Finite Group Databasebeta

← C83×#12167 #35239991

Subgroups — C83×#12167 #35239991

Order 21248.

354 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3259C211yesmaximal, characteristic
2C83×#2185C212yesmaximal, complemented
3C83×#2047C211yesmaximal, characteristic, complemented
4C83×#1349C212yesmaximal, complemented
5C83×#1323C211yesmaximal, characteristic, complemented
6C22×Q16×C83C2211yescharacteristic
7#540×C83—21no—
8C2×SD32×C83—22no—
9#478×C83—22no—
10#469×C83—21no—
11#466×C83C2212yes—
12#459×C83—22no—
13#424×C83C2211yescharacteristic
14#419×C83—21no—
15C2×#101×C83—21no—
16#358×C83—22no—
17C22⋊C16×C83C2212yes—
18#341×C83C2211yescharacteristic
19C22×Q8×C83—21no—
20C2×Q16×C83—22no—
21C2×Q16×C83—42no—
22C2×D8×C83—21no—
23C2×D8×C83D412yes—
24C2×M16×C83—21no—
25C2×M16×C83D411yescharacteristic
26C22×C8×C83C2311yescharacteristic
27#127×C83—21no—
28#125×C83—21no—
29#123×C83—42no—
30#121×C83—22no—
31#121×C83—42no—
32C4×D4×C83—22no—
33#117×C83D411yescharacteristic
34SD32×C83—44no—
35C2×C16×C83—22no—
36#107×C83D412yes—
37#106×C83—21no—
38C4⋊C8×C83—22no—
39#103×C83—42no—
40#102×C83—22no—
41#102×C83—22no—
42#101×C83—42no—
43C2×Q8×C83—22no—
44C2×Q8×C83—41no—
45C2×Q8×C83—41no—
46C2×Q8×C83—41no—
47C2×Q8×C83—41no—
48C2×Q8×C83—41no—
49C2×D4×C83—22no—
50C2×D4×C83—42no—
51C2×D4×C83—42no—
52C22×C4×C83—42no—
53C22×C4×C83—41no—
54C22×C4×C83C2×D411yescharacteristic
55Q16×C83—24no—
56Q16×C83—42no—
57D8×C83—42no—
58D8×C83—81no—
59M16×C83—41no—
60M16×C83—41no—
61M16×C83—41no—
62C2×C8×C83—21no—
63C2×C8×C83—41no—
64C2×C8×C83—41no—
65C2×C8×C83C2×D412yes—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—42no—
68C4⋊C4×C83D812yes—
69C22⋊C4×C83—21no—
70C22⋊C4×C83—42no—
71C22⋊C4×C83—42no—
72C42×C83—21no—
73C16×C83—42no—
74C23×C83—42no—
75C23×C83AGL(1,ℤ/8)11yescharacteristic
76Q8×C83—42no—
77Q8×C83—42no—
78Q8×C83—81no—
79Q8×C83—81no—
80D4×C83—42no—
81D4×C83—42no—
82D4×C83—162no—
83D4×C83—162no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—42no—
88C2×C4×C83—42no—
89C2×C4×C83—81no—
90C2×C4×C83—81no—
91C2×C4×C83—162no—
92C2×C4×C83#12011yescharacteristic
93C2×C4×C83C2×D811yescharacteristic
94C8×C83—22no—
95C8×C83—41no—
96C8×C83—81no—
97C22×C83—21no—
98C22×C83—22no—
99C22×C83—82no—
100C22×C83—82no—
101C22×C83—82no—
102C22×C83#44711yescharacteristic
103C4×C83—21no—
104C4×C83—42no—
105C4×C83—82no—
106C4×C83—82no—
107C4×C83#45411yescharacteristic
108C4×C83B(4,𝔽2)11yescharacteristic
109#12167C8311yesmaximal, characteristic, direct product factor, Sylow
110C2×C83—41no—
111C2×C83—162no—
112C2×C83#159811yescharacteristic
113C2×C83#219412yes—
114#3259C2×C8311yescharacteristic
115#2185C2×C8312yescomplemented
116#2047C2×C8311yescharacteristic, complemented
117#1349C2×C8312yescomplemented
118#1323C2×C8311yescharacteristic, complemented
119C83#1216711yescharacteristic, direct product factor, Sylow
120C22×Q16C22×C8311yescharacteristic
121#540—21no—
122C2×SD32—22no—
123#478—22no—
124#469—21no—
125#466C22×C8312yes—
126#459—22no—
127#424C22×C8311yescharacteristic
128#419—21no—
129C2×#101—21no—
130#358—22no—
131C22⋊C16C22×C8312yes—
132#341C22×C8311yescharacteristic
133C22×Q8—21no—
134C2×Q16—22no—
135C2×Q16—42no—
136C2×D8—21no—
137C2×D8D4×C8312yes—
138C2×M16—21no—
139C2×M16D4×C8311yescharacteristic
140C22×C8C23×C8311yescharacteristic
141#127—21no—
142#125—21no—
143#123—42no—
144#121—22no—
145#121—42no—
146C4×D4—22no—
147#117D4×C8311yescharacteristic
148SD32—44no—
149C2×C16—22no—
150#107D4×C8312yes—
151#106—21no—
152C4⋊C8—22no—
153#103—42no—
154#102—22no—
155#102—22no—
156#101—42no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4C2×D4×C8311yescharacteristic
169Q16—24no—
170Q16—42no—
171D8—42no—
172D8—81no—
173M16—41no—
174M16—41no—
175M16—41no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C2×D4×C8312yes—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4D8×C8312yes—
183C22⋊C4—21no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—21no—
187C16—42no—
188C23—42no—
189C23AGL(1,ℤ/8)×C8311yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—81no—
193Q8—81no—
194D4—42no—
195D4—42no—
196D4—162no—
197D4—162no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—162no—
206C2×C4#120×C8311yescharacteristic
207C2×C4C2×D8×C8311yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—21no—
212C22—22no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22#447×C8311yescharacteristic
217C4—21no—
218C4—42no—
219C4—82no—
220C4—82no—
221C4#454×C8311yescharacteristic
222C4B(4,𝔽2)×C8311yescharacteristic
223C2—41no—
224C2—162no—
225C2C83×#159811yescharacteristic
226C2C83×#219412yes—
subgroupG/H
center, socleC22×C83#447
commutator, Frattini, cosocleC22×C8C23×C83
prime radicalC83×#3259C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C22×C4×C83 ◁ #424×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#2185C2semidirect
3. C83×#2047C2semidirect
4. C83×#1349C2semidirect
5. C83×#1323C2semidirect
109. #12167C83direct (N × H)
115. #2185C2×C83semidirect
116. #2047C2×C83semidirect
117. #1349C2×C83semidirect
118. #1323C2×C83semidirect
119. C83#12167direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#32592
C83×#21852
C83×#20472
C83×#13492
C83×#13232
#1216783