Finite Group Databasebeta

← C83×#9168 #35236992

Subgroups — C83×#9168 #35236992

Order 21248.

432 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3079C211yesmaximal, characteristic
2C83×#2925C211yesmaximal, characteristic
3C83×#1548C211yesmaximal, characteristic, complemented
4C2×#332×C83C211yesmaximal, characteristic, complemented
5C83×#1477C211yesmaximal, characteristic
6C83×#1301C211yesmaximal, characteristic, complemented
7C83×#1292C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C83C2211yescharacteristic
9C4×#41×C83C2211yescharacteristic
10#477×C83C412yes—
11#476×C83C412yes—
12#475×C83—21no—
13#474×C83—21no—
14#433×C83—22no—
15#432×C83—21no—
16#431×C83—21no—
17#431×C83C2211yescharacteristic
18#428×C83—21no—
19#423×C83C2211yescharacteristic
20#423×C83C2211yescharacteristic
21C2×C4⋊C8×C83C2211yescharacteristic
22C2×C4⋊C8×C83C2211yescharacteristic
23#417×C83—21no—
24#408×C83—21no—
25#405×C83—21no—
26#333×C83—22no—
27#332×C83—21no—
28#332×C83—21no—
29Q8⋊C8×C83—21no—
30D4⋊C8×C83—21no—
31C2×#41×C83D411yescharacteristic
32C22×Q8×C83—21no—
33C2×M16×C83—21no—
34C2×M16×C83—21no—
35C22×C8×C83—21no—
36C22×C8×C83—21no—
37C4⋊Q8×C83—21no—
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—21no—
40C4⋊Q8×C83C2×C412yes—
41C4×Q8×C83C2×C411yescharacteristic
42C4×Q8×C83D411yescharacteristic
43C4×D4×C83—21no—
44C4×D4×C83—21no—
45C4×D4×C83C2×C411yescharacteristic
46C4×D4×C83D411yescharacteristic
47#117×C83—21no—
48#117×C83D411yescharacteristic
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C42×C83—21no—
52C2×C42×C83C2311yescharacteristic
53#107×C83—22no—
54#106×C83—22no—
55C4⋊C8×C83—21no—
56C4⋊C8×C83—21no—
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83C2×C412yes—
62C4⋊C8×C83D411yescharacteristic
63C4⋊C8×C83D411yescharacteristic
64C4⋊C8×C83D412yes—
65#103×C83—22no—
66#102×C83—22no—
67C22⋊C8×C83—22no—
68C22⋊C8×C83—22no—
69C22⋊C8×C83—21no—
70#97×C83—22no—
71C4×C8×C83—22no—
72C4×C8×C83—21no—
73#41×C83—21no—
74#41×C83—21no—
75#41×C83—21no—
76#41×C83—21no—
77C2×Q8×C83—22no—
78C2×Q8×C83—41no—
79C2×Q8×C83—81no—
80C2×Q8×C83—81no—
81C2×Q8×C83C22⋊C411yescharacteristic
82C2×D4×C83—21no—
83C2×D4×C83C22⋊C411yescharacteristic
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—41no—
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83#4111yescharacteristic
90C22×C4×C83#4111yescharacteristic
91M16×C83—41no—
92M16×C83—41no—
93M16×C83—41no—
94M16×C83—41no—
95C2×C8×C83—22no—
96C2×C8×C83—21no—
97C2×C8×C83—22no—
98C2×C8×C83—21no—
99C2×C8×C83—41no—
100C2×C8×C83—41no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83—81no—
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—22no—
107C4⋊C4×C83—22no—
108C4⋊C4×C83—41no—
109C4⋊C4×C83—41no—
110C4⋊C4×C83C22⋊C411yescharacteristic
111C4⋊C4×C83C22⋊C411yescharacteristic
112C22⋊C4×C83—21no—
113C22⋊C4×C83—21no—
114C22⋊C4×C83—21no—
115C42×C83—21no—
116C42×C83—21no—
117C42×C83—21no—
118C42×C83C22×C411yescharacteristic
119C42×C83C2×D411yescharacteristic
120C42×C83C2×D411yescharacteristic
121C42×C83C2×D411yescharacteristic
122C23×C83—21no—
123C23×C83#12311yescharacteristic
124Q8×C83—21no—
125Q8×C83—21no—
126Q8×C83—82no—
127Q8×C83—81no—
128Q8×C83—81no—
129D4×C83—21no—
130D4×C83—21no—
131D4×C83—41no—
132D4×C83—41no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—42no—
144C2×C4×C83—81no—
145C2×C4×C83—82no—
146C2×C4×C83C2×C22⋊C411yescharacteristic
147C2×C4×C83C4×D411yescharacteristic
148C2×C4×C83C4×D411yescharacteristic
149C2×C4×C83#12011yescharacteristic
150C2×C4×C83#12111yescharacteristic
151C2×C4×C83#12111yescharacteristic
152C2×C4×C83#13511yescharacteristic
153C2×C4×C83#13511yescharacteristic
154C2×C4×C83#13711yescharacteristic
155C2×C4×C83#13711yescharacteristic
156C8×C83—42no—
157C8×C83—41no—
158C8×C83—41no—
159C8×C83—42no—
160C8×C83—81no—
161C8×C83—81no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83#38611yescharacteristic
168C22×C83#48511yescharacteristic
169C22×C83#48511yescharacteristic
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83—41no—
176C4×C83—82no—
177C4×C83#41611yescharacteristic
178C4×C83#41911yescharacteristic
179C4×C83#45211yescharacteristic
180C4×C83#45211yescharacteristic
181C4×C83#47111yescharacteristic
182C4×C83#47211yescharacteristic
183#9168C8311yesmaximal, characteristic, direct product factor, Sylow
184C2×C83—21no—
185C2×C83—21no—
186C2×C83—81no—
187C2×C83#187311yescharacteristic
188C2×C83#190211yescharacteristic
189C2×C83#190711yescharacteristic
190#3079C2×C8311yescharacteristic
191#2925C2×C8311yescharacteristic
192#1548C2×C8311yescharacteristic, complemented
193C2×#332C2×C8311yescharacteristic, complemented
194#1477C2×C8311yescharacteristic
195#1301C2×C8311yescharacteristic, complemented
196#1292C2×C8311yescharacteristic, complemented
197C83#916811yescharacteristic, direct product factor, Sylow
198C2×C4⋊Q8C22×C8311yescharacteristic
199C4×#41C22×C8311yescharacteristic
200#477C4×C8312yes—
201#476C4×C8312yes—
202#475—21no—
203#474—21no—
204#433—22no—
205#432—21no—
206#431—21no—
207#431C22×C8311yescharacteristic
208#428—21no—
209#423C22×C8311yescharacteristic
210#423C22×C8311yescharacteristic
211C2×C4⋊C8C22×C8311yescharacteristic
212C2×C4⋊C8C22×C8311yescharacteristic
213#417—21no—
214#408—21no—
215#405—21no—
216#333—22no—
217#332—21no—
218#332—21no—
219Q8⋊C8—21no—
220D4⋊C8—21no—
221C2×#41D4×C8311yescharacteristic
222C22×Q8—21no—
223C2×M16—21no—
224C2×M16—21no—
225C22×C8—21no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8C2×C4×C8312yes—
231C4×Q8C2×C4×C8311yescharacteristic
232C4×Q8D4×C8311yescharacteristic
233C4×D4—21no—
234C4×D4—21no—
235C4×D4C2×C4×C8311yescharacteristic
236C4×D4D4×C8311yescharacteristic
237#117—21no—
238#117D4×C8311yescharacteristic
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C42—21no—
242C2×C42C23×C8311yescharacteristic
243#107—22no—
244#106—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8C2×C4×C8312yes—
252C4⋊C8D4×C8311yescharacteristic
253C4⋊C8D4×C8311yescharacteristic
254C4⋊C8D4×C8312yes—
255#103—22no—
256#102—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—21no—
260#97—22no—
261C4×C8—22no—
262C4×C8—21no—
263#41—21no—
264#41—21no—
265#41—21no—
266#41—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8C22⋊C4×C8311yescharacteristic
272C2×D4—21no—
273C2×D4C22⋊C4×C8311yescharacteristic
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4C2×D4×C8311yescharacteristic
279C22×C4#41×C8311yescharacteristic
280C22×C4#41×C8311yescharacteristic
281M16—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—81no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4C22⋊C4×C8311yescharacteristic
301C4⋊C4C22⋊C4×C8311yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42C22×C4×C8311yescharacteristic
309C42C2×D4×C8311yescharacteristic
310C42C2×D4×C8311yescharacteristic
311C42C2×D4×C8311yescharacteristic
312C23—21no—
313C23#123×C8311yescharacteristic
314Q8—21no—
315Q8—21no—
316Q8—82no—
317Q8—81no—
318Q8—81no—
319D4—21no—
320D4—21no—
321D4—41no—
322D4—41no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4C2×C22⋊C4×C8311yescharacteristic
337C2×C4C4×D4×C8311yescharacteristic
338C2×C4C4×D4×C8311yescharacteristic
339C2×C4#120×C8311yescharacteristic
340C2×C4#121×C8311yescharacteristic
341C2×C4#121×C8311yescharacteristic
342C2×C4#135×C8311yescharacteristic
343C2×C4#135×C8311yescharacteristic
344C2×C4#137×C8311yescharacteristic
345C2×C4#137×C8311yescharacteristic
346C8—42no—
347C8—41no—
348C8—41no—
349C8—42no—
350C8—81no—
351C8—81no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22#386×C8311yescharacteristic
358C22#485×C8311yescharacteristic
359C22#485×C8311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—82no—
367C4#416×C8311yescharacteristic
368C4#419×C8311yescharacteristic
369C4#452×C8311yescharacteristic
370C4#452×C8311yescharacteristic
371C4#471×C8311yescharacteristic
372C4#472×C8311yescharacteristic
373C2—21no—
374C2—21no—
375C2—81no—
376C2C83×#187311yescharacteristic
377C2C83×#190211yescharacteristic
378C2C83×#190711yescharacteristic
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×#1548C2semidirect
4. C2×#332×C83C2semidirect
6. C83×#1301C2semidirect
7. C83×#1292C2semidirect
183. #9168C83direct (N × H)
192. #1548C2×C83semidirect
193. C2×#332C2×C83semidirect
195. #1301C2×C83semidirect
196. #1292C2×C83semidirect
197. C83#9168direct (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×#29252
C83×#15482
C2×#332×C832
C83×#14772
C83×#13012
C83×#12922
#916883