Finite Group Databasebeta

← #35323493

Subgroups — #35323493

Order 21248.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#24917987C211yesmaximal, characteristic
2#24909183C212yesmaximal
3#24905570C212yesmaximal
4#24905547C211yesmaximal, characteristic
5C83×#1971C211yesmaximal, characteristic
6#19128258—21no—
7C2×#14581732C2212yes—
8#19128110—24no—
9#19127739—21no—
10#19127739C2212yes—
11#19127505—24no—
12C22×Q16×C83C2211yescharacteristic
13C2×#106×C83C2211yescharacteristic
14C2×#103×C83—21no—
15C2×#97×C83C2211yescharacteristic
16#391×C83—21no—
17#14581792—22no—
18#14581784—22no—
19#14581771—42no—
20C4×Dic166—42no—
21#14581751—22no—
22#14581732—22no—
23#14581732—22no—
24C2×C83⋊C16—22no—
25#14581726—22no—
26#14581726D414yes—
27#14581724—22no—
28#14581724—22no—
29C22×Q8×C83—21no—
30C2×Q16×C83—22no—
31C2×Q16×C83—22no—
32C2×Q16×C83—21no—
33C2×Q16×C83Q812yescomplemented
34C22×C8×C83D411yescharacteristic
35C22×C8×C83C2311yescharacteristic
36C2×C4⋊C4×C83—21no—
37C2×C42×C83D411yescharacteristic
38#106×C83—22no—
39#103×C83—42no—
40#97×C83—22no—
41#95×C83—42no—
42C2×Dic166—42no—
43C4⋊Dic83—24no—
44C4⋊Dic83—42no—
45Dic83⋊C4—44no—
46C22⋊Dic83—42no—
47C4×Dic83—42no—
48C83⋊C16—42no—
49C2×Q8×C83—22no—
50C2×Q8×C83—41no—
51C2×Q8×C83—41no—
52C2×Q8×C83—41no—
53C2×Q8×C83—41no—
54C2×Q8×C83—41no—
55C22×C4×C83—41no—
56C22×C4×C83—41no—
57C22×C4×C83D812yes—
58C22×C4×C83C2×D411yescharacteristic
59Q16×C83—24no—
60Q16×C83—42no—
61C2×C8×C83—21no—
62C2×C8×C83—21no—
63C2×C8×C83SD1612yes—
64C2×C8×C83C2×D412yes—
65C2×C8×C83C2×Q811yescharacteristic
66C2×C8×C83#4112yes—
67C2×C8×C83#4111yescharacteristic
68C4⋊C4×C83—41no—
69C4⋊C4×C83—41no—
70C42×C83—21no—
71C42×C83—21no—
72C23×C83C2×D811yescharacteristic
73Q8×C83—42no—
74Q8×C83—42no—
75Q8×C83—81no—
76Q8×C83—81no—
77C2×C4×C83—22no—
78C2×C4×C83—42no—
79C2×C4×C83—42no—
80C2×C4×C83—81no—
81C2×C4×C83—81no—
82C2×C4×C83—82no—
83C2×C4×C83—81no—
84C2×C4×C83#12011yescharacteristic
85C2×C4×C83C22⋊Q811yescharacteristic
86C2×C4×C83#12411yescharacteristic
87C2×C4×C83C2×SD1611yescharacteristic
88C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C83#13711yescharacteristic
90C2×Dic83—44no—
91Dic166—84no—
92C8×C83—22no—
93C8×C83C22⋊Q812yes—
94C8×C83#12412yes—
95C22×C83—22no—
96C22×C83#44711yescharacteristic
97C22×C83#47411yescharacteristic
98C22×C83#48611yescharacteristic
99C4×C83—42no—
100C4×C83—82no—
101C4×C83—161no—
102C4×C83#39311yescharacteristic
103C4×C83#44811yescharacteristic
104C4×C83#47611yescharacteristic
105C4×C83#48911yescharacteristic
106Dic83—84no—
107#13368—831nomaximal, Sylow
108C2×C83—21no—
109C2×C83—21no—
110C2×C83#202311yescharacteristic
111C2×C83#219112yes—
112#3162—831no—
113#2141—832no—
114#1971D8311yescharacteristic
115#1386—832no—
116#1377—831no—
117C83#1336811yescharacteristic, complemented, Sylow
118C22×Q16C2×D8311yescharacteristic
119#533—1661no—
120C2×M32—832no—
121#501—1664no—
122#427—832no—
123#427—1661no—
124C2×#106C2×D8311yescharacteristic
125C2×#103—21no—
126C2×#97C2×D8311yescharacteristic
127#391—21no—
128#358—1664no—
129C22×Q8—21no—
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—21no—
133C2×Q16Dic16612yescomplemented
134C22×C8C22⋊D8311yescharacteristic
135C22×C8C22×D8311yescharacteristic
136C4⋊Q8—1662no—
137#125—1662no—
138C22⋊Q8—3322no—
139C4×Q8—3322no—
140#117—1662no—
141C2×C4⋊C4—21no—
142C2×C42C22⋊D8311yescharacteristic
143M32—1662no—
144M32—1662no—
145C2×C16—1662no—
146#107—834no—
147#107—1662no—
148#106—22no—
149#106—1662no—
150#106—1662no—
151#103—42no—
152#97—22no—
153#95—42no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—3322no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#1305057512yes—
164C22×C4C2×C22⋊D8311yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1305057812yes—
170C2×C8D4×D8312yes—
171C2×C8C2×Dic16611yescharacteristic
172C2×C8#1305058912yes—
173C2×C8#1305059011yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—3324no—
177C4⋊C4—3322no—
178C4⋊C4—1664no—
179C22⋊C4—3322no—
180C42—21no—
181C42—21no—
182C42—3322no—
183C16—3322no—
184C23C2×#1305057511yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—6644no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—3324no—
198C2×C4#1458176011yescharacteristic
199C2×C4#1458177111yescharacteristic
200C2×C4#1458177911yescharacteristic
201C2×C4C2×#1305057811yescharacteristic
202C2×C4#1458181711yescharacteristic
203C2×C4#1458182211yescharacteristic
204C8—22no—
205C8#1458176912yes—
206C8#1458177812yes—
207C22—22no—
208C22#1912783411yescharacteristic
209C22#1912798111yescharacteristic
210C22#1912804711yescharacteristic
211C4—42no—
212C4—82no—
213C4—161no—
214C4—6644no—
215C4#1912761311yescharacteristic
216C4#1912783911yescharacteristic
217C4#1912799111yescharacteristic
218C4#1912806011yescharacteristic
219C2—21no—
220C2—21no—
221C2#2490859911yescharacteristic
222C2#2490941312yes—
subgroupG/H
centerC22#19127834
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1971C2
socleC22×C83#447
prime radicalC23×C83C2×D8
hypercenter#1971D83
hypocenterC83#13368
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1971

Complements

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

NH ≅ G/Ntype
33. C2×Q16×C83Q8semidirect
117. C83#13368semidirect
133. C2×Q16Dic166semidirect

Maximal subgroups

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

subgroupindex
#249179872
#249091832
#249055702
#249055472
C83×#19712
#1336883