Finite Group Databasebeta

← C83×#12115 #35239939

Subgroups — C83×#12115 #35239939

Order 21248.

422 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3195C211yesmaximal, characteristic, complemented
2C83×#2191C212yesmaximal, complemented
3C83×#2010C211yesmaximal, characteristic
4C83×#1340C212yesmaximal, complemented
5C83×#1320C211yesmaximal, characteristic, complemented
6#575×C83C2211yescharacteristic
7#525×C83—21no—
8C2×C22⋊Q8×C83—21no—
9#510×C83—22no—
10#483×C83—22no—
11#470×C83—21no—
12#470×C83C2212yes—
13#451×C83—22no—
14#421×C83—21no—
15#407×C83C2211yescharacteristic
16#360×C83—22no—
17#349×C83C2212yes—
18H(3,ℤ/4)×C83C2211yescharacteristic
19C2×#41×C83—21no—
20C22×Q8×C83—21no—
21C23×C4×C83D411yescharacteristic
22#137×C83—41no—
23AGL(1,ℤ/8)×C83—41no—
24#135×C83—22no—
25C2×Q16×C83—21no—
26C2×Q16×C83D412yescomplemented
27C2×M16×C83D411yescharacteristic
28C2×M16×C83C2311yescharacteristic
29#123×C83—41no—
30C22⋊Q8×C83—22no—
31C22⋊Q8×C83—22no—
32C22⋊Q8×C83—41no—
33C22⋊Q8×C83—41no—
34C22⋊Q8×C83—41no—
35#121×C83—41no—
36#120×C83—41no—
37C4×D4×C83—42no—
38#117×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—41no—
41C2×C22⋊C4×C83—41no—
42Q32×C83—42no—
43SD32×C83—42no—
44M32×C83—22no—
45#106×C83—21no—
46#106×C83D412yescomplemented
47#104×C83—42no—
48#103×C83—22no—
49#103×C83—22no—
50C22⋊C8×C83—22no—
51C24×C83D811yescharacteristic
52#41×C83—41no—
53#41×C83—41no—
54#41×C83—41no—
55#41×C83—41no—
56C2×Q8×C83—21no—
57C2×Q8×C83—22no—
58C2×Q8×C83—41no—
59C2×Q8×C83—81no—
60C2×Q8×C83—81no—
61C2×D4×C83—21no—
62C2×D4×C83—42no—
63C2×D4×C83—41no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—41no—
67C22×C4×C83—41no—
68C22×C4×C83—41no—
69C22×C4×C83—41no—
70C22×C4×C83—41no—
71C22×C4×C83D811yescharacteristic
72C22×C4×C83C2×D411yescharacteristic
73Q16×C83—22no—
74Q16×C83—42no—
75Q16×C83—81no—
76SD16×C83—42no—
77D8×C83—22no—
78M16×C83—41no—
79M16×C83—41no—
80C2×C8×C83—21no—
81C2×C8×C83C2×D412yes—
82C4⋊C4×C83—22no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—42no—
85C4⋊C4×C83—41no—
86C4⋊C4×C83—42no—
87C4⋊C4×C83—42no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—42no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—41no—
93C42×C83—22no—
94C16×C83—42no—
95C23×C83—21no—
96C23×C83—41no—
97C23×C83—42no—
98C23×C83—41no—
99C23×C83C2×D811yescharacteristic
100Q8×C83—41no—
101Q8×C83—42no—
102Q8×C83—41no—
103Q8×C83—42no—
104Q8×C83—82no—
105D4×C83—41no—
106D4×C83—41no—
107D4×C83—81no—
108D4×C83—84no—
109D4×C83—81no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—42no—
117C2×C4×C83—41no—
118C2×C4×C83—42no—
119C2×C4×C83—42no—
120C2×C4×C83—81no—
121C2×C4×C83—81no—
122C2×C4×C83—82no—
123C2×C4×C83—82no—
124C2×C4×C83—81no—
125C2×C4×C83#12011yescharacteristic
126C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
127C8×C83—22no—
128C8×C83—41no—
129C22×C83—22no—
130C22×C83—21no—
131C22×C83—41no—
132C22×C83—41no—
133C22×C83—42no—
134C22×C83—41no—
135C22×C83—81no—
136C22×C83—82no—
137C22×C83—81no—
138C22×C83#44711yescharacteristic
139C4×C83—21no—
140C4×C83—41no—
141C4×C83—82no—
142C4×C83—82no—
143C4×C83—81no—
144C4×C83—82no—
145C4×C83#45311yescharacteristic
146C4×C83B(4,𝔽2)11yescharacteristic
147#12115C8311yesmaximal, characteristic, direct product factor, Sylow
148C2×C83—21no—
149C2×C83—41no—
150C2×C83—42no—
151C2×C83—161no—
152C2×C83#159511yescharacteristic
153#3195C2×C8311yescharacteristic, complemented
154#2191C2×C8312yescomplemented
155#2010C2×C8311yescharacteristic
156#1340C2×C8312yescomplemented
157#1320C2×C8311yescharacteristic, complemented
158C83#1211511yescharacteristic, direct product factor, Sylow
159#575C22×C8311yescharacteristic
160#525—21no—
161C2×C22⋊Q8—21no—
162#510—22no—
163#483—22no—
164#470—21no—
165#470C22×C8312yes—
166#451—22no—
167#421—21no—
168#407C22×C8311yescharacteristic
169#360—22no—
170#349C22×C8312yes—
171H(3,ℤ/4)C22×C8311yescharacteristic
172C2×#41—21no—
173C22×Q8—21no—
174C23×C4D4×C8311yescharacteristic
175#137—41no—
176AGL(1,ℤ/8)—41no—
177#135—22no—
178C2×Q16—21no—
179C2×Q16D4×C8312yescomplemented
180C2×M16D4×C8311yescharacteristic
181C2×M16C23×C8311yescharacteristic
182#123—41no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188#121—41no—
189#120—41no—
190C4×D4—42no—
191#117—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C22⋊C4—41no—
195Q32—42no—
196SD32—42no—
197M32—22no—
198#106—21no—
199#106D4×C8312yescomplemented
200#104—42no—
201#103—22no—
202#103—22no—
203C22⋊C8—22no—
204C24D8×C8311yescharacteristic
205#41—41no—
206#41—41no—
207#41—41no—
208#41—41no—
209C2×Q8—21no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—81no—
213C2×Q8—81no—
214C2×D4—21no—
215C2×D4—42no—
216C2×D4—41no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4D8×C8311yescharacteristic
225C22×C4C2×D4×C8311yescharacteristic
226Q16—22no—
227Q16—42no—
228Q16—81no—
229SD16—42no—
230D8—22no—
231M16—41no—
232M16—41no—
233C2×C8—21no—
234C2×C8C2×D4×C8312yes—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—41no—
246C42—22no—
247C16—42no—
248C23—21no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23C2×D8×C8311yescharacteristic
253Q8—41no—
254Q8—42no—
255Q8—41no—
256Q8—42no—
257Q8—82no—
258D4—41no—
259D4—41no—
260D4—81no—
261D4—84no—
262D4—81no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4#120×C8311yescharacteristic
279C2×C4AGL(1,ℤ/8)×C8311yescharacteristic
280C8—22no—
281C8—41no—
282C22—22no—
283C22—21no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22#447×C8311yescharacteristic
292C4—21no—
293C4—41no—
294C4—82no—
295C4—82no—
296C4—81no—
297C4—82no—
298C4#453×C8311yescharacteristic
299C4B(4,𝔽2)×C8311yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—161no—
304C2C83×#159511yescharacteristic
subgroupG/H
center, socleC2×C83#1595
commutator, Frattini, cosocleC2×M16C23×C83
prime radicalC83×#2010C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C83 ◁ C2×C4×C83 ◁ C2×M16×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3195C2semidirect
2. C83×#2191C2semidirect
4. C83×#1340C2semidirect
5. C83×#1320C2semidirect
26. C2×Q16×C83D4semidirect
46. #106×C83D4semidirect
147. #12115C83direct (N × H)
153. #3195C2×C83semidirect
154. #2191C2×C83semidirect
156. #1340C2×C83semidirect
157. #1320C2×C83semidirect
158. C83#12115direct (N × H)
179. C2×Q16D4×C83semidirect
199. #106D4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#31952
C83×#21912
C83×#20102
C83×#13402
C83×#13202
#1211583