Finite Group Databasebeta

← C83×#9466 #35237290

Subgroups — C83×#9466 #35237290

Order 21248.

726 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2436C211yesmaximal, characteristic
2C83×#2310C211yesmaximal, characteristic, complemented
3C83×#1784C211yesmaximal, characteristic, complemented
4C83×#1510C212yesmaximal
5C83×#1284C212yesmaximal, complemented
6C23×Q8×C83C2211yescharacteristic
7C2×#124×C83C412yescomplemented
8C2×#121×C83C2211yescharacteristic
9C2×#117×C83—21no—
10C2×#101×C83—22no—
11#407×C83C2212yes—
12#393×C83—22no—
13#391×C83—24no—
14#391×C83C412yescomplemented
15#386×C83—21no—
16#386×C83—21no—
17#381×C83—21no—
18#381×C83C2212yes—
19C4×C22⋊C4×C83C2211yescharacteristic
20#327×C83—24no—
21C22×Q8×C83—22no—
22C22×Q8×C83—22no—
23C22×Q8×C83—44no—
24C22×Q8×C83C2×C412yes—
25C22×Q8×C83D412yes—
26C22×D4×C83—21no—
27C22×D4×C83C2×C411yescharacteristic
28C23×C4×C83D411yescharacteristic
29C23×C4×C83D411yescharacteristic
30C23×C4×C83C2311yescharacteristic
31C2×M16×C83—22no—
32#124×C83—44no—
33#121×C83—22no—
34#121×C83—22no—
35#117×C83—44no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83C2×C411yescharacteristic
38C2×C22⋊C4×C83—22no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—21no—
43C2×C42×C83—22no—
44C2×C42×C83C2×C412yes—
45#101×C83—44no—
46C22⋊C8×C83—22no—
47C22⋊C8×C83—22no—
48#95×C83—24no—
49#95×C83—22no—
50#95×C83—22no—
51#95×C83—21no—
52#95×C83—22no—
53#95×C83D414yes—
54C24×C83—21no—
55C24×C83C2×D411yescharacteristic
56C2×Q8×C83—24no—
57C2×Q8×C83—24no—
58C2×Q8×C83—24no—
59C2×Q8×C83—24no—
60C2×Q8×C83—42no—
61C2×Q8×C83—42no—
62C2×Q8×C83—44no—
63C2×Q8×C83—44no—
64C2×Q8×C83—48no—
65C2×D4×C83—22no—
66C2×D4×C83—42no—
67C2×D4×C83—44no—
68C2×D4×C83—41no—
69C2×D4×C83—41no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—24no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—22no—
77C22×C4×C83—24no—
78C22×C4×C83—42no—
79C22×C4×C83—42no—
80C22×C4×C83C22⋊C412yes—
81C22×C4×C83C22⋊C412yes—
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D412yes—
85C22×C4×C83#4112yes—
86M16×C83—44no—
87C2×C8×C83—42no—
88C4⋊C4×C83—22no—
89C4⋊C4×C83—44no—
90C22⋊C4×C83—44no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—44no—
93C22⋊C4×C83—44no—
94C22⋊C4×C83—42no—
95C42×C83—42no—
96C42×C83—44no—
97C23×C83—22no—
98C23×C83—22no—
99C23×C83—41no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—41no—
103C23×C83—42no—
104C23×C83C2×C22⋊C411yescharacteristic
105C23×C83#12011yescharacteristic
106C23×C83#12311yescharacteristic
107Q8×C83—48no—
108Q8×C83—48no—
109D4×C83—82no—
110D4×C83—82no—
111D4×C83—82no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—24no—
115C2×C4×C83—22no—
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—22no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—44no—
127C2×C4×C83—42no—
128C2×C4×C83—44no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
131C2×C4×C83—41no—
132C2×C4×C83—42no—
133C2×C4×C83—44no—
134C2×C4×C83#9914yes—
135C2×C4×C83C4×D412yes—
136C2×C4×C83#12112yes—
137C8×C83—82no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—44no—
146C22×C83—84no—
147C22×C83—81no—
148C22×C83—81no—
149C22×C83—81no—
150C22×C83#38611yescharacteristic
151C22×C83C2×#9912yes—
152C22×C83#45412yes—
153C4×C83—22no—
154C4×C83—24no—
155C4×C83—22no—
156C4×C83—42no—
157C4×C83—84no—
158C4×C83—82no—
159C4×C83—81no—
160#9466C8311yesmaximal, characteristic, direct product factor, Sylow
161C2×C83—22no—
162C2×C83—22no—
163C2×C83—41no—
164C2×C83—161no—
165C2×C83#188811yescharacteristic
166C2×C83#189811yescharacteristic
167C2×C83#191211yescharacteristic
168#2436C2×C8311yescharacteristic
169#2310C2×C8311yescharacteristic, complemented
170#1784C2×C8311yescharacteristic, complemented
171#1510C2×C8312yes—
172#1284C2×C8312yescomplemented
173C83#946611yescharacteristic, direct product factor, Sylow
174C23×Q8C22×C8311yescharacteristic
175C2×#124C4×C8312yescomplemented
176C2×#121C22×C8311yescharacteristic
177C2×#117—21no—
178C2×#101—22no—
179#407C22×C8312yes—
180#393—22no—
181#391—24no—
182#391C4×C8312yescomplemented
183#386—21no—
184#386—21no—
185#381—21no—
186#381C22×C8312yes—
187C4×C22⋊C4C22×C8311yescharacteristic
188#327—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—44no—
192C22×Q8C2×C4×C8312yes—
193C22×Q8D4×C8312yes—
194C22×D4—21no—
195C22×D4C2×C4×C8311yescharacteristic
196C23×C4D4×C8311yescharacteristic
197C23×C4D4×C8311yescharacteristic
198C23×C4C23×C8311yescharacteristic
199C2×M16—22no—
200#124—44no—
201#121—22no—
202#121—22no—
203#117—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C2×C4×C8311yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42C2×C4×C8312yes—
213#101—44no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—22no—
221#95D4×C8314yes—
222C24—21no—
223C24C2×D4×C8311yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×D4—22no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C22⋊C4×C8312yes—
249C22×C4C22⋊C4×C8312yes—
250C22×C4C22×C4×C8311yescharacteristic
251C22×C4C2×D4×C8311yescharacteristic
252C22×C4C2×D4×C8312yes—
253C22×C4#41×C8312yes—
254M16—44no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C42—42no—
264C42—44no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—41no—
271C23—42no—
272C23C2×C22⋊C4×C8311yescharacteristic
273C23#120×C8311yescharacteristic
274C23#123×C8311yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—82no—
278D4—82no—
279D4—82no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4#99×C8314yes—
303C2×C4C4×D4×C8312yes—
304C2×C4#121×C8312yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—84no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#386×C8311yescharacteristic
319C22C2×#99×C8312yes—
320C22#454×C8312yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—84no—
326C4—82no—
327C4—81no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—161no—
332C2C83×#188811yescharacteristic
333C2C83×#189811yescharacteristic
334C2C83×#191211yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC2×#121×C83C22
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×#2310C2semidirect
3. C83×#1784C2semidirect
5. C83×#1284C2semidirect
7. C2×#124×C83C4semidirect
14. #391×C83C4semidirect
160. #9466C83direct (N × H)
169. #2310C2×C83semidirect
170. #1784C2×C83semidirect
172. #1284C2×C83semidirect
173. C83#9466direct (N × H)
175. C2×#124C4×C83semidirect
182. #391C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#24362
C83×#23102
C83×#17842
C83×#15102
C83×#12842
#946683