Finite Group Databasebeta

← #11744155

Subgroups — #11744155

Order 896.

1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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
1C7×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2#90374C213yesmaximal, complemented
3#90365C216yesmaximal, complemented
4#90361C216yesmaximal, complemented
5#90359C213yesmaximal, complemented
6#90283C216yesmaximal, complemented
7#90281C216yesmaximal, complemented
8C22×C22⋊D7C2216yescomplemented
9C2×#1161C2213yescomplemented
10C2×#1159C2213yescomplemented
11C2×D4×D7C2216yescomplemented
12C7×#124C2213yescomplemented
13C7×#124C2216yescomplemented
14C7×#120C2216yescomplemented
15#6516C2212yescomplemented
16#6510C2216yescomplemented
17#6507C2213yescomplemented
18#6503C2213yes—
19#6503C2216yescomplemented
20#6502C22112yescomplemented
21#6500C2216yescomplemented
22#6448C22112yescomplemented
23#6447C22112yescomplemented
24#6447C22112yescomplemented
25#6445C22112yescomplemented
26#6445C2216yescomplemented
27(C22⋊D7)⋊C4C22112yescomplemented
28#6440C2216yescomplemented
29#6438C2213yescomplemented
30#6437C2212yescomplemented
31#6435C2216yescomplemented
32C42×D7C2211yescharacteristic, complemented
33#3549—71nomaximal, Sylow
34C24×C7C2312yes—
35C23×D7C2316yes—
36C2×C7×Q8C2313yes—
37C2×C7×D4C2313yes—
38C2×C7×D4C2316yes—
39C2×C22⋊D7—412no—
40C2×C22⋊D7C23112yes—
41C2×C22⋊D7C2316yes—
42C2×C22⋊D7C23112yes—
43C22×Dic7C2316yes—
44#1161—46no—
45#1159—46no—
46D4×D7—412no—
47C2×D28C2316yes—
48C2×D28C2316yes—
49C2×D28C2313yes—
50C2×C4×D7C2313yes—
51C2×C4×D7C2316yes—
52C2×Dic14C2313yes—
53C7×C22⋊C4C2316yes—
54C7×C22⋊C4C23112yes—
55C42×C7C2311yescharacteristic
56C42×C7C2312yes—
57C22⋊Dic7C2316yes—
58(C2×D7)⋊C4C23112yes—
59(C2×D7)⋊C4C23112yes—
60Dic7⋊C4C23112yes—
61C4×Dic7C2313yes—
62C4×Dic7C2316yes—
63H(3,𝔽4)—73no—
64H(3,𝔽4)D711yescharacteristic, complemented
65#555—76no—
66D42—76no—
67#535—76no—
68#535—76no—
69#532—73no—
70C23×C7—46no—
71C23×C7C2416yes—
72C22×D7—412no—
73C22×D7—46no—
74C22×D7C2416yes—
75C22×D7C2411yescharacteristic
76C7×Q8—43no—
77C7×D4—46no—
78C7×D4—43no—
79C2×C4×C7C2416yes—
80C2×C4×C7C2413yes—
81C22⋊D7—224no—
82C22⋊D7—46no—
83C22⋊D7—412no—
84C2×Dic7—212no—
85C2×Dic7C2416yes—
86C2×Dic7C2413yes—
87D28—46no—
88D28—43no—
89D28—46no—
90C4×D7—212no—
91C4×D7—26no—
92Dic14—43no—
93C2×#41—73no—
94C2×#41—73no—
95C22×D4—76no—
96C22×D4—76no—
97C4⋊Q8—73no—
98#127—73no—
99#127—76no—
100#127—72no—
101#124—76no—
102#124—712no—
103#124—73no—
104#124C2×D713yescomplemented
105#124C2×D716yescomplemented
106#121—712no—
107#121—712no—
108#121—712no—
109#120—72no—
110#120—76no—
111#120—76no—
112#120—712no—
113#120C2×D716yescomplemented
114C4×D4—712no—
115#117—76no—
116#117—76no—
117C2×C42—71no—
118C22×C7—26no—
119C22×C7—212no—
120C22×C7—84no—
121C22×C7C2511yescharacteristic
122C2×D7—23no—
123C2×D7—26no—
124C2×D7—212no—
125C2×D7—812no—
126C4×C7—23no—
127C4×C7—26no—
128Dic7—26no—
129Dic7H(5,𝔽2)16yes—
130C24—76no—
131C24C22×D712yes—
132#41—286no—
133#41—286no—
134C2×Q8—73no—
135C2×Q8C22×D713yes—
136C2×D4—712no—
137C2×D4—712no—
138C2×D4—76no—
139C2×D4—73no—
140C2×D4—76no—
141C2×D4—76no—
142C2×D4—2812no—
143C2×D4—2812no—
144C2×D4C22×D713yes—
145C2×D4C22×D716yes—
146C22×C4—76no—
147C22×C4—73no—
148C22×C4—76no—
149C4⋊C4—712no—
150C22⋊C4—76no—
151C22⋊C4—712no—
152C22⋊C4—712no—
153C22⋊C4C22×D716yes—
154C22⋊C4C22×D7112yes—
155C42—73no—
156C42—76no—
157C42C22×D711yescharacteristic
158C42C22×D712yes—
159C2×C7—46no—
160C2×C7C2×H(5,𝔽2)13yes—
161D7—41no—
162D7—46no—
163C23—46no—
164C23—76no—
165C23—71no—
166C23—286no—
167C23—2812no—
168C23C23×D716yes—
169Q8—43no—
170Q8—283no—
171D4—46no—
172D4—43no—
173D4—1424no—
174D4—286no—
175D4—286no—
176D4—283no—
177D4—2812no—
178D4—286no—
179C2×C4—73no—
180C2×C4—76no—
181C2×C4—1412no—
182C2×C4—146no—
183C2×C4—1412no—
184C2×C4C23×D713yes—
185C2×C4C23×D716yes—
186C7#354911yescharacteristic, complemented, Sylow
187C22—26no—
188C22—212no—
189C22—84no—
190C22—143no—
191C22—146no—
192C22—1412no—
193C22—5612no—
194C22C24×D711yescharacteristic
195C4—26no—
196C4—23no—
197C4—76no—
198C4—146no—
199C2—46no—
200C2—286no—
201C2—281no—
202C2D7×H(5,𝔽2)13yes—
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×H(3,𝔽4)C2
hypercenterH(3,𝔽4)D7
hypocenterC7#3549
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C7×H(3,𝔽4)C2semidirect
2. #90374C2semidirect
3. #90365C2semidirect
4. #90361C2semidirect
5. #90359C2semidirect
6. #90283C2semidirect
7. #90281C2semidirect
8. C22×C22⋊D7C22semidirect
9. C2×#1161C22semidirect
10. C2×#1159C22semidirect
11. C2×D4×D7C22semidirect
12. C7×#124C22semidirect
13. C7×#124C22semidirect
14. C7×#120C22semidirect
15. #6516C22semidirect
16. #6510C22semidirect
17. #6507C22semidirect
19. #6503C22semidirect
20. #6502C22semidirect
21. #6500C22semidirect
22. #6448C22semidirect
23. #6447C22semidirect
24. #6447C22semidirect
25. #6445C22semidirect
26. #6445C22semidirect
27. (C22⋊D7)⋊C4C22semidirect
28. #6440C22semidirect
29. #6438C22semidirect
30. #6437C22semidirect
31. #6435C22semidirect
32. C42×D7C22semidirect
64. H(3,𝔽4)D7semidirect
104. #124C2×D7semidirect
105. #124C2×D7semidirect
113. #120C2×D7semidirect
186. C7#3549semidirect

Maximal subgroups

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

subgroupindex
C7×H(3,𝔽4)2
#903742
#903652
#903612
#903592
#902832
#902812
#35497