Finite Group Databasebeta

← #35319168

Subgroups — #35319168

Order 21248.

306 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 79 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#24909720C211yesmaximal, characteristic, complemented
2#24909712C211yesmaximal, characteristic
3#24909690C211yesmaximal, characteristic, complemented
4#24909686C211yesmaximal, characteristic, complemented
5#24907055C211yesmaximal, characteristic, complemented
6#24905381C211yesmaximal, characteristic, complemented
7C83×#1709C211yesmaximal, characteristic, complemented
8C2×#14581735—22no—
9C2×#14581733—22no—
10#19128106C2211yescharacteristic
11#19128078C2211yescharacteristic
12#19128074—21no—
13#19128048—21no—
14#19127505—22no—
15#19127502—22no—
16C4×C83⋊C16C2211yescharacteristic
17C4×C83⋊C16C2211yescharacteristic
18#494×C83C2211yescharacteristic
19#493×C83C2211yescharacteristic
20#486×C83—21no—
21#486×C83—21no—
22C82×C83C2211yescharacteristic
23C2×#13050579—42no—
24C2×D664—21no—
25C2×D664—21no—
26#14581792—21no—
27#14581789—21no—
28#14581735—44no—
29#14581733—44no—
30C2×C83⋊C16D412yes—
31C2×C83⋊C16D412yes—
32#14581726—21no—
33#14581726—21no—
34#14581714—42no—
35C2×Q16×C83—21no—
36C2×Q16×C83—21no—
37C2×D8×C83—21no—
38C2×D8×C83—21no—
39C4⋊Q8×C83—21no—
40#127×C83—21no—
41#102×C83—42no—
42#102×C83—42no—
43C4×C8×C83D411yescharacteristic
44C4×C8×C83D411yescharacteristic
45C4×C8×C83C2311yescharacteristic
46C2×Dic166—41no—
47C2×D332—41no—
48C2×D332—41no—
49C2×D332—41no—
50#13050579—84no—
51D664—42no—
52D664—42no—
53C4⋊Dic83—41no—
54C4⋊Dic83—41no—
55C83⋊C16—22no—
56C83⋊C16—22no—
57C2×Q8×C83—41no—
58C2×D4×C83—41no—
59C2×D4×C83—41no—
60C2×D4×C83—41no—
61Q16×C83—42no—
62Q16×C83—42no—
63D8×C83—42no—
64D8×C83—42no—
65C2×C8×C83D812yes—
66C2×C8×C83D812yes—
67C2×C8×C83C2×D411yescharacteristic
68C2×C8×C83C2×D411yescharacteristic
69C4⋊C4×C83—41no—
70C4⋊C4×C83—41no—
71C42×C83C2×D411yescharacteristic
72C23×C83—81no—
73C22×D83—81no—
74Q8×C83—81no—
75Q8×C83—81no—
76D4×C83—81no—
77D4×C83—81no—
78D4×C83—82no—
79D4×C83—82no—
80C2×C4×C83—81no—
81C2×C4×C83#12711yescharacteristic
82C2×C4×C83C2×D811yescharacteristic
83C2×C4×C83C2×D811yescharacteristic
84C2×Dic83—81no—
85D332—81no—
86D332—81no—
87D332—82no—
88D332—82no—
89Dic166—82no—
90C8×C83—22no—
91C8×C83—22no—
92C8×C83AGL(1,ℤ/8)12yes—
93C8×C83AGL(1,ℤ/8)12yes—
94C22×C83—161no—
95C22×C83—161no—
96C22×C83—161no—
97C22×C83#49311yescharacteristic
98C2×D83—161no—
99C2×D83—161no—
100C2×D83—161no—
101C4×C83—161no—
102C4×C83#49611yescharacteristic
103C4×C83#49611yescharacteristic
104C4×C83#50812yes—
105C4×C83#50912yes—
106Dic83—161no—
107#12527—831nomaximal, Sylow
108C2×C83—321no—
109C2×C83#172111yescharacteristic
110C2×C83#224911yescharacteristic
111C2×C83#225011yescharacteristic
112D83—321no—
113#2248—831no—
114#2246—831no—
115#2241—831no—
116#2240—831no—
117#1709D8311yescharacteristic, complemented
118#1706—831no—
119#1312—831no—
120C83#1252711yescharacteristic, complemented, Sylow
121C2×SD32—1662no—
122C2×D16—1662no—
123#500—831no—
124#494C2×D8311yescharacteristic
125#493—831no—
126#493C2×D8311yescharacteristic
127#492—1661no—
128#486—21no—
129#486—21no—
130#486—1661no—
131#358—1662no—
132#357—1662no—
133C4×C16—831no—
134C4×C16—831no—
135C82C2×D8311yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×SD16—3322no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—1661no—
142C2×D8—1661no—
143C4⋊Q8—21no—
144C4⋊Q8—1661no—
145#127—21no—
146#127—1661no—
147SD32—3324no—
148D16—3324no—
149C2×C16—832no—
150C2×C16—832no—
151#107—1661no—
152#107—1661no—
153#102—42no—
154#102—42no—
155#102—3322no—
156C4×C8C22⋊D8311yescharacteristic
157C4×C8C22⋊D8311yescharacteristic
158C4×C8C22×D8311yescharacteristic
159C2×Q8—41no—
160C2×Q8—3321no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—3321no—
165C2×D4—3321no—
166C2×D4—3321no—
167Q16—42no—
168Q16—42no—
169SD16—6644no—
170D8—42no—
171D8—42no—
172D8—3322no—
173D8—3322no—
174C2×C8#1305057512yes—
175C2×C8#1305057512yes—
176C2×C8D4×D8311yescharacteristic
177C2×C8D4×D8311yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—3321no—
181C4⋊C4—3321no—
182C42C2×C22⋊D8311yescharacteristic
183C16—1662no—
184C16—1662no—
185C23—81no—
186C23—6641no—
187Q8—81no—
188Q8—81no—
189Q8—6642no—
190D4—81no—
191D4—81no—
192D4—82no—
193D4—82no—
194D4—6641no—
195D4—6641no—
196D4—6642no—
197D4—6642no—
198C2×C4—81no—
199C2×C4—6641no—
200C2×C4#1458178811yescharacteristic
201C2×C4C2×#1305057511yescharacteristic
202C2×C4C2×#1305057511yescharacteristic
203C8—22no—
204C8—22no—
205C8#1458181612yes—
206C8#1458181612yes—
207C22—161no—
208C22—161no—
209C22—161no—
210C22—13281no—
211C22—13281no—
212C22—13281no—
213C22#1912807711yescharacteristic
214C4—161no—
215C4—13281no—
216C4#1912808911yescharacteristic
217C4#1912808911yescharacteristic
218C4#1912813112yes—
219C4#1912813612yes—
220C2—321no—
221C2—26561no—
222C2#2490713011yescharacteristic
223C2#2490972211yescharacteristic
224C2#2490973011yescharacteristic
subgroupG/H
centerC22#19128077
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1709C2
socleC22×C83#493
prime radical#24909712C2
hypercenter#1709D83
hypocenterC83#12527
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1709

Complements

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

NH ≅ G/Ntype
1. #24909720C2semidirect
3. #24909690C2semidirect
4. #24909686C2semidirect
5. #24907055C2semidirect
6. #24905381C2semidirect
7. C83×#1709C2semidirect
117. #1709D83semidirect
120. C83#12527semidirect

Maximal subgroups

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

subgroupindex
#249097202
#249097122
#249096902
#249096862
#249070552
#249053812
C83×#17092
#1252783