Finite Group Databasebeta

← #72246196

Subgroups — #72246196

Order 4704.

830 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 187 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
1#72245227C213yesmaximal, complemented
2#72245226C212yesmaximal, complemented
3#41×#3862C211yesmaximal, characteristic, complemented
4C2×C4⋊#64425C213yesmaximal, complemented
5D4×#64425C216yesmaximal, complemented
6#13096739C311yesmaximal, characteristic, complemented, Hall
7C22×#64425C2216yescomplemented
8Q8×#3862C2211yescharacteristic, complemented
9D4×#3862C2213yescomplemented
10C2×C4×#3862C2213yescomplemented
11#11924222C2216yescomplemented
12C4⋊#64425C2216yescomplemented
13C4⋊#64425C2213yescomplemented
14C4×#64425C2216yescomplemented
15#11924218C2211yescharacteristic, complemented
16C72×#41C2×C311yescharacteristic, complemented
17#11716133C2×C312yescomplemented
18D4×C7⋊D7C2×C316yescomplemented
19#11716129C2×C313yescomplemented
20C2×C4⋊(C7⋊D7)C2×C313yescomplemented
21#10621518—71nomaximal
22#10621518—71nomaximal
23C22×#3862C2313yes—
24C2×#64425—46no—
25C2×#64425C2316yes—
26C4×#3862C2311yescharacteristic
27C4×#3862C2313yes—
28#10596875C2312yes—
29C22×C7⋊D7C22×C316yescomplemented
30C72×Q8C22×C311yescharacteristic, complemented
31C72×D4C22×C313yescomplemented
32C2×C4×C72C22×C313yescomplemented
33C22⋊(C7⋊D7)C22×C316yescomplemented
34C4⋊(C7⋊D7)C22×C313yescomplemented
35C4⋊(C7⋊D7)C22×C316yescomplemented
36C4×C7⋊D7C22×C316yescomplemented
37#87648C22×C311yescharacteristic, complemented
38#41×ΓL(1,𝔽8)—71no—
39#41×ΓL(1,𝔽8)—141no—
40#66659—72no—
41#66659—72no—
42D4×AGL(1,𝔽7)—76no—
43D4×AGL(1,𝔽7)—76no—
44#1157⋊C3—73no—
45#1157⋊C3—73no—
46C2×C4⋊AGL(1,𝔽7)—73no—
47C2×C4⋊AGL(1,𝔽7)—73no—
48C2×#3862—23no—
49C2×#3862C2411yescharacteristic
50#64425—26no—
51#6553—71no—
52#6553—71no—
53#6553—212no—
54C22×C72C23×C313yes—
55C2×C7⋊D7—46no—
56C2×C7⋊D7C23×C316yes—
57C4×C72C23×C311yescharacteristic
58C4×C72C23×C313yes—
59C7⋊Dic7C23×C312yes—
60C22×AGL(1,𝔽7)—76no—
61C22×AGL(1,𝔽7)—76no—
62Q8×ΓL(1,𝔽8)—71no—
63Q8×ΓL(1,𝔽8)—141no—
64D4×ΓL(1,𝔽8)—73no—
65D4×ΓL(1,𝔽8)—143no—
66C2×C4×ΓL(1,𝔽8)—73no—
67C2×C4×ΓL(1,𝔽8)—143no—
68(C22⋊D7)⋊C3—76no—
69(C22⋊D7)⋊C3—76no—
70C4⋊AGL(1,𝔽7)—73no—
71C4⋊AGL(1,𝔽7)—76no—
72C4⋊AGL(1,𝔽7)—73no—
73C4⋊AGL(1,𝔽7)—76no—
74C4×AGL(1,𝔽7)—76no—
75C4×AGL(1,𝔽7)—76no—
76ΓL(1,𝔽8)⋊Q8—71no—
77ΓL(1,𝔽8)⋊Q8—71no—
78#3862H(5,𝔽2)11yescharacteristic, complemented, Hall
79C7×#41—32no—
80C7×#41AGL(1,𝔽7)11yescharacteristic, complemented
81C7×#41AGL(1,𝔽7)11yescharacteristic, complemented
82#1161—72no—
83#1161—72no—
84#1161—214no—
85D4×D7—76no—
86D4×D7—76no—
87D4×D7—2112no—
88#1157—73no—
89#1157—73no—
90#1157—216no—
91C2×D28—73no—
92C2×D28—73no—
93C2×D28—216no—
94C2×C72—23no—
95C2×C72C24×C311yescharacteristic
96C7⋊D7—26no—
97C3×H(5,𝔽2)—491noHall
98C22×ΓL(1,𝔽8)—73no—
99C22×ΓL(1,𝔽8)—143no—
100C2×AGL(1,𝔽7)—76no—
101C2×AGL(1,𝔽7)—76no—
102C2×AGL(1,𝔽7)—286no—
103C2×AGL(1,𝔽7)—286no—
104C4×ΓL(1,𝔽8)—71no—
105C4×ΓL(1,𝔽8)—73no—
106C4×ΓL(1,𝔽8)—141no—
107C4×ΓL(1,𝔽8)—143no—
108ΓL(1,𝔽8)⋊C4—72no—
109ΓL(1,𝔽8)⋊C4—72no—
110C22×D7—76no—
111C22×D7—76no—
112C22×D7—2112no—
113C7×Q8—32no—
114C7×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
115C7×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
116C7×D4—36no—
117C7×D4C2×AGL(1,𝔽7)13yescomplemented
118C7×D4C2×AGL(1,𝔽7)13yescomplemented
119C2×C4×C7—36no—
120C2×C4×C7C2×AGL(1,𝔽7)13yescomplemented
121C2×C4×C7C2×AGL(1,𝔽7)13yescomplemented
122C22⋊D7—76no—
123C22⋊D7—76no—
124C22⋊D7—2112no—
125D28—73no—
126D28—73no—
127D28—76no—
128D28—76no—
129D28—216no—
130D28—2112no—
131C4×D7—76no—
132C4×D7—76no—
133C4×D7—2112no—
134Dic14—71no—
135Dic14—71no—
136Dic14—212no—
137C72C3×H(5,𝔽2)11yescharacteristic, complemented, Sylow
138C3×#41—492no—
139C3×#41—493no—
140C3×#41—981no—
141C2×C3×D4—496no—
142C2×C3×D4—493no—
143C2×ΓL(1,𝔽8)—71no—
144C2×ΓL(1,𝔽8)—141no—
145C2×ΓL(1,𝔽8)—143no—
146C2×ΓL(1,𝔽8)—283no—
147AGL(1,𝔽7)—146no—
148AGL(1,𝔽7)—146no—
149H(5,𝔽2)—491noSylow
150C22×C7—36no—
151C22×C7C22×AGL(1,𝔽7)13yes—
152C22×C7C22×AGL(1,𝔽7)13yes—
153C2×D7—76no—
154C2×D7—76no—
155C2×D7—2112no—
156C2×D7—286no—
157C2×D7—286no—
158C2×D7—8412no—
159C4×C7—32no—
160C4×C7—36no—
161C4×C7C22×AGL(1,𝔽7)11yescharacteristic
162C4×C7C22×AGL(1,𝔽7)11yescharacteristic
163C4×C7C22×AGL(1,𝔽7)13yes—
164C4×C7C22×AGL(1,𝔽7)13yes—
165Dic7—72no—
166Dic7—72no—
167Dic7—214no—
168C23×C3—496no—
169C3×Q8—491no—
170C3×Q8—981no—
171C3×D4—496no—
172C3×D4—496no—
173C3×D4—493no—
174C3×D4—983no—
175C2×C3×C4—496no—
176C2×C3×C4—983no—
177ΓL(1,𝔽8)—71no—
178ΓL(1,𝔽8)—141no—
179#41—492no—
180#41—493no—
181#41#6442511yescharacteristic, complemented
182C2×D4—496no—
183C2×D4—493no—
184C2×C7—23no—
185C2×C7—23no—
186C2×C7—32no—
187C2×C7—66no—
188C2×C7C23×AGL(1,𝔽7)11yescharacteristic
189C2×C7C23×AGL(1,𝔽7)11yescharacteristic
190D7—146no—
191D7—146no—
192D7—4212no—
193C22×C3—496no—
194C22×C3—983no—
195C22×C3—1966no—
196C3×C4—492no—
197C3×C4—983no—
198C3×C4—981no—
199C23—496no—
200Q8—491no—
201Q8C2×#6442511yescharacteristic, complemented
202D4—496no—
203D4—496no—
204D4—493no—
205D4C2×#6442513yescomplemented
206C2×C4—496no—
207C2×C4C2×#6442513yescomplemented
208C7—32no—
209C7#1062151811yescharacteristic, complemented
210C7#1062151811yescharacteristic, complemented
211C2×C3—986no—
212C2×C3—981no—
213C2×C3—1963no—
214C22—496no—
215C22—1966no—
216C22C22×#6442513yes—
217C4—492no—
218C4C22×#6442513yes—
219C4C22×#6442511yescharacteristic
220C3—981noSylow
221C2—23no—
222C2—986no—
223C2C23×#6442511yescharacteristic
subgroupG/H
center, FrattiniC2C23×#64425
commutator, socle, cosocleC2×C72C24×C3
FittingC72×#41C2×C3
hypercenter#41#64425
hypocenterC72C3×H(5,𝔽2)
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #72245227C2semidirect
2. #72245226C2semidirect
3. #41×#3862C2semidirect
4. C2×C4⋊#64425C2semidirect
5. D4×#64425C2semidirect
6. #13096739C3semidirect
7. C22×#64425C22semidirect
8. Q8×#3862C22semidirect
9. D4×#3862C22semidirect
10. C2×C4×#3862C22semidirect
11. #11924222C22semidirect
12. C4⋊#64425C22semidirect
13. C4⋊#64425C22semidirect
14. C4×#64425C22semidirect
15. #11924218C22semidirect
16. C72×#41C2×C3semidirect
17. #11716133C2×C3semidirect
18. D4×C7⋊D7C2×C3semidirect
19. #11716129C2×C3semidirect
20. C2×C4⋊(C7⋊D7)C2×C3semidirect
29. C22×C7⋊D7C22×C3semidirect
30. C72×Q8C22×C3semidirect
31. C72×D4C22×C3semidirect
32. C2×C4×C72C22×C3semidirect
33. C22⋊(C7⋊D7)C22×C3semidirect
34. C4⋊(C7⋊D7)C22×C3semidirect
35. C4⋊(C7⋊D7)C22×C3semidirect
36. C4×C7⋊D7C22×C3semidirect
37. #87648C22×C3semidirect
78. #3862H(5,𝔽2)semidirect
80. C7×#41AGL(1,𝔽7)semidirect
81. C7×#41AGL(1,𝔽7)semidirect
114. C7×Q8C2×AGL(1,𝔽7)semidirect
115. C7×Q8C2×AGL(1,𝔽7)semidirect
117. C7×D4C2×AGL(1,𝔽7)semidirect
118. C7×D4C2×AGL(1,𝔽7)semidirect
120. C2×C4×C7C2×AGL(1,𝔽7)semidirect
121. C2×C4×C7C2×AGL(1,𝔽7)semidirect
137. C72C3×H(5,𝔽2)semidirect
181. #41#64425semidirect
201. Q8C2×#64425semidirect
205. D4C2×#64425semidirect
207. C2×C4C2×#64425semidirect
209. C7#10621518semidirect
210. C7#10621518semidirect

Maximal subgroups

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

subgroupindex
#722452272
#722452262
#41×#38622
C2×C4⋊#644252
D4×#644252
#130967393
#106215187