Finite Group Databasebeta

← #44568739

Subgroups — #44568739

Order 32340.

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

normal subgroups only

This group has 168 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
1C5⋊(C11⋊#64422)C211yesmaximal, characteristic, complemented
2C11×C5⋊#64425C211yesmaximal, characteristic, complemented
3C5×C11⋊#64422C211yesmaximal, characteristic, complemented
4D77⋊D35C311yesmaximal, characteristic, complemented, Hall
5C5×C11×#3862C2211yescharacteristic, complemented, Hall
6#20266343—51nomaximal, Hall
7C11×C5⋊(C7⋊D7)C2×C311yescharacteristic, complemented
8C7×D385C2×C311yescharacteristic, complemented
9C5×C7×D77C2×C311yescharacteristic, complemented
10D55⋊AGL(1,𝔽7)—71nomaximal
11D11×C5⋊AGL(1,𝔽7)—71nomaximal
12C11⋊#64422—51no—
13C11⋊#64422D511yescharacteristic, complemented
14C11×#64425—51no—
15#64422⋊D5—111nomaximal, Hall
16C5×C72×C11C22×C311yescharacteristic, complemented, Hall
17C5⋊(C11⋊AGL(1,𝔽7))—141no—
18C11×C5⋊AGL(1,𝔽7)—71no—
19C11×C5⋊AGL(1,𝔽7)—71no—
20ΓL(1,𝔽8)×D55—71no—
21C5×C11⋊AGL(1,𝔽7)—141no—
22C5×ΓL(1,𝔽8)×D11—71no—
23(C7⋊D7)⋊D11—51noHall
24C11×#3862C2×D511yescharacteristic, complemented, Hall
25D35⋊D11—71no—
26D11×D35—71no—
27C5⋊#64422—111no—
28C5⋊#64425D1111yescharacteristic, complemented
29C5×#64422—111no—
30C5×C11×ΓL(1,𝔽8)—141no—
31C5×C11×ΓL(1,𝔽8)—141no—
32C11×C7⋊D7—51no—
33C7×D77—51no—
34C7×D77C3×D511yescharacteristic, complemented
35D7×D35—111noHall
36D11×AGL(1,𝔽7)—351no—
37D11×AGL(1,𝔽7)—351no—
38D385AGL(1,𝔽7)11yescharacteristic, complemented
39C11×D35—71no—
40C11×D35—71no—
41C11×D35—421no—
42C7×D55—71no—
43C5×D77AGL(1,𝔽7)11yescharacteristic, complemented
44C5×C7×D11—71no—
45C5×#3862C2×D1111yescharacteristic, complemented, Hall
46C3×D5×D11—491noHall
47#64422⋊C2—551noHall
48C72×C11C2×C3×D511yescharacteristic, complemented, Hall
49C5⋊(C7⋊D7)C3×D1111yescharacteristic, complemented
50C7×D35—111no—
51C5×C7×D7—111no—
52C11⋊AGL(1,𝔽7)—141no—
53C11⋊AGL(1,𝔽7)—701no—
54C11×AGL(1,𝔽7)—351no—
55C11×AGL(1,𝔽7)—351no—
56ΓL(1,𝔽8)×D11—71no—
57ΓL(1,𝔽8)×D11—351no—
58C2×C5⋊AGL(1,𝔽7)—771no—
59D5×AGL(1,𝔽7)—771no—
60C5×C7×C11—61no—
61C5×C7×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
62C5×C7×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
63C3×D55—981no—
64C3×D5×C11—491no—
65C3×C5×D11—981no—
66D7×D11—351no—
67D7×D11—351no—
68#64425—51no—
69#64422—111no—
70#64422—551no—
71C5×C72C2×C3×D1111yescharacteristic, complemented, Hall
72C11×ΓL(1,𝔽8)—141no—
73C11×ΓL(1,𝔽8)—141no—
74D5×D11—491noHall
75C2×C5×ΓL(1,𝔽8)—771no—
76C5⋊AGL(1,𝔽7)—71no—
77C5⋊AGL(1,𝔽7)—71no—
78C5⋊AGL(1,𝔽7)—771no—
79D5×ΓL(1,𝔽8)—771no—
80C5×AGL(1,𝔽7)—1541no—
81D72—551noHall
82C3×C5×C11—1961noHall
83D77—51no—
84D77C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
85C7×D11—71no—
86C7×D11—351no—
87C11×D7—351no—
88C11×D7—351no—
89C11×D7—2101no—
90#3862D5×D1111yescharacteristic, complemented, Hall
91C2×D35—771no—
92D5×D7—771no—
93C2×C3×D11—2451noHall
94D55—71no—
95C5×D11—71no—
96D5×C11—491no—
97C5×ΓL(1,𝔽8)—71no—
98C5×ΓL(1,𝔽8)—141no—
99C7⋊D7—51no—
100C7×D7—111no—
101C7×D7—551no—
102C2×AGL(1,𝔽7)—3851no—
103C2×AGL(1,𝔽7)—3851no—
104C7×C11—61no—
105C7×C11D5×AGL(1,𝔽7)11yescharacteristic, complemented
106C7×C11C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
107C2×C5×C7—771no—
108D35—71no—
109D35—71no—
110D35—111no—
111D35—421no—
112C5×D7—111no—
113C7×D5—771no—
114C2×C3×C11—2451no—
115C3×D11—981no—
116C3×D11—4901no—
117C2×C3×D5—5391noHall
118C5×C11#64422⋊C211yescharacteristic, complemented, Hall
119C72C3×D5×D1111yescharacteristic, complemented, Sylow
120C2×D11—2451noHall
121C2×ΓL(1,𝔽8)—771no—
122C2×ΓL(1,𝔽8)—3851no—
123AGL(1,𝔽7)—7701no—
124AGL(1,𝔽7)—351no—
125AGL(1,𝔽7)—351no—
126AGL(1,𝔽7)—1541no—
127C5×C7—61no—
128C5×C7D11×AGL(1,𝔽7)11yescharacteristic, complemented
129C5×C7D11×AGL(1,𝔽7)11yescharacteristic, complemented
130C3×C11—1961noHall
131C2×C3×C5—10781no—
132C3×D5—491no—
133C3×D5—10781no—
134C2×D7—3851no—
135C2×D7—3851no—
136C2×C11—2451no—
137D11—71no—
138D11—351no—
139ΓL(1,𝔽8)—141no—
140ΓL(1,𝔽8)—141no—
141C2×D5—5391noHall
142C3×C5—1961noHall
143C2×C7—771no—
144C2×C7—3851no—
145D7—111no—
146D7—351no—
147D7—351no—
148D7—551no—
149D7—2101no—
150C22×C3—26951noHall
151C11#64422⋊D511yescharacteristic, complemented, Sylow
152C2×C5—771no—
153D5—491no—
154D5—771no—
155C7—61no—
156C7D11×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
157C7D55⋊AGL(1,𝔽7)11yescharacteristic, complemented
158C2×C3—53901no—
159C2×C3—10781no—
160C2×C3—2451no—
161C5#2026634311yescharacteristic, complemented, Sylow
162C22—26951noSylow
163C3—1961noSylow
164C2—771no—
165C2—3851no—
166C2—2451no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C11C22×C3
LCSG ▷ C5×C72×C11
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊(C11⋊#64422)C2semidirect
2. C11×C5⋊#64425C2semidirect
3. C5×C11⋊#64422C2semidirect
4. D77⋊D35C3semidirect
5. C5×C11×#3862C22semidirect
7. C11×C5⋊(C7⋊D7)C2×C3semidirect
8. C7×D385C2×C3semidirect
9. C5×C7×D77C2×C3semidirect
13. C11⋊#64422D5semidirect
16. C5×C72×C11C22×C3semidirect
24. C11×#3862C2×D5semidirect
28. C5⋊#64425D11semidirect
34. C7×D77C3×D5semidirect
38. D385AGL(1,𝔽7)semidirect
43. C5×D77AGL(1,𝔽7)semidirect
45. C5×#3862C2×D11semidirect
48. C72×C11C2×C3×D5semidirect
49. C5⋊(C7⋊D7)C3×D11semidirect
61. C5×C7×C11C2×AGL(1,𝔽7)semidirect
62. C5×C7×C11C2×AGL(1,𝔽7)semidirect
71. C5×C72C2×C3×D11semidirect
84. D77C5⋊AGL(1,𝔽7)semidirect
90. #3862D5×D11semidirect
105. C7×C11D5×AGL(1,𝔽7)semidirect
106. C7×C11C2×C5⋊AGL(1,𝔽7)semidirect
118. C5×C11#64422⋊C2semidirect
119. C72C3×D5×D11semidirect
128. C5×C7D11×AGL(1,𝔽7)semidirect
129. C5×C7D11×AGL(1,𝔽7)semidirect
151. C11#64422⋊D5semidirect
156. C7D11×C5⋊AGL(1,𝔽7)semidirect
157. C7D55⋊AGL(1,𝔽7)semidirect
161. C5#20266343semidirect

Maximal subgroups

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

subgroupindex
C5⋊(C11⋊#64422)2
C11×C5⋊#644252
C5×C11⋊#644222
D77⋊D353
#202663435
D55⋊AGL(1,𝔽7)7
D11×C5⋊AGL(1,𝔽7)7
#64422⋊D511