Finite Group Databasebeta

← #35323494

Subgroups — #35323494

Order 21248.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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#24909184C212yesmaximal
3#24908334C211yesmaximal, characteristic
4#24905569C212yesmaximal
5C83×#1377C211yesmaximal, characteristic
6C22×Dic332C2211yescharacteristic
7#19128258—21no—
8#19128110—24no—
9#19127739—21no—
10#19127739C2212yes—
11C2×#14581724C2211yescharacteristic
12C2×#14581717—21no—
13#19127607—21no—
14#19127506—24no—
15C2×M32×C83C2212yes—
16C2×#97×C83C2211yescharacteristic
17C22×Dic166—21no—
18C2×Dic332—22no—
19C2×Dic332—22no—
20C2×Dic332—21no—
21C2×Dic332Q812yescomplemented
22#14581792—22no—
23#14581784—22no—
24#14581771—42no—
25C4×Dic166—42no—
26#14581751—22no—
27C2×C4⋊Dic83—21no—
28#14581726—22no—
29#14581726D414yes—
30#14581724—22no—
31#14581724—22no—
32#14581724—22no—
33#14581717—42no—
34C83⋊#95—42no—
35C22×C8×C83D411yescharacteristic
36C22×C8×C83C2311yescharacteristic
37C2×C42×C83D411yescharacteristic
38M32×C83—22no—
39M32×C83—22no—
40C2×C16×C83—22no—
41#97×C83—22no—
42C2×Dic166—22no—
43C2×Dic166—41no—
44C2×Dic166—41no—
45C2×Dic166—41no—
46C2×Dic166—41no—
47C2×Dic166—42no—
48C2×Dic166—41no—
49C22×Dic83—41no—
50C22×Dic83—41no—
51Dic332—24no—
52Dic332—42no—
53C4⋊Dic83—24no—
54C4⋊Dic83—42no—
55C4⋊Dic83—41no—
56C4⋊Dic83—41no—
57Dic83⋊C4—44no—
58C22⋊Dic83—42no—
59C4×Dic83—42no—
60C22×C4×C83D812yes—
61C22×C4×C83C2×D411yescharacteristic
62C2×C8×C83—21no—
63C2×C8×C83—21no—
64C2×C8×C83SD1612yes—
65C2×C8×C83C2×D412yes—
66C2×C8×C83C2×Q811yescharacteristic
67C2×C8×C83#4111yescharacteristic
68C2×C8×C83#4112yes—
69C42×C83—21no—
70C42×C83—21no—
71C16×C83—42no—
72C23×C83C2×D811yescharacteristic
73C2×C4×C83—22no—
74C2×C4×C83—42no—
75C2×C4×C83#12011yescharacteristic
76C2×C4×C83C22⋊Q811yescharacteristic
77C2×C4×C83#12411yescharacteristic
78C2×C4×C83C2×SD1611yescharacteristic
79C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C83#13711yescharacteristic
81C2×Dic83—44no—
82C2×Dic83—42no—
83C2×Dic83—81no—
84C2×Dic83—81no—
85C2×Dic83—82no—
86C2×Dic83—81no—
87Dic166—42no—
88Dic166—42no—
89Dic166—84no—
90Dic166—81no—
91Dic166—81no—
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#39311yescharacteristic
101C4×C83#44811yescharacteristic
102C4×C83#47611yescharacteristic
103C4×C83#48911yescharacteristic
104Dic83—84no—
105Dic83—82no—
106Dic83—161no—
107#13368—831nomaximal, Sylow
108C2×C83—21no—
109C2×C83—21no—
110C2×C83#202311yescharacteristic
111C2×C83#219112yes—
112#3162—831no—
113#2141—832no—
114#1971—831no—
115#1386—832no—
116#1377D8311yescharacteristic
117C83#1336811yescharacteristic, complemented, Sylow
118C22×Q16—831no—
119#533—1661no—
120C2×M32C2×D8312yes—
121#501—1664no—
122#427—832no—
123#427—1661no—
124C2×#106—831no—
125C2×#103—1661no—
126C2×#97C2×D8311yescharacteristic
127#391—1661no—
128#358—1664no—
129C22×Q8—1661no—
130C2×Q16—832no—
131C2×Q16—1662no—
132C2×Q16—1661no—
133C2×Q16—1662no—
134C22×C8D33211yescharacteristic
135C22×C8C22×D8311yescharacteristic
136C4⋊Q8—1662no—
137#125—1662no—
138C22⋊Q8—3322no—
139C4×Q8—3322no—
140#117—1662no—
141C2×C4⋊C4—1661no—
142C2×C42D33211yescharacteristic
143M32—22no—
144M32—22no—
145C2×C16—22no—
146#107—834no—
147#107—1662no—
148#106—1662no—
149#106—1662no—
150#106—1662no—
151#103—3322no—
152#97—22no—
153#95—3322no—
154C2×Q8—3322no—
155C2×Q8—3321no—
156C2×Q8—3321no—
157C2×Q8—3321no—
158C2×Q8—3321no—
159C2×Q8—3321no—
160C2×Q8—1662no—
161C22×C4—3321no—
162C22×C4—3321no—
163C22×C4D66412yes—
164C22×C4C2×D33211yescharacteristic
165Q16—3322no—
166Q16—1664no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1305057912yes—
170C2×C8D4×D8312yes—
171C2×C8Q8×D8311yescharacteristic
172C2×C8#1305058911yescharacteristic
173C2×C8#1305059012yes—
174C4⋊C4—3324no—
175C4⋊C4—3322no—
176C4⋊C4—3321no—
177C4⋊C4—3321no—
178C4⋊C4—1664no—
179C22⋊C4—3322no—
180C42—21no—
181C42—21no—
182C42—3322no—
183C16—42no—
184C23C2×D66411yescharacteristic
185Q8—3322no—
186Q8—3322no—
187Q8—6644no—
188Q8—6641no—
189Q8—6641no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—3324no—
193C2×C4—3322no—
194C2×C4—6642no—
195C2×C4—6641no—
196C2×C4—6641no—
197C2×C4—6641no—
198C2×C4#1458176211yescharacteristic
199C2×C4#1458177211yescharacteristic
200C2×C4#1458178011yescharacteristic
201C2×C4C2×#1305057911yescharacteristic
202C2×C4#1458182011yescharacteristic
203C2×C4#1458182511yescharacteristic
204C8—22no—
205C8#1458177012yes—
206C8#1458177812yes—
207C22—22no—
208C22#1912783711yescharacteristic
209C22#1912798211yescharacteristic
210C22#1912804811yescharacteristic
211C4—42no—
212C4—13281no—
213C4—6642no—
214C4—6644no—
215C4#1912761411yescharacteristic
216C4#1912784211yescharacteristic
217C4#1912799211yescharacteristic
218C4#1912806111yescharacteristic
219C2—21no—
220C2—21no—
221C2#2490860011yescharacteristic
222C2#2490941612yes—
subgroupG/H
centerC22#19127837
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1377C2
socleC22×C83#447
prime radicalC23×C83C2×D8
hypercenter#1377D83
hypocenterC83#13368
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

Complements

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

NH ≅ G/Ntype
21. C2×Dic332Q8semidirect
117. C83#13368semidirect

Maximal subgroups

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

subgroupindex
#249179872
#249091842
#249083342
#249055692
C83×#13772
#1336883