Finite Group Databasebeta

← #35293722

Subgroups — #35293722

Order 21248.

438 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 87 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#24909001C212yesmaximal, complemented
2#24908996C211yesmaximal, characteristic
3#24905289C212yesmaximal, complemented
4#24905284C211yesmaximal, characteristic
5C83×#2804C211yesmaximal, characteristic
6C2×#14581725—22no—
7C2×#14581723—21no—
8C2×#14581716—22no—
9C2×#14581713—21no—
10#19127571C2211yescharacteristic
11#19127456C2212yes—
12#19127456C2211yescharacteristic
13#400×C83—21no—
14#393×C83—22no—
15#391×C83C2212yes—
16#390×C83C2211yescharacteristic
17#386×C83—21no—
18#385×C83—22no—
19C22×C83⋊C8—21no—
20C22×C83⋊C8—22no—
21C2×C4⋊Dic83—21no—
22C2×C4⋊Dic83—22no—
23#14581725—44no—
24#14581723—42no—
25#14581716—44no—
26#14581713—42no—
27C22×Q8×C83D412yes—
28C22×D4×C83D411yescharacteristic
29C23×C4×C83—21no—
30C2×C4⋊C4×C83—21no—
31C2×C4⋊C4×C83D412yes—
32C2×C4⋊C4×C83D411yescharacteristic
33C2×C22⋊C4×C83—22no—
34C2×C22⋊C4×C83—22no—
35C2×C22⋊C4×C83—21no—
36C2×C42×C83C2311yescharacteristic
37#95×C83—22no—
38#95×C83—21no—
39#95×C83—22no—
40#95×C83—22no—
41C22×Dic83—41no—
42C2×C83⋊C8—22no—
43C2×C83⋊C8—24no—
44C2×C83⋊C8—41no—
45C2×C83⋊C8—42no—
46C2×C83⋊C8—42no—
47C2×C83⋊C8—41no—
48C4⋊Dic83—41no—
49C4⋊Dic83—42no—
50C4⋊Dic83—41no—
51C4⋊Dic83—42no—
52C24×C83—21no—
53C2×Q8×C83—22no—
54C2×Q8×C83—22no—
55C2×Q8×C83—42no—
56C2×Q8×C83—42no—
57C2×D4×C83—21no—
58C2×D4×C83—41no—
59C2×D4×C83—41no—
60C2×D4×C83Q1612yes—
61C22×C4×C83—22no—
62C22×C4×C83—21no—
63C22×C4×C83—22no—
64C22×C4×C83—82no—
65C22×C4×C83—81no—
66C22×C4×C83C2×D411yescharacteristic
67C22×C4×C83C2×D412yes—
68C4⋊C4×C83—41no—
69C4⋊C4×C83—42no—
70C4⋊C4×C83—81no—
71C22⋊C4×C83—42no—
72C22⋊C4×C83—42no—
73C22⋊C4×C83—82no—
74C22⋊C4×C83—81no—
75C42×C83—41no—
76C23×C83—22no—
77C23×C83—42no—
78C23×C83—42no—
79C23×C83—41no—
80C23×C83—41no—
81C23×C83#12011yescharacteristic
82Q8×C83—44no—
83Q8×C83—82no—
84D4×C83—42no—
85D4×C83—81no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—42no—
91C2×C4×C83—42no—
92C2×C4×C83—42no—
93C2×C4×C83—42no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83—42no—
97C2×C4×C83—42no—
98C2×C4×C83—84no—
99C2×C4×C83C2×Q1611yescharacteristic
100C2×C4×C83#13512yes—
101C2×C4×C83#13712yes—
102C2×C4×C83#13711yescharacteristic
103C2×Dic83—81no—
104C2×Dic83—82no—
105C83⋊C8—42no—
106C83⋊C8—44no—
107C22×C83—42no—
108C22×C83—44no—
109C22×C83—82no—
110C22×C83—81no—
111C22×C83—81no—
112C22×C83#45111yescharacteristic
113C22×C83#45212yes—
114C22×C83#45311yescharacteristic
115C22×C83#45312yes—
116C22×C83#45811yescharacteristic
117C4×C83—21no—
118C4×C83—21no—
119C4×C83—22no—
120C4×C83—22no—
121C4×C83—82no—
122C4×C83—81no—
123C4×C83—82no—
124Dic83—161no—
125#7833—831nomaximal, Sylow
126C2×C83—82no—
127C2×C83#161411yescharacteristic
128C2×C83#161812yes—
129C2×C83#163011yescharacteristic
130C2×C83#163212yes—
131C2×C83#165511yescharacteristic
132#2804D8311yescharacteristic
133#2095—832no—
134#2094—831no—
135#1272—832no—
136#1270—831no—
137C83#783311yescharacteristic, complemented, Sylow
138C2×#107—1662no—
139C2×#106—1661no—
140C2×#103—1662no—
141C2×#102—1661no—
142#400—21no—
143#393—22no—
144#391C2×D8312yes—
145#390C2×D8311yescharacteristic
146#386—21no—
147#385—22no—
148#384—831no—
149#336—831no—
150#336—832no—
151C22×Q8C22⋊D8312yes—
152C22×D4C22⋊D8311yescharacteristic
153C23×C4—21no—
154C22×C8—1661no—
155C22×C8—1662no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—1661no—
158C2×C4⋊C4—1662no—
159C2×C4⋊C4C22⋊D8312yes—
160C2×C4⋊C4C22⋊D8311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42C22×D8311yescharacteristic
165#107—3324no—
166#106—3322no—
167#103—3324no—
168#102—3322no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95—22no—
173C24—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4#1305058012yes—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—82no—
186C22×C4—81no—
187C22×C4—3321no—
188C22×C4C2×C22⋊D8311yescharacteristic
189C22×C4C2×C22⋊D8312yes—
190C2×C8—3321no—
191C2×C8—3322no—
192C2×C8—3322no—
193C2×C8—3321no—
194C2×C8—1662no—
195C2×C8—1664no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—3321no—
200C4⋊C4—3322no—
201C4⋊C4—3321no—
202C4⋊C4—3322no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1458176111yescharacteristic
214Q8—44no—
215Q8—82no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—6642no—
232C2×C4—6641no—
233C2×C4C2×#1305058011yescharacteristic
234C2×C4#1458181212yes—
235C2×C4#1458182212yes—
236C2×C4#1458182311yescharacteristic
237C8—3324no—
238C8—3322no—
239C22—44no—
240C22—42no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22#1912785711yescharacteristic
245C22#1912786312yes—
246C22#1912786912yes—
247C22#1912786911yescharacteristic
248C22#1912789111yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—13281no—
257C2—82no—
258C2#2490648811yescharacteristic
259C2#2490651112yes—
260C2#2490658411yescharacteristic
261C2#2490659712yes—
262C2#2490675211yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2804C2
socleC23×C83#120
prime radicalC22×D4×C83D4
hypercenter#2804D83
hypocenterC83#7833
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
1. #24909001C2semidirect
3. #24905289C2semidirect
137. C83#7833semidirect

Maximal subgroups

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

subgroupindex
#249090012
#249089962
#249052892
#249052842
C83×#28042
#783383