Finite Group Databasebeta

← #35318651

Subgroups — #35318651

Order 21248.

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

normal subgroups only

This group has 224 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#24909541C211yesmaximal, characteristic, complemented
2#24909536C211yesmaximal, characteristic
3#24909301C211yesmaximal, characteristic, complemented
4#24905539C211yesmaximal, characteristic, complemented
5#24905502C211yesmaximal, characteristic, complemented
6#24905500C211yesmaximal, characteristic, complemented
7C83×#2248C211yesmaximal, characteristic, complemented
8#19128142—21no—
9#19128137—21no—
10#19128117—21no—
11#19127819C2211yescharacteristic
12#19127522C2211yescharacteristic
13#19127522C2211yescharacteristic
14#19127522C2211yescharacteristic
15#19127517—21no—
16#19127514—21no—
17#19127484—21no—
18C2×SD32×C83—21no—
19C2×SD32×C83—21no—
20#494×C83C2211yescharacteristic
21#493×C83C2211yescharacteristic
22C4×C16×C83C2211yescharacteristic
23#14581814D411yescharacteristic
24#14581798—21no—
25#14581738—41no—
26#14581736—41no—
27#14581735—41no—
28#14581733—41no—
29#14581732—21no—
30#14581732—21no—
31#14581732—21no—
32#14581731—41no—
33C16×D83—41no—
34#14581728D411yescharacteristic
35#14581720—21no—
36C2×Q16×C83—21no—
37C2×Q16×C83—21no—
38C2×D8×C83—21no—
39C2×D8×C83—21no—
40C4⋊Q8×C83—21no—
41#127×C83—21no—
42SD32×C83—42no—
43SD32×C83—41no—
44SD32×C83—41no—
45C2×C16×C83D411yescharacteristic
46C2×C16×C83D411yescharacteristic
47C4×C8×C83C2311yescharacteristic
48#13050590—21no—
49Dic332D811yescharacteristic
50#13050579—21no—
51D664D811yescharacteristic
52#13050574—21no—
53#13050573—41no—
54C8×D83—41no—
55C83⋊C16—41no—
56C83⋊C16—41no—
57C83⋊C16—41no—
58C2×Q8×C83—41no—
59C2×D4×C83—41no—
60C2×D4×C83—81no—
61Q16×C83—41no—
62Q16×C83—41no—
63Q16×C83—41no—
64Q16×C83—41no—
65D8×C83—41no—
66D8×C83—41no—
67D8×C83—41no—
68D8×C83—41no—
69C2×C8×C83C2×D411yescharacteristic
70C2×C8×C83C2×D411yescharacteristic
71C4⋊C4×C83—81no—
72C42×C83#4111yescharacteristic
73C16×C83—21no—
74C16×C83D811yescharacteristic
75C16×C83D811yescharacteristic
76C23×C83—81no—
77Q8×C83—81no—
78Q8×C83—81no—
79D4×C83—81no—
80D4×C83—81no—
81D4×C83—162no—
82C2×C4×C83—21no—
83C2×C4×C83—81no—
84C2×C4×C83#12111yescharacteristic
85C22⋊D83—41no—
86D332—21no—
87C4×D83—41no—
88Dic166—21no—
89C8×C83C2×D811yescharacteristic
90C8×C83C2×D811yescharacteristic
91C8×C83#13511yescharacteristic
92C8×C83AGL(1,ℤ/8)11yescharacteristic
93C83⋊C8—41no—
94C22×C83—162no—
95C22×C83—161no—
96C22×C83#47211yescharacteristic
97C2×D83—41no—
98C4×C83—22no—
99C4×C83—161no—
100C4×C83#45911yescharacteristic
101C4×C83#46611yescharacteristic
102Dic83—41no—
103#12438—831nomaximal, Sylow
104C2×C83—21no—
105C2×C83—321no—
106C2×C83#166711yescharacteristic
107D83—81no—
108#2248D8311yescharacteristic, complemented
109#2214—831no—
110#2213—831no—
111#2170—831no—
112#1373—831no—
113#1358—831no—
114#1357—831no—
115C83#1243811yescharacteristic, complemented, Sylow
116#510—1661no—
117#509—1661no—
118C2×SD32—21no—
119C2×SD32—21no—
120#504—1661no—
121#494C2×D8311yescharacteristic
122#493C2×D8311yescharacteristic
123#443—831no—
124#364—831no—
125#364—831no—
126#364—831no—
127#362—1661no—
128#361—1661no—
129#350—1661no—
130C4×C16C2×D8311yescharacteristic
131#135—831no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8—21no—
135C2×D8—21no—
136#131—1661no—
137C4⋊Q8—21no—
138#127—21no—
139Q32—3321no—
140SD32—42no—
141SD32—41no—
142SD32—41no—
143SD32—3321no—
144SD32—3321no—
145D16—3321no—
146M32—3321no—
147M32—1661no—
148M32—1661no—
149M32—1661no—
150C2×C16—3321no—
151C2×C16C22⋊D8311yescharacteristic
152C2×C16C22⋊D8311yescharacteristic
153#108—831no—
154#104—1661no—
155C4×C8C22×D8311yescharacteristic
156#41—1661no—
157C2×Q8—41no—
158C2×D4—41no—
159C2×D4—81no—
160Q16—41no—
161Q16—41no—
162Q16—41no—
163Q16—41no—
164Q16—831no—
165SD16—1661no—
166D8—41no—
167D8—41no—
168D8—41no—
169D8—41no—
170D8—831no—
171M16—3321no—
172M16—1661no—
173C2×C8—3321no—
174C2×C8D4×D8311yescharacteristic
175C2×C8C2×C22⋊D8311yescharacteristic
176C4⋊C4—81no—
177C42#1305058911yescharacteristic
178C16—21no—
179C16—3321no—
180C16—3321no—
181C16—3321no—
182C16#1305057511yescharacteristic
183C16#1305057511yescharacteristic
184C23—81no—
185Q8—81no—
186Q8—81no—
187Q8—1661no—
188D4—81no—
189D4—81no—
190D4—162no—
191D4—3321no—
192D4—1661no—
193C2×C4—21no—
194C2×C4—81no—
195C2×C4—3321no—
196C2×C4#1458176311yescharacteristic
197C8—3321no—
198C8D8×D8311yescharacteristic
199C8C2×#1305057511yescharacteristic
200C8#1458181111yescharacteristic
201C8#1458181811yescharacteristic
202C22—162no—
203C22—161no—
204C22—3321no—
205C22#1912796811yescharacteristic
206C4—22no—
207C4—161no—
208C4—3321no—
209C4#1912789311yescharacteristic
210C4#1912793411yescharacteristic
211C2—21no—
212C2—321no—
213C2—6641no—
214C2#2490681511yescharacteristic
subgroupG/H
centerC2#24906815
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#2248C2
socleC2×C83#1667
prime radical#24909536C2
hypercenter#2248D83
hypocenterC83#12438
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #24909541C2semidirect
3. #24909301C2semidirect
4. #24905539C2semidirect
5. #24905502C2semidirect
6. #24905500C2semidirect
7. C83×#2248C2semidirect
108. #2248D83semidirect
115. C83#12438semidirect

Maximal subgroups

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

subgroupindex
#249095412
#249095362
#249093012
#249055392
#249055022
#249055002
C83×#22482
#1243883