Finite Group Databasebeta

← #31800640

Subgroups — #31800640

Order 17728.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×D4×D277C211yesmaximal, characteristic, complemented
2#22579869C211yesmaximal, characteristic, complemented
3#22579863C212yesmaximal, complemented
4#22579856C211yesmaximal, characteristic, complemented
5#22579854C211yesmaximal, characteristic, complemented
6#22579852C212yesmaximal, complemented
7#22579848C212yesmaximal, complemented
8D4×Dic277C211yesmaximal, characteristic, complemented
9C4×C22⋊D277C211yesmaximal, characteristic, complemented
10C2×#17984540C212yesmaximal, complemented
11C2×#41×C277C211yesmaximal, characteristic, complemented
12C23×D277C2212yes—
13C2×D1108C2211yescharacteristic
14C2×C22⋊D277C2211yescharacteristic
15C2×C22⋊D277C2212yes—
16C2×C22⋊D277C2212yescomplemented
17D4×D277—24no—
18C22×Dic277C2212yescomplemented
19C2×C4×D277C2211yescharacteristic
20C4⋊Dic277C2211yescharacteristic, complemented
21#17984542C2211yescharacteristic, complemented
22#17984542C2212yescomplemented
23C22⋊Dic277C2211yescharacteristic, complemented
24C22⋊Dic277C2212yescomplemented
25#17984540C2211yescharacteristic
26#17984540C2212yes—
27#17984540C2212yescomplemented
28#17984540C2212yescomplemented
29#17984540C2212yescomplemented
30C4×Dic277C2211yescharacteristic, complemented
31#41×C277—24no—
32C2×Q8×C277C2211yescharacteristic, complemented
33C2×D4×C277C2211yescharacteristic
34C2×D4×C277C2212yescomplemented
35C22×C4×C277C2212yescomplemented
36C22×C4×C277C2211yescharacteristic, complemented
37C22×D277—24no—
38C22×D277—24no—
39C22×D277—42no—
40C22×D277C2312yes—
41C22×D277C2311yescharacteristic
42D1108—22no—
43C22⋊D277—24no—
44C22⋊D277—22no—
45C22⋊D277—42no—
46C2×Dic277—42no—
47C2×Dic277C2311yescharacteristic
48C2×Dic277C2311yescharacteristic
49C2×Dic277C2312yes—
50C2×Dic277C2311yescharacteristic
51C4×D277—22no—
52C23×C277C2311yescharacteristic
53C23×C277C2312yes—
54Q8×C277—22no—
55D4×C277—24no—
56D4×C277D414yescomplemented
57C2×C4×C277—24no—
58C2×C4×C277—22no—
59C2×C4×C277C2311yescharacteristic
60C2×C4×C277C2312yes—
61C2×C4×C277C2311yescharacteristic
62C22×C277—22no—
63C22×C277—22no—
64C22×C277—21no—
65C22×C277—21no—
66C22×C277—21no—
67C22×C277C2×D414yes—
68C22×C277C2411yescharacteristic
69C2×D277—21no—
70C2×D277—21no—
71C2×D277—22no—
72C2×D277—22no—
73C2×D277—22no—
74C2×D277—48no—
75C2×D277#4112yescomplemented
76C4×C277—21no—
77C4×C277—22no—
78C4×C277C2×D412yes—
79Dic277—21no—
80Dic277—21no—
81Dic277—21no—
82Dic277—22no—
83C2×C277—24no—
84C2×C277—41no—
85C2×C277C22×D411yescharacteristic
86C2×C277C2×#4111yescharacteristic
87C2×C277H(5,𝔽2)11yescharacteristic
88D277—22no—
89D277—42no—
90C277#54611yescharacteristic, complemented, Sylow
91#546—2771nomaximal, Sylow
92C2×#41D27711yescharacteristic, complemented
93C22×D4—2771no—
94#124—2771no—
95#123—2772no—
96C22⋊Q8—2771no—
97#121—2772no—
98#121—2771no—
99#120—2772no—
100C4×D4—2771no—
101C4×D4—2771no—
102C2×C22⋊C4—2772no—
103C24—2772no—
104#41—24no—
105C2×Q8C2×D27711yescharacteristic, complemented
106C2×D4—2771no—
107C2×D4—2772no—
108C2×D4—2772no—
109C2×D4—2771no—
110C2×D4—5544no—
111C2×D4C2×D27711yescharacteristic
112C2×D4C2×D27712yescomplemented
113C22×C4—2772no—
114C22×C4—2771no—
115C22×C4C2×D27712yescomplemented
116C22×C4C2×D27711yescharacteristic, complemented
117C4⋊C4—2771no—
118C4⋊C4—2772no—
119C4⋊C4—2771no—
120C22⋊C4—2772no—
121C22⋊C4—2771no—
122C22⋊C4—2771no—
123C22⋊C4—2772no—
124C22⋊C4—2772no—
125C22⋊C4—2772no—
126C22⋊C4—2772no—
127C42—2771no—
128C23—2771no—
129C23—2772no—
130C23—5544no—
131C23—5544no—
132C23—11082no—
133C23C22×D27711yescharacteristic
134C23C22×D27712yes—
135Q8—22no—
136D4—24no—
137D4—5542no—
138D4—5544no—
139D4—5542no—
140D4—11082no—
141D4C22⋊D27714yescomplemented
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—2771no—
145C2×C4—2771no—
146C2×C4—2772no—
147C2×C4—2771no—
148C2×C4—5542no—
149C2×C4—11082no—
150C2×C4C22×D27711yescharacteristic
151C2×C4C22×D27712yes—
152C2×C4C22×D27711yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—2772no—
159C22—5541no—
160C22—5541no—
161C22—5542no—
162C22—5542no—
163C22—5542no—
164C22—11088no—
165C22C2×C22⋊D27714yes—
166C22C23×D27711yescharacteristic
167C4—21no—
168C4—22no—
169C4—5541no—
170C4—5541no—
171C4—5541no—
172C4—5542no—
173C4C2×C22⋊D27712yes—
174C2—24no—
175C2—41no—
176C2—5542no—
177C2—11082no—
178C2C22×C22⋊D27711yescharacteristic
179C2#41×D27711yescharacteristic
180C2#2257992611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D277
commutator, socle, cosocleC22×C277C24
FittingC2×#41×C277C2
hypercenterC2×#41D277
hypocenterC277#546
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D4×D277C2semidirect
2. #22579869C2semidirect
3. #22579863C2semidirect
4. #22579856C2semidirect
5. #22579854C2semidirect
6. #22579852C2semidirect
7. #22579848C2semidirect
8. D4×Dic277C2semidirect
9. C4×C22⋊D277C2semidirect
10. C2×#17984540C2semidirect
11. C2×#41×C277C2semidirect
16. C2×C22⋊D277C22semidirect
18. C22×Dic277C22semidirect
20. C4⋊Dic277C22semidirect
21. #17984542C22semidirect
22. #17984542C22semidirect
23. C22⋊Dic277C22semidirect
24. C22⋊Dic277C22semidirect
27. #17984540C22semidirect
28. #17984540C22semidirect
29. #17984540C22semidirect
30. C4×Dic277C22semidirect
32. C2×Q8×C277C22semidirect
34. C2×D4×C277C22semidirect
35. C22×C4×C277C22semidirect
36. C22×C4×C277C22semidirect
56. D4×C277D4semidirect
75. C2×D277#41semidirect
90. C277#546semidirect
92. C2×#41D277semidirect
105. C2×Q8C2×D277semidirect
112. C2×D4C2×D277semidirect
115. C22×C4C2×D277semidirect
116. C22×C4C2×D277semidirect
141. D4C22⋊D277semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2772
#225798692
#225798632
#225798562
#225798542
#225798522
#225798482
D4×Dic2772
C4×C22⋊D2772
C2×#179845402
C2×#41×C2772
#546277