Finite Group Databasebeta

← #31800643

Subgroups — #31800643

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×#17984564C211yesmaximal, characteristic, complemented
2C22×D1108C211yesmaximal, characteristic, complemented
3#22579869C211yesmaximal, characteristic, complemented
4#22579865C212yesmaximal, complemented
5#22579855C211yesmaximal, characteristic, complemented
6#22579855C212yesmaximal, complemented
7#22579850C212yesmaximal, complemented
8#22579842C211yesmaximal, characteristic, complemented
9C4×C22⋊D277C211yesmaximal, characteristic, complemented
10C22⋊C4×D277C212yesmaximal, complemented
11C22⋊Q8×C277C211yesmaximal, characteristic, complemented
12C23×D277C2212yes—
13#17984564—24no—
14C2×D1108—24no—
15C2×D1108C2212yes—
16C2×D1108C2211yescharacteristic
17C2×D1108C2211yescharacteristic
18C2×D1108C2212yescomplemented
19C2×D1108C2212yescomplemented
20C2×C22⋊D277C2211yescharacteristic
21C2×C4×D277C2212yescomplemented
22C2×C4×D277C2211yescharacteristic, complemented
23C2×C4×D277C2212yescomplemented
24#17984542C2211yescharacteristic, complemented
25C22⋊Dic277C2211yescharacteristic
26#17984540C2212yescomplemented
27#17984540C2212yescomplemented
28#17984540C2211yescharacteristic, complemented
29#17984540C2212yescomplemented
30#17984540C2212yescomplemented
31C4×Dic277C2211yescharacteristic, complemented
32C2×Q8×C277C2211yescharacteristic, complemented
33C22×C4×C277C2211yescharacteristic
34C4⋊C4×C277C2211yescharacteristic, complemented
35C4⋊C4×C277C2212yescomplemented
36C22⋊C4×C277C2212yes—
37C22×D277—24no—
38C22×D277—24no—
39C22×D277—42no—
40C22×D277C2312yes—
41C22×D277C2311yescharacteristic
42C22×D277C2312yes—
43D1108—24no—
44D1108—24no—
45D1108—22no—
46D1108—42no—
47D1108D414yescomplemented
48C22⋊D277—22no—
49C2×Dic277C2311yescharacteristic
50C2×Dic277C2311yescharacteristic
51C2×Dic277C2311yescharacteristic
52C4×D277—24no—
53C4×D277—22no—
54C4×D277—42no—
55C23×C277C2311yescharacteristic
56Q8×C277—22no—
57C2×C4×C277—22no—
58C2×C4×C277C2312yes—
59C2×C4×C277C2312yes—
60C2×C4×C277C2311yescharacteristic
61C2×C4×C277C2311yescharacteristic
62C22×C277—21no—
63C22×C277—21no—
64C22×C277#4112yescomplemented
65C22×C277C2411yescharacteristic
66C2×D277—22no—
67C2×D277—22no—
68C2×D277—21no—
69C2×D277—22no—
70C2×D277—21no—
71C2×D277—22no—
72C2×D277—21no—
73C2×D277—22no—
74C2×D277—48no—
75C2×D277C2×D414yes—
76C4×C277—21no—
77C4×C277—22no—
78C4×C277—22no—
79C4×C277C2×D412yes—
80Dic277—21no—
81Dic277—21no—
82Dic277—21no—
83C2×C277—22no—
84C2×C277C22×D411yescharacteristic
85C2×C277C2×#4111yescharacteristic
86C2×C277H(5,𝔽2)11yescharacteristic
87D277—24no—
88D277—41no—
89D277—42no—
90C277#54611yescharacteristic, complemented, Sylow
91#546—2771nomaximal, Sylow
92C2×#41—2771no—
93C22×D4—2771no—
94#124—2771no—
95#123—2772no—
96C22⋊Q8D27711yescharacteristic, complemented
97#121—2771no—
98#121—2772no—
99#120—2772no—
100C4×D4—2771no—
101C4×D4—2771no—
102C2×C22⋊C4—2772no—
103C24—2772no—
104#41—5544no—
105C2×Q8C2×D27711yescharacteristic, complemented
106C2×D4—2771no—
107C2×D4—2771no—
108C2×D4—2772no—
109C2×D4—2771no—
110C2×D4—2772no—
111C2×D4—2772no—
112C2×D4—5544no—
113C22×C4—2772no—
114C22×C4—2771no—
115C22×C4—2772no—
116C22×C4C2×D27711yescharacteristic
117C4⋊C4—2771no—
118C4⋊C4C2×D27711yescharacteristic, complemented
119C4⋊C4C2×D27712yescomplemented
120C22⋊C4—2771no—
121C22⋊C4—2772no—
122C22⋊C4—2772no—
123C22⋊C4—2772no—
124C22⋊C4—2771no—
125C22⋊C4—2772no—
126C22⋊C4C2×D27712yes—
127C42—2771no—
128C23—2772no—
129C23—2772no—
130C23—2771no—
131C23—5544no—
132C23—5544no—
133C23—11082no—
134C23C22×D27711yescharacteristic
135Q8—22no—
136D4—2774no—
137D4—5544no—
138D4—5544no—
139D4—5542no—
140D4—5542no—
141D4—11082no—
142C2×C4—22no—
143C2×C4—2771no—
144C2×C4—2771no—
145C2×C4—2771no—
146C2×C4—5544no—
147C2×C4—5542no—
148C2×C4—11082no—
149C2×C4C22×D27712yes—
150C2×C4C22×D27712yes—
151C2×C4C22×D27711yescharacteristic
152C2×C4C22×D27711yescharacteristic
153C22—21no—
154C22—21no—
155C22—2774no—
156C22—5542no—
157C22—5542no—
158C22—5541no—
159C22—5542no—
160C22—5541no—
161C22—5542no—
162C22—5541no—
163C22—5542no—
164C22—11088no—
165C22#1798456412yescomplemented
166C22C23×D27711yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—5541no—
171C4—5541no—
172C4—5541no—
173C4D4×D27712yes—
174C2—22no—
175C2—5544no—
176C2—11082no—
177C2—11081no—
178C2C2×D4×D27711yescharacteristic
179C2C2×#1798456411yescharacteristic
180C2#2257992611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D277
commutator, socle, cosocleC22×C277C24
FittingC22⋊Q8×C277C2
hypercenterC22⋊Q8D277
hypocenterC277#546
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17984564C2semidirect
2. C22×D1108C2semidirect
3. #22579869C2semidirect
4. #22579865C2semidirect
5. #22579855C2semidirect
6. #22579855C2semidirect
7. #22579850C2semidirect
8. #22579842C2semidirect
9. C4×C22⋊D277C2semidirect
10. C22⋊C4×D277C2semidirect
11. C22⋊Q8×C277C2semidirect
18. C2×D1108C22semidirect
19. C2×D1108C22semidirect
21. C2×C4×D277C22semidirect
22. C2×C4×D277C22semidirect
23. C2×C4×D277C22semidirect
24. #17984542C22semidirect
26. #17984540C22semidirect
27. #17984540C22semidirect
28. #17984540C22semidirect
29. #17984540C22semidirect
30. #17984540C22semidirect
31. C4×Dic277C22semidirect
32. C2×Q8×C277C22semidirect
34. C4⋊C4×C277C22semidirect
35. C4⋊C4×C277C22semidirect
47. D1108D4semidirect
64. C22×C277#41semidirect
90. C277#546semidirect
96. C22⋊Q8D277semidirect
105. C2×Q8C2×D277semidirect
118. C4⋊C4C2×D277semidirect
119. C4⋊C4C2×D277semidirect
165. C22#17984564semidirect

Maximal subgroups

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

subgroupindex
C2×#179845642
C22×D11082
#225798692
#225798652
#225798552
#225798502
#225798422
C4×C22⋊D2772
C22⋊C4×D2772
C22⋊Q8×C2772
#546277