Finite Group Databasebeta

← #34198816

Subgroups — #34198816

Order 20224.

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#23765216C211yesmaximal, characteristic
2#23763155C211yesmaximal, characteristic
3#23761394C212yesmaximal
4#23761388C212yesmaximal
5C79×#3177C211yesmaximal, characteristic
6#19098518—21no—
7#19098384—21no—
8#19098345C2212yes—
9#19098341—21no—
10C79⋊#338C2211yescharacteristic
11#19098320C2211yescharacteristic
12#19098319—21no—
13#541×C79—21no—
14#499×C79C411yescharacteristic, complemented
15#494×C79C411yescharacteristic, complemented
16#491×C79—21no—
17#470×C79C2212yes—
18#467×C79—21no—
19#441×C79C412yescomplemented
20C4×Q16×C79—21no—
21#405×C79C2211yescharacteristic
22C2×#11929608—21no—
23#14579120—21no—
24#14579101—22no—
25#14579071—22no—
26C8×Dic79—22no—
27C2×Q16×C79—21no—
28C2×Q16×C79C2×C412yes—
29C2×M16×C79C2311yescharacteristic
30C22×C8×C79D411yescharacteristic
31C4⋊Q8×C79—21no—
32#126×C79—42no—
33#125×C79—21no—
34#125×C79—42no—
35C22⋊Q8×C79—22no—
36C4×Q8×C79—22no—
37#117×C79D411yescharacteristic
38#107×C79—21no—
39#106×C79C2×C412yes—
40#103×C79—22no—
41#103×C79—22no—
42#97×C79C2×C411yescharacteristic
43#97×C79D411yescharacteristic
44C4×C8×C79C2×C411yescharacteristic
45C4×C8×C79D411yescharacteristic
46#11929608—41no—
47#11929608—41no—
48C2×C79⋊C8—22no—
49C4⋊Dic79—41no—
50C22⋊Dic79—41no—
51C4×Dic79—22no—
52C79⋊C16—42no—
53C2×Q8×C79—22no—
54C22×C4×C79C2×D411yescharacteristic
55Q16×C79—41no—
56Q16×C79—41no—
57Q16×C79—42no—
58M16×C79—41no—
59C2×C8×C79—21no—
60C2×C8×C79—21no—
61C2×C8×C79C22⋊C411yescharacteristic
62C2×C8×C79C22⋊C411yescharacteristic
63C2×C8×C79C22×C411yescharacteristic
64C2×C8×C79C2×D411yescharacteristic
65C4⋊C4×C79—21no—
66C4⋊C4×C79—22no—
67C4⋊C4×C79—42no—
68C4⋊C4×C79—42no—
69C4⋊C4×C79—42no—
70C4⋊C4×C79—42no—
71C22⋊C4×C79—21no—
72C22⋊C4×C79—42no—
73C42×C79—42no—
74C42×C79C22⋊C411yescharacteristic
75C42×C79C22⋊C411yescharacteristic
76C23×C79AGL(1,ℤ/8)11yescharacteristic
77Q8×C79—42no—
78Q8×C79—42no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79—21no—
83C2×C4×C79—42no—
84C2×C4×C79—42no—
85C2×C4×C79C2×C22⋊C411yescharacteristic
86C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
87C2×Dic79—42no—
88C8×C79—21no—
89C8×C79—21no—
90C8×C79—21no—
91C8×C79#10411yescharacteristic
92C8×C79#10411yescharacteristic
93C79⋊C8—42no—
94C22×C79—21no—
95C22×C79—41no—
96C22×C79#41811yescharacteristic
97C4×C79—21no—
98C4×C79—41no—
99C4×C79—41no—
100C4×C79—82no—
101C4×C79—82no—
102C4×C79#41011yescharacteristic
103C4×C79C2×#10411yescharacteristic
104Dic79—82no—
105#11771—791nomaximal, Sylow
106C2×C79—21no—
107C2×C79—41no—
108C2×C79#150511yescharacteristic
109#3177D7911yescharacteristic
110#2166—791no—
111#1747—791no—
112#1342—792no—
113#1340—792no—
114C79#1177111yescharacteristic, complemented, Sylow
115#541—21no—
116#499Dic7911yescharacteristic, complemented
117#494Dic7911yescharacteristic, complemented
118#491—21no—
119#470C2×D7912yes—
120#467—21no—
121#441Dic7912yescomplemented
122C4×Q16—21no—
123#405—1581no—
124#405C2×D7911yescharacteristic
125#364—1581no—
126#349—792no—
127#347—1581no—
128#338—791no—
129H(3,ℤ/4)—791no—
130H(3,ℤ/4)—1581no—
131C2×Q16—21no—
132C2×Q16C2×Dic7912yes—
133C2×M16—1581no—
134C2×M16C22×D7911yescharacteristic
135C22×C8C22⋊D7911yescharacteristic
136C4⋊Q8—21no—
137#126—42no—
138#125—21no—
139#125—42no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142#117—1581no—
143#117C22⋊D7911yescharacteristic
144M32—1582no—
145#107—21no—
146#106C2×Dic7912yes—
147#103—22no—
148#103—22no—
149#97—1582no—
150#97C2×Dic7911yescharacteristic
151#97C22⋊D7911yescharacteristic
152C4×C8—1582no—
153C4×C8C2×Dic7911yescharacteristic
154C4×C8C22⋊D7911yescharacteristic
155C2×Q8—22no—
156C22×C4C2×C22⋊D7911yescharacteristic
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—41no—
161M16—3161no—
162M16—3161no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1582no—
166C2×C8C22⋊Dic7911yescharacteristic
167C2×C8C22⋊Dic7911yescharacteristic
168C2×C8C22×Dic7911yescharacteristic
169C2×C8C2×C22⋊D7911yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—3161no—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C22⋊C4—3161no—
180C42—42no—
181C42—1582no—
182C42C22⋊Dic7911yescharacteristic
183C42C22⋊Dic7911yescharacteristic
184C16—3162no—
185C23#1457918711yescharacteristic
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3162no—
195C2×C4C2×C22⋊Dic7911yescharacteristic
196C2×C4#1457918711yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—3162no—
201C8#1457908811yescharacteristic
202C8#1457908811yescharacteristic
203C22—21no—
204C22—41no—
205C22#1909856811yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—6322no—
212C4#1909853711yescharacteristic
213C4C2×#1457908811yescharacteristic
214C2—21no—
215C2—41no—
216C2#2376190811yescharacteristic
subgroupG/H
centerC2#23761908
commutatorC2×C8×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#3177C2
socleC2×C79#1505
prime radicalC23×C79AGL(1,ℤ/8)
cosocleC2×M16×C79C23
hypercenter#3177D79
hypocenterC79#11771
LCSG ▷ C2×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
14. #499×C79C4semidirect
15. #494×C79C4semidirect
19. #441×C79C4semidirect
114. C79#11771semidirect
116. #499Dic79semidirect
117. #494Dic79semidirect
121. #441Dic79semidirect

Maximal subgroups

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

subgroupindex
#237652162
#237631552
#237613942
#237613882
C79×#31772
#1177179