Finite Group Databasebeta

← C83×#11550 #35239374

Subgroups — C83×#11550 #35239374

Order 21248.

408 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2500C211yesmaximal, characteristic
2C83×#1464C211yesmaximal, characteristic, complemented
3C83×#1458C211yesmaximal, characteristic, complemented
4C83×#1324C212yesmaximal, complemented
5C83×#1318C212yesmaximal, complemented
6C2×#121×C83C411yescharacteristic
7C2×#117×C83C2211yescharacteristic
8#432×C83—21no—
9#431×C83—21no—
10#394×C83—21no—
11#394×C83—21no—
12#390×C83C411yescharacteristic
13#390×C83C2211yescharacteristic
14#388×C83—21no—
15#388×C83—21no—
16#384×C83C411yescharacteristic
17#382×C83C2211yescharacteristic
18C4×C22⋊C4×C83C411yescharacteristic
19#324×C83C2212yes—
20#324×C83C2212yes—
21#323×C83—21no—
22#323×C83—21no—
23C22×D4×C83—21no—
24C22×D4×C83C2×C411yescharacteristic
25C23×C4×C83C2×C411yescharacteristic
26#121×C83—41no—
27#121×C83—41no—
28#117×C83—41no—
29#117×C83—41no—
30C2×C4⋊C4×C83—21no—
31C2×C4⋊C4×C83—21no—
32C2×C4⋊C4×C83C2×C411yescharacteristic
33C2×C4⋊C4×C83C2×C411yescharacteristic
34C2×C4⋊C4×C83C2311yescharacteristic
35C2×C22⋊C4×C83—21no—
36C2×C22⋊C4×C83—21no—
37C2×C22⋊C4×C83—21no—
38C2×C22⋊C4×C83—21no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83D411yescharacteristic
43C2×C42×C83C2×C411yescharacteristic
44C2×C42×C83C2×C411yescharacteristic
45C2×C42×C83D411yescharacteristic
46C2×C42×C83D411yescharacteristic
47C4⋊C8×C83—41no—
48C4⋊C8×C83—41no—
49C22⋊C8×C83—22no—
50C22⋊C8×C83—22no—
51#97×C83—41no—
52C4×C8×C83—41no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56C24×C83—21no—
57C24×C83C22⋊C411yescharacteristic
58C2×D4×C83—22no—
59C2×D4×C83—41no—
60C2×D4×C83—41no—
61C2×D4×C83—42no—
62C2×D4×C83—81no—
63C2×D4×C83—81no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—22no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—41no—
71C22×C4×C83C22⋊C411yescharacteristic
72C22×C4×C83C22⋊C411yescharacteristic
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22×C411yescharacteristic
75C22×C4×C83C2×D411yescharacteristic
76C22×C4×C83C2×D411yescharacteristic
77C2×C8×C83—42no—
78C2×C8×C83—42no—
79C4⋊C4×C83—41no—
80C4⋊C4×C83—41no—
81C4⋊C4×C83—41no—
82C4⋊C4×C83—81no—
83C4⋊C4×C83—81no—
84C22⋊C4×C83—41no—
85C22⋊C4×C83—41no—
86C22⋊C4×C83—42no—
87C22⋊C4×C83—42no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—81no—
92C22⋊C4×C83—81no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—22no—
98C42×C83—41no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83C2×C22⋊C411yescharacteristic
112D4×C83—82no—
113D4×C83—84no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—42no—
124C2×C4×C83—42no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83#9911yescharacteristic
138C2×C4×C83#9911yescharacteristic
139C2×C4×C83#13112yes—
140C8×C83—82no—
141C8×C83—82no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—41no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—81no—
148C22×C83—81no—
149C22×C83—81no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83#40811yescharacteristic
157C22×C83C2×#9911yescharacteristic
158C22×C83#41311yescharacteristic
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—22no—
162C4×C83—42no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—81no—
168C4×C83—81no—
169#11550C8311yesmaximal, characteristic, direct product factor, Sylow
170C2×C83—21no—
171C2×C83—21no—
172C2×C83—81no—
173C2×C83—161no—
174C2×C83#146811yescharacteristic
175C2×C83#212611yescharacteristic
176C2×C83#213411yescharacteristic
177#2500C2×C8311yescharacteristic
178#1464C2×C8311yescharacteristic, complemented
179#1458C2×C8311yescharacteristic, complemented
180#1324C2×C8312yescomplemented
181#1318C2×C8312yescomplemented
182C83#1155011yescharacteristic, direct product factor, Sylow
183C2×#121C4×C8311yescharacteristic
184C2×#117C22×C8311yescharacteristic
185#432—21no—
186#431—21no—
187#394—21no—
188#394—21no—
189#390C4×C8311yescharacteristic
190#390C22×C8311yescharacteristic
191#388—21no—
192#388—21no—
193#384C4×C8311yescharacteristic
194#382C22×C8311yescharacteristic
195C4×C22⋊C4C4×C8311yescharacteristic
196#324C22×C8312yes—
197#324C22×C8312yes—
198#323—21no—
199#323—21no—
200C22×D4—21no—
201C22×D4C2×C4×C8311yescharacteristic
202C23×C4C2×C4×C8311yescharacteristic
203#121—41no—
204#121—41no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×C4×C8311yescharacteristic
210C2×C4⋊C4C2×C4×C8311yescharacteristic
211C2×C4⋊C4C23×C8311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4D4×C8311yescharacteristic
219C2×C22⋊C4D4×C8311yescharacteristic
220C2×C42C2×C4×C8311yescharacteristic
221C2×C42C2×C4×C8311yescharacteristic
222C2×C42D4×C8311yescharacteristic
223C2×C42D4×C8311yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—41no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228#97—41no—
229C4×C8—41no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C24C22⋊C4×C8311yescharacteristic
235C2×D4—22no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—81no—
240C2×D4—81no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4C22⋊C4×C8311yescharacteristic
249C22×C4C22⋊C4×C8311yescharacteristic
250C22×C4C22⋊C4×C8311yescharacteristic
251C22×C4C22×C4×C8311yescharacteristic
252C22×C4C2×D4×C8311yescharacteristic
253C22×C4C2×D4×C8311yescharacteristic
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊C4×C8311yescharacteristic
289D4—82no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4#99×C8311yescharacteristic
315C2×C4#99×C8311yescharacteristic
316C2×C4#131×C8312yes—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#408×C8311yescharacteristic
334C22C2×#99×C8311yescharacteristic
335C22#413×C8311yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2C83×#146811yescharacteristic
351C2C83×#212611yescharacteristic
352C2C83×#213411yescharacteristic
subgroupG/H
center, socleC22×C83C2×#99
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C4⋊C4C23×C83
prime radicalC2×#121×C83C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C22×C4×C83 ◁ C2×#117×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1464C2semidirect
3. C83×#1458C2semidirect
4. C83×#1324C2semidirect
5. C83×#1318C2semidirect
169. #11550C83direct (N × H)
178. #1464C2×C83semidirect
179. #1458C2×C83semidirect
180. #1324C2×C83semidirect
181. #1318C2×C83semidirect
182. C83#11550direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#25002
C83×#14642
C83×#14582
C83×#13242
C83×#13182
#1155083