Finite Group Databasebeta

← #31800838

Subgroups — #31800838

Order 17728.

810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 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#22579928C212yesmaximal, complemented
2#22579927C214yesmaximal, complemented
3#22579926C214yesmaximal, complemented
4#22579925C216yesmaximal, complemented
5C2×#17984563C216yesmaximal, complemented
6#41×D277C218yesmaximal, complemented
7C2×#41×C277C211yesmaximal, characteristic, complemented
8#17984564C2218yescomplemented
9#17984563C22112yescomplemented
10#17984563C2214yescomplemented
11#17984563C22112yescomplemented
12#17984563C22112yescomplemented
13#17984562C22124yescomplemented
14C2×Dic554C2216yescomplemented
15Q8×D277C2218yescomplemented
16C2×D1108C2216yescomplemented
17C2×C22⋊D277C22112yescomplemented
18D4×D277C22124yescomplemented
19C2×C4×D277C22112yescomplemented
20#41×C277C2214yescomplemented
21#41×C277C2214yescomplemented
22C2×Q8×C277C2211yescharacteristic, complemented
23C2×D4×C277C2213yescomplemented
24C22×C4×C277C2213yescomplemented
25C22×D277C23112yes—
26Dic554C23112yes—
27Dic554C2314yes—
28D1108C23112yes—
29D1108C2314yes—
30C22⋊D277C23124yes—
31C22⋊D277C2318yes—
32C2×Dic277C23112yes—
33C4×D277C2318yes—
34C4×D277C23124yes—
35C23×C277C2313yes—
36Q8×C277C2314yes—
37D4×C277C23112yes—
38C2×C4×C277C2311yescharacteristic
39C2×C4×C277C2313yes—
40C2×C4×C277C2316yes—
41C2×C4×C277C2316yes—
42C22×C277—43no—
43C22×C277C2411yescharacteristic
44C22×C277C2416yes—
45C2×D277—412no—
46C2×D277C2418yes—
47C4×C277C2411yescharacteristic
48C4×C277C2416yes—
49C4×C277C2411yescharacteristic
50Dic277C2418yes—
51C2×C277—26no—
52C2×C277—21no—
53C2×C277C2511yescharacteristic
54D277—28no—
55C277#58511yescharacteristic, complemented, Sylow
56#585—2771nomaximal, Sylow
572−1+4—2774no—
582−1+4—2772no—
59H(5,𝔽2)—2774no—
60H(5,𝔽2)—2776no—
61C2×#41—2778no—
62C2×#41—2776no—
63C2×#41D27711yescharacteristic, complemented
64#41—2778no—
65#41—27724no—
66#41—27712no—
67#41—2774no—
68#41—27712no—
69#41—27712no—
70#41C2×D27714yescomplemented
71#41C2×D27714yescomplemented
72C2×Q8—2778no—
73C2×Q8—2776no—
74C2×Q8C2×D27711yescharacteristic, complemented
75C2×D4—27712no—
76C2×D4—27724no—
77C2×D4—2776no—
78C2×D4C2×D27713yescomplemented
79C22×C4—27712no—
80C22×C4C2×D27713yescomplemented
81C23—27712no—
82C23C22×D27713yes—
83Q8—27712no—
84Q8—2774no—
85Q8C22×D27714yes—
86D4—27724no—
87D4—2778no—
88D4—27712no—
89D4—2774no—
90D4C22×D277112yes—
91C2×C4—27712no—
92C2×C4—27724no—
93C2×C4—2778no—
94C2×C4C22×D27716yes—
95C2×C4C22×D27713yes—
96C2×C4C22×D27716yes—
97C2×C4C22×D27711yescharacteristic
98C22—43no—
99C22—2778no—
100C22—110812no—
101C22C23×D27716yes—
102C22C23×D27711yescharacteristic
103C4—2778no—
104C4C23×D27716yes—
105C4C23×D27711yescharacteristic
106C4C23×D27711yescharacteristic
107C2—21no—
108C2—26no—
109C2—5548no—
110C2C24×D27711yescharacteristic
subgroupG/H
centerC4C23×D277
commutator, socle, cosocleC2×C277C25
FrattiniC2C24×D277
FittingC2×#41×C277C2
hypercenterC2×#41D277
hypocenterC277#585
LCSG ▷ C2×C277 ▷ C277
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #22579928C2semidirect
2. #22579927C2semidirect
3. #22579926C2semidirect
4. #22579925C2semidirect
5. C2×#17984563C2semidirect
6. #41×D277C2semidirect
7. C2×#41×C277C2semidirect
8. #17984564C22semidirect
9. #17984563C22semidirect
10. #17984563C22semidirect
11. #17984563C22semidirect
12. #17984563C22semidirect
13. #17984562C22semidirect
14. C2×Dic554C22semidirect
15. Q8×D277C22semidirect
16. C2×D1108C22semidirect
17. C2×C22⋊D277C22semidirect
18. D4×D277C22semidirect
19. C2×C4×D277C22semidirect
20. #41×C277C22semidirect
21. #41×C277C22semidirect
22. C2×Q8×C277C22semidirect
23. C2×D4×C277C22semidirect
24. C22×C4×C277C22semidirect
55. C277#585semidirect
63. C2×#41D277semidirect
70. #41C2×D277semidirect
71. #41C2×D277semidirect
74. C2×Q8C2×D277semidirect
78. C2×D4C2×D277semidirect
80. C22×C4C2×D277semidirect

Maximal subgroups

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

subgroupindex
#225799282
#225799272
#225799262
#225799252
C2×#179845632
#41×D2772
C2×#41×C2772
#585277