Finite Group Databasebeta

← #35317184

Subgroups — #35317184

Order 21248.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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#24908624C211yesmaximal, characteristic
2#24906470C212yesmaximal
3#24905469C212yesmaximal
4#24905397C211yesmaximal, characteristic
5C83×#2734C211yesmaximal, characteristic
6#19128332C2211yescharacteristic
7#19128031—22no—
8#19127710—21no—
9#19127470C2211yescharacteristic
10#19127461—21no—
11#19127437—22no—
12#19127427C2212yes—
13C2×C4⋊Q8×C83—21no—
14#398×C83C2212yes—
15#395×C83—22no—
16#391×C83C2211yescharacteristic
17#389×C83—21no—
18#387×C83—22no—
19C2×#13050574—21no—
20#14581791—41no—
21#14581783—41no—
22#14581771—22no—
23#14581751—21no—
24#14581723—42no—
25#14581716—41no—
26#14581716—41no—
27#14581715—42no—
28C22⋊(C83⋊C8)—22no—
29C22×Q8×C83—21no—
30C22×Q8×C83D411yescharacteristic
31C4⋊Q8×C83—42no—
32C4⋊Q8×C83—42no—
33C2×C4⋊C4×C83—21no—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83—22no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83D412yes—
38C2×C4⋊C4×C83C2311yescharacteristic
39C2×C42×C83—21no—
40C2×C42×C83D411yescharacteristic
41#95×C83—22no—
42#95×C83—22no—
43#95×C83—22no—
44#95×C83D412yes—
45C2×Dic166—22no—
46#13050574—41no—
47#13050574—41no—
48C2×C83⋊C8—41no—
49C2×C83⋊C8—42no—
50C4⋊Dic83—22no—
51C4⋊Dic83—21no—
52C4⋊Dic83—21no—
53Dic83⋊C4—42no—
54C22⋊Dic83—41no—
55C4×Dic83—41no—
56C2×Q8×C83—21no—
57C2×Q8×C83—42no—
58C2×Q8×C83—41no—
59C2×Q8×C83—41no—
60C2×Q8×C83—81no—
61C2×Q8×C83—81no—
62C2×Q8×C83SD1612yes—
63C22×C4×C83—21no—
64C22×C4×C83—22no—
65C22×C4×C83—21no—
66C22×C4×C83—22no—
67C22×C4×C83C2×D411yescharacteristic
68C22×C4×C83C2×D412yes—
69C4⋊C4×C83—24no—
70C4⋊C4×C83—41no—
71C4⋊C4×C83—42no—
72C4⋊C4×C83—44no—
73C4⋊C4×C83—42no—
74C4⋊C4×C83—42no—
75C4⋊C4×C83—81no—
76C42×C83—41no—
77C42×C83—42no—
78C23×C83#12011yescharacteristic
79Q8×C83—42no—
80Q8×C83—81no—
81Q8×C83—84no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—24no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—41no—
89C2×C4×C83—42no—
90C2×C4×C83—42no—
91C2×C4×C83—41no—
92C2×C4×C83—41no—
93C2×C4×C83—41no—
94C2×C4×C83—42no—
95C2×C4×C83—42no—
96C2×C4×C83—42no—
97C2×C4×C83—42no—
98C2×C4×C83C2×SD1611yescharacteristic
99C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
100C2×Dic83—41no—
101C2×Dic83—41no—
102Dic166—82no—
103C83⋊C8—81no—
104C83⋊C8—82no—
105C22×C83—21no—
106C22×C83—21no—
107C22×C83#45011yescharacteristic
108C22×C83#45511yescharacteristic
109C22×C83B(4,𝔽2)11yescharacteristic
110C4×C83—21no—
111C4×C83—21no—
112C4×C83—44no—
113C4×C83—42no—
114C4×C83—42no—
115C4×C83—42no—
116C4×C83—82no—
117Dic83—81no—
118Dic83—81no—
119#12213—831nomaximal, Sylow
120C2×C83—21no—
121C2×C83—21no—
122C2×C83#160111yescharacteristic
123C2×C83#220111yescharacteristic
124C2×C83#220511yescharacteristic
125#2734D8311yescharacteristic
126#2028—831no—
127#1611—832no—
128#1346—832no—
129#1319—831no—
130C83#1221311yescharacteristic, complemented, Sylow
131#544—831no—
132C2×C4⋊Q8—21no—
133#483—1662no—
134#419—1661no—
135#398C2×D8312yes—
136#395—22no—
137#391C2×D8311yescharacteristic
138#389—21no—
139#387—22no—
140#343—831no—
141#339—1661no—
142#328—1662no—
143#324—832no—
144C22×Q8—21no—
145C22×Q8C22⋊D8311yescharacteristic
146C2×M16—1661no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—3321no—
150#125—3321no—
151C22⋊Q8—1662no—
152#117—1661no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D8312yes—
158C2×C4⋊C4C22×D8311yescharacteristic
159C2×C42—21no—
160C2×C42C22⋊D8311yescharacteristic
161#106—3322no—
162#103—3322no—
163#103—3321no—
164#103—3321no—
165C22⋊C8—1662no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95C22⋊D8312yes—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—1662no—
177C2×Q8#1305057812yes—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4C2×C22⋊D8311yescharacteristic
183C22×C4C2×C22⋊D8312yes—
184M16—3321no—
185M16—3321no—
186C2×C8—3322no—
187C2×C8—3321no—
188C4⋊C4—24no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—3322no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1662no—
199C22⋊C4—3321no—
200C42—41no—
201C42—42no—
202C42—3321no—
203C23#1458176111yescharacteristic
204Q8—42no—
205Q8—81no—
206Q8—84no—
207Q8—6642no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—3321no—
225C2×C4—3321no—
226C2×C4C2×#1305057811yescharacteristic
227C2×C4#1458181811yescharacteristic
228C8—6641no—
229C8—6642no—
230C22—21no—
231C22—21no—
232C22#1912785211yescharacteristic
233C22#1912787911yescharacteristic
234C22#1912788811yescharacteristic
235C4—21no—
236C4—21no—
237C4—44no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—6641no—
243C4—6641no—
244C2—21no—
245C2—21no—
246C2#2490641511yescharacteristic
247C2#2490947611yescharacteristic
248C2#2490949311yescharacteristic
subgroupG/H
centerC22#19127888
commutator, cosocleC2×C4⋊C4×C83C23
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2734C2
socleC22×C83B(4,𝔽2)
prime radicalC23×C83#120
hypercenter#2734D83
hypocenterC83#12213
LCSG ▷ C2×C4⋊C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2734

Complements

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

NH ≅ G/Ntype
130. C83#12213semidirect

Maximal subgroups

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

subgroupindex
#249086242
#249064702
#249054692
#249053972
C83×#27342
#1221383