Finite Group Databasebeta

← #35301832

Subgroups — #35301832

Order 21248.

554 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 115 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#24916548C211yesmaximal, characteristic, complemented
2#24912462C211yesmaximal, characteristic
3#24906132C212yesmaximal
4#24905364C212yesmaximal, complemented
5C83×#1801C211yesmaximal, characteristic, complemented
6C2×#14581789C2211yescharacteristic
7C2×#14581781—21no—
8#19127832—24no—
9#19127758—22no—
10#19127754C2212yes—
11#19127618C414yes—
12#19127603—22no—
13#19127602—21no—
14C4×C4⋊Dic83C2211yescharacteristic
15#19127444—24no—
16C2×C4×Q8×C83C2211yescharacteristic
17#423×C83C2212yes—
18C2×#101×C83—21no—
19#343×C83—21no—
20C22×D332—21no—
21C22×D332—21no—
22C22×D332C2×C412yes—
23#14581789—22no—
24#14581789—22no—
25#14581781—44no—
26C2×C4⋊Dic83—21no—
27C2×(C2×D83)⋊C4—24no—
28C2×(C2×D83)⋊C4—24no—
29C2×C4×Dic83—21no—
30C4⋊(C83⋊C8)—24no—
31C4⋊(C83⋊C8)D414yes—
32C22⋊(C83⋊C8)—22no—
33#14581703—24no—
34C4×C83⋊C8—22no—
35C83⋊#95C2×C412yes—
36C22×Q8×C83D411yescharacteristic
37C2×M16×C83—22no—
38C4×Q8×C83—22no—
39C4×Q8×C83—22no—
40C2×C4⋊C4×C83C2×C412yes—
41C2×C4⋊C4×C83D411yescharacteristic
42C2×C42×C83D412yes—
43C2×C42×C83C2311yescharacteristic
44C4⋊C8×C83—22no—
45C4⋊C8×C83—22no—
46#101×C83—42no—
47C23×D83—22no—
48C2×D332—24no—
49C2×D332—44no—
50C2×D332—41no—
51C2×D332—42no—
52C2×D332—41no—
53C2×D332—42no—
54C2×D332—41no—
55C2×D332—41no—
56C2×D332—82no—
57C22×Dic83—22no—
58C2×C83⋊C8—24no—
59C4⋊Dic83—42no—
60(C2×D83)⋊C4—48no—
61(C2×D83)⋊C4—84no—
62C4×Dic83—42no—
63C2×Q8×C83—22no—
64C2×Q8×C83—22no—
65C2×Q8×C83—21no—
66C2×Q8×C83—21no—
67C22×C4×C83C22⋊C412yes—
68C22×C4×C83C22⋊C412yes—
69C22×C4×C83C22×C411yescharacteristic
70C22×C4×C83C2×D411yescharacteristic
71C22×C4×C83C2×D411yescharacteristic
72M16×C83—42no—
73M16×C83—42no—
74C2×C8×C83—42no—
75C4⋊C4×C83—24no—
76C4⋊C4×C83—22no—
77C42×C83—22no—
78C42×C83—22no—
79C42×C83C2×D412yes—
80C42×C83#4112yes—
81C23×C83C2×C22⋊C411yescharacteristic
82C22×D83—44no—
83C22×D83—44no—
84C22×D83—42no—
85C22×D83—42no—
86C22×D83—42no—
87Q8×C83—42no—
88Q8×C83—42no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83C4×D412yes—
97C2×C4×C83#12011yescharacteristic
98C2×C4×C83#12112yes—
99C2×C4×C83#12311yescharacteristic
100C2×C4×C83AGL(1,ℤ/8)12yes—
101C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C83#13711yescharacteristic
103C2×Dic83—42no—
104C2×Dic83—42no—
105C2×Dic83—42no—
106D332—84no—
107D332—82no—
108D332—82no—
109D332—82no—
110D332—82no—
111C8×C83—82no—
112C83⋊C8—44no—
113C22×C83—21no—
114C22×C83—21no—
115C22×C83#38611yescharacteristic
116C22×C83#41711yescharacteristic
117C22×C83#41811yescharacteristic
118C2×D83—84no—
119C2×D83—84no—
120C2×D83—82no—
121C2×D83—82no—
122C2×D83—82no—
123C4×C83—21no—
124C4×C83—22no—
125C4×C83—42no—
126C4×C83—42no—
127C4×C83#45412yes—
128C4×C83#46812yes—
129C4×C83#46912yes—
130Dic83—82no—
131#9406—831nomaximal, Sylow
132C2×C83—21no—
133C2×C83—21no—
134C2×C83#188811yescharacteristic
135C2×C83#189511yescharacteristic
136C2×C83#191411yescharacteristic
137D83—162no—
138#2983—831no—
139#2548—831no—
140#1801D8311yescharacteristic, complemented
141#1545—832no—
142#1303—832no—
143C83#940611yescharacteristic, complemented, Sylow
144C2×#127—831no—
145C2×#124—1661no—
146C2×C4×Q8C2×D8311yescharacteristic
147#446—1664no—
148#432—1662no—
149#431—832no—
150#423C2×D8312yes—
151C2×#101—21no—
152#394—834no—
153#390—1661no—
154#390—1662no—
155C4×C4⋊C4—831no—
156#343—21no—
157#331—1664no—
158C22×Q8C22⋊D8311yescharacteristic
159C22×D4—832no—
160C22×D4—1661no—
161C22×D4—1661no—
162C2×M16—22no—
163#127—1662no—
164#127—1662no—
165#124—3324no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1661no—
169C2×C4⋊C4C4×D8312yes—
170C2×C4⋊C4C22⋊D8311yescharacteristic
171C2×C22⋊C4—1664no—
172C2×C22⋊C4—1664no—
173C2×C42—1661no—
174C2×C42D33212yes—
175C2×C42C22×D8311yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—834no—
179C4⋊C8—1664no—
180#101—42no—
181C22⋊C8—1662no—
182#97—1664no—
183C4×C8—1662no—
184#95—832no—
185C24—1662no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×D4—3324no—
191C2×D4—3321no—
192C2×D4—3322no—
193C2×D4—3321no—
194C2×D4—3322no—
195C2×D4—3321no—
196C2×D4—3321no—
197C2×D4—6642no—
198C2×D4—1664no—
199C22×C4—1662no—
200C22×C4(C2×D83)⋊C412yes—
201C22×C4(C2×D83)⋊C412yes—
202C22×C4C2×C4×D8311yescharacteristic
203C22×C4C2×C22⋊D8311yescharacteristic
204C22×C4C2×D33211yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—42no—
208C2×C8—1664no—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—3322no—
212C22⋊C4—3328no—
213C22⋊C4—6644no—
214C42—22no—
215C42—22no—
216C42—3322no—
217C42D4×D8312yes—
218C42#1305058912yes—
219C23—3324no—
220C23—3324no—
221C23—3322no—
222C23—3322no—
223C23—3322no—
224C23C2×(C2×D83)⋊C411yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—6644no—
228D4—6642no—
229D4—6642no—
230D4—6642no—
231D4—6642no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—3322no—
240C2×C4—3322no—
241C2×C4—3322no—
242C2×C4(C22⋊D83)⋊C412yes—
243C2×C4#1458176211yescharacteristic
244C2×C4#1458176412yes—
245C2×C4#1458177611yescharacteristic
246C2×C4#1458181712yes—
247C2×C4#1458181811yescharacteristic
248C2×C4#1458182311yescharacteristic
249C8—82no—
250C8—3324no—
251C22—21no—
252C22—21no—
253C22—6644no—
254C22—6644no—
255C22—6642no—
256C22—6642no—
257C22—6642no—
258C22#1912757811yescharacteristic
259C22#1912770011yescharacteristic
260C22#1912770511yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—6642no—
266C4#1912787612yes—
267C4#1912794912yes—
268C4#1912795212yes—
269C2—21no—
270C2—21no—
271C2—13282no—
272C2#2490786411yescharacteristic
273C2#2490789911yescharacteristic
274C2#2490800811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1801C2
socleC22×C83#386
prime radicalC2×#14581789C22
cosocleC2×C42×C83C23
hypercenter#1801D83
hypocenterC83#9406
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #24916548C2semidirect
4. #24905364C2semidirect
5. C83×#1801C2semidirect
140. #1801D83semidirect
143. C83#9406semidirect

Maximal subgroups

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

subgroupindex
#249165482
#249124622
#249061322
#249053642
C83×#18012
#940683