Finite Group Databasebeta

← #14544186

Subgroups — #14544186

Order 2176.

242 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 60 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#11763831C211yesmaximal, characteristic, complemented
2C4×C4⋊#595C211yesmaximal, characteristic, complemented
3#11763662C212yesmaximal, complemented
4#11763660C212yesmaximal
5#11763341C211yesmaximal, characteristic
6C2×C4⋊#595—21no—
7C2×C4×#595—21no—
8C4×D4×C17C411yescharacteristic, complemented
9#10587624C2211yescharacteristic
10#10587622C411yescharacteristic, complemented
11C4×C22⋊D17—21no—
12#10587612—21no—
13#10587583C2211yescharacteristic
14#10587576C411yescharacteristic, complemented
15#10587569—21no—
16#10587565—21no—
17C42×D17C2211yescharacteristic
18C4×Dic34C411yescharacteristic, complemented
19(C2×#595)⋊C4C2212yes—
20#10587543C2212yes—
21#10587542—21no—
22C4⋊#3624—21no—
23C22×#595—22no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C2×C4×D17D411yescharacteristic
28C2×C4×D17D411yescharacteristic
29C2×C4×D17C2311yescharacteristic
30C2×Dic34C2×C411yescharacteristic
31C2×#3624—22no—
32C4⋊#595—24no—
33C4×#595—42no—
34C4⋊C4×C17C2×C411yescharacteristic
35C22⋊C4×C17—21no—
36C42×C17C2×C411yescharacteristic
37C22⋊Dic17—21no—
38C22⋊Dic17—21no—
39C22⋊Dic17—21no—
40#63227—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17C2×C411yescharacteristic
44#63225—21no—
45#63225—21no—
46C4×Dic17—21no—
47C4×Dic17C2×C411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C23×C17—21no—
51C22×D17C2×D411yescharacteristic
52C2×#595—24no—
53C2×#595—42no—
54C2×#595—42no—
55D4×C17—41no—
56C2×C4×C17—21no—
57C2×C4×C17—42no—
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22⋊C411yescharacteristic
60C2×C4×C17C22×C411yescharacteristic
61C22⋊D17—41no—
62C22⋊D17—41no—
63C2×Dic17—21no—
64C2×Dic17—21no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68C4×D17—21no—
69C4×D17—21no—
70C4×D17—22no—
71C4×D17—41no—
72Dic34—41no—
73#3624—42no—
74#1509—171nomaximal, Sylow
75C22×C17—41no—
76C22×C17—41no—
77C22×C17—41no—
78C22×C17C2×C22⋊C411yescharacteristic
79C2×D17—21no—
80C2×D17—21no—
81C2×D17AGL(1,ℤ/8)11yescharacteristic
82C2×D17#13711yescharacteristic
83#595—44no—
84C4×C17—21no—
85C4×C17—21no—
86C4×C17—41no—
87C4×C17#10412yes—
88Dic17—21no—
89Dic17—21no—
90Dic17—21no—
91Dic17—41no—
92Dic17—41no—
93#529—171no—
94#432—171no—
95C4×C4⋊C4—171no—
96#328—172no—
97#327—172no—
98C2×C17—81no—
99C2×C17#41011yescharacteristic
100C2×C17#41711yescharacteristic
101C2×C17C2×#10411yescharacteristic
102D17—21no—
103D17—21no—
104#126—341no—
105#125—171no—
106#124—171no—
107#123—341no—
108C22⋊Q8—171no—
109#121—171no—
110C4×Q8—171no—
111C4×D4—341no—
112C4×D4#59511yescharacteristic, complemented
113#117—341no—
114C2×C4⋊C4—341no—
115C2×C42—171no—
116C2×C42—341no—
117C4⋊C8—341no—
118C22⋊C8—172no—
119#97—341no—
120#95—172no—
121C17#150911yescharacteristic, complemented, Sylow
122C2×Q8—171no—
123C2×D4—341no—
124C2×D4C2×#59511yescharacteristic
125C22×C4—21no—
126C22×C4—171no—
127C22×C4—171no—
128C22×C4—171no—
129C22×C4—342no—
130C2×C8—342no—
131C4⋊C4—171no—
132C4⋊C4—341no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4—344no—
136C4⋊C4—341no—
137C4⋊C4C2×#59511yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—341no—
140C22⋊C4—341no—
141C22⋊C4—341no—
142C22⋊C4—341no—
143C42—171no—
144C42—171no—
145C42—171no—
146C42—341no—
147C42—682no—
148C42C2×#59511yescharacteristic
149C23—21no—
150C23—171no—
151Q8—681no—
152D4—41no—
153D4—681no—
154D4—681no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—171no—
158C2×C4—171no—
159C2×C4—171no—
160C2×C4—341no—
161C2×C4—341no—
162C2×C4—342no—
163C2×C4—344no—
164C2×C4—341no—
165C2×C4—341no—
166C2×C4—681no—
167C2×C4—682no—
168C2×C4—682no—
169C2×C4C22⋊#59511yescharacteristic
170C2×C4C22⋊#59511yescharacteristic
171C2×C4C22×#59511yescharacteristic
172C8—682no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—171no—
177C22—171no—
178C22—341no—
179C22—341no—
180C22C2×C22⋊#59511yescharacteristic
181C4—21no—
182C4—21no—
183C4—41no—
184C4—341no—
185C4—341no—
186C4—341no—
187C4—684no—
188C4—681no—
189C4—681no—
190C4#1058754812yes—
191C2—81no—
192C2—341no—
193C2—341no—
194C2#1176378811yescharacteristic
195C2#1176380011yescharacteristic
196C2C2×#1058754811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#10587624C22
cosocleC2×C4×D17C23
hypercenterC4×D4#595
hypocenterC17#1509
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763831C2semidirect
2. C4×C4⋊#595C2semidirect
3. #11763662C2semidirect
8. C4×D4×C17C4semidirect
10. #10587622C4semidirect
14. #10587576C4semidirect
18. C4×Dic34C4semidirect
112. C4×D4#595semidirect
121. C17#1509semidirect

Maximal subgroups

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

subgroupindex
#117638312
C4×C4⋊#5952
#117636622
#117636602
#117633412
#150917