Finite Group Databasebeta

← #35320686

Subgroups — #35320686

Order 21248.

334 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 79 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#24916110C211yesmaximal, characteristic
2#24907564C212yesmaximal
3#24905591C212yesmaximal
4#24905586C211yesmaximal, characteristic
5C83×#1391C211yesmaximal, characteristic
6#41×Dic83C2211yescharacteristic
7#19127759—22no—
8#19127755—22no—
9#19127729C411yescharacteristic, complemented
10#19127727C411yescharacteristic, complemented
11#19127685C2212yes—
12#19127668C2211yescharacteristic
13M16×Dic83C411yescharacteristic, complemented
14C2×#14581702C411yescharacteristic, complemented
15#19127460—22no—
16#19127457—22no—
17#410×C83C2212yes—
18#408×C83C2211yescharacteristic
19C2×#13050574—22no—
20C2×#13050574C2×C411yescharacteristic
21C22×C83⋊C8C2×C411yescharacteristic
22Q8×Dic83—21no—
23D4×Dic83—21no—
24D4×Dic83—22no—
25#14581751C2×C411yescharacteristic
26#14581751D412yes—
27C2×C4×Dic83C2×C411yescharacteristic
28C2×C4×Dic83D412yes—
29Dic83⋊C8—22no—
30Dic83⋊C8—22no—
31#14581711—22no—
32#14581709—22no—
33C22⋊(C83⋊C8)—22no—
34#14581702—21no—
35#14581702—21no—
36#14581702—21no—
37C8×Dic83—21no—
38C2×#41×C83C2311yescharacteristic
39C2×M16×C83C2×C411yescharacteristic
40C22×C8×C83C2×C411yescharacteristic
41#117×C83—22no—
42#99×C83—22no—
43#99×C83—22no—
44C22⋊C8×C83—22no—
45C22×Dic83—21no—
46C22×Dic83—22no—
47#13050574—41no—
48#13050574—42no—
49#13050574—42no—
50C2×C83⋊C8—21no—
51C2×C83⋊C8—21no—
52C2×C83⋊C8—41no—
53C2×C83⋊C8—42no—
54C4⋊Dic83—21no—
55C4⋊Dic83—22no—
56C22⋊Dic83—21no—
57C22⋊Dic83—22no—
58C4×Dic83—21no—
59C4×Dic83—22no—
60C4×Dic83C22⋊C411yescharacteristic
61C4×Dic83C22⋊C411yescharacteristic
62C4×Dic83C22⋊C412yes—
63#41×C83—41no—
64#41×C83—41no—
65C2×Q8×C83#4111yescharacteristic
66C2×D4×C83#4111yescharacteristic
67C2×D4×C83#4112yes—
68C22×C4×C83C22×C411yescharacteristic
69C22×C4×C83C2×D412yes—
70M16×C83—41no—
71C2×C8×C83—21no—
72C2×C8×C83—21no—
73C2×C8×C83—41no—
74C4⋊C4×C83—42no—
75C22⋊C4×C83—42no—
76C42×C83—42no—
77C23×C83#11711yescharacteristic
78C23×C83#12312yes—
79Q8×C83—41no—
80D4×C83—41no—
81D4×C83—42no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—82no—
87C2×C4×C83C2×C22⋊C411yescharacteristic
88C2×C4×C83#11711yescharacteristic
89C2×C4×C83#12312yes—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—81no—
94C2×Dic83—82no—
95C8×C83—41no—
96C8×C83—41no—
97C83⋊C8—41no—
98C83⋊C8—41no—
99C83⋊C8—82no—
100C22×C83—22no—
101C22×C83—21no—
102C22×C83—41no—
103C22×C83—42no—
104C22×C83#38111yescharacteristic
105C4×C83—22no—
106C4×C83—21no—
107C4×C83—162no—
108C4×C83#41011yescharacteristic
109C4×C83#41011yescharacteristic
110Dic83—41no—
111Dic83—42no—
112Dic83—41no—
113#12865—831nomaximal, Sylow
114C2×C83—21no—
115C2×C83—42no—
116C2×C83—41no—
117C2×C83#182011yescharacteristic
118#2923—831no—
119#1827—832no—
120#1395—832no—
121#1392—831no—
122#1391D8311yescharacteristic
123C83#1286511yescharacteristic, complemented, Sylow
124C4×#41—831no—
125#433—1662no—
126#432—1662no—
127#424—831no—
128#424—831no—
129#413—832no—
130#410C2×D8312yes—
131#408—831no—
132#408C2×D8311yescharacteristic
133C4×M16—831no—
134C2×#97—831no—
135#339—1662no—
136H(3,ℤ/4)—1662no—
137C2×#41C22×D8311yescharacteristic
138C2×M16—831no—
139C2×M16—1662no—
140C2×M16C4×D8311yescharacteristic
141C22×C8—831no—
142C22×C8C4×D8311yescharacteristic
143C4×Q8—1661no—
144C4×D4—1661no—
145C4×D4—1662no—
146#117—22no—
147#117—831no—
148#117—832no—
149C2×C42—831no—
150C2×C42—832no—
151C4⋊C8—1662no—
152C4⋊C8—1662no—
153#101—1662no—
154#100—1662no—
155#99—22no—
156#99—22no—
157C22⋊C8—22no—
158C22⋊C8—1662no—
159#97—1661no—
160#97—1661no—
161#97—1661no—
162C4×C8—1661no—
163#41—41no—
164#41—41no—
165C2×Q8#1305059111yescharacteristic
166C2×D4#1305058911yescharacteristic
167C2×D4#1305059012yes—
168C22×C4—1662no—
169C22×C4—1661no—
170C22×C4C2×C4×D8311yescharacteristic
171C22×C4D4×D8312yes—
172M16—41no—
173M16—3321no—
174M16—3322no—
175M16—3322no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—3321no—
180C2×C8—3322no—
181C2×C8—1661no—
182C2×C8—1661no—
183C4⋊C4—42no—
184C4⋊C4—1662no—
185C4⋊C4—1661no—
186C22⋊C4—42no—
187C22⋊C4—1662no—
188C22⋊C4—1661no—
189C42—42no—
190C42—831no—
191C42—831no—
192C42—832no—
193C42—1662no—
194C42—1661no—
195C23#1458174911yescharacteristic
196C23#1458177412yes—
197Q8—41no—
198D4—41no—
199D4—42no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—82no—
205C2×C4—6642no—
206C2×C4—6641no—
207C2×C4—1661no—
208C2×C4—1661no—
209C2×C4—1662no—
210C2×C4C22⋊C4×D8311yescharacteristic
211C2×C4#1458175011yescharacteristic
212C2×C4#1458177712yes—
213C8—41no—
214C8—41no—
215C8—3321no—
216C8—3321no—
217C8—6642no—
218C22—22no—
219C22—21no—
220C22—42no—
221C22—41no—
222C22#1912756311yescharacteristic
223C4—22no—
224C4—21no—
225C4—162no—
226C4—3321no—
227C4—3322no—
228C4—3321no—
229C4#1912767311yescharacteristic
230C4#1912767311yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2490753911yescharacteristic
subgroupG/H
centerC4#19127673
commutatorC22×C4×C83C22×C4
FrattiniC2×#41C22×D83
FittingC83×#1391C2
socleC2×C83#1820
prime radical, cosocleC2×#41×C83C23
hypercenter#1391D83
hypocenterC83#12865
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
9. #19127729C4semidirect
10. #19127727C4semidirect
13. M16×Dic83C4semidirect
14. C2×#14581702C4semidirect
123. C83#12865semidirect

Maximal subgroups

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

subgroupindex
#249161102
#249075642
#249055912
#249055862
C83×#13912
#1286583