Finite Group Databasebeta

← #44472849

Subgroups — #44472849

Order 29988.

184 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C3⋊(C17⋊#64422)C211yesmaximal, characteristic, complemented
2C3⋊(C17⋊#64422)C211yesmaximal, characteristic, complemented
3C3×C17×#64425C211yesmaximal, characteristic, complemented
4#23756260C311yesmaximal, characteristic, complemented
5#23756185—31nomaximal
6C3×C17×#3862C2211yescharacteristic, complemented, Hall
7C7×D357C2×C311yescharacteristic, complemented
8C7×D357C2×C311yescharacteristic, complemented
9C3×C17×C7⋊D7C2×C311yescharacteristic, complemented
10C17⋊#64422—31no—
11C17⋊#64422—31no—
12C17×#64425—21no—
13C17×#64425D311yescharacteristic, complemented
14AGL(1,𝔽7)×D51—71nomaximal
15AGL(1,𝔽7)×D51—71nomaximal
16(C7⋊D7)⋊D17—31noHall
17C17×#3862—21no—
18C17×#3862C2×D311yescharacteristic, complemented
19C3×C72×C17C22×C311yescharacteristic, complemented
20ΓL(1,𝔽8)×D51—71no—
21ΓL(1,𝔽8)×D51—71no—
22C3⋊(C17⋊AGL(1,𝔽7))—141no—
23C3⋊(C17⋊AGL(1,𝔽7))—141no—
24C3×C17×AGL(1,𝔽7)—71no—
25C3×C17×AGL(1,𝔽7)—71no—
26#13183636—171nomaximal, Hall
27C17×C7⋊D7C3×D311yescharacteristic, complemented
28C7×D119—31no—
29C7×D119—31no—
30D7×D51—71no—
31D7×D51—71no—
32D17×AGL(1,𝔽7)—211no—
33D17×AGL(1,𝔽7)—211no—
34C3×C17×ΓL(1,𝔽8)—141no—
35C3×C17×ΓL(1,𝔽8)—141no—
36C3⋊#64422—171no—
37C3⋊#64422—171no—
38C3×#64425D1711yescharacteristic, complemented
39C72×C17C2×C3×D311yescharacteristic, complemented, Hall
40D357AGL(1,𝔽7)11yescharacteristic, complemented
41D357AGL(1,𝔽7)11yescharacteristic, complemented
42C7×D51—71no—
43C7×D51—71no—
44C3×D7×C17—71no—
45C3×D7×C17—71no—
46C3×D7×C17—421no—
47C17⋊AGL(1,𝔽7)—211no—
48C17⋊AGL(1,𝔽7)—421no—
49C17×AGL(1,𝔽7)—71no—
50C17×AGL(1,𝔽7)—71no—
51C17×AGL(1,𝔽7)—141no—
52C17×AGL(1,𝔽7)—141no—
53ΓL(1,𝔽8)×D17—211no—
54ΓL(1,𝔽8)×D17—211no—
55C2×C3×D51—491noHall
56D21⋊D7—171no—
57#64422⋊C2—511no—
58D7×D17—211no—
59D7×D17—211no—
60C3×#3862C2×D1711yescharacteristic, complemented, Hall
61C3×C7×C17—61no—
62C3×C7×C17C2×AGL(1,𝔽7)11yescharacteristic, complemented
63C3×C7×C17C2×AGL(1,𝔽7)11yescharacteristic, complemented
64C17×ΓL(1,𝔽8)—71no—
65C17×ΓL(1,𝔽8)—141no—
66C17×ΓL(1,𝔽8)—141no—
67C17×ΓL(1,𝔽8)—281no—
68C2×C32×C17—491no—
69C3×D51—981no—
70C3×D51—981no—
71C7×D21—171no—
72C7×D21—171no—
73C3×C7⋊D7C3×D1711yescharacteristic, complemented
74#64425—21no—
75#64425D5111yescharacteristic, complemented
76#64422—511no—
77#64422—511no—
78D3×AGL(1,𝔽7)—1191no—
79D3×AGL(1,𝔽7)—1191no—
80D119—31no—
81D119—31no—
82C7×D17—211no—
83C7×D17—211no—
84D7×C17—71no—
85D7×C17—71no—
86D7×C17—421no—
87C2×D51—491no—
88C2×C3×D17—1471no—
89D72—511noHall
90C32×C17—1961noHall
91C3×C72C2×C3×D1711yescharacteristic, complemented
92#3862—21no—
93#3862C2×D5111yescharacteristic, complemented
94C3⋊AGL(1,𝔽7)—1191no—
95C3⋊AGL(1,𝔽7)—2381no—
96D3×ΓL(1,𝔽8)—1191no—
97D3×ΓL(1,𝔽8)—1191no—
98C3×AGL(1,𝔽7)—71no—
99C3×AGL(1,𝔽7)—71no—
100C7×C17—61no—
101C7×C17D3×AGL(1,𝔽7)11yescharacteristic, complemented
102C7×C17D3×AGL(1,𝔽7)11yescharacteristic, complemented
103C2×C3×C17—491no—
104C2×C3×C17—491no—
105C2×C3×C17—981no—
106D51—71no—
107D51—71no—
108C3×D17—2941no—
109C3×D17—2941no—
110C7⋊D7C3×D5111yescharacteristic, complemented
111C7×D7—511no—
112C7×D7—511no—
113D3×D7—1191no—
114D3×D7—1191no—
115C2×AGL(1,𝔽7)—3571no—
116C2×AGL(1,𝔽7)—3571no—
117C2×D17—1471noHall
118C3×ΓL(1,𝔽8)—71no—
119C3×ΓL(1,𝔽8)—141no—
120C3×C17—981no—
121C3×C17—1961no—
122C3×C17#64422⋊C211yescharacteristic, complemented
123C72C2×C3×D5111yescharacteristic, complemented, Sylow
124D21—171no—
125D21—171no—
126C3×D7—71no—
127C3×D7—71no—
128C3×D7—421no—
129D3×C7—1191no—
130D3×C7—1191no—
131C2×ΓL(1,𝔽8)—3571no—
132C2×ΓL(1,𝔽8)—3571no—
133AGL(1,𝔽7)—71no—
134AGL(1,𝔽7)—71no—
135AGL(1,𝔽7)—141no—
136AGL(1,𝔽7)—141no—
137AGL(1,𝔽7)—7141no—
138AGL(1,𝔽7)—7141no—
139C2×C3×D3—8331noHall
140C2×C17—491no—
141D17—211no—
142D17—211no—
143C2×D7—3571no—
144C2×D7—3571no—
145C3×C7—61no—
146C3×C7D17×AGL(1,𝔽7)11yescharacteristic, complemented
147C3×C7D17×AGL(1,𝔽7)11yescharacteristic, complemented
148ΓL(1,𝔽8)—141no—
149ΓL(1,𝔽8)—141no—
150ΓL(1,𝔽8)—281no—
151ΓL(1,𝔽8)—281no—
152C2×C32—491no—
153C3×D3—16661no—
154C3×D3—16661no—
155C17#1318363611yescharacteristic, complemented, Sylow
156C2×C7—3571no—
157C2×C7—3571no—
158D7—71no—
159D7—71no—
160D7—421no—
161D7—511no—
162D7—511no—
163C22×C3—24991no—
164C2×D3—8331no—
165C32—1961noSylow
166C7—61no—
167C7AGL(1,𝔽7)×D5111yescharacteristic, complemented
168C7AGL(1,𝔽7)×D5111yescharacteristic, complemented
169C2×C3—491no—
170C2×C3—491no—
171C2×C3—981no—
172C2×C3—49981no—
173C2×C3—49981no—
174D3—1191no—
175D3—1191no—
176C22—24991noSylow
177C3—1961no—
178C3—1961no—
179C3#2375618511yescharacteristic, complemented
180C2—491no—
181C2—3571no—
182C2—3571no—
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. C3⋊(C17⋊#64422)C2semidirect
2. C3⋊(C17⋊#64422)C2semidirect
3. C3×C17×#64425C2semidirect
4. #23756260C3semidirect
6. C3×C17×#3862C22semidirect
7. C7×D357C2×C3semidirect
8. C7×D357C2×C3semidirect
9. C3×C17×C7⋊D7C2×C3semidirect
13. C17×#64425D3semidirect
18. C17×#3862C2×D3semidirect
19. C3×C72×C17C22×C3semidirect
27. C17×C7⋊D7C3×D3semidirect
38. C3×#64425D17semidirect
39. C72×C17C2×C3×D3semidirect
40. D357AGL(1,𝔽7)semidirect
41. D357AGL(1,𝔽7)semidirect
60. C3×#3862C2×D17semidirect
62. C3×C7×C17C2×AGL(1,𝔽7)semidirect
63. C3×C7×C17C2×AGL(1,𝔽7)semidirect
73. C3×C7⋊D7C3×D17semidirect
75. #64425D51semidirect
91. C3×C72C2×C3×D17semidirect
93. #3862C2×D51semidirect
101. C7×C17D3×AGL(1,𝔽7)semidirect
102. C7×C17D3×AGL(1,𝔽7)semidirect
110. C7⋊D7C3×D51semidirect
122. C3×C17#64422⋊C2semidirect
123. C72C2×C3×D51semidirect
146. C3×C7D17×AGL(1,𝔽7)semidirect
147. C3×C7D17×AGL(1,𝔽7)semidirect
155. C17#13183636semidirect
167. C7AGL(1,𝔽7)×D51semidirect
168. C7AGL(1,𝔽7)×D51semidirect
179. C3#23756185semidirect

Maximal subgroups

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

subgroupindex
C3⋊(C17⋊#64422)2
C3×C17×#644252
#237562603
#237561853
AGL(1,𝔽7)×D517
#1318363617