Finite Group Databasebeta

← C83×#9389 #35237213

Subgroups — C83×#9389 #35237213

Order 21248.

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

normal subgroups only

This group has 496 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×#2921C211yesmaximal, characteristic, complemented
2C83×#2610C211yesmaximal, characteristic
3C83×#1527C212yesmaximal, complemented
4C83×#1464C211yesmaximal, characteristic, complemented
5C83×#1300C212yesmaximal, complemented
6C2×#124×C83C2211yescharacteristic
7C2×#117×C83C2211yescharacteristic
8#433×C83—24no—
9#432×C83C2212yes—
10#423×C83—21no—
11#407×C83—21no—
12C4×M16×C83C2212yes—
13#395×C83C411yescharacteristic
14#394×C83C411yescharacteristic
15#393×C83C411yescharacteristic
16#392×C83C411yescharacteristic
17#390×C83—21no—
18#388×C83—21no—
19#388×C83—21no—
20#386×C83—21no—
21#376×C83C2211yescharacteristic
22#329×C83—24no—
23#324×C83—21no—
24#323×C83—21no—
25C22×Q8×C83C2×C411yescharacteristic
26C22×D4×C83—21no—
27C22×D4×C83C2×C411yescharacteristic
28C23×C4×C83D411yescharacteristic
29C2×M16×C83—22no—
30#124×C83—22no—
31#124×C83—22no—
32#117×C83—22no—
33#117×C83—22no—
34C2×C4⋊C4×C83—21no—
35C2×C4⋊C4×C83C2×C411yescharacteristic
36C2×C4⋊C4×C83D411yescharacteristic
37C2×C22⋊C4×C83—21no—
38C2×C22⋊C4×C83—21no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83C2×C411yescharacteristic
44C2×C22⋊C4×C83D411yescharacteristic
45C2×C42×C83—21no—
46C2×C42×C83D411yescharacteristic
47C2×C42×C83C2311yescharacteristic
48C4⋊C8×C83—24no—
49C4⋊C8×C83—22no—
50C22⋊C8×C83—24no—
51C22⋊C8×C83—22no—
52#97×C83—22no—
53#97×C83D414yescomplemented
54C4×C8×C83—22no—
55#95×C83—21no—
56#95×C83—21no—
57#95×C83—21no—
58#95×C83C2×C411yescharacteristic
59#95×C83C2×C411yescharacteristic
60C24×C83—21no—
61C24×C83C22⋊C411yescharacteristic
62C2×Q8×C83—22no—
63C2×Q8×C83—81no—
64C2×D4×C83—22no—
65C2×D4×C83—42no—
66C2×D4×C83—42no—
67C2×D4×C83—81no—
68C2×D4×C83—81no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—41no—
75C22×C4×C83—41no—
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82M16×C83—42no—
83M16×C83—42no—
84C2×C8×C83—24no—
85C2×C8×C83—42no—
86C4⋊C4×C83—22no—
87C4⋊C4×C83—22no—
88C4⋊C4×C83—81no—
89C22⋊C4×C83—22no—
90C22⋊C4×C83—22no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—81no—
97C22⋊C4×C83—81no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—42no—
101C42×C83C2×D412yes—
102C42×C83#4112yes—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83—41no—
115C23×C83C2×C22⋊C411yescharacteristic
116Q8×C83—82no—
117D4×C83—82no—
118D4×C83—84no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—22no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
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—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83C4×D412yes—
142C2×C4×C83#12011yescharacteristic
143C2×C4×C83#12112yes—
144C2×C4×C83#12311yescharacteristic
145C8×C83—44no—
146C8×C83—82no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—42no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83—81no—
157C22×C83—81no—
158C22×C83—81no—
159C22×C83—81no—
160C22×C83—81no—
161C22×C83#38611yescharacteristic
162C22×C83#42111yescharacteristic
163C22×C83#42111yescharacteristic
164C4×C83—22no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—22no—
168C4×C83—42no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—81no—
173C4×C83—81no—
174#9389C8311yesmaximal, characteristic, direct product factor, Sylow
175C2×C83—21no—
176C2×C83—21no—
177C2×C83—81no—
178C2×C83—161no—
179C2×C83#188411yescharacteristic
180C2×C83#189711yescharacteristic
181C2×C83#191411yescharacteristic
182#2921C2×C8311yescharacteristic, complemented
183#2610C2×C8311yescharacteristic
184#1527C2×C8312yescomplemented
185#1464C2×C8311yescharacteristic, complemented
186#1300C2×C8312yescomplemented
187C83#938911yescharacteristic, direct product factor, Sylow
188C2×#124C22×C8311yescharacteristic
189C2×#117C22×C8311yescharacteristic
190#433—24no—
191#432C22×C8312yes—
192#423—21no—
193#407—21no—
194C4×M16C22×C8312yes—
195#395C4×C8311yescharacteristic
196#394C4×C8311yescharacteristic
197#393C4×C8311yescharacteristic
198#392C4×C8311yescharacteristic
199#390—21no—
200#388—21no—
201#388—21no—
202#386—21no—
203#376C22×C8311yescharacteristic
204#329—24no—
205#324—21no—
206#323—21no—
207C22×Q8C2×C4×C8311yescharacteristic
208C22×D4—21no—
209C22×D4C2×C4×C8311yescharacteristic
210C23×C4D4×C8311yescharacteristic
211C2×M16—22no—
212#124—22no—
213#124—22no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×C4×C8311yescharacteristic
218C2×C4⋊C4D4×C8311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×C4×C8311yescharacteristic
226C2×C22⋊C4D4×C8311yescharacteristic
227C2×C42—21no—
228C2×C42D4×C8311yescharacteristic
229C2×C42C23×C8311yescharacteristic
230C4⋊C8—24no—
231C4⋊C8—22no—
232C22⋊C8—24no—
233C22⋊C8—22no—
234#97—22no—
235#97D4×C8314yescomplemented
236C4×C8—22no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×C4×C8311yescharacteristic
241#95C2×C4×C8311yescharacteristic
242C24—21no—
243C24C22⋊C4×C8311yescharacteristic
244C2×Q8—22no—
245C2×Q8—81no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—81no—
250C2×D4—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊C4×C8311yescharacteristic
259C22×C4C22⋊C4×C8311yescharacteristic
260C22×C4C22⋊C4×C8311yescharacteristic
261C22×C4C22×C4×C8311yescharacteristic
262C22×C4C2×D4×C8311yescharacteristic
263C22×C4C2×D4×C8311yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—24no—
267C2×C8—42no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—81no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42C2×D4×C8312yes—
284C42#41×C8312yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊C4×C8311yescharacteristic
298Q8—82no—
299D4—82no—
300D4—84no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4C4×D4×C8312yes—
324C2×C4#120×C8311yescharacteristic
325C2×C4#121×C8312yes—
326C2×C4#123×C8311yescharacteristic
327C8—44no—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#386×C8311yescharacteristic
344C22#421×C8311yescharacteristic
345C22#421×C8311yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—22no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—161no—
360C2C83×#188411yescharacteristic
361C2C83×#189711yescharacteristic
362C2C83×#191411yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radical#392×C83C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#2921C2semidirect
3. C83×#1527C2semidirect
4. C83×#1464C2semidirect
5. C83×#1300C2semidirect
53. #97×C83D4semidirect
174. #9389C83direct (N × H)
182. #2921C2×C83semidirect
184. #1527C2×C83semidirect
185. #1464C2×C83semidirect
186. #1300C2×C83semidirect
187. C83#9389direct (N × H)
235. #97D4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29212
C83×#26102
C83×#15272
C83×#14642
C83×#13002
#938983