Finite Group Databasebeta

← #35314951

Subgroups — #35314951

Order 21248.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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#24909279C211yesmaximal, characteristic
2#24907218C211yesmaximal, characteristic
3#24905457C212yesmaximal
4#24905451C212yesmaximal
5C83×#3177C211yesmaximal, characteristic
6#19127656—21no—
7#19127522—21no—
8#19127483C2212yes—
9#19127479—21no—
10C83⋊#338C2211yescharacteristic
11#19127458C2211yescharacteristic
12#19127457—21no—
13#541×C83—21no—
14#499×C83C411yescharacteristic, complemented
15#494×C83C411yescharacteristic, complemented
16#491×C83—21no—
17#470×C83C2212yes—
18#467×C83—21no—
19#441×C83C412yescomplemented
20C4×Q16×C83—21no—
21#405×C83C2211yescharacteristic
22C2×#13050574—21no—
23#14581751—21no—
24#14581732—22no—
25#14581702—22no—
26C8×Dic83—22no—
27C2×Q16×C83—21no—
28C2×Q16×C83C2×C412yes—
29C2×M16×C83C2311yescharacteristic
30C22×C8×C83D411yescharacteristic
31C4⋊Q8×C83—21no—
32#126×C83—42no—
33#125×C83—21no—
34#125×C83—42no—
35C22⋊Q8×C83—22no—
36C4×Q8×C83—22no—
37#117×C83D411yescharacteristic
38#107×C83—21no—
39#106×C83C2×C412yes—
40#103×C83—22no—
41#103×C83—22no—
42#97×C83C2×C411yescharacteristic
43#97×C83D411yescharacteristic
44C4×C8×C83C2×C411yescharacteristic
45C4×C8×C83D411yescharacteristic
46#13050574—41no—
47#13050574—41no—
48C2×C83⋊C8—22no—
49C4⋊Dic83—41no—
50C22⋊Dic83—41no—
51C4×Dic83—22no—
52C83⋊C16—42no—
53C2×Q8×C83—22no—
54C22×C4×C83C2×D411yescharacteristic
55Q16×C83—42no—
56Q16×C83—41no—
57Q16×C83—41no—
58M16×C83—41no—
59C2×C8×C83—21no—
60C2×C8×C83—21no—
61C2×C8×C83C22⋊C411yescharacteristic
62C2×C8×C83C22⋊C411yescharacteristic
63C2×C8×C83C22×C411yescharacteristic
64C2×C8×C83C2×D411yescharacteristic
65C4⋊C4×C83—21no—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—42no—
68C4⋊C4×C83—42no—
69C4⋊C4×C83—42no—
70C4⋊C4×C83—42no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—42no—
73C42×C83—42no—
74C42×C83C22⋊C411yescharacteristic
75C42×C83C22⋊C411yescharacteristic
76C23×C83AGL(1,ℤ/8)11yescharacteristic
77Q8×C83—42no—
78Q8×C83—42no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—42no—
84C2×C4×C83—42no—
85C2×C4×C83C2×C22⋊C411yescharacteristic
86C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
87C2×Dic83—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—82no—
101C4×C83—82no—
102C4×C83#41011yescharacteristic
103C4×C83C2×#10411yescharacteristic
104Dic83—82no—
105#11771—831nomaximal, Sylow
106C2×C83—21no—
107C2×C83—41no—
108C2×C83#150511yescharacteristic
109#3177D8311yescharacteristic
110#2166—831no—
111#1747—831no—
112#1342—832no—
113#1340—832no—
114C83#1177111yescharacteristic, complemented, Sylow
115#541—21no—
116#499Dic8311yescharacteristic, complemented
117#494Dic8311yescharacteristic, complemented
118#491—21no—
119#470C2×D8312yes—
120#467—21no—
121#441Dic8312yescomplemented
122C4×Q16—21no—
123#405—1661no—
124#405C2×D8311yescharacteristic
125#364—1661no—
126#349—832no—
127#347—1661no—
128#338—831no—
129H(3,ℤ/4)—831no—
130H(3,ℤ/4)—1661no—
131C2×Q16—21no—
132C2×Q16C2×Dic8312yes—
133C2×M16—1661no—
134C2×M16C22×D8311yescharacteristic
135C22×C8C22⋊D8311yescharacteristic
136C4⋊Q8—21no—
137#126—42no—
138#125—21no—
139#125—42no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142#117—1661no—
143#117C22⋊D8311yescharacteristic
144M32—1662no—
145#107—21no—
146#106C2×Dic8312yes—
147#103—22no—
148#103—22no—
149#97—1662no—
150#97C2×Dic8311yescharacteristic
151#97C22⋊D8311yescharacteristic
152C4×C8—1662no—
153C4×C8C2×Dic8311yescharacteristic
154C4×C8C22⋊D8311yescharacteristic
155C2×Q8—22no—
156C22×C4C2×C22⋊D8311yescharacteristic
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—41no—
161M16—3321no—
162M16—3321no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1662no—
166C2×C8C22⋊Dic8311yescharacteristic
167C2×C8C22⋊Dic8311yescharacteristic
168C2×C8C22×Dic8311yescharacteristic
169C2×C8C2×C22⋊D8311yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—3321no—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C22⋊C4—3321no—
180C42—42no—
181C42—1662no—
182C42C22⋊Dic8311yescharacteristic
183C42C22⋊Dic8311yescharacteristic
184C16—3322no—
185C23#1458181811yescharacteristic
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3322no—
195C2×C4C2×C22⋊Dic8311yescharacteristic
196C2×C4#1458181811yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—3322no—
201C8#1458171911yescharacteristic
202C8#1458171911yescharacteristic
203C22—21no—
204C22—41no—
205C22#1912770611yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—6642no—
212C4#1912767511yescharacteristic
213C4C2×#1458171911yescharacteristic
214C2—21no—
215C2—41no—
216C2#2490597111yescharacteristic
subgroupG/H
centerC2#24905971
commutatorC2×C8×C83C22×C4
FrattiniC2×M16C22×D83
FittingC83×#3177C2
socleC2×C83#1505
prime radicalC23×C83AGL(1,ℤ/8)
cosocleC2×M16×C83C23
hypercenter#3177D83
hypocenterC83#11771
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
14. #499×C83C4semidirect
15. #494×C83C4semidirect
19. #441×C83C4semidirect
114. C83#11771semidirect
116. #499Dic83semidirect
117. #494Dic83semidirect
121. #441Dic83semidirect

Maximal subgroups

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

subgroupindex
#249092792
#249072182
#249054572
#249054512
C83×#31772
#1177183