Finite Group Databasebeta

← C83×#9441 #35237265

Subgroups — C83×#9441 #35237265

Order 21248.

576 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3153C211yesmaximal, characteristic
2C83×#2372C211yesmaximal, characteristic, complemented
3C83×#1533C211yesmaximal, characteristic, complemented
4C83×#1505C212yesmaximal, complemented
5C83×#1300C212yesmaximal, complemented
6C2×#137×C83—21no—
7C2×AGL(1,ℤ/8)×C83—21no—
8#532×C83C2211yescharacteristic
9C2×#124×C83C2211yescharacteristic
10#497×C83C412yescomplemented
11#496×C83C412yescomplemented
12#495×C83—22no—
13#432×C83C2212yes—
14C4×M16×C83C2211yescharacteristic
15#391×C83—21no—
16#388×C83—22no—
17#386×C83—21no—
18C4×C22⋊C4×C83—22no—
19#376×C83C2212yes—
20#329×C83—22no—
21#327×C83—22no—
22#327×C83—22no—
23C2×#41×C83—21no—
24C22×Q8×C83D411yescharacteristic
25C22×D4×C83D411yescharacteristic
26C23×C4×C83—21no—
27#137×C83—42no—
28#137×C83—42no—
29AGL(1,ℤ/8)×C83—42no—
30AGL(1,ℤ/8)×C83—42no—
31C2×Q16×C83—22no—
32C2×SD16×C83—22no—
33C2×SD16×C83—22no—
34C2×D8×C83—22no—
35C2×M16×C83—21no—
36C4⋊Q8×C83C2×C411yescharacteristic, complemented
37#127×C83C2×C411yescharacteristic, complemented
38#124×C83—21no—
39#124×C83—22no—
40#124×C83C2×C412yes—
41#124×C83D412yes—
42#121×C83—22no—
43C4×D4×C83—22no—
44C2×C4⋊C4×C83—21no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—21no—
48C2×C42×C83—22no—
49C2×C42×C83—21no—
50C2×C42×C83C2311yescharacteristic
51C4⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53#97×C83—22no—
54#97×C83C2×C412yescomplemented
55C4×C8×C83—21no—
56#95×C83—22no—
57#95×C83—22no—
58#95×C83D412yes—
59#95×C83D412yes—
60C24×C83—21no—
61#41×C83—41no—
62#41×C83—41no—
63#41×C83—41no—
64#41×C83—41no—
65C2×Q8×C83—21no—
66C2×Q8×C83—21no—
67C2×Q8×C83—41no—
68C2×Q8×C83—41no—
69C2×Q8×C83C22⋊C412yes—
70C2×D4×C83—21no—
71C2×D4×C83—22no—
72C2×D4×C83—21no—
73C2×D4×C83—41no—
74C2×D4×C83—41no—
75C2×D4×C83—41no—
76C2×D4×C83C22⋊C412yes—
77C22×C4×C83—22no—
78C22×C4×C83—21no—
79C22×C4×C83—22no—
80C22×C4×C83—44no—
81C22×C4×C83—41no—
82C22×C4×C83—81no—
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D412yes—
85Q16×C83—44no—
86SD16×C83—44no—
87SD16×C83—44no—
88D8×C83—44no—
89M16×C83—22no—
90M16×C83—22no—
91C2×C8×C83—22no—
92C2×C8×C83—42no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83—81no—
95C22⋊C4×C83—24no—
96C22⋊C4×C83—24no—
97C22⋊C4×C83—44no—
98C22⋊C4×C83—42no—
99C22⋊C4×C83—81no—
100C42×C83—44no—
101C42×C83—81no—
102C42×C83C22×C411yescharacteristic
103C42×C83C2×D411yescharacteristic
104C42×C83#4112yes—
105C23×C83—22no—
106C23×C83—42no—
107C23×C83—42no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83#12011yescharacteristic
112Q8×C83—42no—
113Q8×C83—41no—
114Q8×C83—41no—
115Q8×C83—81no—
116D4×C83—42no—
117D4×C83—41no—
118D4×C83—41no—
119D4×C83—81no—
120D4×C83—82no—
121D4×C83—81no—
122D4×C83—81no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—24no—
126C2×C4×C83—22no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83—41no—
135C2×C4×C83—42no—
136C2×C4×C83—81no—
137C2×C4×C83—81no—
138C2×C4×C83—84no—
139C2×C4×C83C2×C22⋊C411yescharacteristic
140C2×C4×C83C4×D412yes—
141C2×C4×C83#12112yes—
142C2×C4×C83#12311yescharacteristic
143C8×C83—24no—
144C8×C83—82no—
145C22×C83—22no—
146C22×C83—42no—
147C22×C83—44no—
148C22×C83—81no—
149C22×C83—81no—
150C22×C83—82no—
151C22×C83—81no—
152C22×C83—82no—
153C22×C83#38611yescharacteristic
154C22×C83#45311yescharacteristic
155C22×C83#45311yescharacteristic
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—22no—
160C4×C83—44no—
161C4×C83—42no—
162C4×C83—82no—
163C4×C83—81no—
164#9441C8311yesmaximal, characteristic, direct product factor, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—82no—
168C2×C83—161no—
169C2×C83#188711yescharacteristic
170C2×C83#190612yes—
171#3153C2×C8311yescharacteristic
172#2372C2×C8311yescharacteristic, complemented
173#1533C2×C8311yescharacteristic, complemented
174#1505C2×C8312yescomplemented
175#1300C2×C8312yescomplemented
176C83#944111yescharacteristic, direct product factor, Sylow
177C2×#137—21no—
178C2×AGL(1,ℤ/8)—21no—
179#532C22×C8311yescharacteristic
180C2×#124C22×C8311yescharacteristic
181#497C4×C8312yescomplemented
182#496C4×C8312yescomplemented
183#495—22no—
184#432C22×C8312yes—
185C4×M16C22×C8311yescharacteristic
186#391—21no—
187#388—22no—
188#386—21no—
189C4×C22⋊C4—22no—
190#376C22×C8312yes—
191#329—22no—
192#327—22no—
193#327—22no—
194C2×#41—21no—
195C22×Q8D4×C8311yescharacteristic
196C22×D4D4×C8311yescharacteristic
197C23×C4—21no—
198#137—42no—
199#137—42no—
200AGL(1,ℤ/8)—42no—
201AGL(1,ℤ/8)—42no—
202C2×Q16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×D8—22no—
206C2×M16—21no—
207C4⋊Q8C2×C4×C8311yescharacteristic, complemented
208#127C2×C4×C8311yescharacteristic, complemented
209#124—21no—
210#124—22no—
211#124C2×C4×C8312yes—
212#124D4×C8312yes—
213#121—22no—
214C4×D4—22no—
215C2×C4⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42C23×C8311yescharacteristic
222C4⋊C8—22no—
223C22⋊C8—22no—
224#97—22no—
225#97C2×C4×C8312yescomplemented
226C4×C8—21no—
227#95—22no—
228#95—22no—
229#95D4×C8312yes—
230#95D4×C8312yes—
231C24—21no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—41no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8C22⋊C4×C8312yes—
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4C22⋊C4×C8312yes—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—41no—
253C22×C4—81no—
254C22×C4C2×D4×C8311yescharacteristic
255C22×C4C2×D4×C8312yes—
256Q16—44no—
257SD16—44no—
258SD16—44no—
259D8—44no—
260M16—22no—
261M16—22no—
262C2×C8—22no—
263C2×C8—42no—
264C4⋊C4—22no—
265C4⋊C4—81no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—81no—
271C42—44no—
272C42—81no—
273C42C22×C4×C8311yescharacteristic
274C42C2×D4×C8311yescharacteristic
275C42#41×C8312yes—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#120×C8311yescharacteristic
283Q8—42no—
284Q8—41no—
285Q8—41no—
286Q8—81no—
287D4—42no—
288D4—41no—
289D4—41no—
290D4—81no—
291D4—82no—
292D4—81no—
293D4—81no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—84no—
310C2×C4C2×C22⋊C4×C8311yescharacteristic
311C2×C4C4×D4×C8312yes—
312C2×C4#121×C8312yes—
313C2×C4#123×C8311yescharacteristic
314C8—24no—
315C8—82no—
316C22—22no—
317C22—42no—
318C22—44no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—81no—
323C22—82no—
324C22#386×C8311yescharacteristic
325C22#453×C8311yescharacteristic
326C22#453×C8311yescharacteristic
327C4—21no—
328C4—21no—
329C4—22no—
330C4—22no—
331C4—44no—
332C4—42no—
333C4—82no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2—161no—
339C2C83×#188711yescharacteristic
340C2C83×#190612yes—
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC83×#3153C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2372C2semidirect
3. C83×#1533C2semidirect
4. C83×#1505C2semidirect
5. C83×#1300C2semidirect
10. #497×C83C4semidirect
11. #496×C83C4semidirect
36. C4⋊Q8×C83C2×C4semidirect
37. #127×C83C2×C4semidirect
54. #97×C83C2×C4semidirect
164. #9441C83direct (N × H)
172. #2372C2×C83semidirect
173. #1533C2×C83semidirect
174. #1505C2×C83semidirect
175. #1300C2×C83semidirect
176. C83#9441direct (N × H)
181. #497C4×C83semidirect
182. #496C4×C83semidirect
207. C4⋊Q8C2×C4×C83semidirect
208. #127C2×C4×C83semidirect
225. #97C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#31532
C83×#23722
C83×#15332
C83×#15052
C83×#13002
#944183