Finite Group Databasebeta

← #35313846

Subgroups — #35313846

Order 21248.

288 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 75 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#24916108C211yesmaximal, characteristic
2#24909275C211yesmaximal, characteristic, complemented
3#24905415C212yesmaximal, complemented
4#24905400C212yesmaximal
5C83×#2165C211yesmaximal, characteristic, complemented
6C4×#13050590C411yescharacteristic
7#19127763—21no—
8#19127763—21no—
9#19127752—21no—
10#19127752—21no—
11#19127729C411yescharacteristic
12#19127729C2211yescharacteristic
13#19127669C411yescharacteristic
14#19127669C2211yescharacteristic
15C2×C4×C83⋊C8C411yescharacteristic
16#19127522—21no—
17#19127483C2212yes—
18#19127479—21no—
19C4×M16×C83C2211yescharacteristic
20#364×C83—21no—
21#349×C83C2212yes—
22#347×C83—21no—
23C2×#13050590C2×C411yescharacteristic
24C22×C83⋊C8C2×C411yescharacteristic
25C22×C83⋊C8C2×C411yescharacteristic
26C4×Dic166—21no—
27C4×D332—21no—
28C4×C22⋊D83—22no—
29#14581751C2×C411yescharacteristic
30#14581749—21no—
31C42×D83—21no—
32#14581732—22no—
33Dic83⋊C8—21no—
34Dic83⋊C8—21no—
35Dic83⋊C8—21no—
36Dic83⋊C8—21no—
37(C2×D83)⋊C8—21no—
38(C2×D83)⋊C8—21no—
39(C2×D83)⋊C8—21no—
40(C2×D83)⋊C8—21no—
41C4×C83⋊C8—21no—
42C4×C83⋊C8—21no—
43C2×M16×C83C2×C411yescharacteristic
44C2×M16×C83C2311yescharacteristic
45C2×C42×C83C2×C411yescharacteristic
46M32×C83—22no—
47#97×C83D411yescharacteristic
48#97×C83D411yescharacteristic
49C4×C8×C83D411yescharacteristic
50C4×C8×C83D411yescharacteristic
51#13050590—41no—
52#13050590—41no—
53C2×Dic166M1611yescharacteristic
54C2×D332M1611yescharacteristic
55C2×C22⋊D83—21no—
56C2×C4×D83—22no—
57C2×C4×D83—21no—
58C2×C83⋊C8—21no—
59C2×C83⋊C8—21no—
60C2×C83⋊C8—41no—
61C2×C83⋊C8—41no—
62C4⋊Dic83—21no—
63Dic83⋊C4—22no—
64C22⋊Dic83M1612yes—
65(C2×D83)⋊C4—22no—
66C4×Dic83—22no—
67C4×Dic83—21no—
68C83⋊C16—42no—
69C22×C4×C83—21no—
70C22×C4×C83C22×C411yescharacteristic
71M16×C83—41no—
72M16×C83—41no—
73C2×C8×C83C22⋊C411yescharacteristic
74C2×C8×C83C22⋊C411yescharacteristic
75C2×C8×C83C2×D411yescharacteristic
76C2×C8×C83C2×D411yescharacteristic
77C42×C83—21no—
78C42×C83C22⋊C411yescharacteristic
79C42×C83C22⋊C411yescharacteristic
80C16×C83—42no—
81C23×C83C2×M1611yescharacteristic
82C22×D83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—42no—
88C2×C4×C83C2×C22⋊C411yescharacteristic
89C2×C4×C83C2×M1611yescharacteristic
90C22⋊D83—42no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93D332—41no—
94C4×D83—44no—
95C4×D83—41no—
96C4×D83—41no—
97Dic166—41no—
98C8×C83—21no—
99C8×C83—21no—
100C8×C83—21no—
101C8×C83—21no—
102C83⋊C8—41no—
103C83⋊C8—41no—
104C22×C83—21no—
105C22×C83—41no—
106C22×C83#40711yescharacteristic
107C2×D83—42no—
108C2×D83—41no—
109C4×C83—22no—
110C4×C83—21no—
111C4×C83—41no—
112C4×C83#41011yescharacteristic
113C4×C83#41311yescharacteristic
114Dic83—42no—
115Dic83—41no—
116#11524—831nomaximal, Sylow
117C2×C83—21no—
118C2×C83—41no—
119C2×C83#146411yescharacteristic
120D83—81no—
121#2922—831no—
122#2165—831no—
123#2165D8311yescharacteristic, complemented
124#1328—832no—
125#1321—832no—
126C83#1152411yescharacteristic, complemented, Sylow
127C4×#41—831no—
128#433—1661no—
129#433—1661no—
130#431—1661no—
131#431—1661no—
132#424—831no—
133#424—831no—
134#408—831no—
135#408—831no—
136C4×M16C2×D8311yescharacteristic
137C2×C4×C8—831no—
138#364—21no—
139#364—1661no—
140#349—832no—
141#349C2×D8312yes—
142#347—21no—
143#347—1661no—
144C2×#41—831no—
145C2×M16C4×D8311yescharacteristic
146C2×M16C22×D8311yescharacteristic
147C22×C8—831no—
148C22×C8—831no—
149C4×Q8—1661no—
150C4×D4—1661no—
151C4×D4—1662no—
152#117—831no—
153#117—1661no—
154C2×C42—1661no—
155C2×C42C4×D8311yescharacteristic
156M32—22no—
157M32—1662no—
158C4⋊C8—1661no—
159C4⋊C8—1661no—
160C4⋊C8—1661no—
161C4⋊C8—1661no—
162C22⋊C8—1661no—
163C22⋊C8—1661no—
164C22⋊C8—1661no—
165C22⋊C8—1661no—
166#97D33211yescharacteristic
167#97C22⋊D8311yescharacteristic
168C4×C8—1661no—
169C4×C8—1661no—
170C4×C8D33211yescharacteristic
171C4×C8C22⋊D8311yescharacteristic
172#41—3321no—
173#41—3321no—
174C2×Q8—831no—
175C2×D4—831no—
176C2×D4—1661no—
177C22×C4—21no—
178C22×C4—1662no—
179C22×C4—1661no—
180C22×C4C2×C4×D8311yescharacteristic
181M16—41no—
182M16—41no—
183C2×C8—3321no—
184C2×C8—3321no—
185C2×C8—1661no—
186C2×C8—1661no—
187C2×C8(C2×D83)⋊C411yescharacteristic
188C2×C8(C2×D83)⋊C411yescharacteristic
189C2×C8C2×C22⋊D8311yescharacteristic
190C2×C8C2×D33211yescharacteristic
191C4⋊C4—1662no—
192C4⋊C4—1661no—
193C22⋊C4—832no—
194C22⋊C4—1662no—
195C42—21no—
196C42—1662no—
197C42—1661no—
198C42(C2×D83)⋊C411yescharacteristic
199C42(C2×D83)⋊C411yescharacteristic
200C16—42no—
201C16—3322no—
202C23—1661no—
203C23M16×D8311yescharacteristic
204Q8—3321no—
205D4—3321no—
206D4—3322no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—3324no—
213C2×C4—3321no—
214C2×C4—3321no—
215C2×C4—1662no—
216C2×C4—1661no—
217C2×C4C2×(C2×D83)⋊C411yescharacteristic
218C2×C4M16×D8311yescharacteristic
219C8—21no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—3321no—
224C8—3321no—
225C22—21no—
226C22—41no—
227C22—3322no—
228C22—3321no—
229C22#1912766411yescharacteristic
230C4—22no—
231C4—21no—
232C4—41no—
233C4—3322no—
234C4—3321no—
235C4#1912767411yescharacteristic
236C4#1912768311yescharacteristic
237C2—21no—
238C2—41no—
239C2—6641no—
240C2#2490579511yescharacteristic
subgroupG/H
centerC4#19127674
commutatorC22×C4×C83C22×C4
FrattiniC2×M16C22×D83
FittingC83×#2165C2
socleC2×C83#1464
prime radicalC2×#13050590C2×C4
cosocleC2×M16×C83C23
hypercenter#2165D83
hypocenterC83#11524
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #24909275C2semidirect
3. #24905415C2semidirect
5. C83×#2165C2semidirect
123. #2165D83semidirect
126. C83#11524semidirect

Maximal subgroups

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

subgroupindex
#249161082
#249092752
#249054152
#249054002
C83×#21652
#1152483