Finite Group Databasebeta

← C83×#9354 #35237178

Subgroups — C83×#9354 #35237178

Order 21248.

470 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3039C211yesmaximal, characteristic
2C83×#2991C211yesmaximal, characteristic
3C83×#1997C211yesmaximal, characteristic, complemented
4C83×#1548C211yesmaximal, characteristic, complemented
5C83×#1489C211yesmaximal, characteristic
6C83×#1309C211yesmaximal, characteristic, complemented
7C83×#1299C211yesmaximal, characteristic, complemented
8C2×#137×C83C412yes—
9C2×C4⋊Q8×C83C2211yescharacteristic
10C4×#41×C83C2211yescharacteristic
11#462×C83—21no—
12#462×C83—21no—
13#461×C83—21no—
14#461×C83—21no—
15C8×Q8×C83—21no—
16#436×C83—22no—
17#436×C83—21no—
18#431×C83—21no—
19#431×C83C2211yescharacteristic
20#423×C83—21no—
21#423×C83C2211yescharacteristic
22#423×C83C2211yescharacteristic
23#423×C83C2211yescharacteristic
24#419×C83C412yes—
25C4×M16×C83C2211yescharacteristic
26#343×C83—22no—
27#333×C83—21no—
28#333×C83—21no—
29Q8⋊C8×C83—21no—
30D4⋊C8×C83—21no—
31C2×#41×C83C2×C411yescharacteristic
32C22×Q8×C83C2×C412yes—
33#137×C83—44no—
34C2×Q16×C83—22no—
35C2×SD16×C83—22no—
36C2×M16×C83—22no—
37C2×M16×C83—21no—
38C2×M16×C83—21no—
39C2×M16×C83C2×C412yes—
40C2×M16×C83D412yes—
41C4⋊Q8×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83—21no—
45C4×Q8×C83D411yescharacteristic
46C4×Q8×C83D411yescharacteristic
47C4×D4×C83—21no—
48C4×D4×C83—21no—
49C4×D4×C83D411yescharacteristic
50C4×D4×C83D411yescharacteristic
51#117×C83—21no—
52#117×C83C2×C411yescharacteristic
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C42×C83—21no—
56C2×C42×C83C2311yescharacteristic
57C4⋊C8×C83—22no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83D411yescharacteristic
65C4⋊C8×C83D411yescharacteristic
66#103×C83—22no—
67#103×C83—22no—
68C22⋊C8×C83—22no—
69C22⋊C8×C83—21no—
70#97×C83—21no—
71C4×C8×C83—22no—
72C4×C8×C83—21no—
73C4×C8×C83—21no—
74#41×C83—22no—
75#41×C83—41no—
76C2×Q8×C83—22no—
77C2×Q8×C83—22no—
78C2×Q8×C83—42no—
79C2×Q8×C83—42no—
80C2×Q8×C83C22⋊C411yescharacteristic
81C2×D4×C83—21no—
82C2×D4×C83C22⋊C411yescharacteristic
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—22no—
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83#4111yescharacteristic
90Q16×C83—44no—
91SD16×C83—44no—
92M16×C83—22no—
93M16×C83—41no—
94M16×C83—42no—
95M16×C83—42no—
96M16×C83—82no—
97M16×C83—81no—
98C2×C8×C83—22no—
99C2×C8×C83—21no—
100C2×C8×C83—22no—
101C2×C8×C83—21no—
102C2×C8×C83—41no—
103C2×C8×C83—41no—
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—41no—
107C4⋊C4×C83—41no—
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×C83C2×D411yescharacteristic
119C42×C83C2×D411yescharacteristic
120C42×C83C2×D411yescharacteristic
121C42×C83#4111yescharacteristic
122C23×C83—21no—
123C23×C83C4×D411yescharacteristic
124Q8×C83—21no—
125Q8×C83—42no—
126Q8×C83—42no—
127Q8×C83—82no—
128Q8×C83#10012yes—
129D4×C83—21no—
130D4×C83—21no—
131D4×C83—42no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—42no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—42no—
147C2×C4×C83—82no—
148C2×C4×C83C2×C22⋊C411yescharacteristic
149C2×C4×C83C4×D411yescharacteristic
150C2×C4×C83#12011yescharacteristic
151C2×C4×C83#12111yescharacteristic
152C2×C4×C83#12111yescharacteristic
153C2×C4×C83#12311yescharacteristic
154C8×C83—42no—
155C8×C83—42no—
156C8×C83—42no—
157C8×C83—41no—
158C8×C83—41no—
159C8×C83—81no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83#38611yescharacteristic
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—22no—
170C4×C83—21no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—82no—
174C4×C83C2×#10011yescharacteristic
175C4×C83#41311yescharacteristic
176C4×C83#45511yescharacteristic
177C4×C83#45611yescharacteristic
178C4×C83#47111yescharacteristic
179C4×C83#47311yescharacteristic
180#9354C8311yesmaximal, characteristic, direct product factor, Sylow
181C2×C83—41no—
182C2×C83—81no—
183C2×C83#188211yescharacteristic
184C2×C83#190011yescharacteristic
185C2×C83#190811yescharacteristic
186#3039C2×C8311yescharacteristic
187#2991C2×C8311yescharacteristic
188#1997C2×C8311yescharacteristic, complemented
189#1548C2×C8311yescharacteristic, complemented
190#1489C2×C8311yescharacteristic
191#1309C2×C8311yescharacteristic, complemented
192#1299C2×C8311yescharacteristic, complemented
193C83#935411yescharacteristic, direct product factor, Sylow
194C2×#137C4×C8312yes—
195C2×C4⋊Q8C22×C8311yescharacteristic
196C4×#41C22×C8311yescharacteristic
197#462—21no—
198#462—21no—
199#461—21no—
200#461—21no—
201C8×Q8—21no—
202#436—22no—
203#436—21no—
204#431—21no—
205#431C22×C8311yescharacteristic
206#423—21no—
207#423C22×C8311yescharacteristic
208#423C22×C8311yescharacteristic
209#423C22×C8311yescharacteristic
210#419C4×C8312yes—
211C4×M16C22×C8311yescharacteristic
212#343—22no—
213#333—21no—
214#333—21no—
215Q8⋊C8—21no—
216D4⋊C8—21no—
217C2×#41C2×C4×C8311yescharacteristic
218C22×Q8C2×C4×C8312yes—
219#137—44no—
220C2×Q16—22no—
221C2×SD16—22no—
222C2×M16—22no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16C2×C4×C8312yes—
226C2×M16D4×C8312yes—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4×Q8D4×C8311yescharacteristic
232C4×Q8D4×C8311yescharacteristic
233C4×D4—21no—
234C4×D4—21no—
235C4×D4D4×C8311yescharacteristic
236C4×D4D4×C8311yescharacteristic
237#117—21no—
238#117C2×C4×C8311yescharacteristic
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C42—21no—
242C2×C42C23×C8311yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8D4×C8311yescharacteristic
251C4⋊C8D4×C8311yescharacteristic
252#103—22no—
253#103—22no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256#97—21no—
257C4×C8—22no—
258C4×C8—21no—
259C4×C8—21no—
260#41—22no—
261#41—41no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8C22⋊C4×C8311yescharacteristic
267C2×D4—21no—
268C2×D4C22⋊C4×C8311yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4C22×C4×C8311yescharacteristic
274C22×C4C2×D4×C8311yescharacteristic
275C22×C4#41×C8311yescharacteristic
276Q16—44no—
277SD16—44no—
278M16—22no—
279M16—41no—
280M16—42no—
281M16—42no—
282M16—82no—
283M16—81no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4C22⋊C4×C8311yescharacteristic
297C4⋊C4C22⋊C4×C8311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42C2×D4×C8311yescharacteristic
305C42C2×D4×C8311yescharacteristic
306C42C2×D4×C8311yescharacteristic
307C42#41×C8311yescharacteristic
308C23—21no—
309C23C4×D4×C8311yescharacteristic
310Q8—21no—
311Q8—42no—
312Q8—42no—
313Q8—82no—
314Q8#100×C8312yes—
315D4—21no—
316D4—21no—
317D4—42no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4C2×C22⋊C4×C8311yescharacteristic
335C2×C4C4×D4×C8311yescharacteristic
336C2×C4#120×C8311yescharacteristic
337C2×C4#121×C8311yescharacteristic
338C2×C4#121×C8311yescharacteristic
339C2×C4#123×C8311yescharacteristic
340C8—42no—
341C8—42no—
342C8—42no—
343C8—41no—
344C8—41no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#386×C8311yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—82no—
360C4C2×#100×C8311yescharacteristic
361C4#413×C8311yescharacteristic
362C4#455×C8311yescharacteristic
363C4#456×C8311yescharacteristic
364C4#471×C8311yescharacteristic
365C4#473×C8311yescharacteristic
366C2—41no—
367C2—81no—
368C2C83×#188211yescharacteristic
369C2C83×#190011yescharacteristic
370C2C83×#190811yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC2×#41×C83C2×C4
LCSG ▷ C22×C4 ▷ 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×#1997C2semidirect
4. C83×#1548C2semidirect
6. C83×#1309C2semidirect
7. C83×#1299C2semidirect
180. #9354C83direct (N × H)
188. #1997C2×C83semidirect
189. #1548C2×C83semidirect
191. #1309C2×C83semidirect
192. #1299C2×C83semidirect
193. C83#9354direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#30392
C83×#29912
C83×#19972
C83×#15482
C83×#14892
C83×#13092
C83×#12992
#935483