Finite Group Databasebeta

← C83×#9413 #35237237

Subgroups — C83×#9413 #35237237

Order 21248.

720 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2539C211yesmaximal, characteristic
2C83×#2312C211yesmaximal, characteristic, complemented
3C83×#1507C212yesmaximal, complemented
4C83×#1462C211yesmaximal, characteristic, complemented
5C83×#1297C212yesmaximal, complemented
6C2×#123×C83—21no—
7C2×#121×C83C2211yescharacteristic
8C2×#120×C83—21no—
9C22×C22⋊C4×C83C2211yescharacteristic
10#407×C83C2212yes—
11#399×C83C412yescomplemented
12#392×C83C412yescomplemented
13#390×C83—22no—
14#388×C83—24no—
15#380×C83—21no—
16#379×C83—21no—
17C4×C22⋊C4×C83C2211yescharacteristic
18C4×C22⋊C4×C83C2212yes—
19#327×C83—24no—
20#323×C83—22no—
21C25×C83D411yescharacteristic
22C22×D4×C83—21no—
23C22×D4×C83—22no—
24C22×D4×C83C2×C411yescharacteristic
25C23×C4×C83D411yescharacteristic
26C23×C4×C83C2311yescharacteristic
27C2×M16×C83—22no—
28#123×C83—44no—
29#121×C83—22no—
30#121×C83—22no—
31#120×C83—44no—
32C2×C4⋊C4×C83—22no—
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83C2×C411yescharacteristic
35C2×C22⋊C4×C83—21no—
36C2×C22⋊C4×C83—22no—
37C2×C22⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—22no—
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83C2×C412yes—
45C2×C22⋊C4×C83D412yes—
46C2×C42×C83—22no—
47C2×C42×C83—21no—
48C22⋊C8×C83—22no—
49C22⋊C8×C83—22no—
50#95×C83—22no—
51#95×C83C2×C412yes—
52#95×C83D414yescomplemented
53C24×C83—21no—
54C24×C83—21no—
55C24×C83—21no—
56C24×C83—42no—
57C24×C83—42no—
58C24×C83—81no—
59C24×C83C22⋊C412yes—
60C24×C83C2×D411yescharacteristic
61C2×D4×C83—22no—
62C2×D4×C83—44no—
63C2×D4×C83—42no—
64C2×D4×C83—42no—
65C2×D4×C83—44no—
66C2×D4×C83—41no—
67C2×D4×C83—41no—
68C2×D4×C83—82no—
69C2×D4×C83—81no—
70C22×C4×C83—22no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—24no—
75C22×C4×C83—42no—
76C22×C4×C83—81no—
77C22×C4×C83C22⋊C412yes—
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D412yes—
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83#4112yes—
82M16×C83—82no—
83C2×C8×C83—42no—
84C4⋊C4×C83—22no—
85C4⋊C4×C83—44no—
86C4⋊C4×C83—82no—
87C22⋊C4×C83—24no—
88C22⋊C4×C83—24no—
89C22⋊C4×C83—24no—
90C22⋊C4×C83—44no—
91C22⋊C4×C83—44no—
92C22⋊C4×C83—44no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—82no—
95C22⋊C4×C83—82no—
96C42×C83—81no—
97C42×C83—82no—
98C23×C83—22no—
99C23×C83—22no—
100C23×C83—22no—
101C23×C83—22no—
102C23×C83—22no—
103C23×C83—41no—
104C23×C83—42no—
105C23×C83—42no—
106C23×C83—42no—
107C23×C83—42no—
108C23×C83—42no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83—42no—
112C23×C83—42no—
113C23×C83—41no—
114C23×C83—42no—
115C23×C83—81no—
116C23×C83—81no—
117C23×C83—88no—
118C23×C83C2×C22⋊C411yescharacteristic
119C23×C83#12011yescharacteristic
120C23×C83#12311yescharacteristic
121D4×C83—82no—
122D4×C83—88no—
123D4×C83—82no—
124D4×C83—82no—
125C2×C4×C83—24no—
126C2×C4×C83—22no—
127C2×C4×C83—41no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—41no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83—42no—
135C2×C4×C83—44no—
136C2×C4×C83—41no—
137C2×C4×C83—44no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—44no—
141C2×C4×C83—81no—
142C2×C4×C83C4×D412yes—
143C2×C4×C83#12112yes—
144C8×C83—82no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—44no—
153C22×C83—44no—
154C22×C83—42no—
155C22×C83—42no—
156C22×C83—42no—
157C22×C83—82no—
158C22×C83—84no—
159C22×C83—82no—
160C22×C83—82no—
161C22×C83—84no—
162C22×C83—81no—
163C22×C83—81no—
164C22×C83—81no—
165C22×C83#38611yescharacteristic
166C22×C83#42112yes—
167C22×C83#45312yes—
168C4×C83—22no—
169C4×C83—22no—
170C4×C83—44no—
171C4×C83—42no—
172C4×C83—84no—
173C4×C83—81no—
174#9413C8311yesmaximal, characteristic, direct product factor, Sylow
175C2×C83—22no—
176C2×C83—22no—
177C2×C83—41no—
178C2×C83—82no—
179C2×C83—161no—
180C2×C83#188711yescharacteristic
181C2×C83#189711yescharacteristic
182C2×C83#191311yescharacteristic
183#2539C2×C8311yescharacteristic
184#2312C2×C8311yescharacteristic, complemented
185#1507C2×C8312yescomplemented
186#1462C2×C8311yescharacteristic, complemented
187#1297C2×C8312yescomplemented
188C83#941311yescharacteristic, direct product factor, Sylow
189C2×#123—21no—
190C2×#121C22×C8311yescharacteristic
191C2×#120—21no—
192C22×C22⋊C4C22×C8311yescharacteristic
193#407C22×C8312yes—
194#399C4×C8312yescomplemented
195#392C4×C8312yescomplemented
196#390—22no—
197#388—24no—
198#380—21no—
199#379—21no—
200C4×C22⋊C4C22×C8311yescharacteristic
201C4×C22⋊C4C22×C8312yes—
202#327—24no—
203#323—22no—
204C25D4×C8311yescharacteristic
205C22×D4—21no—
206C22×D4—22no—
207C22×D4C2×C4×C8311yescharacteristic
208C23×C4D4×C8311yescharacteristic
209C23×C4C23×C8311yescharacteristic
210C2×M16—22no—
211#123—44no—
212#121—22no—
213#121—22no—
214#120—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C2×C4×C8311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C2×C4×C8312yes—
228C2×C22⋊C4D4×C8312yes—
229C2×C42—22no—
230C2×C42—21no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233#95—22no—
234#95C2×C4×C8312yes—
235#95D4×C8314yescomplemented
236C24—21no—
237C24—21no—
238C24—21no—
239C24—42no—
240C24—42no—
241C24—81no—
242C24C22⋊C4×C8312yes—
243C24C2×D4×C8311yescharacteristic
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—82no—
252C2×D4—81no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4C22⋊C4×C8312yes—
261C22×C4C22×C4×C8311yescharacteristic
262C22×C4C2×D4×C8312yes—
263C22×C4C2×D4×C8311yescharacteristic
264C22×C4#41×C8312yes—
265M16—82no—
266C2×C8—42no—
267C4⋊C4—22no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—82no—
278C22⋊C4—82no—
279C42—81no—
280C42—82no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—41no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—88no—
301C23C2×C22⋊C4×C8311yescharacteristic
302C23#120×C8311yescharacteristic
303C23#123×C8311yescharacteristic
304D4—82no—
305D4—88no—
306D4—82no—
307D4—82no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—81no—
325C2×C4C4×D4×C8312yes—
326C2×C4#121×C8312yes—
327C8—82no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#386×C8311yescharacteristic
349C22#421×C8312yes—
350C22#453×C8312yes—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—42no—
355C4—84no—
356C4—81no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—161no—
362C2C83×#188711yescharacteristic
363C2C83×#189711yescharacteristic
364C2C83×#191311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC83×#2539C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2312C2semidirect
3. C83×#1507C2semidirect
4. C83×#1462C2semidirect
5. C83×#1297C2semidirect
11. #399×C83C4semidirect
12. #392×C83C4semidirect
52. #95×C83D4semidirect
174. #9413C83direct (N × H)
184. #2312C2×C83semidirect
185. #1507C2×C83semidirect
186. #1462C2×C83semidirect
187. #1297C2×C83semidirect
188. C83#9413direct (N × H)
194. #399C4×C83semidirect
195. #392C4×C83semidirect
235. #95D4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#25392
C83×#23122
C83×#15072
C83×#14622
C83×#12972
#941383