Finite Group Databasebeta

← C2×C83×#1317 #35239248

Subgroups — C2×C83×#1317 #35239248

Order 21248.

726 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#381×C83C211yesmaximal, characteristic
2C2×#323×C83C212yesmaximal
3C83×#1317C214yesmaximal, direct product factor
4C24×C4×C83C411yescharacteristic
5C22×C22⋊C4×C83C411yescharacteristic
6C22×C22⋊C4×C83C2211yescharacteristic
7C2×C22⋊C8×C83—22no—
8#381×C83—22no—
9#381×C83C412yes—
10#381×C83C2212yes—
11C2×#95×C83—22no—
12#323×C83—22no—
13#323×C83C2214yes—
14C25×C83C2×C411yescharacteristic
15C23×C4×C83—21no—
16C23×C4×C83—21no—
17C23×C4×C83—22no—
18C23×C4×C83—21no—
19C23×C4×C83—41no—
20C23×C4×C83—41no—
21C23×C4×C83—42no—
22C23×C4×C83C814yescomplemented
23C23×C4×C83C2×C412yes—
24C23×C4×C83D412yes—
25C22×C8×C83—42no—
26C2×C22⋊C4×C83—21no—
27C2×C22⋊C4×C83—41no—
28C2×C22⋊C4×C83—41no—
29C2×C22⋊C4×C83—42no—
30C2×C22⋊C4×C83C2×C412yes—
31C2×C22⋊C4×C83C2×C411yescharacteristic
32C2×C22⋊C4×C83D412yes—
33C2×C22⋊C4×C83C2311yescharacteristic
34C22⋊C8×C83—24no—
35C22⋊C8×C83—42no—
36#95×C83—24no—
37#95×C83—24no—
38C24×C83—21no—
39C24×C83—21no—
40C24×C83—22no—
41C24×C83—44no—
42C24×C83C22⋊C411yescharacteristic
43C24×C83C22⋊C411yescharacteristic
44C24×C83C2×C812yes—
45C24×C83M1612yes—
46C24×C83C22×C411yescharacteristic
47C22×C4×C83—24no—
48C22×C4×C83—24no—
49C22×C4×C83—22no—
50C22×C4×C83—22no—
51C22×C4×C83—22no—
52C22×C4×C83—22no—
53C22×C4×C83—22no—
54C22×C4×C83—44no—
55C22×C4×C83—44no—
56C22×C4×C83—42no—
57C22×C4×C83—42no—
58C22×C4×C83—41no—
59C22×C4×C83—41no—
60C22×C4×C83—42no—
61C22×C4×C83—44no—
62C22×C4×C83—44no—
63C22×C4×C83—42no—
64C22×C4×C83—42no—
65C22×C4×C83—44no—
66C22×C4×C83—44no—
67C22×C4×C83—42no—
68C22×C4×C83C22⋊C412yes—
69C22×C4×C83C2×C814yescomplemented
70C22×C4×C83C2×D412yes—
71C2×C8×C83—44no—
72C2×C8×C83—84no—
73C22⋊C4×C83—41no—
74C22⋊C4×C83—42no—
75C22⋊C4×C83—41no—
76C22⋊C4×C83—42no—
77C22⋊C4×C83—42no—
78C23×C83—24no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—21no—
84C23×C83—21no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—42no—
88C23×C83—44no—
89C23×C83—44no—
90C23×C83—42no—
91C23×C83—48no—
92C23×C83—44no—
93C23×C83—44no—
94C23×C83C22⋊C812yes—
95C23×C83C22⋊C812yes—
96C23×C83#9911yescharacteristic
97C23×C83#9911yescharacteristic
98C23×C83#10012yes—
99C23×C83C2×C22⋊C411yescharacteristic
100C23×C83C22×C811yescharacteristic
101C23×C83C2×M1611yescharacteristic
102C2×C4×C83—24no—
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83—48no—
106C2×C4×C83—44no—
107C2×C4×C83—44no—
108C2×C4×C83—44no—
109C2×C4×C83—44no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83—44no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—44no—
116C2×C4×C83—42no—
117C2×C4×C83—82no—
118C2×C4×C83—84no—
119C8×C83—84no—
120C22×C83—24no—
121C22×C83—24no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—22no—
125C22×C83—21no—
126C22×C83—21no—
127C22×C83—42no—
128C22×C83—41no—
129C22×C83—41no—
130C22×C83—42no—
131C22×C83—42no—
132C22×C83—44no—
133C22×C83—44no—
134C22×C83—44no—
135C22×C83—44no—
136C22×C83—42no—
137C22×C83—42no—
138C22×C83#32312yes—
139C22×C83#32312yes—
140C22×C83#35214yes—
141C22×C83C2×C22⋊C811yescharacteristic
142C22×C83C2×#9911yescharacteristic
143C22×C83C2×#10011yescharacteristic
144C4×C83—42no—
145C4×C83—42no—
146C4×C83—44no—
147C4×C83—82no—
148C2×#1317C8311yesmaximal, characteristic, direct product factor, Sylow
149C2×C83—21no—
150C2×C83—22no—
151C2×C83—21no—
152C2×C83—42no—
153C2×C83—41no—
154C2×C83—41no—
155C2×C83#131714yesdirect product factor
156C2×C83C2×#32311yescharacteristic
157C2×C83C2×#35212yes—
158C2×#381C2×C8311yescharacteristic
159C2×#323C2×C8312yes—
160#1317C2×C8314yesdirect product factor
161C83C2×#131711yescharacteristic, direct product factor, Sylow
162C24×C4C4×C8311yescharacteristic
163C22×C22⋊C4C4×C8311yescharacteristic
164C22×C22⋊C4C22×C8311yescharacteristic
165C2×C22⋊C8—22no—
166#381—22no—
167#381C4×C8312yes—
168#381C22×C8312yes—
169C2×#95—22no—
170#323—22no—
171#323C22×C8314yes—
172C25C2×C4×C8311yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—42no—
180C23×C4C8×C8314yescomplemented
181C23×C4C2×C4×C8312yes—
182C23×C4D4×C8312yes—
183C22×C8—42no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4C2×C4×C8312yes—
189C2×C22⋊C4C2×C4×C8311yescharacteristic
190C2×C22⋊C4D4×C8312yes—
191C2×C22⋊C4C23×C8311yescharacteristic
192C22⋊C8—24no—
193C22⋊C8—42no—
194#95—24no—
195#95—24no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—44no—
200C24C22⋊C4×C8311yescharacteristic
201C24C22⋊C4×C8311yescharacteristic
202C24C2×C8×C8312yes—
203C24M16×C8312yes—
204C24C22×C4×C8311yescharacteristic
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4C22⋊C4×C8312yes—
227C22×C4C2×C8×C8314yescomplemented
228C22×C4C2×D4×C8312yes—
229C2×C8—44no—
230C2×C8—84no—
231C22⋊C4—41no—
232C22⋊C4—42no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C23—24no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—42no—
246C23—44no—
247C23—44no—
248C23—42no—
249C23—48no—
250C23—44no—
251C23—44no—
252C23C22⋊C8×C8312yes—
253C23C22⋊C8×C8312yes—
254C23#99×C8311yescharacteristic
255C23#99×C8311yescharacteristic
256C23#100×C8312yes—
257C23C2×C22⋊C4×C8311yescharacteristic
258C23C22×C8×C8311yescharacteristic
259C23C2×M16×C8311yescharacteristic
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—84no—
277C8—84no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22#323×C8312yes—
297C22#323×C8312yes—
298C22#352×C8314yes—
299C22C2×C22⋊C8×C8311yescharacteristic
300C22C2×#99×C8311yescharacteristic
301C22C2×#100×C8311yescharacteristic
302C4—42no—
303C4—42no—
304C4—44no—
305C4—82no—
306C2—21no—
307C2—22no—
308C2—21no—
309C2—42no—
310C2—41no—
311C2—41no—
312C2C83×#131714yesdirect product factor
313C2C2×#323×C8311yescharacteristic
314C2C2×#352×C8312yes—
subgroupG/H
center, socleC23×C83#99
commutatorC23C22×C8×C83
Frattini, cosocleC2×C22⋊C4C23×C83
prime radicalC25×C83C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C83 ◁ C24×C83 ◁ C22×C22⋊C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1317C2direct (N × H)
22. C23×C4×C83C8semidirect
69. C22×C4×C83C2×C8semidirect
148. C2×#1317C83direct (N × H)
155. C2×C83#1317direct (N × H)
160. #1317C2×C83direct (N × H)
161. C83C2×#1317direct (N × H)
180. C23×C4C8×C83semidirect
227. C22×C4C2×C8×C83semidirect
312. C2C83×#1317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#381×C832
C2×#323×C832
C83×#13172
C2×#131783