Finite Group Databasebeta

← #34277595

Subgroups — #34277595

Order 20224.

736 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 151 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#23774144C213yesmaximal, complemented
2#23773771C213yesmaximal
3#23763061C213yesmaximal, complemented
4#23763057C213yesmaximal, complemented
5#23763012C211yesmaximal, characteristic, complemented
6#23761702C211yesmaximal, characteristic, complemented
7C79×#1711C211yesmaximal, characteristic, complemented
8#19099118C2213yes—
9#127×D79C2211yescharacteristic
10#19098959C2213yescomplemented
11#19098942C2213yescomplemented
12#19098926—26no—
13#19098914C2213yescomplemented
14#19098911C2213yes—
15#19098910C2213yes—
16#19098909C2213yes—
17#19098820—26no—
18#19098667—26no—
19#19098566—26no—
20C4×#11929607C2213yes—
21#19098284C2213yescomplemented
22#487×C79C2213yescomplemented
23#486×C79C2213yes—
24C82×C79C2211yescharacteristic, complemented
25C2×D4×D79—23no—
26C2×#11929614—26no—
27C2×#11929607D416yes—
28#14579161C2313yes—
29#14579158C2311yescharacteristic
30#14579157—23no—
31#14579152—23no—
32#14579141—23no—
33#14579137—23no—
34Q8×Dic79—23no—
35#14579126—23no—
36#14579119—23no—
37C42×D79C2311yescharacteristic
38#14579092—26no—
39#14579086—26no—
40#14579083—26no—
41#14579082—26no—
42#14579081—26no—
43#14579071D416yes—
44C4×C79⋊C8C2313yes—
45C4⋊Q8×C79C2313yes—
46#127×C79C2311yescharacteristic
47#103×C79—26no—
48#102×C79—26no—
49C4×C8×C79C2313yes—
50C23×D79—41no—
51C2×Dic158—23no—
52C2×D316—23no—
53C2×C22⋊D79—43no—
54D4×D79—412no—
55C2×C4×D79C2×D413yes—
56#11929614—412no—
57#11929607—46no—
58C2×C79⋊C8C2×D416yes—
59C4⋊Dic79—23no—
60C4⋊Dic79—23no—
61C4⋊Dic79—43no—
62Dic79⋊C4—43no—
63(C2×D79)⋊C4—43no—
64C4×Dic79—43no—
65C4×Dic79C2×D413yes—
66C2×Q8×C79—23no—
67C2×D4×C79—23no—
68C2×C8×C79C2×D416yes—
69C4⋊C4×C79—23no—
70C4⋊C4×C79—23no—
71C42×C79C2411yescharacteristic
72C23×C79—41no—
73C22×D79—41no—
74C22×D79—86no—
75C22×D79#12711yescharacteristic
76Q8×C79—46no—
77D4×C79—46no—
78C2×C4×C79—43no—
79C2×C4×C79C22×D413yes—
80C22⋊D79—86no—
81C2×Dic79—43no—
82C2×Dic79#12713yes—
83D316—46no—
84C4×D79—26no—
85Dic158—46no—
86C8×C79—26no—
87C79⋊C8—26no—
88C22×C79—83no—
89C22×C79C2×#12711yescharacteristic
90C2×D79—23no—
91C2×D79—83no—
92C2×D79—164no—
93C4×C79—83no—
94C4×C79#57516yes—
95Dic79—23no—
96Dic79—83no—
97#19753—791nomaximal, Sylow
98C2×C79—161no—
99C2×C79#315313yes—
100D79—41no—
101D79—161no—
102#3185—793no—
103#3138—793no—
104#1720—793no—
105#1720—793no—
106#1711—791no—
107#1711D7911yescharacteristic, complemented
108#1455—791no—
109C79#1975311yescharacteristic, complemented, Sylow
110#533—793no—
111C2×#127—791no—
112#498—793no—
113#494—793no—
114#490—1586no—
115#487—793no—
116#487C2×D7913yescomplemented
117#486—793no—
118#486—793no—
119#486—793no—
120#486C2×D7913yes—
121#470—1586no—
122#441—1586no—
123#418—1586no—
124C4×M16—793no—
125#322—793no—
126C82C2×D7911yescharacteristic, complemented
127C22×D4—1583no—
128C2×Q16—1586no—
129C2×M16—796no—
130C4⋊Q8—793no—
131C4⋊Q8C22×D7913yes—
132#127—791no—
133#127—1583no—
134#127C22×D7911yescharacteristic
135#125—1583no—
136C22⋊Q8—1583no—
137C22⋊Q8—1583no—
138C4×Q8—1583no—
139C4×Q8—1583no—
140#117—1583no—
141C2×C42—791no—
142#106—1586no—
143#103—26no—
144#103—1586no—
145#102—26no—
146#102—1586no—
147#102—1586no—
148#102—1586no—
149#97—796no—
150C4×C8—793no—
151C4×C8C22×D7913yes—
152C24—3161no—
153C2×Q8—23no—
154C2×Q8—1583no—
155C2×D4—23no—
156C2×D4—3163no—
157C2×D4—31612no—
158C2×D4—1583no—
159C22×C4—793no—
160Q16—31612no—
161M16—3166no—
162C2×C8—796no—
163C2×C8D4×D7916yes—
164C4⋊C4—23no—
165C4⋊C4—23no—
166C4⋊C4—3163no—
167C4⋊C4—3163no—
168C4⋊C4—1583no—
169C4⋊C4—1583no—
170C22⋊C4—3163no—
171C42—3163no—
172C42—793no—
173C42C23×D7911yescharacteristic
174C23—41no—
175C23—3161no—
176C23—791no—
177C23—6326no—
178Q8—46no—
179Q8—3166no—
180D4—46no—
181D4—3166no—
182D4—6326no—
183C2×C4—43no—
184C2×C4—3163no—
185C2×C4—793no—
186C2×C4—1586no—
187C2×C4C2×D4×D7913yes—
188C8—26no—
189C8—1586no—
190C22—83no—
191C22—6323no—
192C22—1583no—
193C22—12644no—
194C22#127×D7911yescharacteristic
195C4—83no—
196C4—6323no—
197C4—1583no—
198C4#1909936016yes—
199C2—161no—
200C2—3161no—
201C2—12641no—
202C2#2377386113yes—
subgroupG/H
centerC22#127×D79
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#1711C2
socleC22×C79C2×#127
prime radical#127×D79C22
hypercenter#1711D79
hypocenterC79#19753
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ #1711

Complements

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

NH ≅ G/Ntype
1. #23774144C2semidirect
3. #23763061C2semidirect
4. #23763057C2semidirect
5. #23763012C2semidirect
6. #23761702C2semidirect
7. C79×#1711C2semidirect
10. #19098959C22semidirect
11. #19098942C22semidirect
13. #19098914C22semidirect
21. #19098284C22semidirect
22. #487×C79C22semidirect
24. C82×C79C22semidirect
107. #1711D79semidirect
109. C79#19753semidirect
116. #487C2×D79semidirect
126. C82C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237741442
#237737712
#237630612
#237630572
#237630122
#237617022
C79×#17112
#1975379