Finite Group Databasebeta

← #28413317

Subgroups — #28413317

Order 14700.

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

normal subgroups only

This group has 252 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⋊(C5⋊#64423)C211yesmaximal, characteristic, complemented
2C5×C5⋊#64423C211yesmaximal, characteristic, complemented
3C5×C5⋊#64424C211yesmaximal, characteristic, complemented
4D35⋊D35C311yesmaximal, characteristic, complemented, Hall
5C52×#3861C2211yescharacteristic, complemented, Hall
6#64424⋊D5—51nomaximal
7#64423⋊D5—51nomaximal
8C7×C5⋊D35C2×C311yescharacteristic, complemented
9C5×C5⋊(C7⋊D7)C2×C311yescharacteristic, complemented
10C5×C7×D35C2×C311yescharacteristic, complemented
11(C5⋊AGL(1,𝔽7))⋊D5—71nomaximal
12D5×C5⋊AGL(1,𝔽7)—71nomaximal
13C5⋊#64423—51no—
14C5⋊#64423—51no—
15C5⋊#64423—102no—
16C5⋊#64423D511yescharacteristic, complemented
17C5⋊#64424D511yescharacteristic, complemented
18C5×#64423—51no—
19C5×#64424—51no—
20C52×C72C22×C311yescharacteristic, complemented, Hall
21C5⋊(C5⋊AGL(1,𝔽7))—71no—
22ΓL(1,𝔽8)×C5⋊D5—71no—
23C5×C5⋊AGL(1,𝔽7)—71no—
24C5×C5⋊AGL(1,𝔽7)—71no—
25C5×C5⋊AGL(1,𝔽7)—71no—
26C5×C5⋊AGL(1,𝔽7)—143no—
27C5×D5×ΓL(1,𝔽8)—71no—
28D35⋊D7—51no—
29D7×D35—51no—
30C5×#3861—22no—
31C5×#3861C2×D511yescharacteristic, complemented
32C5×#3861C2×D511yescharacteristic, complemented
33(C5⋊D5)⋊D7—71no—
34D5×D35—71no—
35#64423⋊C2—251noHall
36C52×ΓL(1,𝔽8)—71no—
37C52×ΓL(1,𝔽8)—71no—
38C52×ΓL(1,𝔽8)—143no—
39C5⋊(C7⋊D7)C3×D511yescharacteristic, complemented
40C7×D35—51no—
41C7×D35—51no—
42C7×D35—102no—
43C7×D35C3×D511yescharacteristic, complemented
44C5×C7⋊D7—51no—
45C5×C7×D7—51no—
46C2×C5⋊AGL(1,𝔽7)—351no—
47D5×AGL(1,𝔽7)—351no—
48D5×AGL(1,𝔽7)—351no—
49D5×AGL(1,𝔽7)—351no—
50C5⋊D35AGL(1,𝔽7)11yescharacteristic, complemented
51C7×C5⋊D5—71no—
52C5×D35—71no—
53C5×D35—71no—
54C5×D35—143no—
55C5×D35AGL(1,𝔽7)11yescharacteristic, complemented
56C5×C7×D5—71no—
57C3×D52—491noHall
58#64424—51no—
59#64423—51no—
60#64423—251no—
61C5×C72—22no—
62C5×C72C2×C3×D511yescharacteristic, complemented
63C5×C72C2×C3×D511yescharacteristic, complemented
64C2×C5×ΓL(1,𝔽8)—351no—
65C5⋊AGL(1,𝔽7)—71no—
66C5⋊AGL(1,𝔽7)—71no—
67C5⋊AGL(1,𝔽7)—71no—
68C5⋊AGL(1,𝔽7)—143no—
69C5⋊AGL(1,𝔽7)—351no—
70C5⋊AGL(1,𝔽7)—351no—
71C5⋊AGL(1,𝔽7)—702no—
72D5×ΓL(1,𝔽8)—71no—
73D5×ΓL(1,𝔽8)—351no—
74D5×ΓL(1,𝔽8)—351no—
75D5×ΓL(1,𝔽8)—702no—
76C5×AGL(1,𝔽7)—351no—
77C5×AGL(1,𝔽7)—351no—
78C5×AGL(1,𝔽7)—351no—
79C5×AGL(1,𝔽7)—703no—
80D72—251noHall
81C52×C7—23no—
82C52×C7C2×AGL(1,𝔽7)11yescharacteristic, complemented
83C52×C7C2×AGL(1,𝔽7)11yescharacteristic, complemented
84C3×C5⋊D5—491no—
85C3×C5×D5—491no—
86C3×C5×D5—491no—
87#3861D5211yescharacteristic, complemented, Hall
88C2×D35—351no—
89D5×D7—351no—
90D5×D7—351no—
91D5×D7—351no—
92C5×ΓL(1,𝔽8)—71no—
93C5×ΓL(1,𝔽8)—71no—
94C5×ΓL(1,𝔽8)—71no—
95C5×ΓL(1,𝔽8)—71no—
96C5×ΓL(1,𝔽8)—143no—
97C5×ΓL(1,𝔽8)—143no—
98C5×ΓL(1,𝔽8)—142no—
99C5×ΓL(1,𝔽8)—142no—
100C5×ΓL(1,𝔽8)—286no—
101D52—491noHall
102C7⋊D7—51no—
103C7×D7—51no—
104C7×D7—251no—
105C2×AGL(1,𝔽7)—1751no—
106C2×AGL(1,𝔽7)—1751no—
107C3×C52—491noHall
108C2×C5×C7—351no—
109D35—51no—
110D35—51no—
111D35—71no—
112D35—71no—
113D35—102no—
114D35—143no—
115D35C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
116C5×D7—51no—
117C5×D7—351no—
118C5×D7—351no—
119C5×D7—703no—
120C7×D5—71no—
121C7×D5—351no—
122C7×D5—351no—
123C7×D5—702no—
124C2×C3×D5—2451no—
125C2×C3×D5—2451no—
126C5⋊D5—71no—
127C5×D5—71no—
128C5×D5—491no—
129C72C3×D5211yescharacteristic, complemented, Sylow
130C2×ΓL(1,𝔽8)—351no—
131C2×ΓL(1,𝔽8)—1751no—
132AGL(1,𝔽7)—351no—
133AGL(1,𝔽7)—351no—
134AGL(1,𝔽7)—351no—
135AGL(1,𝔽7)—703no—
136AGL(1,𝔽7)—1751no—
137C5×C7—22no—
138C5×C7—22no—
139C5×C7—23no—
140C5×C7—23no—
141C5×C7—46no—
142C5×C7D5×AGL(1,𝔽7)11yescharacteristic, complemented
143C5×C7D5×AGL(1,𝔽7)11yescharacteristic, complemented
144C5×C7D5×AGL(1,𝔽7)11yescharacteristic, complemented
145C5×C7C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
146C2×C3×C5—2451no—
147C2×C3×C5—2451no—
148C3×D5—491no—
149C3×D5—491no—
150C3×D5—4902no—
151C3×D5—2451no—
152C3×D5—2451no—
153C2×D7—1751no—
154C2×D7—1751no—
155C52#64423⋊C211yescharacteristic, complemented, Sylow
156ΓL(1,𝔽8)—71no—
157ΓL(1,𝔽8)—71no—
158ΓL(1,𝔽8)—143no—
159C2×D5—2451no—
160C2×D5—2451no—
161C3×C5—491no—
162C3×C5—491no—
163C3×C5—982no—
164C2×C7—351no—
165C2×C7—1751no—
166D7—51no—
167D7—251no—
168D7—351no—
169D7—351no—
170D7—703no—
171C22×C3—12251noHall
172C2×C5—351no—
173C2×C5—2451no—
174D5—71no—
175D5—351no—
176D5—351no—
177D5—491no—
178D5—702no—
179C7—23no—
180C7D5×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
181C7(C5⋊AGL(1,𝔽7))⋊D511yescharacteristic, complemented
182C2×C3—12251no—
183C2×C3—2451no—
184C2×C3—2451no—
185C5—22no—
186C5#64423⋊D511yescharacteristic, complemented
187C5#64424⋊D511yescharacteristic, complemented
188C22—12251noSylow
189C3—491noSylow
190C2—351no—
191C2—1751no—
192C2—2451no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C72C22×C3
LCSG ▷ C52×C72
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊(C5⋊#64423)C2semidirect
2. C5×C5⋊#64423C2semidirect
3. C5×C5⋊#64424C2semidirect
4. D35⋊D35C3semidirect
5. C52×#3861C22semidirect
8. C7×C5⋊D35C2×C3semidirect
9. C5×C5⋊(C7⋊D7)C2×C3semidirect
10. C5×C7×D35C2×C3semidirect
16. C5⋊#64423D5semidirect
17. C5⋊#64424D5semidirect
20. C52×C72C22×C3semidirect
31. C5×#3861C2×D5semidirect
32. C5×#3861C2×D5semidirect
39. C5⋊(C7⋊D7)C3×D5semidirect
43. C7×D35C3×D5semidirect
50. C5⋊D35AGL(1,𝔽7)semidirect
55. C5×D35AGL(1,𝔽7)semidirect
62. C5×C72C2×C3×D5semidirect
63. C5×C72C2×C3×D5semidirect
82. C52×C7C2×AGL(1,𝔽7)semidirect
83. C52×C7C2×AGL(1,𝔽7)semidirect
87. #3861D52semidirect
115. D35C5⋊AGL(1,𝔽7)semidirect
129. C72C3×D52semidirect
142. C5×C7D5×AGL(1,𝔽7)semidirect
143. C5×C7D5×AGL(1,𝔽7)semidirect
144. C5×C7D5×AGL(1,𝔽7)semidirect
145. C5×C7C2×C5⋊AGL(1,𝔽7)semidirect
155. C52#64423⋊C2semidirect
180. C7D5×C5⋊AGL(1,𝔽7)semidirect
181. C7(C5⋊AGL(1,𝔽7))⋊D5semidirect
186. C5#64423⋊D5semidirect
187. C5#64424⋊D5semidirect

Maximal subgroups

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

subgroupindex
C5⋊(C5⋊#64423)2
C5×C5⋊#644232
C5×C5⋊#644242
D35⋊D353
#64424⋊D55
#64423⋊D55
(C5⋊AGL(1,𝔽7))⋊D57
D5×C5⋊AGL(1,𝔽7)7