Finite Group Databasebeta

← #35317751

Subgroups — #35317751

Order 21248.

310 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 59 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#24909680C212yesmaximal, complemented
2#24909016C211yesmaximal, characteristic
3#24905451C212yesmaximal, complemented
4#24905401C211yesmaximal, characteristic
5C83×#3202C211yesmaximal, characteristic
6#19127750C2211yescharacteristic
7#19127727—21no—
8#19127718—21no—
9#19127523—22no—
10#19127517—22no—
11#19127483C2212yes—
12#19127458C2211yescharacteristic
13#536×C83—21no—
14#485×C83—22no—
15#470×C83C2212yes—
16#467×C83—21no—
17#451×C83—22no—
18#408×C83C2211yescharacteristic
19C2×#13050574—21no—
20#14581751—21no—
21#14581732—22no—
22#14581728—22no—
23Dic83⋊C8—42no—
24#14581719—42no—
25C2×#41×C83D411yescharacteristic
26C22×Q8×C83—21no—
27C2×Q16×C83—21no—
28C2×Q16×C83D412yes—
29C2×M16×C83C2311yescharacteristic
30C22×C8×C83D411yescharacteristic
31C4⋊Q8×C83—41no—
32#124×C83—41no—
33#123×C83—41no—
34#123×C83—41no—
35C22⋊Q8×C83—22no—
36C22⋊Q8×C83—22no—
37#117×C83—21no—
38#107×C83—21no—
39#106×C83D412yes—
40#103×C83—22no—
41#103×C83—22no—
42C22⋊C8×C83—22no—
43#13050574—41no—
44#13050574—41no—
45C2×C83⋊C8—41no—
46C4⋊Dic83—41no—
47C22⋊Dic83—41no—
48C4×Dic83—22no—
49C83⋊C16—42no—
50#41×C83—41no—
51#41×C83—41no—
52C2×Q8×C83—22no—
53C2×Q8×C83—41no—
54C2×Q8×C83—81no—
55C2×Q8×C83—81no—
56C2×Q8×C83SD1611yescharacteristic
57C2×D4×C83—21no—
58C2×D4×C83SD1611yescharacteristic
59C22×C4×C83—21no—
60C22×C4×C83—41no—
61C22×C4×C83C2×D411yescharacteristic
62Q16×C83—42no—
63Q16×C83—81no—
64M16×C83—41no—
65C2×C8×C83—21no—
66C2×C8×C83—41no—
67C2×C8×C83C2×D412yes—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—42no—
70C4⋊C4×C83—42no—
71C22⋊C4×C83—41no—
72C22⋊C4×C83—42no—
73C22⋊C4×C83—42no—
74C42×C83—41no—
75C23×C83—21no—
76C23×C83AGL(1,ℤ/8)11yescharacteristic
77Q8×C83—41no—
78Q8×C83—42no—
79Q8×C83—42no—
80Q8×C83—82no—
81D4×C83—41no—
82D4×C83—81no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—41no—
87C2×C4×C83—41no—
88C2×C4×C83—42no—
89C2×C4×C83—42no—
90C2×C4×C83—82no—
91C2×C4×C83#12011yescharacteristic
92C2×C4×C83C2×SD1611yescharacteristic
93C2×Dic83—42no—
94C8×C83—22no—
95C8×C83—41no—
96C83⋊C8—81no—
97C22×C83—21no—
98C22×C83—41no—
99C22×C83—42no—
100C22×C83—41no—
101C22×C83#45011yescharacteristic
102C4×C83—21no—
103C4×C83—41no—
104C4×C83—82no—
105C4×C83—82no—
106C4×C83#45311yescharacteristic
107C4×C83#45811yescharacteristic
108Dic83—82no—
109#12302—831nomaximal, Sylow
110C2×C83—21no—
111C2×C83—41no—
112C2×C83—81no—
113C2×C83#161311yescharacteristic
114#3202D8311yescharacteristic
115#2239—832no—
116#2098—831no—
117#1340—832no—
118#1321—831no—
119C83#1230211yescharacteristic, complemented, Sylow
120#536—21no—
121#485—22no—
122#470C2×D8312yes—
123#467—21no—
124#451—22no—
125#430—831no—
126#424—1661no—
127#421—1661no—
128#408C2×D8311yescharacteristic
129#365—1662no—
130#362—1662no—
131#349—832no—
132H(3,ℤ/4)—831no—
133C2×#41C22⋊D8311yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16C22⋊D8312yes—
137C2×M16—1661no—
138C2×M16C22×D8311yescharacteristic
139C22×C8C22⋊D8311yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—1661no—
148M32—1662no—
149#108—1662no—
150#107—21no—
151#106C22⋊D8312yes—
152C4⋊C8—3322no—
153#104—3322no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8#1305057811yescharacteristic
164C2×D4—21no—
165C2×D4#1305057811yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×C22⋊D8311yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—3321no—
173M16—3321no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—3321no—
177C2×C8C2×C22⋊D8312yes—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—3321no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—3321no—
186C42—41no—
187C42—1662no—
188C16—3322no—
189C23—21no—
190C23#1458181811yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—3322no—
206C2×C4#1458176111yescharacteristic
207C2×C4C2×#1305057811yescharacteristic
208C8—22no—
209C8—41no—
210C8—6641no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#1912785211yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—6642no—
221C4#1912786911yescharacteristic
222C4#1912789111yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#2490648311yescharacteristic
subgroupG/H
centerC2#24906483
commutator, cosocleC2×M16×C83C23
FrattiniC2×M16C22×D83
FittingC83×#3202C2
socleC2×C83#1613
prime radicalC2×#41×C83D4
hypercenter#3202D83
hypocenterC83#12302
LCSG ▷ C2×M16×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #24909680C2semidirect
3. #24905451C2semidirect
119. C83#12302semidirect

Maximal subgroups

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

subgroupindex
#249096802
#249090162
#249054512
#249054012
C83×#32022
#1230283