Finite Group Databasebeta

← #35318086

Subgroups — #35318086

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#24909686C211yesmaximal, characteristic
3#24906820C211yesmaximal, characteristic
4#24905517C211yesmaximal, characteristic, complemented
5#24905441C211yesmaximal, characteristic, complemented
6#24905427C211yesmaximal, characteristic, complemented
7C83×#2207C211yesmaximal, characteristic
8#19128430—21no—
9#19128106C2211yescharacteristic
10#19128102—21no—
11#19128010—22no—
12#19127986—21no—
13#19127898—21no—
14C8×Dic166C2211yescharacteristic
15Q16×Dic83—21no—
16#19127502—22no—
17C4×C83⋊C16C2211yescharacteristic
18#19127454C2211yescharacteristic
19#19127445—21no—
20C2×SD32×C83—21no—
21#493×C83C2211yescharacteristic
22C4×Q16×C83C2211yescharacteristic
23C4⋊C16×C83C2211yescharacteristic
24#357×C83—21no—
25C2×#13050578—41no—
26#14581792—21no—
27#14581792—41no—
28#14581783—22no—
29#14581783—42no—
30Q8×Dic83—22no—
31C4×Dic166—41no—
32C4×Dic166D411yescharacteristic
33C2×C83⋊C16—21no—
34#14581726—21no—
35#14581726D412yes—
36#14581723—22no—
37C4⋊(C83⋊C8)—21no—
38C4⋊(C83⋊C8)D411yescharacteristic
39Dic83⋊C8—21no—
40#14581716—22no—
41#14581713—41no—
42C8×Dic83—21no—
43C2×Q16×C83D411yescharacteristic
44C2×D8×C83—21no—
45C2×D8×C83—21no—
46#127×C83—21no—
47C4×Q8×C83—21no—
48SD32×C83—42no—
49C2×C16×C83—21no—
50#107×C83D411yescharacteristic
51#103×C83—21no—
52C4×C8×C83C2311yescharacteristic
53C2×Dic166—41no—
54C2×Dic166D811yescharacteristic
55#13050578—82no—
56C2×C83⋊C8—21no—
57C2×C83⋊C8—41no—
58C4⋊Dic83—22no—
59C4⋊Dic83—42no—
60C4⋊Dic83—42no—
61C4⋊Dic83—41no—
62C4⋊Dic83D811yescharacteristic
63Dic83⋊C4—21no—
64Dic83⋊C4—42no—
65C4×Dic83—21no—
66C4×Dic83—42no—
67C83⋊C16—41no—
68C2×Q8×C83—21no—
69C2×D4×C83—41no—
70C2×D4×C83—41no—
71C2×D4×C83—41no—
72Q16×C83—21no—
73Q16×C83SD1612yescomplemented
74D8×C83—42no—
75D8×C83—41no—
76D8×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—
84C23×C83—81no—
85Q8×C83—22no—
86Q8×C83—41no—
87D4×C83—81no—
88D4×C83—81no—
89D4×C83—81no—
90D4×C83—81no—
91D4×C83—161no—
92C2×C4×C83—41no—
93C2×C4×C83—41no—
94C2×C4×C83#12011yescharacteristic
95C2×C4×C83C2×D811yescharacteristic
96C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—42no—
100Dic166—21no—
101Dic166—21no—
102Dic166—81no—
103Dic166—81no—
104C8×C83—21no—
105C8×C83C2×SD1611yescharacteristic
106C8×C83#13711yescharacteristic
107C83⋊C8—41no—
108C83⋊C8—81no—
109C22×C83—161no—
110C22×C83—161no—
111C22×C83—161no—
112C22×C83#44711yescharacteristic
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—321no—
125C2×C83#162711yescharacteristic
126C2×C83#218611yescharacteristic
127C2×C83#219311yescharacteristic
128#3383—831no—
129#2240—831no—
130#2207D8311yescharacteristic
131#1668—831no—
132#1364—831no—
133Q8⋊C16—831no—
134#1332—831no—
135C83#1235311yescharacteristic, complemented, Sylow
136#557—1661no—
137C2×SD32—21no—
138#500—831no—
139#499—1661no—
140#493C2×D8311yescharacteristic
141#479—1662no—
142#475—1661no—
143#460—1661no—
144C8×Q8—831no—
145C4×Q16—1661no—
146C4×Q16C2×D8311yescharacteristic
147C4⋊C16C2×D8311yescharacteristic
148#357—21no—
149#357—1662no—
150C4×C16—831no—
151#335—831no—
152#331—1661no—
153C2×Q16C22⋊D8311yescharacteristic
154C2×SD16—3321no—
155C2×D8—21no—
156C2×D8—21no—
157C4⋊Q8—3321no—
158C4⋊Q8—1661no—
159#127—21no—
160#125—3322no—
161#125—1662no—
162C4×Q8—21no—
163C4×Q8—3321no—
164C4×Q8—831no—
165C4×Q8—1662no—
166SD32—42no—
167C2×C16—21no—
168C2×C16—1661no—
169#107—832no—
170#107—1661no—
171#107C22⋊D8311yescharacteristic
172#106—1662no—
173C4⋊C8—831no—
174C4⋊C8—1661no—
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—41no—
185C2×D4—41no—
186C2×D4—41no—
187Q16—21no—
188Q16#1305057812yescomplemented
189SD16—6642no—
190D8—42no—
191D8—41no—
192D8—41no—
193C2×C8—3321no—
194C2×C8—1661no—
195C2×C8D4×D8311yescharacteristic
196C2×C8C2×C22⋊D8311yescharacteristic
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—81no—
213Q8—22no—
214Q8—41no—
215Q8—6641no—
216Q8—6641no—
217Q8—1661no—
218Q8—1661no—
219D4—81no—
220D4—81no—
221D4—81no—
222D4—81no—
223D4—161no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—3322no—
227C2×C4—1661no—
228C2×C4—1661no—
229C2×C4#1458176011yescharacteristic
230C2×C4D8×D8311yescharacteristic
231C2×C4#1458181611yescharacteristic
232C8—21no—
233C8—3321no—
234C8—6641no—
235C8C2×#1305057811yescharacteristic
236C8#1458182211yescharacteristic
237C22—161no—
238C22—161no—
239C22—161no—
240C22#1912783311yescharacteristic
241C4—21no—
242C4—42no—
243C4—81no—
244C4—3321no—
245C4—3321no—
246C4—6642no—
247C4#1912783911yescharacteristic
248C4#1912787211yescharacteristic
249C4#1912813211yescharacteristic
250C4#1912813811yescharacteristic
251C2—321no—
252C2#2490656511yescharacteristic
253C2#2490938511yescharacteristic
254C2#2490942411yescharacteristic
subgroupG/H
centerC22#19127833
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#2207C2
socleC22×C83#447
prime radical#493×C83C22
hypercenter#2207D83
hypocenterC83#12353
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #24920352C2semidirect
4. #24905517C2semidirect
5. #24905441C2semidirect
6. #24905427C2semidirect
73. Q16×C83SD16semidirect
135. C83#12353semidirect
188. Q16#13050578semidirect

Maximal subgroups

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

subgroupindex
#249203522
#249096862
#249068202
#249055172
#249054412
#249054272
C83×#22072
#1235383