Finite Group Databasebeta

← #1711×D83 #35392078

Subgroups — #1711×D83 #35392078

Order 21248.

802 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 179 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#487×D83C213yesmaximal, complemented
2#486×D83C213yesmaximal
3#24907076C213yesmaximal, complemented
4#24907075C211yesmaximal, characteristic, complemented
5#24907074C213yesmaximal, complemented
6C82×D83C211yesmaximal, characteristic, complemented
7C83×#1711C211yesmaximal, characteristic, complemented
8C4⋊Q8×D83C2213yes—
9#127×D83C2211yescharacteristic
10#19128053C2213yescomplemented
11#19128052C2213yescomplemented
12#19128051C2213yescomplemented
13#19128050—26no—
14#19128049C2213yes—
15#19128048C2213yes—
16#19128047C2213yes—
17#19128046—26no—
18#103×D83—26no—
19#102×D83—26no—
20C4×C8×D83C2213yes—
21C8×C83⋊C8C2213yescomplemented
22#487×C83C2213yescomplemented
23#486×C83C2213yes—
24C82×C83C2211yescharacteristic, complemented
25C2×Q8×D83—23no—
26C2×D4×D83—23no—
27C2×C8×D83D416yes—
28#14581792C2313yes—
29#14581791—23no—
30#14581791—23no—
31#14581790—23no—
32#14581789C2311yescharacteristic
33#14581788—23no—
34C4⋊C4×D83—23no—
35C4⋊C4×D83—23no—
36C42×D83C2311yescharacteristic
37#14581717—26no—
38#14581716—26no—
39#14581715—26no—
40#14581714—26no—
41#14581713—26no—
42#14581712—26no—
43C4×C83⋊C8C2313yes—
44C8×Dic83D416yes—
45C4⋊Q8×C83C2313yes—
46#127×C83C2311yescharacteristic
47#103×C83—26no—
48#102×C83—26no—
49C4×C8×C83C2313yes—
50C23×D83—41no—
51C2×Dic166—23no—
52C2×Dic166—43no—
53Q8×D83—412no—
54C2×D332—23no—
55C2×C22⋊D83—43no—
56D4×D83—412no—
57C2×C4×D83—43no—
58C2×C4×D83C2×D413yes—
59C2×C83⋊C8C2×D416yes—
60C8×D83—212no—
61C4⋊Dic83—23no—
62C4⋊Dic83—23no—
63Dic83⋊C4—43no—
64Dic83⋊C4—43no—
65C4×Dic83C2×D413yes—
66C2×Q8×C83—23no—
67C2×D4×C83—23no—
68C2×C8×C83C2×D416yes—
69C4⋊C4×C83—23no—
70C4⋊C4×C83—23no—
71C42×C83C2411yescharacteristic
72C23×C83—41no—
73C22×D83—41no—
74C22×D83—86no—
75C22×D83#12711yescharacteristic
76Q8×C83—46no—
77D4×C83—46no—
78C2×C4×C83—43no—
79C2×C4×C83C22×D413yes—
80C22⋊D83—86no—
81C2×Dic83—43no—
82C2×Dic83#12713yes—
83D332—46no—
84C4×D83—86no—
85C4×D83#135112yes—
86Dic166—46no—
87Dic166—86no—
88C8×C83—26no—
89C83⋊C8—26no—
90C22×C83—83no—
91C22×C83C2×#12711yescharacteristic
92C2×D83—83no—
93C2×D83—164no—
94C2×D83#49516yes—
95C4×C83—83no—
96C4×C83C2×#13516yes—
97Dic83—83no—
98Dic83#49516yes—
99C2×#1711—831nomaximal, Sylow
100C2×C83—161no—
101C2×C83C2×#49513yes—
102D83—161no—
103D83#171114yesdirect product factor
104C2×#487—833no—
105C2×#486—833no—
106#1711—833no—
107#1711—833no—
108#1711—831no—
109#1711D8311yescharacteristic, direct product factor
110C2×C82—831no—
111C83C2×#171111yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—833no—
113C2×#127—831no—
114#487—833no—
115#487—833no—
116#487—833no—
117#487—1666no—
118#487C2×D8313yescomplemented
119#486—833no—
120#486—833no—
121#486—833no—
122#486—1666no—
123#486C2×D8313yes—
124C2×#103—1666no—
125C2×#102—1666no—
126C2×C4×C8—833no—
127C82—833no—
128C82C2×D8311yescharacteristic, complemented
129C22×Q8—1663no—
130C22×D4—1663no—
131C22×C8—836no—
132C4⋊Q8—833no—
133C4⋊Q8—1663no—
134C4⋊Q8—1663no—
135C4⋊Q8—1663no—
136C4⋊Q8C22×D8313yes—
137#127—831no—
138#127—1663no—
139#127C22×D8311yescharacteristic
140C2×C4⋊C4—1663no—
141C2×C4⋊C4—1663no—
142C2×C42—831no—
143#103—26no—
144#103—1666no—
145#103—1666no—
146#103—1666no—
147#102—26no—
148#102—1666no—
149#102—1666no—
150#102—1666no—
151C4×C8—836no—
152C4×C8—833no—
153C4×C8C22×D8313yes—
154C24—3321no—
155C2×Q8—23no—
156C2×Q8—3323no—
157C2×Q8—33212no—
158C2×Q8—1663no—
159C2×D4—23no—
160C2×D4—33212no—
161C2×D4—3323no—
162C2×D4—1663no—
163C22×C4—3323no—
164C22×C4—833no—
165C2×C8—836no—
166C2×C8—16612no—
167C2×C8D4×D8316yes—
168C4⋊C4—23no—
169C4⋊C4—23no—
170C4⋊C4—3323no—
171C4⋊C4—3323no—
172C4⋊C4—1663no—
173C4⋊C4—1663no—
174C42—833no—
175C42C23×D8311yescharacteristic
176C23—41no—
177C23—3321no—
178C23—831no—
179C23—6646no—
180Q8—46no—
181Q8—3326no—
182Q8—6646no—
183D4—46no—
184D4—3326no—
185D4—6646no—
186C2×C4—43no—
187C2×C4—3323no—
188C2×C4—833no—
189C2×C4—8312no—
190C2×C4—6646no—
191C2×C4C2×D4×D8313yes—
192C8—26no—
193C8—1666no—
194C22—83no—
195C22—13284no—
196C22—836no—
197C22—6643no—
198C22#127×D8311yescharacteristic
199C4—83no—
200C4—836no—
201C4—6643no—
202C4#135×D8316yes—
203C2—161no—
204C2—13281no—
205C2—834no—
206C2#495×D8313yes—
subgroupG/H
centerC22#127×D83
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC83×#1711C2
socleC22×C83C2×#127
prime radical#127×D83C22
hypercenter#1711D83
hypocenterC83C2×#1711
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #487×D83C2semidirect
3. #24907076C2semidirect
4. #24907075C2semidirect
5. #24907074C2semidirect
6. C82×D83C2semidirect
7. C83×#1711C2semidirect
10. #19128053C22semidirect
11. #19128052C22semidirect
12. #19128051C22semidirect
21. C8×C83⋊C8C22semidirect
22. #487×C83C22semidirect
24. C82×C83C22semidirect
103. D83#1711direct (N × H)
109. #1711D83direct (N × H)
111. C83C2×#1711semidirect
118. #487C2×D83semidirect
128. C82C2×D83semidirect

Maximal subgroups

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

subgroupindex
#487×D832
#486×D832
#249070762
#249070752
#249070742
C82×D832
C83×#17112
C2×#171183