Finite Group Databasebeta

← C2×#72228755 #72259850

Subgroups — C2×#72228755 #72259850

Order 4704.

1640 conjugacy classes of subgroups; only the 311 normal subgroups are shown.

normal subgroups only

This group has 1640 conjugacy classes of subgroups — too many to draw. Only its 311 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
1C2×#11924213C211yesmaximal, characteristic, complemented
2#72228755C218yesmaximal, direct product factor
3C22×C4×#3861C211yesmaximal, characteristic, complemented
4C2×C4×#64424C212yesmaximal, complemented
5C2×C4⋊#64424C211yesmaximal, characteristic, complemented
6C2×#11924217C212yesmaximal, complemented
7C2×#11716129C311yesmaximal, characteristic, complemented, Hall
8C23×#3861C2211yescharacteristic
9C22×#64424C2212yes—
10C2×C4×#3861C2211yescharacteristic
11C2×C4×#3861C2211yescharacteristic
12C2×C4×#3861C2214yescomplemented
13#11924217C2218yescomplemented
14C2×#10596874C2212yes—
15C4⋊#64424C2214yescomplemented
16C4×#64424C2218yescomplemented
17#11924213C2214yescomplemented
18C22×C4×C72C2×C311yescharacteristic, complemented
19C2×C22⋊(C7⋊D7)C2×C312yescomplemented
20#11716129C2×C318yescomplemented
21C2×C4⋊(C7⋊D7)C2×C311yescharacteristic, complemented
22C2×C4×C7⋊D7C2×C312yescomplemented
23C2×#87648C2×C311yescharacteristic, complemented
24C2×#1157⋊C3—78nomaximal
25C22×#3861—22no—
26C22×#3861C2312yes—
27C22×#3861C2311yescharacteristic
28C2×#64424—24no—
29C2×#64424C2314yes—
30C4×#3861C2312yes—
31C4×#3861C2312yes—
32#10596874C2314yes—
33C23×C72C22×C311yescharacteristic
34C22×C7⋊D7C22×C312yes—
35C2×C4×C72C22×C311yescharacteristic
36C2×C4×C72C22×C311yescharacteristic
37C2×C4×C72C22×C314yescomplemented
38C22⋊(C7⋊D7)C22×C318yescomplemented
39C2×C7⋊Dic7C22×C312yes—
40C4⋊(C7⋊D7)C22×C314yescomplemented
41C4×C7⋊D7C22×C318yescomplemented
42#87648C22×C314yescomplemented
43C22×C4×ΓL(1,𝔽8)—148no—
44C2×(C22⋊D7)⋊C3—716no—
45#1157⋊C3—764no—
46C2×C4⋊AGL(1,𝔽7)—78no—
47C2×C4×AGL(1,𝔽7)—716no—
48C2×ΓL(1,𝔽8)⋊Q8—78no—
49C2×#3861—22no—
50C2×#3861#4112yescomplemented
51C2×#3861C2411yescharacteristic
52#64424—24no—
53C2×#1157—78no—
54C22×C72—22no—
55C22×C72C23×C312yes—
56C22×C72C23×C311yescharacteristic
57C2×C7⋊D7—24no—
58C2×C7⋊D7C23×C314yes—
59C4×C72C23×C312yes—
60C4×C72C23×C312yes—
61C7⋊Dic7C23×C314yes—
62C23×ΓL(1,𝔽8)—148no—
63C22×AGL(1,𝔽7)—716no—
64C2×C4×ΓL(1,𝔽8)—148no—
65C2×C4×ΓL(1,𝔽8)—148no—
66C2×C4×ΓL(1,𝔽8)—1432no—
67(C22⋊D7)⋊C3—764no—
68C2×ΓL(1,𝔽8)⋊C4—716no—
69C4⋊AGL(1,𝔽7)—732no—
70C4×AGL(1,𝔽7)—764no—
71ΓL(1,𝔽8)⋊Q8—732no—
72#3861C2×#4111yescharacteristic, complemented, Hall
73C22×C4×C7AGL(1,𝔽7)18yescomplemented
74C2×C22⋊D7—716no—
75#1157—764no—
76C2×D28—78no—
77C2×C4×D7—716no—
78C2×Dic14—78no—
79C2×C72—22no—
80C2×C72C3×#4112yescomplemented
81C2×C72C24×C311yescharacteristic
82C7⋊D7—24no—
83C2×C3×#41—491noHall
84C22×ΓL(1,𝔽8)—1416no—
85C22×ΓL(1,𝔽8)—1416no—
86C22×ΓL(1,𝔽8)—148no—
87C2×AGL(1,𝔽7)—732no—
88C2×AGL(1,𝔽7)—1432no—
89C4×ΓL(1,𝔽8)—1416no—
90C4×ΓL(1,𝔽8)—1416no—
91ΓL(1,𝔽8)⋊C4—732no—
92C23×C7C2×AGL(1,𝔽7)18yes—
93C22×D7—716no—
94C2×C4×C7C2×AGL(1,𝔽7)18yes—
95C2×C4×C7C2×AGL(1,𝔽7)18yes—
96C2×C4×C7C2×AGL(1,𝔽7)132yescomplemented
97C22⋊D7—764no—
98C2×Dic7—716no—
99D28—732no—
100C4×D7—764no—
101Dic14—732no—
102C72C2×C3×#4111yescharacteristic, complemented, Sylow
103C3×#41—498no—
104C2×C3×Q8—491no—
105C2×C3×D4—492no—
106C2×C3×D4—491no—
107C22×C3×C4—492no—
108C22×C3×C4—981no—
109C2×ΓL(1,𝔽8)—1416no—
110C2×ΓL(1,𝔽8)—1416no—
111C2×ΓL(1,𝔽8)—148no—
112AGL(1,𝔽7)—1432no—
113C2×#41—491noSylow
114C22×C7—216no—
115C22×C7C22×AGL(1,𝔽7)116yes—
116C22×C7C22×AGL(1,𝔽7)18yes—
117C2×D7—732no—
118C2×D7—1432no—
119C4×C7C22×AGL(1,𝔽7)116yes—
120C4×C7C22×AGL(1,𝔽7)116yes—
121Dic7—732no—
122C23×C3—492no—
123C23×C3—981no—
124C3×Q8—494no—
125C3×D4—498no—
126C3×D4—494no—
127C2×C3×C4—492no—
128C2×C3×C4—498no—
129C2×C3×C4—981no—
130C2×C3×C4—981no—
131C2×C3×C4—984no—
132ΓL(1,𝔽8)—148no—
133#41—498no—
134C2×Q8—491no—
135C2×D4—492no—
136C2×D4—491no—
137C22×C4—492no—
138C22×C4#6442411yescharacteristic, complemented
139C2×C7—216no—
140C2×C7#1157⋊C3116yescomplemented
141C2×C7C23×AGL(1,𝔽7)18yes—
142D7—1432no—
143C22×C3—494no—
144C22×C3—982no—
145C22×C3—984no—
146C22×C3—982no—
147C22×C3—981no—
148C3×C4—494no—
149C3×C4—982no—
150C3×C4—982no—
151C23—492no—
152C23C2×#6442411yescharacteristic
153Q8—494no—
154D4—498no—
155D4—494no—
156C2×C4—492no—
157C2×C4—498no—
158C2×C4C2×#6442411yescharacteristic
159C2×C4C2×#6442411yescharacteristic
160C2×C4C2×#6442414yescomplemented
161C7C2×#1157⋊C318yescomplemented
162C2×C3—982no—
163C2×C3—984no—
164C2×C3—982no—
165C2×C3—981no—
166C22—22no—
167C22—494no—
168C22—984no—
169C22C22×#6442412yes—
170C22C22×#6442411yescharacteristic
171C4—494no—
172C4C22×#6442412yes—
173C4C22×#6442412yes—
174C3—981noSylow
175C2—22no—
176C2—984no—
177C2C23×#6442411yescharacteristic
178C2#7222875512yesdirect product factor
subgroupG/H
centerC2×C4C2×#64424
commutator, cosocleC2×C72C24×C3
FrattiniC2C23×#64424
FittingC22×C4×C72C2×C3
socleC22×C72C23×C3
hypercenterC22×C4#64424
hypocenterC72C2×C3×#41
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#11924213C2semidirect
2. #72228755C2direct (N × H)
3. C22×C4×#3861C2semidirect
4. C2×C4×#64424C2semidirect
5. C2×C4⋊#64424C2semidirect
6. C2×#11924217C2semidirect
7. C2×#11716129C3semidirect
12. C2×C4×#3861C22semidirect
13. #11924217C22semidirect
15. C4⋊#64424C22semidirect
16. C4×#64424C22semidirect
17. #11924213C22semidirect
18. C22×C4×C72C2×C3semidirect
19. C2×C22⋊(C7⋊D7)C2×C3semidirect
20. #11716129C2×C3semidirect
21. C2×C4⋊(C7⋊D7)C2×C3semidirect
22. C2×C4×C7⋊D7C2×C3semidirect
23. C2×#87648C2×C3semidirect
37. C2×C4×C72C22×C3semidirect
38. C22⋊(C7⋊D7)C22×C3semidirect
40. C4⋊(C7⋊D7)C22×C3semidirect
41. C4×C7⋊D7C22×C3semidirect
42. #87648C22×C3semidirect
50. C2×#3861#41semidirect
72. #3861C2×#41semidirect
73. C22×C4×C7AGL(1,𝔽7)semidirect
80. C2×C72C3×#41semidirect
96. C2×C4×C7C2×AGL(1,𝔽7)semidirect
102. C72C2×C3×#41semidirect
138. C22×C4#64424semidirect
140. C2×C7#1157⋊C3semidirect
160. C2×C4C2×#64424semidirect
161. C7C2×#1157⋊C3semidirect
178. C2#72228755direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#119242132
#722287552
C22×C4×#38612
C2×C4×#644242
C2×C4⋊#644242
C2×#119242172
C2×#117161293
C2×#1157⋊C37