Finite Group Databasebeta

← #35318544

Subgroups — #35318544

Order 21248.

240 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 63 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#24909612C211yesmaximal, characteristic, complemented
2#24909504C211yesmaximal, characteristic
3#24907056C211yesmaximal, characteristic
4#24906269C211yesmaximal, characteristic, complemented
5#24905526C211yesmaximal, characteristic, complemented
6#24905508C211yesmaximal, characteristic
7C83×#1365C211yesmaximal, characteristic, complemented
8C2×#14581737—21no—
9#19128078C2211yescharacteristic
10#19128075—21no—
11#19128049—21no—
12#19127825—21no—
13C4×Dic332C2211yescharacteristic
14#19127526—21no—
15C4⋊(C83⋊C16)C2211yescharacteristic
16#19127504—21no—
17#19127503—21no—
18#19127431—21no—
19C8×C83⋊C8C2211yescharacteristic
20#500×C83C2211yescharacteristic
21C4⋊C16×C83C2211yescharacteristic
22#335×C83C2211yescharacteristic
23C2×Dic332D411yescharacteristic
24C2×#13050577—42no—
25C2×D664—21no—
26C2×D664—21no—
27#14581789—21no—
28C4×Dic166—21no—
29#14581737—42no—
30C2×C83⋊C16—21no—
31#14581726D411yescharacteristic
32Dic83⋊C8—41no—
33#14581717—21no—
34#14581712—41no—
35#14581712—41no—
36C4×C83⋊C8D411yescharacteristic
37C4×C83⋊C8D411yescharacteristic
38C8×Dic83—41no—
39C4⋊Q8×C83—21no—
40C2×C16×C83—21no—
41#107×C83—21no—
42#107×C83—21no—
43C4⋊C8×C83—21no—
44C4×C8×C83C2311yescharacteristic
45C2×Dic166—21no—
46C2×D332—41no—
47C2×D332—41no—
48C2×D332—41no—
49Dic332—21no—
50Dic332SD1612yescomplemented
51#13050577—84no—
52D664—41no—
53D664—41no—
54D664—81no—
55C2×C83⋊C8—21no—
56C2×C83⋊C8D811yescharacteristic
57C2×C83⋊C8D811yescharacteristic
58C4⋊Dic83—21no—
59Dic83⋊C4—41no—
60C4×Dic83—41no—
61C83⋊C16—41no—
62C2×Q8×C83—41no—
63C2×C8×C83—41no—
64C2×C8×C83C2×D411yescharacteristic
65C2×C8×C83#4111yescharacteristic
66C4⋊C4×C83—41no—
67C4⋊C4×C83—41no—
68C42×C83C2×D411yescharacteristic
69C16×C83—41no—
70C22×D83—81no—
71Q8×C83—82no—
72C2×C4×C83—81no—
73C2×C4×C83#12111yescharacteristic
74C2×C4×C83C2×D811yescharacteristic
75C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
76C2×Dic83—41no—
77C2×Dic83—41no—
78D332—82no—
79D332—81no—
80D332—81no—
81D332—81no—
82D332—81no—
83Dic166—22no—
84Dic166—41no—
85C8×C83—21no—
86C8×C83—81no—
87C8×C83C2×SD1611yescharacteristic
88C8×C83#13511yescharacteristic
89C83⋊C8—22no—
90C83⋊C8—21no—
91C83⋊C8—21no—
92C22×C83#45911yescharacteristic
93C2×D83—161no—
94C2×D83—161no—
95C2×D83—161no—
96C4×C83—21no—
97C4×C83—161no—
98C4×C83#46511yescharacteristic
99C4×C83#47111yescharacteristic
100C4×C83#50811yescharacteristic
101C4×C83#51011yescharacteristic
102Dic83—42no—
103Dic83—81no—
104#12419—831nomaximal, Sylow
105C2×C83#166511yescharacteristic
106C2×C83#221111yescharacteristic
107C2×C83#221311yescharacteristic
108D83—321no—
109#2227—831no—
110#2207—831no—
111#1706—831no—
112C8×Q16—831no—
113#1367—831no—
114#1365D8311yescharacteristic, complemented
115#1361—831no—
116C83#1241911yescharacteristic, complemented, Sylow
117C2×SD32—1661no—
118#500C2×D8311yescharacteristic
119#493—831no—
120#492—1661no—
121#486—1661no—
122C8×Q8—1661no—
123C4×Q16—831no—
124#366—1661no—
125C4⋊C16—831no—
126C4⋊C16C2×D8311yescharacteristic
127#358—1661no—
128#357—1661no—
129#335C2×D8311yescharacteristic
130Q8⋊C8—1661no—
131C82—831no—
132C2×Q16—831no—
133C2×SD16—3322no—
134C2×D8—1661no—
135C2×D8—1661no—
136C4⋊Q8—21no—
137#127—1661no—
138C4×Q8—1661no—
139SD32—3322no—
140C2×C16—21no—
141C2×C16—1661no—
142#107—21no—
143#107—21no—
144#107—831no—
145C4⋊C8—21no—
146C4⋊C8—3321no—
147#103—1661no—
148#102—3321no—
149#102—3321no—
150C4×C8—3321no—
151C4×C8—831no—
152C4×C8—831no—
153C4×C8C22×D8311yescharacteristic
154C2×Q8—41no—
155C2×Q8—1661no—
156C2×D4—3321no—
157C2×D4—3321no—
158C2×D4—3321no—
159Q16—832no—
160Q16—1661no—
161SD16—6644no—
162D8—3321no—
163D8—3321no—
164D8—6641no—
165C2×C8—41no—
166C2×C8—831no—
167C2×C8—831no—
168C2×C8—1661no—
169C2×C8D4×D8311yescharacteristic
170C2×C8#1305059011yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—3321no—
174C4⋊C4—1661no—
175C42—3321no—
176C42D4×D8311yescharacteristic
177C16—41no—
178C16—3321no—
179C23—6641no—
180Q8—82no—
181Q8—3321no—
182Q8—1662no—
183D4—6642no—
184D4—6641no—
185D4—6641no—
186D4—6641no—
187D4—6641no—
188C2×C4—81no—
189C2×C4—3321no—
190C2×C4—3321no—
191C2×C4#1458176511yescharacteristic
192C2×C4D8×D8311yescharacteristic
193C2×C4#1458181911yescharacteristic
194C8—21no—
195C8—81no—
196C8—1662no—
197C8—1661no—
198C8—1661no—
199C8SD16×D8311yescharacteristic
200C8#1458181511yescharacteristic
201C22—13281no—
202C22—13281no—
203C22—13281no—
204C22#1912789511yescharacteristic
205C4—21no—
206C4—161no—
207C4—3322no—
208C4—6641no—
209C4#1912793311yescharacteristic
210C4#1912796711yescharacteristic
211C4#1912813211yescharacteristic
212C4#1912814311yescharacteristic
213C2—26561no—
214C2#2490680711yescharacteristic
215C2#2490952811yescharacteristic
216C2#2490953811yescharacteristic
subgroupG/H
centerC22#19127895
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1365C2
socleC22×C83#459
prime radical#19128078C22
hypercenter#1365D83
hypocenterC83#12419
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
1. #24909612C2semidirect
4. #24906269C2semidirect
5. #24905526C2semidirect
7. C83×#1365C2semidirect
50. Dic332SD16semidirect
114. #1365D83semidirect
116. C83#12419semidirect

Maximal subgroups

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

subgroupindex
#249096122
#249095042
#249070562
#249062692
#249055262
#249055082
C83×#13652
#1241983