Finite Group Databasebeta

← #14544191

Subgroups — #14544191

Order 2176.

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

normal subgroups only

This group has 326 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#11763781C211yesmaximal, characteristic, complemented
2#11763752C211yesmaximal, characteristic, complemented
3#11763661C214yesmaximal, complemented
4C2×#10587581C211yesmaximal, characteristic
5C22×D68—21no—
6C22×D68C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×#63247—21no—
10C2×C4⋊C4×C17C411yescharacteristic, complemented
11C2×#63227—21no—
12#10587581—21no—
13#10587581—21no—
14#10587581C412yescomplemented
15#10587581C2212yes—
16(C2×#595)⋊C4—21no—
17(C2×#595)⋊C4C2212yescomplemented
18#10587543—21no—
19#10587543C2212yescomplemented
20C23×D17—21no—
21C23×D17D411yescharacteristic
22C22×#595—22no—
23C22×C4×C17—21no—
24C22×C4×C17C2×C411yescharacteristic
25C22×Dic17D411yescharacteristic
26C2×D68—22no—
27C2×D68—22no—
28C2×D68—21no—
29C2×D68—42no—
30C2×D68—44no—
31C2×D68C2×C412yes—
32C2×C4×D17—42no—
33C2×C4×D17C2×C411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17C2311yescharacteristic
37C22⋊#595—42no—
38#63247—42no—
39C2×#3624—22no—
40C4⋊C4×C17—21no—
41C4⋊C4×C17C2×C412yes—
42#63227—22no—
43#63227—22no—
44C23×C17C22⋊C411yescharacteristic
45C22×D17—22no—
46C22×D17—22no—
47C22×D17—22no—
48C22×D17—41no—
49C22×D17—42no—
50C22×D17—42no—
51C22×D17—42no—
52C22×D17C22⋊C411yescharacteristic
53C22×D17C2×D411yescharacteristic
54C2×#595—42no—
55C2×#595—42no—
56C2×#595—42no—
57C2×C4×C17—22no—
58C2×C4×C17—22no—
59C2×C4×C17—42no—
60C2×C4×C17C22⋊C411yescharacteristic
61C2×C4×C17C22×C411yescharacteristic
62C2×Dic17—22no—
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C2×D411yescharacteristic
65D68—42no—
66D68—42no—
67D68—44no—
68D68—44no—
69C4×D17—41no—
70C4×D17—41no—
71C4×D17—41no—
72C4×D17—41no—
73#3624—42no—
74#1510—171nomaximal, Sylow
75C22×C17—21no—
76C22×C17—21no—
77C22×C17#9912yes—
78C22×C17C2×C22⋊C411yescharacteristic
79C2×D17—21no—
80C2×D17—21no—
81C2×D17—21no—
82C2×D17—21no—
83C2×D17—21no—
84C2×D17—44no—
85C2×D17—42no—
86C2×D17—42no—
87C2×D17—42no—
88C2×D17—84no—
89C2×D17AGL(1,ℤ/8)12yes—
90#595—82no—
91C4×C17—21no—
92C4×C17—21no—
93C4×C17—42no—
94Dic17—21no—
95Dic17—21no—
96C2×#121—171no—
97#407—171no—
98#381—171no—
99#327—174no—
100C2×C17—22no—
101C2×C17C2×#9911yescharacteristic
102C2×C17#41811yescharacteristic
103C2×C17#42111yescharacteristic
104D17—22no—
105D17—41no—
106D17—82no—
107C22×D4—171no—
108C22×D4—341no—
109C23×C4—171no—
110C2×M16—341no—
111#121—172no—
112#121—172no—
113#121—341no—
114#121—341no—
115C2×C4⋊C4#59511yescharacteristic, complemented
116C2×C22⋊C4—341no—
117C2×C22⋊C4—341no—
118C22⋊C8—172no—
119C22⋊C8—341no—
120#95—172no—
121#95—341no—
122C17#151011yescharacteristic, complemented, Sylow
123C24—171no—
124C24—341no—
125C2×D4—172no—
126C2×D4—342no—
127C2×D4—342no—
128C2×D4—341no—
129C2×D4—682no—
130C2×D4—684no—
131C22×C4—21no—
132C22×C4—171no—
133C22×C4—171no—
134C22×C4—171no—
135C22×C4—171no—
136C22×C4—171no—
137C22×C4—342no—
138C22×C4—682no—
139C22×C4C2×#59511yescharacteristic
140M16—682no—
141C2×C8—342no—
142C4⋊C4—21no—
143C4⋊C4C2×#59512yes—
144C22⋊C4—342no—
145C22⋊C4—342no—
146C22⋊C4—682no—
147C23—171no—
148C23—171no—
149C23—342no—
150C23—342no—
151C23—342no—
152C23—681no—
153C23—682no—
154C23—682no—
155C23—682no—
156C23C22⋊#59511yescharacteristic
157D4—682no—
158D4—682no—
159D4—684no—
160D4—684no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—171no—
165C2×C4—171no—
166C2×C4—342no—
167C2×C4—681no—
168C2×C4—681no—
169C2×C4—681no—
170C2×C4—681no—
171C2×C4—682no—
172C2×C4—682no—
173C2×C4—682no—
174C2×C4C22⋊#59511yescharacteristic
175C2×C4C22×#59511yescharacteristic
176C8—682no—
177C22—21no—
178C22—21no—
179C22—172no—
180C22—341no—
181C22—341no—
182C22—341no—
183C22—341no—
184C22—341no—
185C22—684no—
186C22—682no—
187C22—682no—
188C22—682no—
189C22—1364no—
190C22#1058753912yes—
191C22C2×C22⋊#59511yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—341no—
196C4—341no—
197C4—1362no—
198C2—22no—
199C2—342no—
200C2—681no—
201C2—1362no—
202C2C2×#1058753911yescharacteristic
203C2#1176380111yescharacteristic
204C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×C4⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#10587581C2
cosocleC2×C4×D17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1510
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11763781C2semidirect
2. #11763752C2semidirect
3. #11763661C2semidirect
6. C22×D68C4semidirect
10. C2×C4⋊C4×C17C4semidirect
14. #10587581C4semidirect
17. (C2×#595)⋊C4C22semidirect
19. #10587543C22semidirect
115. C2×C4⋊C4#595semidirect
122. C17#1510semidirect

Maximal subgroups

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

subgroupindex
#117637812
#117637522
#117636612
C2×#105875812
#151017