Finite Group Databasebeta

← C83×#9429 #35237253

Subgroups — C83×#9429 #35237253

Order 21248.

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

normal subgroups only

This group has 508 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×#3152C211yesmaximal, characteristic
2C83×#2381C211yesmaximal, characteristic, complemented
3C83×#1558C211yesmaximal, characteristic, complemented
4C83×#1514C212yesmaximal
5C83×#1288C212yesmaximal, complemented
6C2×#137×C83—22no—
7#532×C83C2211yescharacteristic
8C2×C4⋊Q8×C83C2211yescharacteristic
9#497×C83—22no—
10#494×C83C412yescomplemented
11#492×C83C412yescomplemented
12#432×C83C2212yes—
13C4×M16×C83C2211yescharacteristic
14#391×C83—22no—
15#387×C83—24no—
16#383×C83C2212yes—
17#333×C83—22no—
18#327×C83—24no—
19C2×#41×C83—21no—
20C22×Q8×C83D412yes—
21#137×C83—44no—
22#137×C83—44no—
23C2×Q16×C83—24no—
24C2×SD16×C83—24no—
25C2×M16×C83—21no—
26C4⋊Q8×C83—22no—
27C4⋊Q8×C83C2×C411yescharacteristic
28C4⋊Q8×C83C2×C412yes—
29C4⋊Q8×C83D412yes—
30#127×C83C2×C411yescharacteristic
31#124×C83—21no—
32#121×C83—22no—
33C4×D4×C83—22no—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83—22no—
36C2×C42×C83—22no—
37C2×C42×C83C2311yescharacteristic
38C4⋊C8×C83—22no—
39C22⋊C8×C83—22no—
40#97×C83—21no—
41#97×C83—22no—
42C4×C8×C83C2×C412yes—
43#95×C83—24no—
44#95×C83—22no—
45#95×C83—22no—
46#95×C83D414yes—
47#41×C83—42no—
48#41×C83—42no—
49C2×Q8×C83—22no—
50C2×Q8×C83—21no—
51C2×Q8×C83—42no—
52C2×Q8×C83—42no—
53C2×Q8×C83C22⋊C414yes—
54C2×D4×C83—21no—
55C2×D4×C83—22no—
56C2×D4×C83—41no—
57C22×C4×C83—22no—
58C22×C4×C83—24no—
59C22×C4×C83—41no—
60C22×C4×C83C2×D411yescharacteristic
61C22×C4×C83C2×D412yes—
62Q16×C83—48no—
63SD16×C83—48no—
64M16×C83—24no—
65C2×C8×C83—22no—
66C2×C8×C83—42no—
67C4⋊C4×C83—24no—
68C4⋊C4×C83—24no—
69C4⋊C4×C83—22no—
70C4⋊C4×C83—82no—
71C22⋊C4×C83—42no—
72C42×C83—82no—
73C42×C83C22×C411yescharacteristic
74C42×C83C2×D411yescharacteristic
75C42×C83#4112yes—
76C23×C83—41no—
77C23×C83#12011yescharacteristic
78Q8×C83—44no—
79Q8×C83—42no—
80Q8×C83—82no—
81D4×C83—42no—
82D4×C83—82no—
83D4×C83—82no—
84C2×C4×C83—22no—
85C2×C4×C83—24no—
86C2×C4×C83—42no—
87C2×C4×C83—42no—
88C2×C4×C83—42no—
89C2×C4×C83—48no—
90C2×C4×C83—41no—
91C2×C4×C83—44no—
92C2×C4×C83—81no—
93C2×C4×C83—81no—
94C2×C4×C83C2×C22⋊C411yescharacteristic
95C2×C4×C83C4×D412yes—
96C2×C4×C83#12112yes—
97C2×C4×C83#12311yescharacteristic
98C8×C83—24no—
99C8×C83—82no—
100C22×C83—22no—
101C22×C83—81no—
102C22×C83—82no—
103C22×C83#38611yescharacteristic
104C22×C83#45312yes—
105C4×C83—21no—
106C4×C83—21no—
107C4×C83—22no—
108C4×C83—22no—
109C4×C83—44no—
110C4×C83—84no—
111C4×C83—81no—
112#9429C8311yesmaximal, characteristic, direct product factor, Sylow
113C2×C83—22no—
114C2×C83—161no—
115C2×C83#188711yescharacteristic
116C2×C83#190512yes—
117#3152C2×C8311yescharacteristic
118#2381C2×C8311yescharacteristic, complemented
119#1558C2×C8311yescharacteristic, complemented
120#1514C2×C8312yes—
121#1288C2×C8312yescomplemented
122C83#942911yescharacteristic, direct product factor, Sylow
123C2×#137—22no—
124#532C22×C8311yescharacteristic
125C2×C4⋊Q8C22×C8311yescharacteristic
126#497—22no—
127#494C4×C8312yescomplemented
128#492C4×C8312yescomplemented
129#432C22×C8312yes—
130C4×M16C22×C8311yescharacteristic
131#391—22no—
132#387—24no—
133#383C22×C8312yes—
134#333—22no—
135#327—24no—
136C2×#41—21no—
137C22×Q8D4×C8312yes—
138#137—44no—
139#137—44no—
140C2×Q16—24no—
141C2×SD16—24no—
142C2×M16—21no—
143C4⋊Q8—22no—
144C4⋊Q8C2×C4×C8311yescharacteristic
145C4⋊Q8C2×C4×C8312yes—
146C4⋊Q8D4×C8312yes—
147#127C2×C4×C8311yescharacteristic
148#124—21no—
149#121—22no—
150C4×D4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C23×C8311yescharacteristic
155C4⋊C8—22no—
156C22⋊C8—22no—
157#97—21no—
158#97—22no—
159C4×C8C2×C4×C8312yes—
160#95—24no—
161#95—22no—
162#95—22no—
163#95D4×C8314yes—
164#41—42no—
165#41—42no—
166C2×Q8—22no—
167C2×Q8—21no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8C22⋊C4×C8314yes—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—41no—
177C22×C4C2×D4×C8311yescharacteristic
178C22×C4C2×D4×C8312yes—
179Q16—48no—
180SD16—48no—
181M16—24no—
182C2×C8—22no—
183C2×C8—42no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—82no—
188C22⋊C4—42no—
189C42—82no—
190C42C22×C4×C8311yescharacteristic
191C42C2×D4×C8311yescharacteristic
192C42#41×C8312yes—
193C23—41no—
194C23#120×C8311yescharacteristic
195Q8—44no—
196Q8—42no—
197Q8—82no—
198D4—42no—
199D4—82no—
200D4—82no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4C2×C22⋊C4×C8311yescharacteristic
212C2×C4C4×D4×C8312yes—
213C2×C4#121×C8312yes—
214C2×C4#123×C8311yescharacteristic
215C8—24no—
216C8—82no—
217C22—22no—
218C22—81no—
219C22—82no—
220C22#386×C8311yescharacteristic
221C22#453×C8312yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—84no—
228C4—81no—
229C2—22no—
230C2—161no—
231C2C83×#188711yescharacteristic
232C2C83×#190512yes—
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radical#532×C83C22
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×#2381C2semidirect
3. C83×#1558C2semidirect
5. C83×#1288C2semidirect
10. #494×C83C4semidirect
11. #492×C83C4semidirect
112. #9429C83direct (N × H)
118. #2381C2×C83semidirect
119. #1558C2×C83semidirect
121. #1288C2×C83semidirect
122. C83#9429direct (N × H)
127. #494C4×C83semidirect
128. #492C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#31522
C83×#23812
C83×#15582
C83×#15142
C83×#12882
#942983