Finite Group Databasebeta

← #35317686

Subgroups — #35317686

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#24909649C212yesmaximal, complemented
3#24908956C211yesmaximal, characteristic, complemented
4#24905473C212yesmaximal, complemented
5C83×#1322C211yesmaximal, characteristic, complemented
6#19128251—21no—
7#19127982—22no—
8#19127937—21no—
9#19127937C2212yes—
10#19127897—22no—
11C2×#14581726C2211yescharacteristic
12#19127599—21no—
13#19127529—22no—
14#19127503—22no—
15#19127464—21no—
16#19127464C2211yescharacteristic
17C2×C4⋊C8×C83C2211yescharacteristic
18C22⋊C16×C83C2212yes—
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×C4⋊Dic83—21no—
30C2×C4⋊Dic83—21no—
31C2×Dic83⋊C4—41no—
32#14581726—21no—
33#14581726—22no—
34#14581726D412yes—
35#14581714—22no—
36#14581714—22no—
37C83⋊#95—41no—
38C22×C8×C83D411yescharacteristic
39C22×C8×C83C2311yescharacteristic
40C2×C42×C83D411yescharacteristic
41C2×C16×C83—22no—
42C4⋊C8×C83—22no—
43#13050590—82no—
44#13050590—82no—
45C2×Dic166—41no—
46C2×D332—22no—
47C2×D332—41no—
48C2×D332—41no—
49C2×C22⋊D83—42no—
50C22×Dic83—41no—
51C22×Dic83—41no—
52C2×C4×D83—42no—
53D664—42no—
54D664—81no—
55C4⋊Dic83—22no—
56C4⋊Dic83—22no—
57C4⋊Dic83—41no—
58C4⋊Dic83—41no—
59C4⋊Dic83—41no—
60C4⋊Dic83—41no—
61Dic83⋊C4—82no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64C22×C4×C83D811yescharacteristic
65C22×C4×C83D811yescharacteristic
66C22×C4×C83C2×D411yescharacteristic
67C2×C8×C83—21no—
68C2×C8×C83—41no—
69C2×C8×C83—41no—
70C2×C8×C83C2×D412yes—
71C42×C83—21no—
72C42×C83—21no—
73C16×C83—42no—
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×C83SD3212yes—
83C2×C4×C83#12011yescharacteristic
84C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
85C22⋊D83—82no—
86C22⋊D83—82no—
87C2×Dic83—42no—
88C2×Dic83—42no—
89C2×Dic83—81no—
90C2×Dic83—81no—
91C2×Dic83—81no—
92C2×Dic83—81no—
93C2×Dic83—81no—
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#45611yescharacteristic
114C4×C83#45811yescharacteristic
115Dic83—82no—
116Dic83—82no—
117Dic83—161no—
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#2087—831no—
128#1347—832no—
129#1322D8311yescharacteristic, complemented
130C83#1229111yescharacteristic, complemented, Sylow
131#532—1661no—
132#474—1662no—
133#466—832no—
134#466—1661no—
135#459—1662no—
136C2×#107—831no—
137C2×C4⋊C8C2×D8311yescharacteristic
138#389—1661no—
139#367—1662no—
140#357—1662no—
141C22⋊C16C2×D8312yes—
142#340—831no—
143#340—1661no—
144C2×#41—3321no—
145C2×D8—832no—
146C2×D8—1661no—
147C22×C8D33211yescharacteristic
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—3321no—
157C2×C4⋊C4—1661no—
158C2×C4⋊C4—1661no—
159C2×C42D33211yescharacteristic
160C2×C16—22no—
161#107—832no—
162#107—1662no—
163#107—1661no—
164C4⋊C8—22no—
165#102—1662no—
166#102—1662no—
167#95—3321no—
168#41—6642no—
169#41—6642no—
170C2×Q8—3321no—
171C2×D4—3321no—
172C2×D4—3321no—
173C2×D4—3322no—
174C2×D4—1662no—
175C22×C4—3322no—
176C22×C4—3321no—
177C22×C4—3321no—
178C22×C4D66411yescharacteristic
179C22×C4D66411yescharacteristic
180C22×C4C2×D33211yescharacteristic
181D8—3322no—
182D8—6641no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D8312yes—
187C4⋊C4—3321no—
188C4⋊C4—3321no—
189C4⋊C4—3321no—
190C4⋊C4—3321no—
191C4⋊C4—6642no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C22⋊C4—3322no—
195C22⋊C4—3322no—
196C42—21no—
197C42—21no—
198C16—42no—
199C23—3322no—
200C23C2×D66411yescharacteristic
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—3322no—
215C2×C4—3322no—
216C2×C4—6644no—
217C2×C4—6641no—
218C2×C4—6641no—
219C2×C4—6641no—
220C2×C4—6641no—
221C2×C4—6641no—
222C2×C4#1458173712yes—
223C2×C4#1458176211yescharacteristic
224C2×C4#1458182011yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—6642no—
230C22—6642no—
231C22—6642no—
232C22#1912783711yescharacteristic
233C22C2×#1458173711yescharacteristic
234C22#1912813911yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—13281no—
239C4—6642no—
240C4—6642no—
241C4#1912788611yescharacteristic
242C4#1912789211yescharacteristic
243C2—21no—
244C2—21no—
245C2—13282no—
246C2#2490646811yescharacteristic
247C2#2490939411yescharacteristic
248C2#2490941111yescharacteristic
subgroupG/H
centerC22#19127837
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1322C2
socleC22×C83#447
prime radical#24918773C2
hypercenter#1322D83
hypocenterC83#12291
LCSG ▷ C22×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #24909649C2semidirect
3. #24908956C2semidirect
4. #24905473C2semidirect
5. C83×#1322C2semidirect
129. #1322D83semidirect
130. C83#12291semidirect

Maximal subgroups

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

subgroupindex
#249187732
#249096492
#249089562
#249054732
C83×#13222
#1229183