Finite Group Databasebeta

← #35316359

Subgroups — #35316359

Order 21248.

234 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 69 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#24909337C211yesmaximal, characteristic
2#24906376C211yesmaximal, characteristic, complemented
3#24906287C211yesmaximal, characteristic, complemented
4#24905528C211yesmaximal, characteristic, complemented
5#24905431C211yesmaximal, characteristic, complemented
6#24905383C211yesmaximal, characteristic, complemented
7C83×#1333C211yesmaximal, characteristic, complemented
8C2×#14581733C411yescharacteristic
9C4×D664C2211yescharacteristic
10#19127778—21no—
11D4×C83⋊C8—21no—
12#19127528C411yescharacteristic
13#19127502C411yescharacteristic
14#19127501C411yescharacteristic
15#19127477C2211yescharacteristic
16#19127452C2211yescharacteristic
17D4⋊(C83⋊C8)—21no—
18#19127428—21no—
19#19127422C2211yescharacteristic
20C4×D8×C83C2211yescharacteristic
21#346×C83C2211yescharacteristic
22#334×C83C2211yescharacteristic
23C2×D664C2×C411yescharacteristic
24C2×#13050573—41no—
25C22×C83⋊C8—41no—
26C4×D332—21no—
27#14581733—22no—
28C2×C83⋊C16C2×C411yescharacteristic
29C2×C83⋊C16C2×C411yescharacteristic
30#14581726C2×C411yescharacteristic
31C4⋊(C83⋊C8)—21no—
32#14581714—21no—
33C22⋊(C83⋊C8)—41no—
34(C2×D83)⋊C8—41no—
35C4×C83⋊C8D411yescharacteristic
36C4×C83⋊C8D411yescharacteristic
37C2×D8×C83C2×C411yescharacteristic
38C4×D4×C83—21no—
39C2×C16×C83—21no—
40#107×C83C2×C411yescharacteristic
41C4⋊C8×C83—21no—
42#102×C83—21no—
43C4×C8×C83C2311yescharacteristic
44C2×D332—21no—
45C2×C4×D83—41no—
46D664—21no—
47D664—21no—
48#13050573—82no—
49C2×C83⋊C8—21no—
50C2×C83⋊C8—21no—
51C2×C83⋊C8—41no—
52C2×C83⋊C8—82no—
53C4⋊Dic83—21no—
54(C2×D83)⋊C4—41no—
55C83⋊C16—21no—
56C83⋊C16—21no—
57C2×D4×C83—21no—
58C22×C4×C83—41no—
59D8×C83—21no—
60D8×C83M1612yescomplemented
61C2×C8×C83—41no—
62C2×C8×C83C22×C411yescharacteristic
63C2×C8×C83#4111yescharacteristic
64C4⋊C4×C83—21no—
65C22⋊C4×C83—41no—
66C42×C83C2×D411yescharacteristic
67C16×C83—22no—
68C23×C83—41no—
69C22×D83—41no—
70D4×C83—22no—
71D4×C83—41no—
72C2×C4×C83—41no—
73C2×C4×C83—81no—
74C2×C4×C83—81no—
75C2×C4×C83C4×D411yescharacteristic
76C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
78C2×Dic83—41no—
79D332—41no—
80D332—41no—
81C4×D83—81no—
82C4×D83—81no—
83C8×C83—21no—
84C8×C83—81no—
85C8×C83C2×M1611yescharacteristic
86C8×C83#13111yescharacteristic
87C83⋊C8—41no—
88C83⋊C8—41no—
89C83⋊C8—81no—
90C22×C83—42no—
91C22×C83—81no—
92C22×C83—81no—
93C22×C83#44211yescharacteristic
94C2×D83—81no—
95C2×D83—81no—
96C2×D83—81no—
97C4×C83—21no—
98C4×C83—21no—
99C4×C83—81no—
100C4×C83#43511yescharacteristic
101C4×C83#44311yescharacteristic
102Dic83—81no—
103#12078—831nomaximal, Sylow
104C2×C83—82no—
105C2×C83#158611yescharacteristic
106C2×C83AGL(1,ℤ/16)11yescharacteristic
107C2×C83AGL(1,ℤ/16)11yescharacteristic
108D83—161no—
109#2177—831no—
110#1594—831no—
111#1580—831no—
112#1368—831no—
113#1333—831no—
114#1333D8311yescharacteristic, complemented
115#1313—831no—
116C83#1207811yescharacteristic, complemented, Sylow
117C2×D16—831no—
118C4×D8—831no—
119C4×D8C2×D8311yescharacteristic
120#436—1661no—
121C8×D4—1661no—
122#367—831no—
123#357—831no—
124#357—831no—
125#346—831no—
126#346C2×D8311yescharacteristic
127#334—831no—
128#334C2×D8311yescharacteristic
129D4⋊C8—1661no—
130D4⋊C8—1661no—
131#322—831no—
132C2×D8—831no—
133C2×D8C4×D8311yescharacteristic
134C2×M16—3321no—
135C22×C8—3321no—
136C4×D4—21no—
137C4×D4—1661no—
138D16—1662no—
139C2×C16—21no—
140C2×C16—831no—
141C2×C16—831no—
142#107—831no—
143#107C4×D8311yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1661no—
146#102—21no—
147#102—1661no—
148C22⋊C8—3321no—
149C22⋊C8—3321no—
150C4×C8—831no—
151C4×C8—831no—
152C4×C8C22×D8311yescharacteristic
153C2×D4—21no—
154C2×D4—1661no—
155C22×C4—41no—
156C22×C4—3321no—
157D8—21no—
158D8—1661no—
159D8—1661no—
160D8#1305057312yescomplemented
161M16—6642no—
162C2×C8—41no—
163C2×C8—3321no—
164C2×C8—6642no—
165C2×C8—1661no—
166C2×C8—1661no—
167C2×C8C2×C4×D8311yescharacteristic
168C2×C8#1305058911yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1661no—
171C22⋊C4—41no—
172C22⋊C4—3321no—
173C42D4×D8311yescharacteristic
174C16—22no—
175C16—1661no—
176C16—1661no—
177C23—41no—
178C23—3321no—
179D4—22no—
180D4—41no—
181D4—3321no—
182D4—3321no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—3321no—
187C2×C4—6641no—
188C2×C4—6641no—
189C2×C4(C22⋊D83)⋊C411yescharacteristic
190C2×C4#1458181611yescharacteristic
191C2×C4#1458181911yescharacteristic
192C8—21no—
193C8—81no—
194C8—3321no—
195C8—3321no—
196C8—6641no—
197C8C2×#1305057311yescharacteristic
198C8#1458180011yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22—6641no—
203C22—6641no—
204C22—6641no—
205C22#1912781011yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—6641no—
210C4#1912776911yescharacteristic
211C4#1912781511yescharacteristic
212C2—82no—
213C2—13281no—
214C2#2490632411yescharacteristic
215C2#2490935811yescharacteristic
216C2#2490936411yescharacteristic
subgroupG/H
centerC22#19127810
commutatorC2×C8×C83C22×C4
FrattiniC4×C8C22×D83
FittingC83×#1333C2
socleC22×C83#442
prime radicalC2×#14581733C4
cosocleC4×C8×C83C23
hypercenter#1333D83
hypocenterC83#12078
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #24906376C2semidirect
3. #24906287C2semidirect
4. #24905528C2semidirect
5. #24905431C2semidirect
6. #24905383C2semidirect
7. C83×#1333C2semidirect
60. D8×C83M16semidirect
114. #1333D83semidirect
116. C83#12078semidirect
160. D8#13050573semidirect

Maximal subgroups

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

subgroupindex
#249093372
#249063762
#249062872
#249055282
#249054312
#249053832
C83×#13332
#1207883