Finite Group Databasebeta

← #35307078

Subgroups — #35307078

Order 21248.

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

normal subgroups only

This group has 566 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#24918388C212yesmaximal
2#24908291C211yesmaximal, characteristic
3#24906144C212yesmaximal, complemented
4#24905296C211yesmaximal, characteristic, complemented
5C83×#2862C211yesmaximal, characteristic, complemented
6#19128251C2211yescharacteristic
7#19128075—22no—
8#19128051—22no—
9#19127903—24no—
10#19127840—24no—
11#19127754C2212yes—
12#19127725C2211yescharacteristic
13C2×#14581720—22no—
14#19127450—24no—
15C2×C4⋊Q8×C83C2212yes—
16#391×C83—24no—
17#382×C83—22no—
18C43×C83C2211yescharacteristic
19C2×#13050590—21no—
20C2×#13050577—28no—
21C2×#13050574—21no—
22#14581792D411yescharacteristic
23#14581789D411yescharacteristic
24#14581780—21no—
25#14581766—22no—
26C4×D332—22no—
27C4⋊(C83⋊C8)—22no—
28C4⋊(C83⋊C8)D414yes—
29#14581720—44no—
30#14581716—28no—
31C22⋊(C83⋊C8)—22no—
32C4×C83⋊C8—22no—
33C22×Q8×C83D414yes—
34C4⋊Q8×C83—22no—
35C4⋊Q8×C83—22no—
36C4⋊Q8×C83—22no—
37C4⋊Q8×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83—22no—
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#13050577—88no—
52#13050574—42no—
53C2×C83⋊C8—24no—
54C2×C83⋊C8—41no—
55C4⋊Dic83—22no—
56(C2×D83)⋊C4—42no—
57C2×Q8×C83—24no—
58C2×Q8×C83—24no—
59C2×Q8×C83—48no—
60C22×C4×C83—22no—
61C22×C4×C83—24no—
62C22×C4×C83C2×D412yes—
63C22×C4×C83C2×D411yescharacteristic
64C4⋊C4×C83—42no—
65C4⋊C4×C83—42no—
66C4⋊C4×C83—42no—
67C4⋊C4×C83—42no—
68C42×C83—22no—
69C42×C83—44no—
70C42×C83SD1614yescomplemented
71C42×C83C2×D411yescharacteristic
72C42×C83C2×D412yes—
73C42×C83#4111yescharacteristic
74C23×C83#12011yescharacteristic
75C22×D83—41no—
76Q8×C83—48no—
77Q8×C83—84no—
78C2×C4×C83—24no—
79C2×C4×C83—24no—
80C2×C4×C83—44no—
81C2×C4×C83—48no—
82C2×C4×C83—44no—
83C2×C4×C83#12012yes—
84C2×C4×C83#12111yescharacteristic
85C2×C4×C83#12112yes—
86C2×C4×C83#12711yescharacteristic
87C2×C4×C83C2×SD1612yes—
88C2×C4×C83#13712yes—
89C22⋊D83—82no—
90C2×Dic83—41no—
91D332—44no—
92D332—42no—
93C4×D83—81no—
94C4×D83—81no—
95Dic166—42no—
96C83⋊C8—44no—
97C83⋊C8—81no—
98C22×C83—22no—
99C22×C83#39211yescharacteristic
100C22×C83#44812yes—
101C2×D83—82no—
102C2×D83—81no—
103C4×C83—22no—
104C4×C83—44no—
105C4×C83—84no—
106C4×C83#45514yes—
107C4×C83#46112yes—
108C4×C83#49211yescharacteristic
109C4×C83#49711yescharacteristic
110Dic83—81no—
111#10287—831nomaximal, Sylow
112C2×C83—22no—
113C2×C83#200412yes—
114C2×C83#201311yescharacteristic
115D83—161no—
116#3206—832no—
117#2862D8311yescharacteristic, complemented
118#1962—831no—
119#1548—832no—
120#1276—831no—
121C83#1028711yescharacteristic, complemented, Sylow
122#532—831no—
123C2×C4⋊Q8C2×D8312yes—
124#492—1662no—
125#487—1662no—
126#461—1664no—
127#448—1664no—
128#431—832no—
129#423—831no—
130C2×#104—1662no—
131#391—24no—
132#382—22no—
133C43C2×D8311yescharacteristic
134#333—1664no—
135C2×#41—1661no—
136C22×Q8C22⋊D8314yes—
137C2×SD16—1668no—
138C2×M16—1661no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—831no—
144#127—831no—
145#124—1661no—
146#121—1662no—
147C4×D4—1662no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C42—22no—
151C2×C42C22⋊D8312yes—
152C2×C42C22×D8311yescharacteristic
153C4⋊C8—834no—
154C4⋊C8—1662no—
155#104—3324no—
156#103—1668no—
157C22⋊C8—1662no—
158C4×C8—1662no—
159#95—28no—
160#41—3322no—
161#41—3322no—
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—48no—
165C2×Q8—1661no—
166C2×D4—3321no—
167C2×D4—1662no—
168C2×D4—1661no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—3321no—
172C22×C4C2×C22⋊D8312yes—
173C22×C4C2×C22⋊D8311yescharacteristic
174SD16—6648no—
175M16—3322no—
176C2×C8—3321no—
177C2×C8—1664no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—1662no—
183C22⋊C4—3322no—
184C42—22no—
185C42—44no—
186C42#1305057714yescomplemented
187C42D4×D8311yescharacteristic
188C42D4×D8312yes—
189C42#1305058911yescharacteristic
190C23—3321no—
191C23#1458176111yescharacteristic
192Q8—48no—
193Q8—84no—
194Q8—3322no—
195D4—3324no—
196D4—3322no—
197D4—6642no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—3321no—
204C2×C4—6641no—
205C2×C4—6641no—
206C2×C4#1458176012yes—
207C2×C4#1458176311yescharacteristic
208C2×C4#1458176412yes—
209C2×C4#1458178811yescharacteristic
210C2×C4C2×#1305057712yes—
211C2×C4#1458182312yes—
212C8—3324no—
213C8—6641no—
214C22—22no—
215C22—6641no—
216C22—6642no—
217C22#1912760911yescharacteristic
218C22#1912784012yes—
219C4—22no—
220C4—44no—
221C4—84no—
222C4—6641no—
223C4#1912787814yes—
224C4#1912790312yes—
225C4#1912807511yescharacteristic
226C4#1912809311yescharacteristic
227C2—22no—
228C2—13281no—
229C2#2490849212yes—
230C2#2490854611yescharacteristic
subgroupG/H
centerC22#19127609
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2862C2
socleC22×C83#392
prime radical#19128251C22
hypercenter#2862D83
hypocenterC83#10287
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. #24906144C2semidirect
4. #24905296C2semidirect
5. C83×#2862C2semidirect
70. C42×C83SD16semidirect
117. #2862D83semidirect
121. C83#10287semidirect
186. C42#13050577semidirect

Maximal subgroups

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

subgroupindex
#249183882
#249082912
#249061442
#249052962
C83×#28622
#1028783