Finite Group Databasebeta

← #35317683

Subgroups — #35317683

Order 21248.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#24918773C211yesmaximal, characteristic
2#24909648C212yesmaximal, complemented
3#24905471C212yesmaximal, complemented
4#24905403C211yesmaximal, characteristic, complemented
5C83×#2087C211yesmaximal, characteristic, complemented
6#19128251—21no—
7#19127982—22no—
8#19127937—21no—
9#19127937C2212yes—
10#19127897—22no—
11#19127528—22no—
12#19127501—22no—
13C22⋊(C83⋊C16)C2212yes—
14C2×#107×C83C2211yescharacteristic
15C2×C4⋊C8×C83C2211yescharacteristic
16#389×C83—21no—
17#340×C83—21no—
18#340×C83C2211yescharacteristic
19C2×#13050590—41no—
20C2×D664—21no—
21C2×D664D412yes—
22#14581792—21no—
23#14581789—21no—
24#14581780—41no—
25#14581766—22no—
26#14581766—41no—
27C4×D332—22no—
28C4×D332—41no—
29C2×C83⋊C16—22no—
30#14581726—21no—
31#14581726D412yes—
32#14581714—22no—
33#14581714—22no—
34C22×C8×C83D411yescharacteristic
35C22×C8×C83C2311yescharacteristic
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—41no—
39C2×C42×C83D411yescharacteristic
40#107×C83—22no—
41C4⋊C8×C83—22no—
42#95×C83—41no—
43#13050590—82no—
44#13050590—82no—
45C2×Dic166—41no—
46C2×D332—22no—
47C2×D332—41no—
48C2×D332—41no—
49C2×C22⋊D83—42no—
50C2×C4×D83—42no—
51D664—42no—
52D664—81no—
53C4⋊Dic83—22no—
54C4⋊Dic83—41no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—42no—
57C83⋊C16—42no—
58C22×C4×C83—41no—
59C22×C4×C83—41no—
60C22×C4×C83D811yescharacteristic
61C22×C4×C83D811yescharacteristic
62C22×C4×C83C2×D411yescharacteristic
63C2×C8×C83—21no—
64C2×C8×C83—41no—
65C2×C8×C83—41no—
66C2×C8×C83C2×D412yes—
67C4⋊C4×C83—22no—
68C4⋊C4×C83—41no—
69C4⋊C4×C83—41no—
70C4⋊C4×C83—41no—
71C4⋊C4×C83—82no—
72C42×C83—21no—
73C42×C83—21no—
74C23×C83C2×D811yescharacteristic
75C22×D83—42no—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—41no—
82C2×C4×C83—42no—
83C2×C4×C83—81no—
84C2×C4×C83—81no—
85C2×C4×C83—81no—
86C2×C4×C83—81no—
87C2×C4×C83—81no—
88C2×C4×C83SD3212yes—
89C2×C4×C83#12011yescharacteristic
90C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
91C22⋊D83—82no—
92C22⋊D83—82no—
93C2×Dic83—42no—
94D332—42no—
95D332—42no—
96D332—82no—
97D332—82no—
98C4×D83—84no—
99Dic166—82no—
100C8×C83—22no—
101C8×C83—41no—
102C22×C83—21no—
103C22×C83—21no—
104C22×C83#44711yescharacteristic
105C22×C83C2×SD3211yescharacteristic
106C22×C83#50911yescharacteristic
107C2×D83—82no—
108C2×D83—82no—
109C2×D83—82no—
110C4×C83—21no—
111C4×C83—22no—
112C4×C83—41no—
113C4×C83—82no—
114C4×C83—161no—
115C4×C83#45611yescharacteristic
116C4×C83#45811yescharacteristic
117Dic83—82no—
118#12291—831nomaximal, Sylow
119C2×C83—21no—
120C2×C83—21no—
121C2×C83#161011yescharacteristic
122C2×C83#218711yescharacteristic
123C2×C83#219011yescharacteristic
124D83—162no—
125#3245—831no—
126#2233—832no—
127#2087D8311yescharacteristic, complemented
128#1347—832no—
129#1322—831no—
130C83#1229111yescharacteristic, complemented, Sylow
131#532—1661no—
132#474—1662no—
133#466—832no—
134#466—1661no—
135#459—1662no—
136C2×#107C2×D8311yescharacteristic
137C2×C4⋊C8C2×D8311yescharacteristic
138#389—21no—
139#367—1662no—
140#357—1662no—
141C22⋊C16—832no—
142#340—21no—
143#340C2×D8311yescharacteristic
144C2×#41—3321no—
145C2×D8—832no—
146C2×D8—1661no—
147C22×C8C22⋊D8311yescharacteristic
148C22×C8C22×D8311yescharacteristic
149C4⋊Q8—1661no—
150#127—1661no—
151#124—3321no—
152#121—3321no—
153#121—1662no—
154C4×D4—3321no—
155C4×D4—1662no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42C22⋊D8311yescharacteristic
160C2×C16—1662no—
161#107—22no—
162#107—832no—
163#107—1661no—
164C4⋊C8—22no—
165#102—1662no—
166#102—1662no—
167#95—41no—
168#41—6642no—
169#41—6642no—
170C2×Q8—3321no—
171C2×D4—3321no—
172C2×D4—3321no—
173C2×D4—3322no—
174C2×D4—1662no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—3322no—
178C22×C4#1305057511yescharacteristic
179C22×C4#1305057511yescharacteristic
180C22×C4C2×C22⋊D8311yescharacteristic
181D8—3322no—
182D8—6641no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D8312yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—3321no—
193C4⋊C4—1662no—
194C22⋊C4—3322no—
195C22⋊C4—3322no—
196C42—21no—
197C42—21no—
198C16—3322no—
199C23—3322no—
200C23C2×#1305057511yescharacteristic
201Q8—6642no—
202D4—3322no—
203D4—3322no—
204D4—6642no—
205D4—6642no—
206D4—6642no—
207D4—6642no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—3322no—
221C2×C4—6644no—
222C2×C4#1458173512yes—
223C2×C4#1458176011yescharacteristic
224C2×C4#1458181711yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—6642no—
230C22—6642no—
231C22—6642no—
232C22#1912783411yescharacteristic
233C22C2×#1458173511yescharacteristic
234C22#1912813611yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—161no—
240C4—6642no—
241C4#1912788311yescharacteristic
242C4#1912789011yescharacteristic
243C2—21no—
244C2—21no—
245C2—13282no—
246C2#2490646511yescharacteristic
247C2#2490939111yescharacteristic
248C2#2490940811yescharacteristic
subgroupG/H
centerC22#19127834
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#2087C2
socleC22×C83#447
prime radical#24918773C2
hypercenter#2087D83
hypocenterC83#12291
LCSG ▷ C22×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2087

Complements

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

NH ≅ G/Ntype
2. #24909648C2semidirect
3. #24905471C2semidirect
4. #24905403C2semidirect
5. C83×#2087C2semidirect
127. #2087D83semidirect
130. C83#12291semidirect

Maximal subgroups

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

subgroupindex
#249187732
#249096482
#249054712
#249054032
C83×#20872
#1229183