Finite Group Databasebeta

← #1711×D79 #34275943

Subgroups — #1711×D79 #34275943

Order 20224.

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×D79C213yesmaximal, complemented
2#486×D79C213yesmaximal
3#23763013C213yesmaximal, complemented
4#23763012C211yesmaximal, characteristic, complemented
5#23763011C213yesmaximal, complemented
6C82×D79C211yesmaximal, characteristic, complemented
7C79×#1711C211yesmaximal, characteristic, complemented
8C4⋊Q8×D79C2213yes—
9#127×D79C2211yescharacteristic
10#19098915C2213yescomplemented
11#19098914C2213yescomplemented
12#19098913C2213yescomplemented
13#19098912—26no—
14#19098911C2213yes—
15#19098910C2213yes—
16#19098909C2213yes—
17#19098908—26no—
18#103×D79—26no—
19#102×D79—26no—
20C4×C8×D79C2213yes—
21C8×C79⋊C8C2213yescomplemented
22#487×C79C2213yescomplemented
23#486×C79C2213yes—
24C82×C79C2211yescharacteristic, complemented
25C2×Q8×D79—23no—
26C2×D4×D79—23no—
27C2×C8×D79D416yes—
28#14579161C2313yes—
29#14579160—23no—
30#14579160—23no—
31#14579159—23no—
32#14579158C2311yescharacteristic
33#14579157—23no—
34C4⋊C4×D79—23no—
35C4⋊C4×D79—23no—
36C42×D79C2311yescharacteristic
37#14579086—26no—
38#14579085—26no—
39#14579084—26no—
40#14579083—26no—
41#14579082—26no—
42#14579081—26no—
43C4×C79⋊C8C2313yes—
44C8×Dic79D416yes—
45C4⋊Q8×C79C2313yes—
46#127×C79C2311yescharacteristic
47#103×C79—26no—
48#102×C79—26no—
49C4×C8×C79C2313yes—
50C23×D79—41no—
51C2×Dic158—23no—
52C2×Dic158—43no—
53Q8×D79—412no—
54C2×D316—23no—
55C2×C22⋊D79—43no—
56D4×D79—412no—
57C2×C4×D79—43no—
58C2×C4×D79C2×D413yes—
59C2×C79⋊C8C2×D416yes—
60C8×D79—212no—
61C4⋊Dic79—23no—
62C4⋊Dic79—23no—
63Dic79⋊C4—43no—
64Dic79⋊C4—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—86no—
85C4×D79#135112yes—
86Dic158—46no—
87Dic158—86no—
88C8×C79—26no—
89C79⋊C8—26no—
90C22×C79—83no—
91C22×C79C2×#12711yescharacteristic
92C2×D79—83no—
93C2×D79—164no—
94C2×D79#49516yes—
95C4×C79—83no—
96C4×C79C2×#13516yes—
97Dic79—83no—
98Dic79#49516yes—
99C2×#1711—791nomaximal, Sylow
100C2×C79—161no—
101C2×C79C2×#49513yes—
102D79—161no—
103D79#171114yesdirect product factor
104C2×#487—793no—
105C2×#486—793no—
106#1711—793no—
107#1711—793no—
108#1711—791no—
109#1711D7911yescharacteristic, direct product factor
110C2×C82—791no—
111C79C2×#171111yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—793no—
113C2×#127—791no—
114#487—793no—
115#487—793no—
116#487—793no—
117#487—1586no—
118#487C2×D7913yescomplemented
119#486—793no—
120#486—793no—
121#486—793no—
122#486—1586no—
123#486C2×D7913yes—
124C2×#103—1586no—
125C2×#102—1586no—
126C2×C4×C8—793no—
127C82—793no—
128C82C2×D7911yescharacteristic, complemented
129C22×Q8—1583no—
130C22×D4—1583no—
131C22×C8—796no—
132C4⋊Q8—793no—
133C4⋊Q8—1583no—
134C4⋊Q8—1583no—
135C4⋊Q8—1583no—
136C4⋊Q8C22×D7913yes—
137#127—791no—
138#127—1583no—
139#127C22×D7911yescharacteristic
140C2×C4⋊C4—1583no—
141C2×C4⋊C4—1583no—
142C2×C42—791no—
143#103—26no—
144#103—1586no—
145#103—1586no—
146#103—1586no—
147#102—26no—
148#102—1586no—
149#102—1586no—
150#102—1586no—
151C4×C8—796no—
152C4×C8—793no—
153C4×C8C22×D7913yes—
154C24—3161no—
155C2×Q8—23no—
156C2×Q8—3163no—
157C2×Q8—31612no—
158C2×Q8—1583no—
159C2×D4—23no—
160C2×D4—31612no—
161C2×D4—3163no—
162C2×D4—1583no—
163C22×C4—3163no—
164C22×C4—793no—
165C2×C8—796no—
166C2×C8—15812no—
167C2×C8D4×D7916yes—
168C4⋊C4—23no—
169C4⋊C4—23no—
170C4⋊C4—3163no—
171C4⋊C4—3163no—
172C4⋊C4—1583no—
173C4⋊C4—1583no—
174C42—793no—
175C42C23×D7911yescharacteristic
176C23—41no—
177C23—3161no—
178C23—791no—
179C23—6326no—
180Q8—46no—
181Q8—3166no—
182Q8—6326no—
183D4—46no—
184D4—3166no—
185D4—6326no—
186C2×C4—43no—
187C2×C4—3163no—
188C2×C4—793no—
189C2×C4—7912no—
190C2×C4—6326no—
191C2×C4C2×D4×D7913yes—
192C8—26no—
193C8—1586no—
194C22—83no—
195C22—796no—
196C22—6323no—
197C22—12644no—
198C22#127×D7911yescharacteristic
199C4—83no—
200C4—796no—
201C4—6323no—
202C4#135×D7916yes—
203C2—161no—
204C2—794no—
205C2—12641no—
206C2#495×D7913yes—
subgroupG/H
centerC22#127×D79
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#1711C2
socleC22×C79C2×#127
prime radical#127×D79C22
hypercenter#1711D79
hypocenterC79C2×#1711
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. #487×D79C2semidirect
3. #23763013C2semidirect
4. #23763012C2semidirect
5. #23763011C2semidirect
6. C82×D79C2semidirect
7. C79×#1711C2semidirect
10. #19098915C22semidirect
11. #19098914C22semidirect
12. #19098913C22semidirect
21. C8×C79⋊C8C22semidirect
22. #487×C79C22semidirect
24. C82×C79C22semidirect
103. D79#1711direct (N × H)
109. #1711D79direct (N × H)
111. C79C2×#1711semidirect
118. #487C2×D79semidirect
128. C82C2×D79semidirect

Maximal subgroups

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

subgroupindex
#487×D792
#486×D792
#237630132
#237630122
#237630112
C82×D792
C79×#17112
C2×#171179