Finite Group Databasebeta

← #14544174

Subgroups — #14544174

Order 2176.

272 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 58 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#11763832C211yesmaximal, characteristic, complemented
2#11763745C211yesmaximal, characteristic, complemented
3#11763661C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763356C211yesmaximal, characteristic
6C2×#63252—21no—
7C2×C4×#595—21no—
8C4×D4×C17C411yescharacteristic, complemented
9#10587626C411yescharacteristic, complemented
10#10587624C2211yescharacteristic
11C4×C22⋊D17—21no—
12#10587581C2211yescharacteristic
13#10587574C411yescharacteristic, complemented
14#10587571—21no—
15#10587570—21no—
16C4×D68C411yescharacteristic, complemented
17C42×D17C2211yescharacteristic
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587544—21no—
21#10587543C2212yes—
22#10587542—21no—
23C22×#595—22no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C2×C22⋊D17—21no—
28C2×C22⋊D17—21no—
29#63252—44no—
30C2×D68—21no—
31C2×D68C2×C411yescharacteristic
32C2×C4×D17—21no—
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17C2311yescharacteristic
36C2×Dic34—21no—
37C2×#3624—22no—
38C4×#595—24no—
39C4⋊C4×C17C2×C411yescharacteristic
40C22⋊C4×C17—21no—
41C42×C17C2×C411yescharacteristic
42C22⋊Dic17—21no—
43#63227—21no—
44#63227—21no—
45C4⋊Dic17C2×C411yescharacteristic
46#63225—21no—
47C4×Dic17C2×C411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C23×C17—21no—
51C22×D17—21no—
52C22×D17C2×D411yescharacteristic
53C2×#595—24no—
54C2×#595—42no—
55C2×#595—42no—
56D4×C17—41no—
57C2×C4×C17—21no—
58C2×C4×C17—42no—
59C2×C4×C17C22⋊C411yescharacteristic
60C2×C4×C17C22⋊C411yescharacteristic
61C2×C4×C17C22×C411yescharacteristic
62C22⋊D17—42no—
63C22⋊D17—42no—
64C22⋊D17—42no—
65C2×Dic17—21no—
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C22⋊C411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69D68—41no—
70D68—42no—
71C4×D17—22no—
72C4×D17—22no—
73C4×D17—41no—
74C4×D17—42no—
75Dic34—42no—
76#3624—42no—
77#1505—171nomaximal, Sylow
78C22×C17—41no—
79C22×C17—41no—
80C22×C17—41no—
81C22×C17C2×C22⋊C411yescharacteristic
82C2×D17—21no—
83C2×D17—21no—
84C2×D17—41no—
85C2×D17—41no—
86C2×D17—41no—
87C2×D17AGL(1,ℤ/8)12yes—
88#595—44no—
89C4×C17—21no—
90C4×C17—21no—
91C4×C17—21no—
92C4×C17—41no—
93Dic17—21no—
94Dic17—21no—
95Dic17—41no—
96Dic17#10412yes—
97#532—171no—
98#432—171no—
99#376—171no—
100#327—172no—
101#327—172no—
102C2×C17—81no—
103C2×C17#41011yescharacteristic
104C2×C17#41811yescharacteristic
105C2×C17C2×#10411yescharacteristic
106D17—22no—
107D17—81no—
108C2×#41—341no—
109C4⋊Q8—171no—
110#127—171no—
111#124—341no—
112#121—171no—
113#121—171no—
114#121—341no—
115C4×D4—171no—
116C4×D4—341no—
117C4×D4#59511yescharacteristic, complemented
118C2×C42—171no—
119C2×C42—341no—
120C4⋊C8—341no—
121C22⋊C8—172no—
122#97—341no—
123#95—172no—
124#95—341no—
125C17#150511yescharacteristic, complemented, Sylow
126#41—684no—
127C2×Q8—341no—
128C2×D4—171no—
129C2×D4—341no—
130C2×D4—341no—
131C2×D4—341no—
132C2×D4—341no—
133C2×D4C2×#59511yescharacteristic
134C22×C4—21no—
135C22×C4—171no—
136C22×C4—171no—
137C22×C4—171no—
138C22×C4—342no—
139C22×C4—341no—
140C2×C8—342no—
141C4⋊C4—171no—
142C4⋊C4—341no—
143C4⋊C4C2×#59511yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—341no—
146C22⋊C4—341no—
147C22⋊C4—341no—
148C42—171no—
149C42—171no—
150C42—171no—
151C42—344no—
152C42C2×#59511yescharacteristic
153C23—21no—
154C23—171no—
155C23—341no—
156Q8—682no—
157D4—41no—
158D4—681no—
159D4—682no—
160D4—682no—
161D4—682no—
162D4—682no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—171no—
166C2×C4—171no—
167C2×C4—171no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—344no—
171C2×C4—341no—
172C2×C4—681no—
173C2×C4—682no—
174C2×C4—682no—
175C2×C4—682no—
176C2×C4C22⋊#59511yescharacteristic
177C2×C4C22⋊#59511yescharacteristic
178C2×C4C22×#59511yescharacteristic
179C8—682no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—172no—
184C22—341no—
185C22—341no—
186C22—681no—
187C22—681no—
188C22—681no—
189C22C2×C22⋊#59511yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—41no—
194C4—172no—
195C4—341no—
196C4—341no—
197C4—684no—
198C4—681no—
199C2—81no—
200C2—342no—
201C2—1361no—
202C2#1176378811yescharacteristic
203C2#1176380111yescharacteristic
204C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#11763356C2
cosocleC2×C4×D17C23
hypercenterC4×D4#595
hypocenterC17#1505
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. #11763832C2semidirect
2. #11763745C2semidirect
3. #11763661C2semidirect
4. #11763660C2semidirect
8. C4×D4×C17C4semidirect
9. #10587626C4semidirect
13. #10587574C4semidirect
16. C4×D68C4semidirect
117. C4×D4#595semidirect
125. C17#1505semidirect

Maximal subgroups

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

subgroupindex
#117638322
#117637452
#117636612
#117636602
#117633562
#150517