Finite Group Databasebeta

← #44472850

Subgroups — #44472850

Order 29988.

184 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 35 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
2C17×C3⋊#64422C211yesmaximal, characteristic, complemented
3C3×C17⋊#64425C211yesmaximal, characteristic, complemented
4D21⋊D119C311yesmaximal, characteristic, complemented
5#64422⋊D17—31nomaximal
6C3×C17×#3862C2211yescharacteristic, complemented, Hall
7C7×D357C2×C311yescharacteristic, complemented
8C7×C17×D21C2×C311yescharacteristic, complemented
9C3×C7⋊D119C2×C311yescharacteristic, complemented
10C17⋊#64422—31no—
11C17×#64422—31no—
12C17⋊#64425—21no—
13C17⋊#64425D311yescharacteristic, complemented
14D51⋊AGL(1,𝔽7)—71nomaximal
15D3×C17⋊AGL(1,𝔽7)—71nomaximal
16D7×D119—31noHall
17C17×#3862—21no—
18C17×#3862C2×D311yescharacteristic, complemented
19C3×C72×C17C22×C311yescharacteristic, complemented
20ΓL(1,𝔽8)×D51—71no—
21D3×C17×ΓL(1,𝔽8)—71no—
22C3⋊(C17⋊AGL(1,𝔽7))—141no—
23C17×C3⋊AGL(1,𝔽7)—141no—
24C3×C17⋊AGL(1,𝔽7)—71no—
25C3×C17⋊AGL(1,𝔽7)—71no—
26#13183636—171nomaximal, Hall
27C7⋊D119C3×D311yescharacteristic, complemented
28C7×D119—31no—
29C7×D7×C17—31no—
30D21⋊D17—71no—
31D3×D119—71no—
32C2×C17⋊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⋊#64422D1711yescharacteristic, complemented
38C3×#64425—171no—
39C72×C17C2×C3×D311yescharacteristic, complemented, Hall
40D357AGL(1,𝔽7)11yescharacteristic, complemented
41C17×D21AGL(1,𝔽7)11yescharacteristic, complemented
42C7×D51—71no—
43D3×C7×C17—71no—
44C3×D119—71no—
45C3×D119—71no—
46C3×D119—421no—
47C2×C17×ΓL(1,𝔽8)—211no—
48C17⋊AGL(1,𝔽7)—71no—
49C17⋊AGL(1,𝔽7)—71no—
50C17⋊AGL(1,𝔽7)—141no—
51C17⋊AGL(1,𝔽7)—141no—
52C17⋊AGL(1,𝔽7)—211no—
53C17×AGL(1,𝔽7)—421no—
54ΓL(1,𝔽8)×D17—211no—
55C3×D3×D17—491noHall
56D21⋊D7—171no—
57#64422⋊C2—511no—
58C2×D119—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)—281no—
67C17×ΓL(1,𝔽8)—281no—
68C3×D51—981no—
69C3×D3×C17—981no—
70C32×D17—491no—
71C7×D21—171no—
72C7×D21C3×D1711yescharacteristic, complemented
73C3×C7⋊D7—171no—
74#64425—171no—
75#64425—341no—
76#64422—31no—
77#64422—511no—
78D3×AGL(1,𝔽7)—1191no—
79D3×AGL(1,𝔽7)—1191no—
80C2×C7×C17—211no—
81D119—31no—
82D119—71no—
83D119—71no—
84D119—421no—
85C7×D17—211no—
86D7×C17—31no—
87C2×C3×D17—1471no—
88D3×D17—491no—
89D72—511noHall
90C32×C17—981noHall
91C3×C72C2×C3×D1711yescharacteristic, complemented
92#3862—21no—
93#3862D3×D1711yescharacteristic, complemented
94C3⋊AGL(1,𝔽7)—141no—
95C3⋊AGL(1,𝔽7)—2381no—
96D3×ΓL(1,𝔽8)—71no—
97D3×ΓL(1,𝔽8)—1191no—
98C3×AGL(1,𝔽7)—1191no—
99C3×AGL(1,𝔽7)—1191no—
100C7×C17—61no—
101C7×C17D3×AGL(1,𝔽7)11yescharacteristic, complemented
102C7×C17D3×AGL(1,𝔽7)11yescharacteristic, complemented
103C2×C3×C17—2941no—
104D51—71no—
105C3×D17—2941no—
106C3×D17—491no—
107C3×D17—491no—
108C3×D17—981no—
109D3×C17—71no—
110C7⋊D7—171no—
111C7×D7—31no—
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)—141no—
119C3×ΓL(1,𝔽8)—141no—
120C3×C17—1961no—
121C3×C17—1961no—
122C3×C17#64422⋊C211yescharacteristic, complemented
123C72C3×D3×D1711yescharacteristic, complemented, Sylow
124D21—171no—
125D21C17⋊AGL(1,𝔽7)11yescharacteristic, complemented
126C3×D7—1191no—
127C3×D7—1191no—
128C3×D7—7141no—
129D3×C7—71no—
130D3×C7—1191no—
131C2×ΓL(1,𝔽8)—211no—
132C2×ΓL(1,𝔽8)—3571no—
133AGL(1,𝔽7)—421no—
134AGL(1,𝔽7)—3571no—
135AGL(1,𝔽7)—1191no—
136AGL(1,𝔽7)—1191no—
137AGL(1,𝔽7)—2381no—
138AGL(1,𝔽7)—2381no—
139C2×C3×D3—8331noHall
140C2×C17—211no—
141D17—211no—
142D17—491no—
143C2×D7—3571no—
144C2×D7—3571no—
145C3×C7—61no—
146C3×C7D17×AGL(1,𝔽7)11yescharacteristic, complemented
147C3×C7C2×C17⋊AGL(1,𝔽7)11yescharacteristic, complemented
148ΓL(1,𝔽8)—71no—
149ΓL(1,𝔽8)—141no—
150ΓL(1,𝔽8)—281no—
151ΓL(1,𝔽8)—281no—
152C2×C32—8331no—
153C3×D3—981no—
154C3×D3—16661no—
155C17#1318363611yescharacteristic, complemented, Sylow
156C2×C7—211no—
157C2×C7—3571no—
158D7—31no—
159D7—511no—
160D7—1191no—
161D7—1191no—
162D7—7141no—
163C22×C3—24991no—
164C2×D3—8331no—
165C32—1961noSylow
166C7—61no—
167C7D3×C17⋊AGL(1,𝔽7)11yescharacteristic, complemented
168C7D51⋊AGL(1,𝔽7)11yescharacteristic, complemented
169C2×C3—2941no—
170C2×C3—8331no—
171C2×C3—8331no—
172C2×C3—16661no—
173C2×C3—49981no—
174D3—71no—
175D3—1191no—
176C22—24991noSylow
177C3—1961no—
178C3—1961no—
179C3#64422⋊D1711yescharacteristic, complemented
180C2—211no—
181C2—8331no—
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. C17×C3⋊#64422C2semidirect
3. C3×C17⋊#64425C2semidirect
4. D21⋊D119C3semidirect
6. C3×C17×#3862C22semidirect
7. C7×D357C2×C3semidirect
8. C7×C17×D21C2×C3semidirect
9. C3×C7⋊D119C2×C3semidirect
13. C17⋊#64425D3semidirect
18. C17×#3862C2×D3semidirect
19. C3×C72×C17C22×C3semidirect
27. C7⋊D119C3×D3semidirect
37. C3⋊#64422D17semidirect
39. C72×C17C2×C3×D3semidirect
40. D357AGL(1,𝔽7)semidirect
41. C17×D21AGL(1,𝔽7)semidirect
60. C3×#3862C2×D17semidirect
62. C3×C7×C17C2×AGL(1,𝔽7)semidirect
63. C3×C7×C17C2×AGL(1,𝔽7)semidirect
72. C7×D21C3×D17semidirect
91. C3×C72C2×C3×D17semidirect
93. #3862D3×D17semidirect
101. C7×C17D3×AGL(1,𝔽7)semidirect
102. C7×C17D3×AGL(1,𝔽7)semidirect
122. C3×C17#64422⋊C2semidirect
123. C72C3×D3×D17semidirect
125. D21C17⋊AGL(1,𝔽7)semidirect
146. C3×C7D17×AGL(1,𝔽7)semidirect
147. C3×C7C2×C17⋊AGL(1,𝔽7)semidirect
155. C17#13183636semidirect
167. C7D3×C17⋊AGL(1,𝔽7)semidirect
168. C7D51⋊AGL(1,𝔽7)semidirect
179. C3#64422⋊D17semidirect

Maximal subgroups

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

subgroupindex
C3⋊(C17⋊#64422)2
C17×C3⋊#644222
C3×C17⋊#644252
D21⋊D1193
#64422⋊D173
D51⋊AGL(1,𝔽7)7
D3×C17⋊AGL(1,𝔽7)7
#1318363617