Finite Group Databasebeta

← #50304178

Subgroups — #50304178

Order 38220.

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⋊#16838462C211yesmaximal, characteristic, complemented
2C5⋊#16838452C211yesmaximal, characteristic, complemented
3C5×#16838465C211yesmaximal, characteristic, complemented
4D7×D455C311yesmaximal, characteristic, complemented, Hall
5C5×C13⋊#3862C2211yescharacteristic, complemented, Hall
6#21439618—51nomaximal, Hall
7C5⋊(C7⋊D91)C2×C311yescharacteristic, complemented
8C7×D455C2×C311yescharacteristic, complemented
9C5×C7×C13×D7C2×C311yescharacteristic, complemented
10C2×C5⋊#10587722—71nomaximal
11#10587721⋊D5—71nomaximal
12#16838465D511yescharacteristic, complemented
13#16838462—51no—
14#16838452—51no—
15C5×C72×C13C22×C311yescharacteristic, complemented, Hall
16#64422⋊D5—131nomaximal, Hall
17C5⋊#10587723—71no—
18C5⋊#10587722—71no—
19C5⋊#10587722—71no—
20C5⋊#10587717—71no—
21C5×#10587721—71no—
22C2×C5×#63270—71no—
23D7×D91—51noHall
24C13⋊#3862C2×D511yescharacteristic, complemented, Hall
25C2×D455—71no—
26D7×D65—71no—
27C5⋊#64422—131no—
28C5⋊#64425—131no—
29C5×#64422—131no—
30C5×#63270—71no—
31C5×#63271—71no—
32C7⋊D91—51no—
33C7×D91—51no—
34C7×C13×D7C3×D511yescharacteristic, complemented
35C2×#10587722—351no—
36#10587717⋊C2—351no—
37D7×D35—131noHall
38C2×C5×C7×C13—71no—
39D455—71no—
40D455—71no—
41D455—421no—
42D455AGL(1,𝔽7)11yescharacteristic, complemented
43C7×D65—71no—
44C5×C13×D7AGL(1,𝔽7)11yescharacteristic, complemented
45C2×C5⋊(D13⋊C3)—491noHall
46C5×#3862—131noHall
47C72×C13C2×C3×D511yescharacteristic, complemented, Hall
48#64422⋊C2—651noHall
49C2×#63270—71no—
50#10587723—351no—
51#10587722—351no—
52#10587722—351no—
53#10587721—71no—
54#10587717—351no—
55C5⋊(C7⋊D7)—131no—
56C7×D35—131no—
57C5×C7×D7D13⋊C311yescharacteristic, complemented
58C5×C7×C13—61no—
59C5×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
60C5×C7×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
61C2×C5⋊AGL(1,𝔽7)—911no—
62D5×AGL(1,𝔽7)—911no—
63C2×C5×C13⋊C3—491no—
64C5⋊(D13⋊C3)—491no—
65C5⋊(D13⋊C3)—491no—
66C2×D91—351no—
67D7×D13—351no—
68#64425—651no—
69#64422—131no—
70#64422—651no—
71#63271—71no—
72#63270—71no—
73C2×D65—491noHall
74C5×C72C2×D13⋊C311yescharacteristic, complemented, Hall
75C2×C5×ΓL(1,𝔽8)—911no—
76C5⋊AGL(1,𝔽7)—911no—
77C5⋊AGL(1,𝔽7)—911no—
78C5⋊AGL(1,𝔽7)—911no—
79D5×ΓL(1,𝔽8)—911no—
80C5×AGL(1,𝔽7)—911no—
81D72—651noHall
82C5×C13⋊C3—491noHall
83C2×C7×C13—71no—
84D91—51no—
85D91—351no—
86D91—351no—
87D91—2101no—
88C7×D13—351no—
89C13×D7C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
90C2×D13⋊C3—2451noHall
91#3862—131noHall
92C2×D35—911no—
93D5×D7—911no—
94C2×C5×C13—71no—
95D65—71no—
96D65—491no—
97C5×ΓL(1,𝔽8)—911no—
98C5×ΓL(1,𝔽8)—911no—
99C7⋊D7—651no—
100C7×D7—651no—
101C7×D7C5⋊(D13⋊C3)11yescharacteristic, complemented
102C7×C13—61no—
103C7×C13D5×AGL(1,𝔽7)11yescharacteristic, complemented
104C7×C13C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
105C2×AGL(1,𝔽7)—4551no—
106C2×AGL(1,𝔽7)—4551no—
107C2×C13⋊C3—491no—
108D13⋊C3—2451no—
109D13⋊C3—2451no—
110C2×C5×C7—71no—
111D35—131no—
112D35—5461no—
113D35—911no—
114D35—911no—
115C5×D7#1058772211yescharacteristic, complemented
116C7×D5—911no—
117C5×C13#64422⋊C211yescharacteristic, complemented, Hall
118C2×C3×D5—6371noHall
119C2×D13—2451noHall
120C72C2×C5⋊(D13⋊C3)11yescharacteristic, complemented, Sylow
121C2×ΓL(1,𝔽8)—911no—
122C2×ΓL(1,𝔽8)—4551no—
123AGL(1,𝔽7)—911no—
124AGL(1,𝔽7)—4551no—
125AGL(1,𝔽7)—4551no—
126AGL(1,𝔽7)—4551no—
127C13⋊C3—491noHall
128C5×C7—61no—
129C5×C7#10587717⋊C211yescharacteristic, complemented
130C5×C7C2×#1058772211yescharacteristic, complemented
131C2×C3×C5—6371no—
132C3×D5—6371no—
133C3×D5—6371no—
134C2×D7—4551no—
135C2×D7—4551no—
136C2×C13—71no—
137D13—351no—
138D13—2451no—
139ΓL(1,𝔽8)—911no—
140ΓL(1,𝔽8)—911no—
141C2×D5—6371noHall
142C3×C5—6371noHall
143C2×C7—71no—
144C2×C7—4551no—
145D7—651no—
146D7—27301no—
147D7—4551no—
148D7—4551no—
149D7C5⋊#1058772211yescharacteristic, complemented
150C13#64422⋊D511yescharacteristic, complemented, Sylow
151C22×C3—31851noHall
152C2×C5—71no—
153D5—911no—
154D5—6371no—
155C7—61no—
156C7#10587721⋊D511yescharacteristic, complemented
157C7C2×C5⋊#1058772211yescharacteristic, complemented
158C2×C3—31851no—
159C2×C3—31851no—
160C2×C3—6371no—
161C5#2143961811yescharacteristic, complemented, Sylow
162C22—31851noSylow
163C3—6371noSylow
164C2—71no—
165C2—31851no—
166C2—4551no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C13C22×C3
LCSG ▷ C5×C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊#16838462C2semidirect
2. C5⋊#16838452C2semidirect
3. C5×#16838465C2semidirect
4. D7×D455C3semidirect
5. C5×C13⋊#3862C22semidirect
7. C5⋊(C7⋊D91)C2×C3semidirect
8. C7×D455C2×C3semidirect
9. C5×C7×C13×D7C2×C3semidirect
12. #16838465D5semidirect
15. C5×C72×C13C22×C3semidirect
24. C13⋊#3862C2×D5semidirect
34. C7×C13×D7C3×D5semidirect
42. D455AGL(1,𝔽7)semidirect
44. C5×C13×D7AGL(1,𝔽7)semidirect
47. C72×C13C2×C3×D5semidirect
57. C5×C7×D7D13⋊C3semidirect
59. C5×C7×C13C2×AGL(1,𝔽7)semidirect
60. C5×C7×C13C2×AGL(1,𝔽7)semidirect
74. C5×C72C2×D13⋊C3semidirect
89. C13×D7C5⋊AGL(1,𝔽7)semidirect
101. C7×D7C5⋊(D13⋊C3)semidirect
103. C7×C13D5×AGL(1,𝔽7)semidirect
104. C7×C13C2×C5⋊AGL(1,𝔽7)semidirect
115. C5×D7#10587722semidirect
117. C5×C13#64422⋊C2semidirect
120. C72C2×C5⋊(D13⋊C3)semidirect
129. C5×C7#10587717⋊C2semidirect
130. C5×C7C2×#10587722semidirect
149. D7C5⋊#10587722semidirect
150. C13#64422⋊D5semidirect
156. C7#10587721⋊D5semidirect
157. C7C2×C5⋊#10587722semidirect
161. C5#21439618semidirect

Maximal subgroups

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

subgroupindex
C5⋊#168384622
C5⋊#168384522
C5×#168384652
D7×D4553
#214396185
C2×C5⋊#105877227
#10587721⋊D57
#64422⋊D513