Finite Group Databasebeta

← C22×#11763755 #17978877

Subgroups — C22×#11763755 #17978877

Order 4352.

5502 conjugacy classes of subgroups; only the 2056 normal subgroups are shown.

normal subgroups only

This group has 5502 conjugacy classes of subgroups — too many to draw. Only its 2056 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
1C23×C4⋊#595C216yesmaximal
2C2×#11763755C2124yesmaximal, direct product factor
3C22×C42×D17C211yesmaximal, characteristic
4C24×#595C2214yes—
5C22×C4⋊#595—248no—
6C22×C4⋊#595C22172yes—
7#11763755C22164yesdirect product factor
8C23×C4×D17C2213yes—
9C22×C4×Dic17C413yescomplemented
10C2×C42×D17C4112yescomplemented
11C2×C42×D17C22112yes—
12C22×C42×C17C411yescharacteristic, complemented
13C24×D17C2311yescharacteristic
14C23×#595—248no—
15C23×#595C23124yes—
16C23×C4×C17C2×C413yes—
17C23×Dic17C2×C413yes—
18C22×C4×D17C2×C4118yes—
19C22×C4×D17D4124yes—
20C22×C4×D17Q8124yes—
21C22×C4×D17C23118yes—
22C2×C4⋊#595—2288no—
23C2×C4⋊#595C23196yes—
24C2×C42×C17C2×C4112yescomplemented
25C2×C4×Dic17C2×C4136yescomplemented
26C42×D17C2×C4148yescomplemented
27C42×D17C23116yes—
28C24×C17C22×C411yescharacteristic
29C23×D17C22×C413yes—
30C23×D17C2×D4112yes—
31C23×D17C2×Q8112yes—
32C23×D17C2413yes—
33C22×#595—2144no—
34C22×#595—464no—
35C22×#595C24116yes—
36C22×C4×C17C4⋊C4124yescomplemented
37C22×C4×C17C22×C4118yes—
38C22×Dic17C4⋊C4124yescomplemented
39C22×Dic17C22×C4118yes—
40C2×C4×D17C4⋊C41144yescomplemented
41C2×C4×D17C22×C4136yes—
42C2×C4×D17C2×D4172yes—
43C2×C4×D17C2×Q8172yes—
44C2×C4×D17C24112yes—
45C4⋊#595—2192no—
46C42×C17C22×C4116yescomplemented
47C4×Dic17C22×C4148yescomplemented
48C22×#384—171nomaximal, Sylow
49C23×C17C2×C4⋊C4112yes—
50C23×C17C23×C413yes—
51C22×D17C2×C4⋊C4136yes—
52C22×D17#127116yes—
53C22×D17C4⋊Q8148yes—
54C22×D17C23×C413yes—
55C22×D17C22×D4118yes—
56C22×D17C22×Q8118yes—
57C22×D17C2511yescharacteristic
58C2×#595—248no—
59C2×#595—496no—
60C2×C4×C17C2×C4⋊C4172yescomplemented
61C2×C4×C17C23×C4112yes—
62C2×Dic17C2×C4⋊C4172yescomplemented
63C2×Dic17C23×C4112yes—
64C4×D17C2×C4⋊C41144yescomplemented
65C4×D17C22×D4124yes—
66C4×D17C22×Q8124yes—
67C23×C4⋊C4—176no—
68C23×C42—171no—
69C2×#384—1724no—
70C22×C17#384116yescomplemented
71C22×C17C22×C4⋊C4118yes—
72C22×C17C24×C411yescharacteristic
73C2×D17#384148yescomplemented
74C2×D17C22×C4⋊C4118yes—
75C2×D17C2×#127112yes—
76C2×D17C2×C4⋊Q8136yes—
77C2×D17C23×D413yes—
78C2×D17C23×Q813yes—
79#595—416no—
80C4×C17C22×C4⋊C4124yescomplemented
81Dic17C22×C4⋊C4124yescomplemented
82C24×C4—173no—
83C24×C4—174no—
84C22×C4⋊C4—1772no—
85C22×C4⋊C4—3448no—
86C22×C42—173no—
87C22×C42—1712no—
88C22×C42—1712no—
89C22×C42#59511yescharacteristic, complemented
90#384—1764no—
91C2×C17C2×#384112yescomplemented
92C2×C17C23×C4⋊C413yes—
93D17C2×#384112yescomplemented
94D17C22×#12711yescharacteristic
95D17C22×C4⋊Q813yes—
96C25—171no—
97C23×C4—173no—
98C23×C4—1724no—
99C23×C4—1724no—
100C23×C4—1718no—
101C23×C4—1718no—
102C23×C4—1724no—
103C23×C4—3448no—
104C23×C4C2×#59513yes—
105C2×C4⋊C4—1796no—
106C2×C4⋊C4—34288no—
107C2×C42—1736no—
108C2×C42—1716no—
109C2×C42—1748no—
110C2×C42C2×#595112yescomplemented
111C17C22×#38411yescharacteristic, complemented, Sylow
112C24—1712no—
113C24—1712no—
114C24—173no—
115C24—173no—
116C24C22×#59511yescharacteristic
117C22×C4—1724no—
118C22×C4—1718no—
119C22×C4—17144no—
120C22×C4—1772no—
121C22×C4—1772no—
122C22×C4—1736no—
123C22×C4—1712no—
124C22×C4—1716no—
125C22×C4—34144no—
126C22×C4—6864no—
127C22×C4C4⋊#595124yescomplemented
128C22×C4C22×#595118yes—
129C4⋊C4—34192no—
130C42—1748no—
131C42C22×#595116yescomplemented
132C23—1716no—
133C23—1748no—
134C23—1718no—
135C23—1736no—
136C23—1718no—
137C23—171no—
138C23—173no—
139C23C2×C4⋊#595112yes—
140C23C23×#59513yes—
141C2×C4—1772no—
142C2×C4—1712no—
143C2×C4—1724no—
144C2×C4—17144no—
145C2×C4—1724no—
146C2×C4—3448no—
147C2×C4—6896no—
148C2×C4C2×C4⋊#595172yescomplemented
149C2×C4C23×#595112yes—
150C22—1712no—
151C22—1736no—
152C22—1748no—
153C22—173no—
154C22—1718no—
155C22—173no—
156C22#11763755116yesdirect product factor
157C22C22×C4⋊#595118yes—
158C22C24×#59511yescharacteristic
159C4—1724no—
160C4—6816no—
161C4C22×C4⋊#595124yescomplemented
162C2—171no—
163C2—173no—
164C2—1712no—
165C2C2×#11763755112yesdirect product factor
166C2C23×C4⋊#59513yes—
subgroupG/H
centerC24C22×#595
commutatorC22×C17C24×C4
FrattiniC22C24×#595
FittingC22×C42×C17C4
socleC24×C17C22×C4
prime radicalC24×D17C23
cosocleC22×D17C25
hypercenterC22×C42#595
hypocenterC17C22×#384
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#11763755C2direct (N × H)
7. #11763755C22direct (N × H)
9. C22×C4×Dic17C4semidirect
10. C2×C42×D17C4semidirect
12. C22×C42×C17C4semidirect
24. C2×C42×C17C2×C4semidirect
25. C2×C4×Dic17C2×C4semidirect
26. C42×D17C2×C4semidirect
36. C22×C4×C17C4⋊C4semidirect
38. C22×Dic17C4⋊C4semidirect
40. C2×C4×D17C4⋊C4semidirect
46. C42×C17C22×C4semidirect
47. C4×Dic17C22×C4semidirect
60. C2×C4×C17C2×C4⋊C4semidirect
62. C2×Dic17C2×C4⋊C4semidirect
64. C4×D17C2×C4⋊C4semidirect
70. C22×C17#384semidirect
73. C2×D17#384semidirect
80. C4×C17C22×C4⋊C4semidirect
81. Dic17C22×C4⋊C4semidirect
89. C22×C42#595semidirect
91. C2×C17C2×#384semidirect
93. D17C2×#384semidirect
110. C2×C42C2×#595semidirect
111. C17C22×#384semidirect
127. C22×C4C4⋊#595semidirect
131. C42C22×#595semidirect
148. C2×C4C2×C4⋊#595semidirect
156. C22#11763755direct (N × H)
161. C4C22×C4⋊#595semidirect
165. C2C2×#11763755direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊#5952
C2×#117637552
C22×C42×D172
C22×#38417