Finite Group Databasebeta

← #35318090

Subgroups — #35318090

Order 21248.

298 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 69 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#24920352C211yesmaximal, characteristic, complemented
2#24909687C211yesmaximal, characteristic
3#24909503C211yesmaximal, characteristic
4#24906824C211yesmaximal, characteristic
5#24905519C211yesmaximal, characteristic, complemented
6#24905443C211yesmaximal, characteristic, complemented
7C83×#1332C211yesmaximal, characteristic, complemented
8#19128430—21no—
9C2×#14581735—21no—
10#19128106C2211yescharacteristic
11#19128102—21no—
12#19128078C2211yescharacteristic
13#19128010—22no—
14#19127986—21no—
15#19127902—21no—
16C8×Dic166C2211yescharacteristic
17Q16×Dic83—21no—
18C4⋊(C83⋊C16)C2211yescharacteristic
19#19127503—22no—
20#19127501—21no—
21#19127444—21no—
22C4×Q16×C83C2211yescharacteristic
23C4×C16×C83C2211yescharacteristic
24#335×C83C2211yescharacteristic
25C2×#13050579—41no—
26C2×D664—21no—
27C2×D664—21no—
28#14581792—21no—
29#14581792—41no—
30#14581789—21no—
31#14581783—22no—
32#14581783—42no—
33Q8×Dic83—22no—
34C4×Dic166—41no—
35C4×Dic166D411yescharacteristic
36#14581735—42no—
37C2×C83⋊C16—21no—
38#14581726—21no—
39#14581726D412yes—
40#14581723—22no—
41C4⋊(C83⋊C8)D411yescharacteristic
42Dic83⋊C8—21no—
43#14581716—22no—
44#14581714—41no—
45C8×Dic83—21no—
46C2×Q16×C83D411yescharacteristic
47C4×Q8×C83—21no—
48C2×C16×C83—21no—
49#107×C83D411yescharacteristic
50C4⋊C8×C83—21no—
51#103×C83—21no—
52C4×C8×C83C2311yescharacteristic
53C2×Dic166—41no—
54C2×Dic166D811yescharacteristic
55C2×D332—41no—
56C2×D332—41no—
57C2×D332—41no—
58#13050579—82no—
59D664—42no—
60D664—41no—
61D664—41no—
62C2×C83⋊C8—21no—
63C4⋊Dic83—22no—
64C4⋊Dic83—42no—
65C4⋊Dic83—42no—
66C4⋊Dic83—41no—
67C4⋊Dic83D811yescharacteristic
68Dic83⋊C4—21no—
69Dic83⋊C4—42no—
70C4×Dic83—21no—
71C4×Dic83—42no—
72C83⋊C16—41no—
73C2×Q8×C83—21no—
74Q16×C83—21no—
75Q16×C83SD1612yescomplemented
76C2×C8×C83—41no—
77C2×C8×C83C2×D411yescharacteristic
78C2×C8×C83C2×D411yescharacteristic
79C4⋊C4×C83—21no—
80C4⋊C4×C83—41no—
81C42×C83—41no—
82C42×C83C2×D411yescharacteristic
83C16×C83—41no—
84C22×D83—81no—
85Q8×C83—22no—
86Q8×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83#12011yescharacteristic
90C2×C4×C83C2×D811yescharacteristic
91C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—42no—
95D332—81no—
96D332—81no—
97D332—81no—
98D332—81no—
99D332—161no—
100Dic166—21no—
101Dic166—21no—
102Dic166—81no—
103Dic166—81no—
104C8×C83—21no—
105C8×C83—81no—
106C8×C83C2×SD1611yescharacteristic
107C8×C83#13711yescharacteristic
108C83⋊C8—41no—
109C22×C83#44711yescharacteristic
110C2×D83—161no—
111C2×D83—161no—
112C2×D83—161no—
113C4×C83—21no—
114C4×C83—42no—
115C4×C83—81no—
116C4×C83#44811yescharacteristic
117C4×C83#45411yescharacteristic
118C4×C83#50811yescharacteristic
119C4×C83#50911yescharacteristic
120Dic83—41no—
121Dic83—41no—
122Dic83—82no—
123#12353—831nomaximal, Sylow
124C2×C83#162711yescharacteristic
125C2×C83#218611yescharacteristic
126C2×C83#219311yescharacteristic
127D83—321no—
128#3383—831no—
129#2240—831no—
130#2207—831no—
131#1668—831no—
132#1364—831no—
133Q8⋊C16—831no—
134#1332D8311yescharacteristic, complemented
135C83#1235311yescharacteristic, complemented, Sylow
136#557—1661no—
137C2×SD32—1661no—
138#500—831no—
139#499—1661no—
140#493—831no—
141#479—1662no—
142#475—1661no—
143#460—1661no—
144C8×Q8—831no—
145C4×Q16—1661no—
146C4×Q16C2×D8311yescharacteristic
147C4⋊C16—831no—
148#357—1662no—
149#357—1661no—
150C4×C16C2×D8311yescharacteristic
151#335C2×D8311yescharacteristic
152#331—1661no—
153C2×Q16D33211yescharacteristic
154C2×SD16—3321no—
155C2×D8—1661no—
156C2×D8—1661no—
157C4⋊Q8—3321no—
158C4⋊Q8—1661no—
159#127—1661no—
160#125—3322no—
161#125—1662no—
162C4×Q8—21no—
163C4×Q8—3321no—
164C4×Q8—831no—
165C4×Q8—1662no—
166SD32—3322no—
167C2×C16—21no—
168C2×C16—1661no—
169#107—832no—
170#107—1661no—
171#107D33211yescharacteristic
172#106—1662no—
173C4⋊C8—21no—
174C4⋊C8—831no—
175C4⋊C8—1661no—
176#103—21no—
177#103—1662no—
178#102—3321no—
179C4×C8—1661no—
180C4×C8C22×D8311yescharacteristic
181C2×Q8—21no—
182C2×Q8—3321no—
183C2×Q8—831no—
184C2×D4—3321no—
185C2×D4—3321no—
186C2×D4—3321no—
187Q16—21no—
188Q16#1305057912yescomplemented
189SD16—6642no—
190D8—3322no—
191D8—3321no—
192D8—3321no—
193C2×C8—41no—
194C2×C8—1661no—
195C2×C8D4×D8311yescharacteristic
196C2×C8C2×D33211yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—3322no—
200C4⋊C4—3322no—
201C4⋊C4—3322no—
202C4⋊C4—3321no—
203C4⋊C4—831no—
204C4⋊C4—1662no—
205C4⋊C4—1661no—
206C42—41no—
207C42—3322no—
208C42—1661no—
209C42D4×D8311yescharacteristic
210C16—41no—
211C16—3321no—
212C23—6641no—
213Q8—22no—
214Q8—41no—
215Q8—6641no—
216Q8—6641no—
217Q8—1661no—
218Q8—1661no—
219D4—13281no—
220D4—6641no—
221D4—6641no—
222D4—6641no—
223D4—6641no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—3322no—
227C2×C4—1661no—
228C2×C4—1661no—
229C2×C4#1458176211yescharacteristic
230C2×C4D8×D8311yescharacteristic
231C2×C4#1458181911yescharacteristic
232C8—21no—
233C8—81no—
234C8—3321no—
235C8C2×#1305057911yescharacteristic
236C8#1458182511yescharacteristic
237C22—13281no—
238C22—13281no—
239C22—13281no—
240C22#1912783611yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—3321no—
245C4—3321no—
246C4—6642no—
247C4#1912784211yescharacteristic
248C4#1912787511yescharacteristic
249C4#1912813411yescharacteristic
250C4#1912813811yescharacteristic
251C2—26561no—
252C2#2490656911yescharacteristic
253C2#2490938911yescharacteristic
254C2#2490942811yescharacteristic
subgroupG/H
centerC22#19127836
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1332C2
socleC22×C83#447
prime radical#19128078C22
hypercenter#1332D83
hypocenterC83#12353
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
1. #24920352C2semidirect
5. #24905519C2semidirect
6. #24905443C2semidirect
7. C83×#1332C2semidirect
75. Q16×C83SD16semidirect
134. #1332D83semidirect
135. C83#12353semidirect
188. Q16#13050579semidirect

Maximal subgroups

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

subgroupindex
#249203522
#249096872
#249095032
#249068242
#249055192
#249054432
C83×#13322
#1235383