Finite Group Databasebeta

← #35307066

Subgroups — #35307066

Order 21248.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#24918591C212yesmaximal
2#24908291C211yesmaximal, characteristic
3#24906148C212yesmaximal, complemented
4#24905296C211yesmaximal, characteristic, complemented
5C83×#2862C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19128097—22no—
8#19128049—22no—
9#19128019—24no—
10#19127988—24no—
11#19127754C2212yes—
12#19127725C2211yescharacteristic
13C2×#14581720—22no—
14#19127447—24no—
15C2×C4⋊Q8×C83C2212yes—
16#391×C83—24no—
17#382×C83—22no—
18C43×C83C2211yescharacteristic
19C2×#13050590—21no—
20C2×#13050574—21no—
21#14581792D411yescharacteristic
22#14581789D411yescharacteristic
23#14581780—21no—
24#14581766—22no—
25C4×D332—22no—
26#14581723—28no—
27C4⋊(C83⋊C8)—22no—
28C4⋊(C83⋊C8)D414yes—
29#14581720—44no—
30#14581712—28no—
31C22⋊(C83⋊C8)—22no—
32C4×C83⋊C8—22no—
33C22×Q8×C83—22no—
34C4⋊Q8×C83—22no—
35C4⋊Q8×C83—22no—
36C4⋊Q8×C83—22no—
37C4⋊Q8×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83D414yes—
40C2×C42×C83—22no—
41C2×C42×C83D412yes—
42C2×C42×C83C2311yescharacteristic
43#95×C83—28no—
44#13050590—42no—
45#13050590—42no—
46C2×Dic166—21no—
47C2×D332—22no—
48C2×D332—21no—
49C2×C22⋊D83—41no—
50C2×C4×D83—41no—
51#13050574—42no—
52C2×C83⋊C8—24no—
53C2×C83⋊C8—41no—
54C4⋊Dic83—22no—
55(C2×D83)⋊C4—42no—
56C2×Q8×C83—48no—
57C2×Q8×C83—42no—
58C2×Q8×C83—42no—
59C22×C4×C83—22no—
60C22×C4×C83—24no—
61C22×C4×C83C2×D412yes—
62C22×C4×C83C2×D411yescharacteristic
63C4⋊C4×C83—24no—
64C4⋊C4×C83—24no—
65C4⋊C4×C83—42no—
66C4⋊C4×C83—42no—
67C42×C83—22no—
68C42×C83—44no—
69C42×C83SD1614yes—
70C42×C83C2×D411yescharacteristic
71C42×C83C2×D412yes—
72C42×C83#4111yescharacteristic
73C23×C83#12011yescharacteristic
74C22×D83—41no—
75Q8×C83—84no—
76Q8×C83—84no—
77C2×C4×C83—24no—
78C2×C4×C83—24no—
79C2×C4×C83—44no—
80C2×C4×C83—48no—
81C2×C4×C83—44no—
82C2×C4×C83#12012yes—
83C2×C4×C83#12111yescharacteristic
84C2×C4×C83#12112yes—
85C2×C4×C83#12711yescharacteristic
86C2×C4×C83C2×SD1612yes—
87C2×C4×C83#13712yes—
88C22⋊D83—82no—
89C2×Dic83—41no—
90D332—44no—
91D332—42no—
92C4×D83—81no—
93C4×D83—81no—
94Dic166—42no—
95C83⋊C8—44no—
96C83⋊C8—81no—
97C22×C83—22no—
98C22×C83#39211yescharacteristic
99C22×C83#44812yes—
100C2×D83—82no—
101C2×D83—81no—
102C4×C83—22no—
103C4×C83—44no—
104C4×C83—84no—
105C4×C83#45414yes—
106C4×C83#46112yes—
107C4×C83#49211yescharacteristic
108C4×C83#49711yescharacteristic
109Dic83—81no—
110#10285—831nomaximal, Sylow
111C2×C83—22no—
112C2×C83#200312yes—
113C2×C83#201311yescharacteristic
114D83—161no—
115#3227—832no—
116#2862D8311yescharacteristic, complemented
117#1962—831no—
118#1549—832no—
119#1276—831no—
120C83#1028511yescharacteristic, complemented, Sylow
121#532—831no—
122C2×C4⋊Q8C2×D8312yes—
123#498—1662no—
124#486—1662no—
125#481—1664no—
126#476—1664no—
127#431—832no—
128#423—831no—
129C2×#104—1662no—
130#391—24no—
131#382—22no—
132C43C2×D8311yescharacteristic
133#332—1664no—
134C2×#41—1661no—
135C22×Q8—22no—
136C2×M16—1661no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—831no—
142#127—831no—
143#124—1661no—
144#121—1662no—
145C4×D4—1662no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4C22⋊D8314yes—
148C2×C42—22no—
149C2×C42C22⋊D8312yes—
150C2×C42C22×D8311yescharacteristic
151#106—1668no—
152C4⋊C8—834no—
153C4⋊C8—1662no—
154#104—3324no—
155#102—1668no—
156C22⋊C8—1662no—
157C4×C8—1662no—
158#95—28no—
159#41—3322no—
160#41—3322no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—1661no—
165C2×D4—3321no—
166C2×D4—1662no—
167C2×D4—1661no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—3321no—
171C22×C4C2×C22⋊D8312yes—
172C22×C4C2×C22⋊D8311yescharacteristic
173M16—3322no—
174C2×C8—3321no—
175C2×C8—1664no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—1662no—
181C22⋊C4—3322no—
182C42—22no—
183C42—44no—
184C42#1305057714yes—
185C42D4×D8311yescharacteristic
186C42D4×D8312yes—
187C42#1305058911yescharacteristic
188C23—3321no—
189C23#1458176111yescharacteristic
190Q8—84no—
191Q8—84no—
192Q8—3322no—
193D4—3324no—
194D4—3322no—
195D4—6642no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—44no—
201C2×C4—3321no—
202C2×C4—6641no—
203C2×C4—6641no—
204C2×C4#1458176012yes—
205C2×C4#1458176311yescharacteristic
206C2×C4#1458176412yes—
207C2×C4#1458178811yescharacteristic
208C2×C4C2×#1305057712yes—
209C2×C4#1458182312yes—
210C8—3324no—
211C8—6641no—
212C22—22no—
213C22—6641no—
214C22—6642no—
215C22#1912760911yescharacteristic
216C22#1912784012yes—
217C4—22no—
218C4—44no—
219C4—84no—
220C4—6641no—
221C4#1912787314yes—
222C4#1912790312yes—
223C4#1912807511yescharacteristic
224C4#1912809311yescharacteristic
225C2—22no—
226C2—13281no—
227C2#2490848512yes—
228C2#2490854611yescharacteristic
subgroupG/H
centerC22#19127609
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2862C2
socleC22×C83#392
prime radical#19128251C22
hypercenter#2862D83
hypocenterC83#10285
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
3. #24906148C2semidirect
4. #24905296C2semidirect
5. C83×#2862C2semidirect
116. #2862D83semidirect
120. C83#10285semidirect

Maximal subgroups

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

subgroupindex
#249185912
#249082912
#249061482
#249052962
C83×#28622
#1028583