Finite Group Databasebeta

← #35303985

Subgroups — #35303985

Order 21248.

426 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#19128097C211yesmaximal, characteristic
2#24908451C211yesmaximal, characteristic
3#24908308C211yesmaximal, characteristic
4#24906218C211yesmaximal, characteristic
5#24906147C211yesmaximal, characteristic
6#24905744C211yesmaximal, characteristic
7C83×#1546C211yesmaximal, characteristic
8C2×#14581792C2211yescharacteristic
9#19128097—21no—
10#19128097—21no—
11#19128097—22no—
12C2×#14581723C414yes—
13#19127725C2211yescharacteristic
14C2×#14581711—22no—
15C4×#13050574C2211yescharacteristic
16C2×C4×C83⋊C8C2211yescharacteristic
17#19127469—22no—
18#19127451—22no—
19#19127448—22no—
20#19127423—21no—
21C8×C83⋊C8—21no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23#431×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25#332×C83—22no—
26C22×Dic166—21no—
27C2×#13050574—21no—
28C2×#13050574D412yes—
29C22×C83⋊C8C2×C412yes—
30#14581792—21no—
31#14581792—21no—
32#14581792—21no—
33#14581792—21no—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83C2×C412yes—
36#14581723—24no—
37#14581723—24no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40#14581711—44no—
41#14581703—21no—
42C4×C83⋊C8—21no—
43C4×C83⋊C8D411yescharacteristic
44C4×C83⋊C8D411yescharacteristic
45C4×C83⋊C8D412yes—
46C22×Q8×C83—21no—
47C2×M16×C83—21no—
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83C2×C412yes—
54C2×C42×C83C2311yescharacteristic
55C4⋊C8×C83—21no—
56C4⋊C8×C83—21no—
57C4⋊C8×C83D412yes—
58C22⋊C8×C83—21no—
59C4×C8×C83—21no—
60C2×Dic166—41no—
61C2×Dic166—41no—
62C2×Dic166—82no—
63C22×Dic83—22no—
64#13050574—22no—
65#13050574—42no—
66#13050574SD1614yes—
67C2×C83⋊C8—22no—
68C2×C83⋊C8—24no—
69C2×C83⋊C8—41no—
70C2×C83⋊C8C22⋊C412yes—
71C2×C83⋊C8C22⋊C412yes—
72C4⋊Dic83—22no—
73C4⋊Dic83—22no—
74C4⋊Dic83—41no—
75C4⋊Dic83—41no—
76C2×Q8×C83—41no—
77C2×Q8×C83—41no—
78C2×Q8×C83—42no—
79C2×Q8×C83—42no—
80C22×C4×C83—22no—
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83#4111yescharacteristic
84M16×C83—42no—
85C2×C8×C83—22no—
86C2×C8×C83—41no—
87C4⋊C4×C83—22no—
88C4⋊C4×C83—22no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C42×C83C2×D411yescharacteristic
92C42×C83C2×D411yescharacteristic
93C42×C83C2×D411yescharacteristic
94C42×C83#4111yescharacteristic
95C23×C83C4×D411yescharacteristic
96Q8×C83—82no—
97Q8×C83—81no—
98Q8×C83—81no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
101C2×C4×C83—42no—
102C2×C4×C83—44no—
103C2×C4×C83C2×C22⋊C411yescharacteristic
104C2×C4×C83C4×D411yescharacteristic
105C2×C4×C83#12111yescharacteristic
106C2×C4×C83#12111yescharacteristic
107C2×C4×C83#12411yescharacteristic
108C2×C4×C83#12711yescharacteristic
109C2×C4×C83C2×SD1612yes—
110C2×C4×C83#13512yes—
111C2×Dic83—42no—
112C2×Dic83—82no—
113Dic166—82no—
114Dic166—82no—
115C8×C83—42no—
116C8×C83—81no—
117C83⋊C8—22no—
118C83⋊C8—24no—
119C83⋊C8—81no—
120C83⋊C8#10114yescomplemented
121C22×C83—21no—
122C22×C83—21no—
123C22×C83#39011yescharacteristic
124C22×C83C4×SD1612yes—
125C4×C83—21no—
126C4×C83—21no—
127C4×C83—82no—
128C4×C83C2×#10112yes—
129C4×C83#46512yes—
130C4×C83#47112yes—
131C4×C83#49211yescharacteristic
132C4×C83#49511yescharacteristic
133Dic83—82no—
134#9794—831nomaximal, Sylow
135C2×C83—22no—
136C2×C83#196611yescharacteristic
137C2×C83#196811yescharacteristic
138C2×C83#197711yescharacteristic
139C2×#498—831no—
140#1997—831no—
141#1966—831no—
142#1566—831no—
143#1549—831no—
144#1546D8311yescharacteristic
145C8×M16—831no—
146C83#979411yescharacteristic, complemented, Sylow
147C2×C4⋊Q8—831no—
148C2×C4⋊Q8C2×D8311yescharacteristic
149#498—1661no—
150#498—1661no—
151#498—1662no—
152#431C2×D8311yescharacteristic
153C2×#106—834no—
154#423—831no—
155#423C2×D8311yescharacteristic
156C2×#101—1662no—
157C4×M16—831no—
158C2×C4×C8—831no—
159#343—1662no—
160#334—1662no—
161#332—22no—
162#332—1662no—
163#322—1661no—
164C82—1661no—
165C22×Q8—21no—
166C22×Q8—1661no—
167C2×M16—21no—
168C2×M16—832no—
169C2×M16—1661no—
170C22×C8—832no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—1661no—
176C4⋊Q8—1661no—
177C4⋊Q8—1661no—
178C4⋊Q8—1661no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—832no—
181C2×C4⋊C4—1661no—
182C2×C4⋊C4C4×D8312yes—
183C2×C42C22×D8311yescharacteristic
184#106—1664no—
185#106—1664no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—1661no—
189C4⋊C8—1661no—
190C4⋊C8C22⋊D8312yes—
191#101—3324no—
192C22⋊C8—21no—
193#97—1661no—
194C4×C8—21no—
195C4×C8—831no—
196C4×C8—831no—
197C4×C8—832no—
198C4×C8—1661no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—3321no—
204C2×Q8—3321no—
205C2×Q8—6642no—
206C22×C4—22no—
207C22×C4—1662no—
208C22×C4C2×C4×D8311yescharacteristic
209C22×C4D4×D8311yescharacteristic
210C22×C4#1305058911yescharacteristic
211M16—42no—
212M16—3322no—
213M16—834no—
214M16—1662no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—3321no—
218C2×C8—832no—
219C2×C8—832no—
220C2×C8—1664no—
221C2×C8—1662no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—3321no—
227C4⋊C4—3321no—
228C4⋊C4—1662no—
229C4⋊C4—1662no—
230C42D4×D8311yescharacteristic
231C42D4×D8311yescharacteristic
232C42C2×C22⋊D8311yescharacteristic
233C42#1305059011yescharacteristic
234C23(C22⋊D83)⋊C411yescharacteristic
235Q8—81no—
236Q8—81no—
237Q8—82no—
238Q8—6642no—
239Q8—6642no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—3322no—
245C2×C4—6642no—
246C2×C4C22⋊C4×D8311yescharacteristic
247C2×C4C4×C22⋊D8311yescharacteristic
248C2×C4#1458176411yescharacteristic
249C2×C4#1458176511yescharacteristic
250C2×C4#1458177811yescharacteristic
251C2×C4#1458178811yescharacteristic
252C2×C4SD16×D8312yes—
253C2×C4#1458181112yes—
254C8—42no—
255C8—81no—
256C8—834no—
257C8—6641no—
258C8—1664no—
259C8—1662no—
260C22—21no—
261C22—21no—
262C22#1912760111yescharacteristic
263C22#1912779212yes—
264C4—21no—
265C4—21no—
266C4—82no—
267C4—6642no—
268C4#101×D8312yes—
269C4#1912792912yes—
270C4#1912796612yes—
271C4#1912807211yescharacteristic
272C4#1912808411yescharacteristic
273C2—22no—
274C2#2490830511yescharacteristic
275C2#2490831811yescharacteristic
276C2#2490836511yescharacteristic
subgroupG/H
centerC22#19127601
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1546C2
socleC22×C83#390
prime radicalC23×C83C4×D4
cosocleC2×C42×C83C23
hypercenter#1546D83
hypocenterC83#9794
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
120. C83⋊C8#101semidirect
146. C83#9794semidirect

Maximal subgroups

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

subgroupindex
C2×#191280972
#249084512
#249083082
#249062182
#249061472
#249057442
C83×#15462
#979483