Finite Group Databasebeta

← C83×#11580 #35239404

Subgroups — C83×#11580 #35239404

Order 21248.

552 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2409C211yesmaximal, characteristic
2C83×#1465C211yesmaximal, characteristic
3C83×#1460C211yesmaximal, characteristic
4C83×#1326C214yesmaximal, complemented
5C23×Q8×C83C411yescharacteristic
6C2×C22⋊Q8×C83—21no—
7C2×C4×D4×C83C2211yescharacteristic
8#431×C83—21no—
9#407×C83—21no—
10#395×C83—21no—
11#393×C83—21no—
12#391×C83—21no—
13#386×C83C412yes—
14#385×C83C2212yes—
15#380×C83C411yescharacteristic
16#324×C83—21no—
17#323×C83—21no—
18#323×C83C2212yes—
19#323×C83C2212yes—
20C22×Q8×C83—21no—
21C22×Q8×C83—21no—
22C22×Q8×C83—41no—
23C22×Q8×C83—41no—
24C22×Q8×C83—44no—
25C22×D4×C83C2×C411yescharacteristic
26C23×C4×C83C2×C411yescharacteristic
27C23×C4×C83C2×C412yes—
28C23×C4×C83D411yescharacteristic
29C2×M16×C83—41no—
30C22⋊Q8×C83—44no—
31C4×D4×C83—41no—
32C4×D4×C83—41no—
33C2×C4⋊C4×C83—21no—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83D411yescharacteristic
36C2×C22⋊C4×C83—22no—
37C2×C22⋊C4×C83C2×C412yes—
38C2×C22⋊C4×C83D411yescharacteristic
39C2×C22⋊C4×C83C2311yescharacteristic
40C2×C42×C83D411yescharacteristic
41C4⋊C8×C83—41no—
42C22⋊C8×C83—22no—
43C22⋊C8×C83—22no—
44C22⋊C8×C83—41no—
45C4×C8×C83—41no—
46#95×C83—21no—
47#95×C83—22no—
48#95×C83—22no—
49C24×C83C22⋊C411yescharacteristic
50C24×C83C22×C411yescharacteristic
51C2×Q8×C83—24no—
52C2×Q8×C83—42no—
53C2×Q8×C83—44no—
54C2×Q8×C83—42no—
55C2×Q8×C83—44no—
56C2×Q8×C83—48no—
57C2×Q8×C83—44no—
58C2×Q8×C83—41no—
59C2×Q8×C83—41no—
60C2×D4×C83—22no—
61C2×D4×C83—22no—
62C2×D4×C83—41no—
63C22×C4×C83—22no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—22no—
69C22×C4×C83—42no—
70C22×C4×C83—44no—
71C22×C4×C83—42no—
72C22×C4×C83—42no—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C2×D411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78M16×C83—82no—
79C2×C8×C83—42no—
80C2×C8×C83—42no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—42no—
83C4⋊C4×C83—44no—
84C22⋊C4×C83—24no—
85C22⋊C4×C83—41no—
86C22⋊C4×C83—41no—
87C22⋊C4×C83—44no—
88C42×C83—21no—
89C42×C83—21no—
90C23×C83—21no—
91C23×C83—21no—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—41no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83C2×C22⋊C411yescharacteristic
100C23×C83#13112yes—
101Q8×C83—44no—
102Q8×C83—48no—
103Q8×C83—44no—
104D4×C83—44no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—24no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83—44no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—44no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—81no—
126C2×C4×C83#9911yescharacteristic
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83#10112yes—
129C8×C83—82no—
130C8×C83—82no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83—22no—
134C22×C83—42no—
135C22×C83—42no—
136C22×C83—42no—
137C22×C83—41no—
138C22×C83—41no—
139C22×C83—41no—
140C22×C83—41no—
141C22×C83—41no—
142C22×C83—81no—
143C22×C83#40811yescharacteristic
144C22×C83C2×#9911yescharacteristic
145C22×C83C2×#10111yescharacteristic
146C4×C83—21no—
147C4×C83—22no—
148C4×C83—41no—
149C4×C83—44no—
150C4×C83—41no—
151C4×C83—41no—
152C4×C83—82no—
153C4×C83AΣL(1,𝔽16)12yes—
154#11580C8311yesmaximal, characteristic, direct product factor, Sylow
155C2×C83—21no—
156C2×C83—21no—
157C2×C83—42no—
158C2×C83—81no—
159C2×C83#147211yescharacteristic
160C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
161C2×C83#212311yescharacteristic
162#2409C2×C8311yescharacteristic
163#1465C2×C8311yescharacteristic
164#1460C2×C8311yescharacteristic
165#1326C2×C8314yescomplemented
166C83#1158011yescharacteristic, direct product factor, Sylow
167C23×Q8C4×C8311yescharacteristic
168C2×C22⋊Q8—21no—
169C2×C4×D4C22×C8311yescharacteristic
170#431—21no—
171#407—21no—
172#395—21no—
173#393—21no—
174#391—21no—
175#386C4×C8312yes—
176#385C22×C8312yes—
177#380C4×C8311yescharacteristic
178#324—21no—
179#323—21no—
180#323C22×C8312yes—
181#323C22×C8312yes—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—44no—
187C22×D4C2×C4×C8311yescharacteristic
188C23×C4C2×C4×C8311yescharacteristic
189C23×C4C2×C4×C8312yes—
190C23×C4D4×C8311yescharacteristic
191C2×M16—41no—
192C22⋊Q8—44no—
193C4×D4—41no—
194C4×D4—41no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4D4×C8311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×C4×C8312yes—
200C2×C22⋊C4D4×C8311yescharacteristic
201C2×C22⋊C4C23×C8311yescharacteristic
202C2×C42D4×C8311yescharacteristic
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207C4×C8—41no—
208#95—21no—
209#95—22no—
210#95—22no—
211C24C22⋊C4×C8311yescharacteristic
212C24C22×C4×C8311yescharacteristic
213C2×Q8—24no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C22⋊C4×C8311yescharacteristic
236C22×C4C22⋊C4×C8311yescharacteristic
237C22×C4C22⋊C4×C8311yescharacteristic
238C22×C4C2×D4×C8311yescharacteristic
239C22×C4C2×D4×C8311yescharacteristic
240M16—82no—
241C2×C8—42no—
242C2×C8—42no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—44no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×C22⋊C4×C8311yescharacteristic
262C23#131×C8312yes—
263Q8—44no—
264Q8—48no—
265Q8—44no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4#99×C8311yescharacteristic
289C2×C4#99×C8311yescharacteristic
290C2×C4#101×C8312yes—
291C8—82no—
292C8—82no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22#408×C8311yescharacteristic
306C22C2×#99×C8311yescharacteristic
307C22C2×#101×C8311yescharacteristic
308C4—21no—
309C4—22no—
310C4—41no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—82no—
315C4AΣL(1,𝔽16)×C8312yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—81no—
320C2C83×#147211yescharacteristic
321C2C2×AΣL(1,𝔽16)×C8311yescharacteristic
322C2C83×#212311yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC24C22×C4×C83
Frattini, cosocleC2×C22⋊C4C23×C83
prime radicalC22×D4×C83C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C22×C4×C83 ◁ C2×C4×D4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
4. C83×#1326C2semidirect
154. #11580C83direct (N × H)
165. #1326C2×C83semidirect
166. C83#11580direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#24092
C83×#14652
C83×#14602
C83×#13262
#1158083