Finite Group Databasebeta

← #17953533

Subgroups — #17953533

Order 4352.

530 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 112 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#14544675C212yesmaximal
2#14543622C214yesmaximal
3C2×#11762674C211yesmaximal, characteristic
4C2×C22⋊#3624C2214yes—
5#11763675—22no—
6#11763675—22no—
7C22×C4⋊Dic17C411yescharacteristic
8C22×C4×Dic17C2211yescharacteristic
9#11762674—22no—
10#11762674—22no—
11C2×#10587502C412yes—
12C2×#10587502C2212yes—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C23×C4×C17C2×C412yes—
16C23×Dic17C2×C411yescharacteristic
17C23×Dic17C2×C412yes—
18C23×Dic17C2311yescharacteristic
19C22×#3624—24no—
20C2×C4⋊C4×C17—21no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—42no—
23C2×C4⋊Dic17—21no—
24C2×C4⋊Dic17—21no—
25C2×C4⋊Dic17—42no—
26C2×C4×Dic17—24no—
27C2×C4×Dic17D412yes—
28C2×C4×Dic17D412yes—
29C22⋊#3624—28no—
30#10587502—24no—
31#10587502—24no—
32C24×C17C22×C411yescharacteristic
33C22×C4×C17—22no—
34C22×C4×C17—22no—
35C22×C4×C17—22no—
36C22×C4×C17—22no—
37C22×C4×C17—24no—
38C22×C4×C17—42no—
39C22×C4×C17—42no—
40C22×C4×C17C22⋊C411yescharacteristic
41C22×C4×C17C22⋊C411yescharacteristic
42C22×Dic17—22no—
43C22×Dic17—22no—
44C22×Dic17—22no—
45C22×Dic17—22no—
46C22×Dic17—24no—
47C22×Dic17—41no—
48C22×Dic17—41no—
49C22×Dic17—42no—
50C22×Dic17—42no—
51C22×Dic17C22⋊C412yes—
52C22×Dic17C2×D412yes—
53C22×Dic17#4114yes—
54C2×#3624—44no—
55C2×#3624—44no—
56C2×#3624—44no—
57C4⋊C4×C17—42no—
58C4⋊C4×C17—42no—
59C4⋊Dic17—42no—
60C4⋊Dic17—42no—
61C4×Dic17—24no—
62C4×Dic17—24no—
63#8703—171nomaximal, Sylow
64C23×C17—21no—
65C23×C17—21no—
66C23×C17—21no—
67C23×C17—21no—
68C23×C17C2×C22⋊C411yescharacteristic
69C23×C17#11712yes—
70C23×C17#13112yes—
71C23×C17#13112yes—
72C2×C4×C17—28no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—42no—
78C2×C4×C17—42no—
79C2×C4×C17—42no—
80C2×C4×C17—84no—
81C2×C4×C17#10012yes—
82C2×C4×C17#10112yes—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—44no—
89C2×Dic17—44no—
90C2×Dic17—42no—
91C2×Dic17—42no—
92C2×Dic17—42no—
93C2×Dic17—84no—
94C2×Dic17#12314yes—
95#3624—84no—
96C2×#382—171no—
97#1817—172no—
98#1280—174no—
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—22no—
102C22×C17—22no—
103C22×C17—22no—
104C22×C17—22no—
105C22×C17#38111yescharacteristic
106C22×C17#40811yescharacteristic
107C22×C17#40811yescharacteristic
108C22×C17#41011yescharacteristic
109C22×C17#41011yescharacteristic
110C22×C17C2×#10011yescharacteristic
111C22×C17C2×#10111yescharacteristic
112C22×C17#43314yes—
113C4×C17—22no—
114C4×C17—22no—
115C4×C17—82no—
116Dic17—24no—
117Dic17—42no—
118Dic17—82no—
119C22×C4⋊C4—171no—
120C22×C4⋊C4#59511yescharacteristic
121C22×C42—171no—
122C2×C22⋊C8—174no—
123#382—342no—
124#382—342no—
125C2×#95—172no—
126C2×#95—172no—
127#336—342no—
128#336—342no—
129C2×C17—24no—
130C2×C17#146911yescharacteristic
131C2×C17#146911yescharacteristic
132C2×C17#147011yescharacteristic
133C2×C17#147011yescharacteristic
134C2×C17#181811yescharacteristic
135C2×C17#182011yescharacteristic
136C2×C17#182111yescharacteristic
137C23×C4—171no—
138C23×C4—171no—
139C23×C4—172no—
140C23×C4C2×#59512yes—
141C23×C4C2×#59511yescharacteristic
142C22×C8—344no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—341no—
147C2×C4⋊C4—341no—
148C2×C4⋊C4—682no—
149C2×C42—172no—
150C2×C42—172no—
151C2×C42—344no—
152C22⋊C8—348no—
153#95—344no—
154#95—344no—
155C17#870311yescharacteristic, complemented, Sylow
156C24C22×#59511yescharacteristic
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—172no—
165C22×C4—174no—
166C22×C4—172no—
167C22×C4—342no—
168C22×C4—342no—
169C22×C4—342no—
170C22×C4—342no—
171C22×C4—344no—
172C22×C4—681no—
173C22×C4—681no—
174C22×C4—682no—
175C22×C4—682no—
176C22×C4C22⋊#59511yescharacteristic
177C22×C4C22⋊#59511yescharacteristic
178C2×C8—684no—
179C2×C8—684no—
180C2×C8—684no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—682no—
184C4⋊C4—682no—
185C42—344no—
186C42—344no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23#1058767712yes—
192C23#1058767812yes—
193C23#1058768012yes—
194C23C2×C22⋊#59511yescharacteristic
195C2×C4—28no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—84no—
204C2×C4—174no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—684no—
211C2×C4—684no—
212C2×C4—682no—
213C2×C4—682no—
214C2×C4—682no—
215C2×C4—1364no—
216C2×C4#1058754512yes—
217C2×C4#1058755312yes—
218C8—1364no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22#1176375211yescharacteristic
226C22#1176378311yescharacteristic
227C22#1176378411yescharacteristic
228C22#1176378911yescharacteristic
229C22#1176379011yescharacteristic
230C22C2×#1058755311yescharacteristic
231C22C2×#1058754511yescharacteristic
232C22#1176383414yes—
233C4—22no—
234C4—22no—
235C4—82no—
236C4—344no—
237C4—682no—
238C4—1362no—
239C2—24no—
240C2#1454405411yescharacteristic
241C2#1454405511yescharacteristic
242C2#1454405711yescharacteristic
243C2#1454405811yescharacteristic
244C2#1454467811yescharacteristic
245C2#1454468211yescharacteristic
246C2#1454468311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutator, prime radicalC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8703
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
155. C17#8703semidirect

Maximal subgroups

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

subgroupindex
#145446752
#145436222
C2×#117626742
#870317