Finite Group Databasebeta

← #14539984

Subgroups — #14539984

Order 2176.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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#11763366C211yesmaximal, characteristic, complemented
2#11763359C211yesmaximal, characteristic
3#11763218C212yesmaximal, complemented
4#11763174C212yesmaximal, complemented
5#11763134C214yesmaximal, complemented
6#11763100C212yesmaximal, complemented
7#11763090C212yesmaximal, complemented
8C17×#432C211yesmaximal, characteristic, complemented
9C2×#63252—21no—
10C2×C42×C17C2211yescharacteristic
11#10587616—21no—
12#10587616C2212yes—
13#10587612C2212yes—
14#10587610—21no—
15#10587610—21no—
16#10587561—21no—
17#10587560C2211yescharacteristic
18C4×D68—21no—
19C4×D68C2212yes—
20#10587556C2212yes—
21#10587555C2211yescharacteristic
22#10587555C2211yescharacteristic
23#10587555C2211yescharacteristic
24C4×Dic34—21no—
25C4×Dic34—21no—
26C17×C4⋊C8C2212yes—
27C17×C22⋊C8C2212yes—
28C17×#97C2212yes—
29#10587492C2214yes—
30#10587492C2214yes—
31#10587489C2214yes—
32#10587488C2214yes—
33C22×C4×C17D411yescharacteristic
34C22×C4×C17D411yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C2×C22⋊D17—22no—
37#63252—44no—
38C2×D68—21no—
39C2×D68—21no—
40C2×D68C2312yes—
41C2×C4×D17—22no—
42C2×Dic34—21no—
43C2×Dic34—21no—
44C2×Dic34—21no—
45C2×C8×C17C2314yes—
46C42×C17D411yescharacteristic
47C42×C17D411yescharacteristic
48C42×C17C2311yescharacteristic
49C42×C17C2311yescharacteristic
50C22⋊Dic17—22no—
51#63227—22no—
52#63227—22no—
53C4⋊Dic17—21no—
54C4⋊Dic17—21no—
55C4⋊Dic17—21no—
56C4⋊Dic17C2312yes—
57C4⋊Dic17C2312yes—
58C4⋊Dic17C2312yes—
59#63225—22no—
60#63225—22no—
61C4×Dic17—22no—
62C23×C17C2×D411yescharacteristic
63C22×D17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—41no—
67C2×C4×C17C2×D411yescharacteristic
68C2×C4×C17C2×D411yescharacteristic
69C2×C4×C17C2×D411yescharacteristic
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2411yescharacteristic
73C22⋊D17—44no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77D68—24no—
78D68—42no—
79D68—42no—
80C4×D17—44no—
81Dic34—42no—
82Dic34—42no—
83Dic34—42no—
84C8×C17—24no—
85#3235—171nomaximal, Sylow
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17C22×D411yescharacteristic
90C2×D17—42no—
91C2×D17—42no—
92C2×D17—42no—
93C4×C17—21no—
94C4×C17—21no—
95C4×C17—21no—
96C4×C17—21no—
97C4×C17AGL(1,ℤ/8)12yes—
98C4×C172−1+412yes—
99Dic17—42no—
100Dic17—42no—
101Dic17—42no—
102#533—171no—
103#532—171no—
104#501—172no—
105#490—172no—
106#482—174no—
107#476—172no—
108#474—172no—
109#432D1711yescharacteristic, complemented
110C2×C17—41no—
111C2×C17#53611yescharacteristic
112C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
113C2×C17#57511yescharacteristic
114D17—82no—
115C2×#41—341no—
116C4⋊Q8—171no—
117C4⋊Q8—171no—
118C4⋊Q8—171no—
119#127—171no—
120#125—172no—
121#124—341no—
122C22⋊Q8—341no—
123C22⋊Q8—341no—
124#121—172no—
125#121—341no—
126C4×Q8—341no—
127C4×Q8—341no—
128C4×D4—172no—
129C4×D4—341no—
130#117—172no—
131C2×C42C2×D1711yescharacteristic
132#107—174no—
133#106—174no—
134C4⋊C8C2×D1712yes—
135#102—174no—
136#102—174no—
137C22⋊C8C2×D1712yes—
138#97C2×D1712yes—
139C17#323511yescharacteristic, complemented, Sylow
140#41—684no—
141C2×Q8—341no—
142C2×Q8—341no—
143C2×Q8—341no—
144C2×D4—172no—
145C2×D4—342no—
146C2×D4—341no—
147C2×D4—341no—
148C22×C4—342no—
149C22×C4D6811yescharacteristic
150C22×C4D6811yescharacteristic
151C22×C4C22×D1711yescharacteristic
152C2×C8C22×D1714yes—
153C4⋊C4—172no—
154C4⋊C4—172no—
155C4⋊C4—172no—
156C4⋊C4—341no—
157C4⋊C4—342no—
158C4⋊C4—342no—
159C4⋊C4—341no—
160C4⋊C4—341no—
161C22⋊C4—342no—
162C22⋊C4—342no—
163C22⋊C4—342no—
164C42—342no—
165C42D6811yescharacteristic
166C42D6811yescharacteristic
167C42C22×D1711yescharacteristic
168C42C22×D1711yescharacteristic
169C23—342no—
170C23C2×D6811yescharacteristic
171Q8—682no—
172Q8—682no—
173Q8—682no—
174D4—344no—
175D4—682no—
176D4—682no—
177D4—684no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—684no—
185C2×C4C2×D6811yescharacteristic
186C2×C4C2×D6811yescharacteristic
187C2×C4C2×D6811yescharacteristic
188C2×C4C2×D6811yescharacteristic
189C2×C4C2×D6811yescharacteristic
190C2×C4C23×D1711yescharacteristic
191C8—24no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—682no—
196C22—682no—
197C22—682no—
198C22C22×D6811yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—682no—
204C4—682no—
205C4—682no—
206C4#1058759412yes—
207C4#1058769712yes—
208C2—41no—
209C2—1362no—
210C2#1176338511yescharacteristic
211C2C2×#1058759411yescharacteristic
212C2#1176360911yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#432C2
socleC22×C17C22×D4
prime radical#11763359C2
hypercenter#432D17
hypocenterC17#3235
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
3. #11763218C2semidirect
4. #11763174C2semidirect
5. #11763134C2semidirect
6. #11763100C2semidirect
7. #11763090C2semidirect
8. C17×#432C2semidirect
109. #432D17semidirect
139. C17#3235semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117633592
#117632182
#117631742
#117631342
#117631002
#117630902
C17×#4322
#323517