Finite Group Databasebeta

← #35314953

Subgroups — #35314953

Order 21248.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#24918122C211yesmaximal, characteristic
2#24907218C211yesmaximal, characteristic
3#24905458C212yesmaximal
4#24905452C212yesmaximal
5C83×#2166C211yesmaximal, characteristic
6#19128313—21no—
7#19128103C411yescharacteristic, complemented
8#19128081C411yescharacteristic, complemented
9#19128070—21no—
10#19127959C2212yes—
11#19127942—21no—
12#19127809C412yescomplemented
13C4×Dic332—21no—
14#19127656—21no—
15C83⋊#338C2211yescharacteristic
16#19127458C2211yescharacteristic
17#19127457—21no—
18#405×C83C2211yescharacteristic
19#364×C83—21no—
20#349×C83C2212yes—
21#347×C83—21no—
22C2×Dic332—21no—
23C2×Dic332C2×C412yes—
24C2×#13050574—21no—
25#14581792—21no—
26#14581785—42no—
27#14581784—21no—
28#14581782—42no—
29#14581771—22no—
30C4×Dic166—22no—
31#14581751—21no—
32#14581726—21no—
33#14581724C2×C412yes—
34#14581717—22no—
35#14581717—22no—
36#14581702—22no—
37C8×Dic83—22no—
38C2×M16×C83C2311yescharacteristic
39C22×C8×C83D411yescharacteristic
40#117×C83D411yescharacteristic
41M32×C83—22no—
42#97×C83C2×C411yescharacteristic
43#97×C83D411yescharacteristic
44C4×C8×C83C2×C411yescharacteristic
45C4×C8×C83D411yescharacteristic
46C2×Dic166—22no—
47Dic332—42no—
48Dic332—41no—
49Dic332—41no—
50#13050574—41no—
51#13050574—41no—
52C2×C83⋊C8—22no—
53C4⋊Dic83—22no—
54C4⋊Dic83—42no—
55C4⋊Dic83—41no—
56Dic83⋊C4—42no—
57Dic83⋊C4—42no—
58Dic83⋊C4—42no—
59C22⋊Dic83—42no—
60C22⋊Dic83—41no—
61C4×Dic83—22no—
62C4×Dic83—42no—
63C22×C4×C83C2×D411yescharacteristic
64M16×C83—41no—
65C2×C8×C83—21no—
66C2×C8×C83—21no—
67C2×C8×C83C22⋊C411yescharacteristic
68C2×C8×C83C22⋊C411yescharacteristic
69C2×C8×C83C22×C411yescharacteristic
70C2×C8×C83C2×D411yescharacteristic
71C4⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C42×C83C22⋊C411yescharacteristic
74C42×C83C22⋊C411yescharacteristic
75C16×C83—42no—
76C23×C83AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83C2×C22⋊C411yescharacteristic
82C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
83C2×Dic83—42no—
84C2×Dic83—42no—
85C2×Dic83—42no—
86Dic166—42no—
87Dic166—42no—
88C8×C83—21no—
89C8×C83—21no—
90C8×C83—21no—
91C8×C83#10411yescharacteristic
92C8×C83#10411yescharacteristic
93C83⋊C8—42no—
94C22×C83—21no—
95C22×C83—41no—
96C22×C83#41811yescharacteristic
97C4×C83—21no—
98C4×C83—41no—
99C4×C83—41no—
100C4×C83#41011yescharacteristic
101C4×C83C2×#10411yescharacteristic
102Dic83—82no—
103Dic83—82no—
104Dic83—82no—
105#11771—831nomaximal, Sylow
106C2×C83—21no—
107C2×C83—41no—
108C2×C83#150511yescharacteristic
109#3177—831no—
110#2166D8311yescharacteristic
111#1747—831no—
112#1342—832no—
113#1340—832no—
114C83#1177111yescharacteristic, complemented, Sylow
115#541—1661no—
116#499—831no—
117#494—831no—
118#491—1661no—
119#470—832no—
120#467—1661no—
121#441—832no—
122C4×Q16—1661no—
123#405—1661no—
124#405C2×D8311yescharacteristic
125#364—21no—
126#349C2×D8312yes—
127#347—21no—
128#338—831no—
129H(3,ℤ/4)—831no—
130H(3,ℤ/4)—1661no—
131C2×Q16—832no—
132C2×Q16—1661no—
133C2×M16—1661no—
134C2×M16C22×D8311yescharacteristic
135C22×C8D33211yescharacteristic
136C4⋊Q8—1661no—
137#126—3322no—
138#125—3322no—
139#125—1661no—
140C22⋊Q8—1662no—
141C4×Q8—1662no—
142#117—1661no—
143#117D33211yescharacteristic
144M32—22no—
145#107—1661no—
146#106—832no—
147#103—1662no—
148#103—1662no—
149#97—1662no—
150#97C4×D8311yescharacteristic
151#97C22⋊D8311yescharacteristic
152C4×C8—1662no—
153C4×C8C4×D8311yescharacteristic
154C4×C8C22⋊D8311yescharacteristic
155C2×Q8—1662no—
156C22×C4C2×D33211yescharacteristic
157Q16—3322no—
158Q16—3321no—
159Q16—3321no—
160M16—41no—
161M16—3321no—
162M16—3321no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1662no—
166C2×C8(C2×D83)⋊C411yescharacteristic
167C2×C8(C2×D83)⋊C411yescharacteristic
168C2×C8C2×C4×D8311yescharacteristic
169C2×C8C2×C22⋊D8311yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—3321no—
172C4⋊C4—3322no—
173C4⋊C4—3322no—
174C4⋊C4—3322no—
175C4⋊C4—3322no—
176C4⋊C4—1662no—
177C22⋊C4—21no—
178C22⋊C4—3321no—
179C22⋊C4—3322no—
180C42—3322no—
181C42—1662no—
182C42(C2×D83)⋊C411yescharacteristic
183C42(C2×D83)⋊C411yescharacteristic
184C16—42no—
185C23#1458182011yescharacteristic
186Q8—3322no—
187Q8—3322no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—3322no—
193C2×C4—3322no—
194C2×C4—3322no—
195C2×C4C2×(C2×D83)⋊C411yescharacteristic
196C2×C4#1458182011yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—3322no—
201C8#1458171811yescharacteristic
202C8#1458171811yescharacteristic
203C22—21no—
204C22—41no—
205C22#1912770711yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—6642no—
210C4—6642no—
211C4—6642no—
212C4#1912767411yescharacteristic
213C4C2×#1458171811yescharacteristic
214C2—21no—
215C2—41no—
216C2#2490597211yescharacteristic
subgroupG/H
centerC2#24905972
commutatorC2×C8×C83C22×C4
FrattiniC2×M16C22×D83
FittingC83×#2166C2
socleC2×C83#1505
prime radicalC23×C83AGL(1,ℤ/8)
cosocleC2×M16×C83C23
hypercenter#2166D83
hypocenterC83#11771
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
7. #19128103C4semidirect
8. #19128081C4semidirect
12. #19127809C4semidirect
114. C83#11771semidirect

Maximal subgroups

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

subgroupindex
#249181222
#249072182
#249054582
#249054522
C83×#21662
#1177183