Finite Group Databasebeta

← #31800648

Subgroups — #31800648

Order 17728.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#17984563C211yesmaximal, characteristic, complemented
2C22×D1108C211yesmaximal, characteristic, complemented
3#22579868C211yesmaximal, characteristic, complemented
4#22579865C212yesmaximal, complemented
5#22579859C211yesmaximal, characteristic, complemented
6#22579854C211yesmaximal, characteristic, complemented
7#22579853C212yesmaximal, complemented
8#22579848C212yesmaximal, complemented
9C4×D1108C211yesmaximal, characteristic, complemented
10C22⋊C4×D277C212yesmaximal, complemented
11C4×D4×C277C211yesmaximal, characteristic, complemented
12C23×D277C2212yes—
13#17984563—24no—
14C2×Dic554C2211yescharacteristic, complemented
15C2×D1108—24no—
16C2×D1108C2212yes—
17C2×D1108C2211yescharacteristic
18C2×D1108C2211yescharacteristic
19C2×C22⋊D277C2212yescomplemented
20C2×C22⋊D277C2212yescomplemented
21C2×C4×D277C2212yescomplemented
22C2×C4×D277C2212yescomplemented
23C4⋊Dic277C2211yescharacteristic, complemented
24#17984542C2212yescomplemented
25C22⋊Dic277C2212yes—
26#17984540C2212yescomplemented
27#17984540C2212yescomplemented
28#17984540C2212yescomplemented
29#17984540C2212yescomplemented
30C2×D4×C277C2211yescharacteristic
31C22×C4×C277C2211yescharacteristic
32C22×C4×C277C2211yescharacteristic, complemented
33C4⋊C4×C277C2211yescharacteristic, complemented
34C22⋊C4×C277C2211yescharacteristic
35C22⋊C4×C277C2211yescharacteristic, complemented
36C42×C277C2211yescharacteristic, complemented
37C22×D277—24no—
38C22×D277—24no—
39C22×D277—42no—
40C22×D277C2312yes—
41C22×D277C2312yes—
42Dic554—22no—
43D1108—24no—
44D1108—22no—
45D1108D414yescomplemented
46C22⋊D277—24no—
47C22⋊D277—42no—
48C2×Dic277C2312yes—
49C2×Dic277C2312yes—
50C4×D277—24no—
51C4×D277—42no—
52C23×C277C2311yescharacteristic
53C23×C277C2311yescharacteristic
54D4×C277—22no—
55C2×C4×C277—22no—
56C2×C4×C277—22no—
57C2×C4×C277C2311yescharacteristic
58C2×C4×C277C2311yescharacteristic
59C2×C4×C277C2311yescharacteristic
60C2×C4×C277C2311yescharacteristic
61C2×C4×C277C2311yescharacteristic
62C22×C277—21no—
63C22×C277—21no—
64C22×C277—21no—
65C22×C277—21no—
66C22×C277—21no—
67C22×C277#4112yescomplemented
68C22×C277C2411yescharacteristic
69C2×D277—22no—
70C2×D277—22no—
71C2×D277—22no—
72C2×D277—22no—
73C2×D277—22no—
74C2×D277—48no—
75C2×D277C2×D414yes—
76C4×C277—21no—
77C4×C277—21no—
78C4×C277—21no—
79C4×C277—21no—
80C4×C277C2×D412yes—
81Dic277—22no—
82Dic277—22no—
83C2×C277—22no—
84C2×C277—41no—
85C2×C277C22×D411yescharacteristic
86C2×C277C2×#4111yescharacteristic
87C2×C277H(5,𝔽2)11yescharacteristic
88D277—24no—
89D277—42no—
90C277#54611yescharacteristic, complemented, Sylow
91#546—2771nomaximal, Sylow
92C2×#41—2771no—
93C22×D4—2771no—
94#124—2771no—
95#123—2772no—
96C22⋊Q8—2771no—
97#121—2771no—
98#121—2772no—
99#120—2772no—
100C4×D4—2771no—
101C4×D4D27711yescharacteristic, complemented
102C2×C22⋊C4—2772no—
103C24—2772no—
104#41—5544no—
105C2×Q8—2771no—
106C2×D4—2772no—
107C2×D4—2772no—
108C2×D4—2771no—
109C2×D4—2772no—
110C2×D4—2771no—
111C2×D4—5544no—
112C2×D4C2×D27711yescharacteristic
113C22×C4—2772no—
114C22×C4—2772no—
115C22×C4C2×D27711yescharacteristic
116C22×C4C2×D27711yescharacteristic, complemented
117C4⋊C4—2772no—
118C4⋊C4—2771no—
119C4⋊C4C2×D27711yescharacteristic, complemented
120C22⋊C4—2772no—
121C22⋊C4—2772no—
122C22⋊C4—2772no—
123C22⋊C4—2772no—
124C22⋊C4—2772no—
125C22⋊C4C2×D27711yescharacteristic
126C22⋊C4C2×D27711yescharacteristic, complemented
127C42C2×D27711yescharacteristic, complemented
128C23—2772no—
129C23—2772no—
130C23—5544no—
131C23—5544no—
132C23—11082no—
133C23C22×D27711yescharacteristic
134C23C22×D27711yescharacteristic
135Q8—5542no—
136D4—22no—
137D4—2774no—
138D4—5544no—
139D4—5544no—
140D4—5542no—
141D4—11082no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—2772no—
145C2×C4—2772no—
146C2×C4—5544no—
147C2×C4—11082no—
148C2×C4C22×D27711yescharacteristic
149C2×C4C22×D27711yescharacteristic
150C2×C4C22×D27711yescharacteristic
151C2×C4C22×D27711yescharacteristic
152C2×C4C22×D27711yescharacteristic
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—2774no—
159C22—5542no—
160C22—5542no—
161C22—5542no—
162C22—5542no—
163C22—5542no—
164C22—11088no—
165C22#1798456312yescomplemented
166C22C23×D27711yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—5542no—
172C4—5542no—
173C4D4×D27712yes—
174C2—22no—
175C2—41no—
176C2—5544no—
177C2—11082no—
178C2C2×D4×D27711yescharacteristic
179C2C2×#1798456311yescharacteristic
180C2#2257992611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D277
commutator, socle, cosocleC22×C277C24
FittingC4×D4×C277C2
hypercenterC4×D4D277
hypocenterC277#546
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#17984563C2semidirect
2. C22×D1108C2semidirect
3. #22579868C2semidirect
4. #22579865C2semidirect
5. #22579859C2semidirect
6. #22579854C2semidirect
7. #22579853C2semidirect
8. #22579848C2semidirect
9. C4×D1108C2semidirect
10. C22⋊C4×D277C2semidirect
11. C4×D4×C277C2semidirect
14. C2×Dic554C22semidirect
19. C2×C22⋊D277C22semidirect
20. C2×C22⋊D277C22semidirect
21. C2×C4×D277C22semidirect
22. C2×C4×D277C22semidirect
23. C4⋊Dic277C22semidirect
24. #17984542C22semidirect
26. #17984540C22semidirect
27. #17984540C22semidirect
28. #17984540C22semidirect
29. #17984540C22semidirect
32. C22×C4×C277C22semidirect
33. C4⋊C4×C277C22semidirect
35. C22⋊C4×C277C22semidirect
36. C42×C277C22semidirect
45. D1108D4semidirect
67. C22×C277#41semidirect
90. C277#546semidirect
101. C4×D4D277semidirect
116. C22×C4C2×D277semidirect
119. C4⋊C4C2×D277semidirect
126. C22⋊C4C2×D277semidirect
127. C42C2×D277semidirect
165. C22#17984563semidirect

Maximal subgroups

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

subgroupindex
C2×#179845632
C22×D11082
#225798682
#225798652
#225798592
#225798542
#225798532
#225798482
C4×D11082
C22⋊C4×D2772
C4×D4×C2772
#546277