Finite Group Databasebeta

← D51×#64425 #44472856

Subgroups — D51×#64425 #44472856

Order 29988.

216 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 37 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
1D51×#3862C211yesmaximal, characteristic, complemented
2C3⋊(C17⋊#64425)C211yesmaximal, characteristic, complemented
3C3×C17×#64425C211yesmaximal, characteristic, complemented
4C7⋊D7×D51C311yesmaximal, characteristic, complemented
5D17×#64425—31nomaximal
6C3×C17×#3862C2211yescharacteristic, complemented, Hall
7C3⋊(C7⋊D119)C2×C311yescharacteristic, complemented
8C72×D51C2×C311yescharacteristic, complemented
9C3×C17×C7⋊D7C2×C311yescharacteristic, complemented
10D17×#3862—31no—
11C17⋊#64425—31no—
12C17×#64425—21no—
13C17×#64425D311yescharacteristic, complemented
14AGL(1,𝔽7)×D51—71nomaximal
15AGL(1,𝔽7)×D51—71nomaximal
16D17×C7⋊D7—31noHall
17C17×#3862—21no—
18C17×#3862C2×D311yescharacteristic, complemented
19C3×C72×C17C22×C311yescharacteristic, complemented
20ΓL(1,𝔽8)×D51—141no—
21ΓL(1,𝔽8)×D51—141no—
22C3⋊(C17⋊AGL(1,𝔽7))—71no—
23C3⋊(C17⋊AGL(1,𝔽7))—71no—
24C3×C17×AGL(1,𝔽7)—71no—
25C3×C17×AGL(1,𝔽7)—71no—
26D3×#64425—171nomaximal, Hall
27C7⋊D119—31no—
28C17×C7⋊D7C3×D311yescharacteristic, complemented
29C72×D17—31no—
30D7×D51—71no—
31D7×D51—71no—
32D7×D51—212no—
33D17×AGL(1,𝔽7)—211no—
34D17×AGL(1,𝔽7)—211no—
35C3×C17×ΓL(1,𝔽8)—141no—
36C3×C17×ΓL(1,𝔽8)—141no—
37C3⋊#64425—171no—
38D3×#3862—171no—
39C3×#64425D1711yescharacteristic, complemented
40C72×C17C2×C3×D311yescharacteristic, complemented, Hall
41D357—71no—
42D357—71no—
43D357—212no—
44C7×D51—32no—
45C7×D51AGL(1,𝔽7)11yescharacteristic, complemented
46C7×D51AGL(1,𝔽7)11yescharacteristic, complemented
47C3×D7×C17—71no—
48C3×D7×C17—71no—
49C3×D7×C17—212no—
50C17⋊AGL(1,𝔽7)—211no—
51C17⋊AGL(1,𝔽7)—211no—
52C17×AGL(1,𝔽7)—71no—
53C17×AGL(1,𝔽7)—71no—
54C17×AGL(1,𝔽7)—141no—
55C17×AGL(1,𝔽7)—141no—
56ΓL(1,𝔽8)×D17—421no—
57ΓL(1,𝔽8)×D17—421no—
58C2×C3×D51—491noHall
59D3×C7⋊D7—171no—
60C2×#64425—511no—
61D7×D17—211no—
62D7×D17—211no—
63D7×D17—632no—
64C3×#3862C2×D1711yescharacteristic, complemented, Hall
65C3×C7×C17—32no—
66C3×C7×C17C2×AGL(1,𝔽7)11yescharacteristic, complemented
67C3×C7×C17C2×AGL(1,𝔽7)11yescharacteristic, complemented
68C17×ΓL(1,𝔽8)—141no—
69C17×ΓL(1,𝔽8)—141no—
70C17×ΓL(1,𝔽8)—281no—
71C17×ΓL(1,𝔽8)—281no—
72C2×C32×C17—491no—
73C3×D51—491no—
74C3×D51—981no—
75C3⋊(C7⋊D7)—171no—
76D3×C72—171no—
77C3×C7⋊D7C3×D1711yescharacteristic, complemented
78C2×#3862—511no—
79#64425—21no—
80#64425—511no—
81#64425D5111yescharacteristic, direct product factor
82D3×AGL(1,𝔽7)—1191no—
83D3×AGL(1,𝔽7)—1191no—
84D119—211no—
85D119—211no—
86D119—632no—
87C7×D17—31no—
88C7×D17—31no—
89C7×D17—92no—
90D7×C17—71no—
91D7×C17—71no—
92D7×C17—212no—
93C2×D51—491no—
94C2×C3×D17—1471no—
95C2×C7⋊D7—511noHall
96C32×C17—981noHall
97C3×C72C2×C3×D1711yescharacteristic, complemented
98#3862—21no—
99#3862C2×D5111yescharacteristic, complemented
100C3⋊AGL(1,𝔽7)—1191no—
101C3⋊AGL(1,𝔽7)—1191no—
102D3×ΓL(1,𝔽8)—1191no—
103D3×ΓL(1,𝔽8)—2381no—
104C3×AGL(1,𝔽7)—71no—
105C3×AGL(1,𝔽7)—71no—
106C7×C17—32no—
107C7×C17D3×AGL(1,𝔽7)11yescharacteristic, complemented
108C7×C17D3×AGL(1,𝔽7)11yescharacteristic, complemented
109C2×C3×C17—491no—
110C2×C3×C17—491no—
111C2×C3×C17—981no—
112D51—491no—
113D51#6442511yescharacteristic, direct product factor
114C3×D17—2941no—
115C3×D17—1471no—
116C2×C72—511no—
117C7⋊D7—511no—
118C7⋊D7C3×D5111yescharacteristic, complemented
119D3×D7—3572no—
120D3×D7—1191no—
121D3×D7—1191no—
122C2×AGL(1,𝔽7)—3571no—
123C2×AGL(1,𝔽7)—3571no—
124C2×D17—1471noHall
125C3×ΓL(1,𝔽8)—71no—
126C3×ΓL(1,𝔽8)—141no—
127C3×C17—981no—
128C3×C17—1961no—
129C3×C17C2×#6442511yescharacteristic, complemented
130C72C2×C3×D5111yescharacteristic, complemented, Sylow
131D21—3572no—
132D21—1191no—
133D21—1191no—
134C3×D7—71no—
135C3×D7—71no—
136C3×D7—212no—
137D3×C7—171no—
138D3×C7—171no—
139D3×C7—512no—
140C2×ΓL(1,𝔽8)—7141no—
141C2×ΓL(1,𝔽8)—7141no—
142AGL(1,𝔽7)—71no—
143AGL(1,𝔽7)—71no—
144AGL(1,𝔽7)—141no—
145AGL(1,𝔽7)—141no—
146AGL(1,𝔽7)—3571no—
147AGL(1,𝔽7)—3571no—
148C2×C3×D3—8331noHall
149C2×C17—491no—
150D17—31no—
151D17—1471no—
152C2×D7—10712no—
153C2×D7—3571no—
154C2×D7—3571no—
155C3×C7—32no—
156C3×C7D17×AGL(1,𝔽7)11yescharacteristic, complemented
157C3×C7D17×AGL(1,𝔽7)11yescharacteristic, complemented
158ΓL(1,𝔽8)—141no—
159ΓL(1,𝔽8)—141no—
160ΓL(1,𝔽8)—281no—
161ΓL(1,𝔽8)—281no—
162C2×C32—491no—
163C3×D3—8331no—
164C3×D3—16661no—
165C17D3×#6442511yescharacteristic, complemented, Sylow
166C2×C7—511no—
167C2×C7—511no—
168C2×C7—1532no—
169D7—71no—
170D7—71no—
171D7—212no—
172D7—10712no—
173D7—3571no—
174D7—3571no—
175C22×C3—24991no—
176C2×D3—8331no—
177C32—981noSylow
178C7—32no—
179C7AGL(1,𝔽7)×D5111yescharacteristic, complemented
180C7AGL(1,𝔽7)×D5111yescharacteristic, complemented
181C2×C3—491no—
182C2×C3—491no—
183C2×C3—981no—
184C2×C3—49981no—
185C2×C3—24991no—
186D3—171no—
187D3—8331no—
188C22—24991noSylow
189C3—981no—
190C3—1961no—
191C3D17×#6442511yescharacteristic, complemented
192C2—491no—
193C2—511no—
194C2—24991no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C72×C17C22×C3
LCSG ▷ C3×C72×C17
UCSC1

Complements

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

NH ≅ G/Ntype
1. D51×#3862C2semidirect
2. C3⋊(C17⋊#64425)C2semidirect
3. C3×C17×#64425C2semidirect
4. C7⋊D7×D51C3semidirect
6. C3×C17×#3862C22semidirect
7. C3⋊(C7⋊D119)C2×C3semidirect
8. C72×D51C2×C3semidirect
9. C3×C17×C7⋊D7C2×C3semidirect
13. C17×#64425D3semidirect
18. C17×#3862C2×D3semidirect
19. C3×C72×C17C22×C3semidirect
28. C17×C7⋊D7C3×D3semidirect
39. C3×#64425D17semidirect
40. C72×C17C2×C3×D3semidirect
45. C7×D51AGL(1,𝔽7)semidirect
46. C7×D51AGL(1,𝔽7)semidirect
64. C3×#3862C2×D17semidirect
66. C3×C7×C17C2×AGL(1,𝔽7)semidirect
67. C3×C7×C17C2×AGL(1,𝔽7)semidirect
77. C3×C7⋊D7C3×D17semidirect
81. #64425D51direct (N × H)
97. C3×C72C2×C3×D17semidirect
99. #3862C2×D51semidirect
107. C7×C17D3×AGL(1,𝔽7)semidirect
108. C7×C17D3×AGL(1,𝔽7)semidirect
113. D51#64425direct (N × H)
118. C7⋊D7C3×D51semidirect
129. C3×C17C2×#64425semidirect
130. C72C2×C3×D51semidirect
156. C3×C7D17×AGL(1,𝔽7)semidirect
157. C3×C7D17×AGL(1,𝔽7)semidirect
165. C17D3×#64425semidirect
179. C7AGL(1,𝔽7)×D51semidirect
180. C7AGL(1,𝔽7)×D51semidirect
191. C3D17×#64425semidirect

Maximal subgroups

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

subgroupindex
D51×#38622
C3⋊(C17⋊#64425)2
C3×C17×#644252
C7⋊D7×D513
D17×#644253
AGL(1,𝔽7)×D517
D3×#6442517