Finite Group Databasebeta

← #35318686

Subgroups — #35318686

Order 21248.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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#24909718C211yesmaximal, characteristic
2#24909372C211yesmaximal, characteristic, complemented
3#24906833C211yesmaximal, characteristic, complemented
4#24905534C211yesmaximal, characteristic, complemented
5#24905511C211yesmaximal, characteristic, complemented
6#24905509C211yesmaximal, characteristic
7C83×#1668C211yesmaximal, characteristic
8C2×#14581736—21no—
9C2×#14581736—21no—
10#19128081C2211yescharacteristic
11#19127986—21no—
12C4⋊(C83⋊C16)—21no—
13C4⋊(C83⋊C16)C2211yescharacteristic
14#19127477—21no—
15C4×C83⋊C16C2211yescharacteristic
16#19127454C2211yescharacteristic
17#19127448—21no—
18#493×C83C2211yescharacteristic
19#460×C83—21no—
20C8×Q8×C83C2211yescharacteristic
21#335×C83C2211yescharacteristic
22#331×C83—21no—
23C2×Dic332—21no—
24C2×Dic332—21no—
25#14581792—21no—
26#14581736—42no—
27#14581736—42no—
28C2×C83⋊C16—21no—
29C2×C83⋊C16D411yescharacteristic
30C2×C83⋊C16D411yescharacteristic
31#14581723—41no—
32C4⋊(C83⋊C8)—21no—
33#14581716—41no—
34C2×SD16×C83—41no—
35C2×D8×C83—21no—
36C2×D8×C83—21no—
37#127×C83—21no—
38C4×Q8×C83D411yescharacteristic
39C4⋊C8×C83—21no—
40C4⋊C8×C83—21no—
41C4⋊C8×C83D411yescharacteristic
42#102×C83—41no—
43C4×C8×C83—21no—
44C4×C8×C83C2311yescharacteristic
45C2×Dic166—41no—
46Dic332—41no—
47Dic332—41no—
48Dic332—81no—
49C2×C83⋊C8—41no—
50C4⋊Dic83—41no—
51C4⋊Dic83—41no—
52C83⋊C16—21no—
53C83⋊C16—41no—
54C83⋊C16SD1612yescomplemented
55C2×Q8×C83D811yescharacteristic
56C2×D4×C83—41no—
57C2×D4×C83—41no—
58C2×D4×C83—41no—
59SD16×C83—82no—
60D8×C83—41no—
61D8×C83—41no—
62D8×C83—81no—
63C2×C8×C83—21no—
64C2×C8×C83—41no—
65C2×C8×C83C2×D411yescharacteristic
66C2×C8×C83#4111yescharacteristic
67C4⋊C4×C83—21no—
68C4⋊C4×C83D811yescharacteristic
69C42×C83—21no—
70C42×C83C2×D411yescharacteristic
71C23×C83—81no—
72Q8×C83—21no—
73Q8×C83—21no—
74D4×C83—81no—
75D4×C83—81no—
76D4×C83—81no—
77D4×C83—81no—
78D4×C83—161no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83#12111yescharacteristic
82C2×C4×C83C2×D811yescharacteristic
83C2×C4×C83#13511yescharacteristic
84C2×Dic83—81no—
85Dic166—81no—
86Dic166—81no—
87C8×C83—21no—
88C8×C83—41no—
89C8×C83—81no—
90C8×C83C2×SD1611yescharacteristic
91C8×C83AGL(1,ℤ/8)11yescharacteristic
92C83⋊C8—81no—
93C22×C83—161no—
94C22×C83—161no—
95C22×C83—161no—
96C22×C83#46611yescharacteristic
97C4×C83—21no—
98C4×C83—41no—
99C4×C83—41no—
100C4×C83#46011yescharacteristic
101C4×C83#47211yescharacteristic
102C4×C83#50911yescharacteristic
103C4×C83#51011yescharacteristic
104Dic83—161no—
105#12444—831nomaximal, Sylow
106C2×C83—321no—
107C2×C83#166811yescharacteristic
108C2×C83#222011yescharacteristic
109C2×C83#222311yescharacteristic
110#2248—831no—
111#2183—831no—
112#1670—831no—
113#1668D8311yescharacteristic
114#1371—831no—
115#1362—831no—
116#1361—831no—
117C83#1244411yescharacteristic, complemented, Sylow
118C2×SD32—1661no—
119C2×SD32—1661no—
120#494—831no—
121#493C2×D8311yescharacteristic
122#475—1661no—
123#460—21no—
124C8×Q8C2×D8311yescharacteristic
125C4⋊C16—831no—
126C4⋊C16—1661no—
127#346—1661no—
128C4×C16—831no—
129#335—831no—
130#335C2×D8311yescharacteristic
131#332—1661no—
132#331—21no—
133C2×Q16—1661no—
134C2×Q16—1661no—
135C2×SD16—41no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—1661no—
139#127—21no—
140C4×Q8C22⋊D8311yescharacteristic
141SD32—3322no—
142SD32—3322no—
143C2×C16—831no—
144C2×C16—831no—
145C2×C16—1661no—
146#106—3321no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—1661no—
150C4⋊C8C22⋊D8311yescharacteristic
151#103—3321no—
152#102—41no—
153C4×C8—21no—
154C4×C8C22×D8311yescharacteristic
155C2×Q8—3321no—
156C2×Q8#1305057511yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—3321no—
161Q16—3321no—
162Q16—6641no—
163SD16—82no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—3321no—
170C2×C8D4×D8311yescharacteristic
171C2×C8#1305058911yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—3321no—
174C4⋊C4—3321no—
175C4⋊C4#1305057511yescharacteristic
176C42—21no—
177C42C2×C22⋊D8311yescharacteristic
178C16—3321no—
179C16—832no—
180C16—1661no—
181C23—81no—
182Q8—21no—
183Q8—21no—
184Q8—6641no—
185Q8—6641no—
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—6641no—
194C2×C4#1458176411yescharacteristic
195C2×C4C2×#1305057511yescharacteristic
196C2×C4#1458181211yescharacteristic
197C8—21no—
198C8—41no—
199C8—81no—
200C8—6641no—
201C8SD16×D8311yescharacteristic
202C8#1458181911yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#1912793511yescharacteristic
207C4—21no—
208C4—41no—
209C4—41no—
210C4—13281no—
211C4#1912789911yescharacteristic
212C4#1912796911yescharacteristic
213C4#1912813711yescharacteristic
214C4#1912814111yescharacteristic
215C2—321no—
216C2#2490682111yescharacteristic
217C2#2490957411yescharacteristic
218C2#2490959111yescharacteristic
subgroupG/H
centerC22#19127935
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1668C2
socleC22×C83#466
prime radical#493×C83C22
hypercenter#1668D83
hypocenterC83#12444
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. #24909372C2semidirect
3. #24906833C2semidirect
4. #24905534C2semidirect
5. #24905511C2semidirect
54. C83⋊C16SD16semidirect
117. C83#12444semidirect

Maximal subgroups

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

subgroupindex
#249097182
#249093722
#249068332
#249055342
#249055112
#249055092
C83×#16682
#1244483