×C83—21no—
108C4⋊C4×C83—21no—
109C4⋊C4×C83—22no—
110C4⋊C4×C83—22no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C4⋊C4×C83C22⋊C411yescharacteristic
114C4⋊C4×C83C22⋊C411yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C42×C83—21no—
119C42×C83—21no—
120C42×C83—21no—
121C42×C83C22×C411yescharacteristic
122C42×C83C2×D411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C42×C83C2×D411yescharacteristic
125C23×C83—21no—
126C23×C83#12311yescharacteristic
127Q8×C83—21no—
128Q8×C83—82no—
129Q8×C83—82no—
130Q8×C83#10312yes—
131D4×C83—22no—
132D4×C83—22no—
133D4×C83—21no—
134D4×C83#10312yes—
135C2×C4×C83—21no—
136C2×C4×C83—22no—
137C2×C4×C83—22no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—42no—
146C2×C4×C83—81no—
147C2×C4×C83—82no—
148C2×C4×C83C2×C22⋊C411yescharacteristic
149C2×C4×C83C4×D411yescharacteristic
150C2×C4×C83C4×D411yescharacteristic
151C2×C4×C83#12011yescharacteristic
152C2×C4×C83#12111yescharacteristic
153C2×C4×C83#12111yescharacteristic
154C2×C4×C83C2×SD1611yescharacteristic
155C2×C4×C83C2×Q1611yescharacteristic
156C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C83#13711yescharacteristic
158C8×C83—42no—
159C8×C83—41no—
160C8×C83—42no—
161C8×C83—41no—
162C8×C83—81no—
163C8×C83—81no—
164C22×C83—21no—
165C22×C83—21no—
166C22×C83—22no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83#38611yescharacteristic
170C22×C83#48311yescharacteristic
171C22×C83#48411yescharacteristic
172C4×C83—22no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—82no—
179C4×C83C2×#10311yescharacteristic
180C4×C83#41711yescharacteristic
181C4×C83#45011yescharacteristic
182C4×C83#45111yescharacteristic
183C4×C83#47111yescharacteristic
184C4×C83#47311yescharacteristic
185#9169C8311yesmaximal, characteristic, direct product factor, Sylow
186C2×C83—21no—
187C2×C83—21no—
188C2×C83—42no—
189C2×C83#187211yescharacteristic
190C2×C83#190311yescharacteristic
191C2×C83#190811yescharacteristic
192#3079C2×C8311yescharacteristic
193#2906C2×C8311yescharacteristic
194#1549C2×C8311yescharacteristic, complemented
195C2×#333C2×C8311yescharacteristic, complemented
196#1477C2×C8311yescharacteristic
197#1301C2×C8311yescharacteristic, complemented
198#1292C2×C8311yescharacteristic, complemented
199C83#916911yescharacteristic, direct product factor, Sylow
200C2×#131—21no—
201C2×C4⋊Q8C22×C8311yescharacteristic
202C4×#41C22×C8311yescharacteristic
203#477—21no—
204#476—21no—
205#475C4×C8312yes—
206#474C4×C8312yes—
207#431C22×C8311yescharacteristic
208#428—21no—
209#424—21no—
210#424—22no—
211#423—21no—
212#423C22×C8311yescharacteristic
213#423C22×C8311yescharacteristic
214C2×C4⋊C8—21no—
215C2×C4⋊C8C22×C8311yescharacteristic
216C2×C4⋊C8C22×C8311yescharacteristic
217#417—21no—
218#333—21no—
219#333—21no—
220#332—22no—
221Q8⋊C8—21no—
222D4⋊C8—21no—
223C2×#41D4×C8311yescharacteristic
224C22×Q8—21no—
225#131—42no—
226#131—42no—
227C2×M16—22no—
228C2×M16—21no—
229C2×M16—21no—
230C22×C8—22no—
231C22×C8—21no—
232C22×C8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8C2×C4×C8312yes—
237C4×Q8C2×C4×C8311yescharacteristic
238C4×Q8D4×C8311yescharacteristic
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C2×C4×C8311yescharacteristic
242C4×D4D4×C8311yescharacteristic
243#117—21no—
244#117D4×C8311yescharacteristic
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C42—21no—
248C2×C42C23×C8311yescharacteristic
249#107—22no—
250#106—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8C2×C4×C8312yes—
260C4⋊C8D4×C8312yes—
261C4⋊C8D4×C8311yescharacteristic
262C4⋊C8D4×C8311yescharacteristic
263#103—22no—
264#102—22no—
265C22⋊C8—21no—
266C4×C8—21no—
267#41—21no—
268#41—21no—
269#41SD16×C8312yes—
270#41Q16×C8312yes—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—82no—
274C2×Q8C22⋊C4×C8311yescharacteristic
275C2×D4—21no—
276C2×D4C22⋊C4×C8311yescharacteristic
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4C2×D4×C8311yescharacteristic
282C22×C4#41×C8311yescharacteristic
283C22×C4#41×C8311yescharacteristic
284M16—44no—
285M16—41no—
286M16—41no—
287M16—41no—
288M16—41no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—44no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—81no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4C22⋊C4×C8311yescharacteristic
306C4⋊C4C22⋊C4×C8311yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42C22×C4×C8311yescharacteristic
314C42C2×D4×C8311yescharacteristic
315C42C2×D4×C8311yescharacteristic
316C42C2×D4×C8311yescharacteristic
317C23—21no—
318C23#123×C8311yescharacteristic
319Q8—21no—
320Q8—82no—
321Q8—82no—
322Q8#103×C8312yes—
323D4—22no—
324D4—22no—
325D4—21no—
326D4#103×C8312yes—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4C2×C22⋊C4×C8311yescharacteristic
341C2×C4C4×D4×C8311yescharacteristic
342C2×C4C4×D4×C8311yescharacteristic
343C2×C4#120×C8311yescharacteristic
344C2×C4#121×C8311yescharacteristic
345C2×C4#121×C8311yescharacteristic
346C2×C4C2×SD16×C8311yescharacteristic
347C2×C4C2×Q16×C8311yescharacteristic
348C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
349C2×C4#137×C8311yescharacteristic
350C8—42no—
351C8—41no—
352C8—42no—
353C8—41no—
354C8—81no—
355C8—81no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22#386×C8311yescharacteristic
362C22#483×C8311yescharacteristic
363C22#484×C8311yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—82no—
371C4C2×#103×C8311yescharacteristic
372C4#417×C8311yescharacteristic
373C4#450×C8311yescharacteristic
374C4#451×C8311yescharacteristic
375C4#471×C8311yescharacteristic
376C4#473×C8311yescharacteristic
377C2—21no—
378C2—21no—
379C2—42no—
380C2C83×#187211yescharacteristic
381C2C83×#190311yescharacteristic
382C2C83×#190811yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#41×C83D4
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×#1549C2semidirect
4. C2×#333×C83C2semidirect
6. C83×#1301C2semidirect
7. C83×#1292C2semidirect
185. #9169C83direct (N × H)
194. #1549C2×C83semidirect
195. C2×#333C2×C83semidirect
197. #1301C2×C83semidirect
198. #1292C2×C83semidirect
199. C83#9169direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#30792
C83×#29062
C83×#15492
C2×#333×C832
C83×#14772
C83×#13012
C83×#12922
#916983