Finite Group Databasebeta

← #35292178

Subgroups — #35292178

Order 21248.

290 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 59 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#24909148C211yesmaximal, characteristic
2#24908292C211yesmaximal, characteristic
3C83×#2126C211yesmaximal, characteristic
4#19127725C411yescharacteristic, complemented
5C2×#14581719—21no—
6C2×#14581719—21no—
7#19127685—21no—
8C42×Dic83C411yescharacteristic, complemented
9#19127500C412yescomplemented
10#19127498—21no—
11#532×C83C2211yescharacteristic
12#410×C83—21no—
13#354×C83—21no—
14#353×C83C412yescomplemented
15C2×#13050574—21no—
16C2×#13050574—41no—
17C2×C4×Dic83—21no—
18C2×C4×Dic83—41no—
19C4⋊(C83⋊C8)—41no—
20#14581719—42no—
21#14581719—42no—
22#14581711—22no—
23#14581709—41no—
24C2×#41×C83D411yescharacteristic
25C2×#41×C83D411yescharacteristic
26C4⋊Q8×C83C2×C411yescharacteristic
27#127×C83C2×C411yescharacteristic
28#124×C83D411yescharacteristic
29#124×C83Q811yescharacteristic
30#121×C83—41no—
31C4×D4×C83—41no—
32#117×C83—41no—
33C2×C42×C83C2×C411yescharacteristic
34#99×C83—22no—
35#99×C83—41no—
36C22×Dic83—41no—
37#13050574—41no—
38#13050574—41no—
39#13050574—42no—
40#13050574—81no—
41C2×C83⋊C8—41no—
42C2×C83⋊C8—81no—
43C4×Dic83—22no—
44C4×Dic83—41no—
45C4×Dic83—44no—
46#41×C83—41no—
47#41×C83—41no—
48#41×C83—41no—
49#41×C83—41no—
50C2×Q8×C83C22⋊C411yescharacteristic
51C2×Q8×C83C4⋊C411yescharacteristic
52C2×D4×C83—21no—
53C2×D4×C83—21no—
54C2×D4×C83—41no—
55C2×D4×C83C22⋊C411yescharacteristic
56C2×D4×C83C4⋊C411yescharacteristic
57C22×C4×C83—21no—
58C22×C4×C83—21no—
59C22×C4×C83—21no—
60C22×C4×C83C22⋊C411yescharacteristic
61C4⋊C4×C83—41no—
62C4⋊C4×C83—81no—
63C22⋊C4×C83—41no—
64C22⋊C4×C83—41no—
65C22⋊C4×C83—42no—
66C42×C83—21no—
67C42×C83—81no—
68C42×C83C4211yescharacteristic
69C42×C83C4⋊C411yescharacteristic
70C23×C83—21no—
71C23×C83—21no—
72C23×C83#9911yescharacteristic
73Q8×C83—41no—
74Q8×C83—41no—
75D4×C83—41no—
76D4×C83—41no—
77D4×C83—82no—
78D4×C83—81no—
79D4×C83—81no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—42no—
87C2×C4×C83—41no—
88C2×C4×C83—41no—
89C2×C4×C83—41no—
90C2×C4×C83—41no—
91C2×C4×C83—82no—
92C2×C4×C83#9511yescharacteristic
93C2×C4×C83#9911yescharacteristic
94C2×Dic83—42no—
95C2×Dic83—44no—
96C83⋊C8—82no—
97C83⋊C8—81no—
98C22×C83—21no—
99C22×C83—41no—
100C22×C83—41no—
101C22×C83—41no—
102C22×C83—81no—
103C22×C83—81no—
104C22×C83#34211yescharacteristic
105C4×C83—22no—
106C4×C83—21no—
107C4×C83—41no—
108C4×C83—41no—
109C4×C83—41no—
110C4×C83—162no—
111C4×C83#35311yescharacteristic
112C4×C83#35411yescharacteristic
113Dic83—44no—
114#7497—831nomaximal, Sylow
115C2×C83—21no—
116C2×C83—41no—
117C2×C83—81no—
118C2×C83—81no—
119C2×C83#139311yescharacteristic
120#2132—831no—
121#2126D8311yescharacteristic
122#1962—831no—
123C83#749711yescharacteristic, complemented, Sylow
124#532C2×D8311yescharacteristic
125#423—831no—
126C2×#104—1661no—
127C2×#104—1661no—
128#413—1661no—
129#410—21no—
130C43—831no—
131#356—832no—
132#355—1661no—
133#354—21no—
134#353Dic8312yescomplemented
135C2×#41D33211yescharacteristic
136C2×#41C22⋊D8311yescharacteristic
137C2×M16—3321no—
138C2×M16—1661no—
139C4⋊Q8C4×D8311yescharacteristic
140#127C2×Dic8311yescharacteristic
141#124Dic16611yescharacteristic
142#124C22⋊D8311yescharacteristic
143#121—41no—
144C4×D4—41no—
145#117—41no—
146C2×C42—3321no—
147C2×C42—1661no—
148C2×C42C4×D8311yescharacteristic
149C4⋊C8—3321no—
150#104—3322no—
151#104—3322no—
152#101—1662no—
153#100—3321no—
154#99—22no—
155#99—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160C2×Q8(C2×D83)⋊C411yescharacteristic
161C2×Q8Dic83⋊C411yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—41no—
165C2×D4C22⋊Dic8311yescharacteristic
166C2×D4C4⋊Dic8311yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—3321no—
171C22×C4(C2×D83)⋊C411yescharacteristic
172M16—3322no—
173M16—3321no—
174M16—3321no—
175M16—6641no—
176C2×C8—3321no—
177C2×C8—6641no—
178C4⋊C4—41no—
179C4⋊C4—81no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—42no—
183C42—21no—
184C42—81no—
185C42—3321no—
186C42—3324no—
187C42—1662no—
188C42C4×Dic8311yescharacteristic
189C42Dic83⋊C411yescharacteristic
190C23—21no—
191C23—21no—
192C23#1458170611yescharacteristic
193Q8—41no—
194Q8—41no—
195D4—41no—
196D4—41no—
197D4—82no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—82no—
212C2×C4—3322no—
213C2×C4—3324no—
214C2×C4C83⋊#9511yescharacteristic
215C2×C4#1458170611yescharacteristic
216C8—6642no—
217C8—6641no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—81no—
223C22—81no—
224C22#1912746711yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—162no—
231C4—3324no—
232C4#1912749311yescharacteristic
233C4#1912749511yescharacteristic
234C2—21no—
235C2—41no—
236C2—81no—
237C2—81no—
238C2#2490558711yescharacteristic
subgroupG/H
centerC4#19127493
commutatorC42×C83C42
Frattini#532C2×D83
FittingC83×#2126C2
socleC2×C83#1393
prime radical, cosocle#532×C83C22
hypercenter#2126D83
hypocenterC83#7497
LCSG ▷ C42×C83 ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #19127725C4semidirect
8. C42×Dic83C4semidirect
9. #19127500C4semidirect
14. #353×C83C4semidirect
123. C83#7497semidirect
134. #353Dic83semidirect

Maximal subgroups

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

subgroupindex
#249091482
#249082922
C83×#21262
#749783