Finite Group Databasebeta

← #17953592

Subgroups — #17953592

Order 4352.

630 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 140 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#14544640C212yesmaximal
2#14543662C214yesmaximal, complemented
3C2×#11762678C211yesmaximal, characteristic
4C2×#10587544C2214yes—
5C2×C22⋊#3624—22no—
6#11763675—22no—
7C22×C4⋊Dic17C411yescharacteristic
8C22×#63225—22no—
9C22×C4×Dic17C2211yescharacteristic
10#11762678—22no—
11#11762678C412yes—
12#11762678C2212yes—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17—21no—
15C23×C4×C17C2×C411yescharacteristic
16C23×Dic17—21no—
17C23×Dic17D412yes—
18C22×#3624—24no—
19C2×C4⋊C4×C17—21no—
20C2×C4⋊C4×C17—42no—
21C2×C4⋊C4×C17C2×C412yes—
22C2×C4⋊Dic17—21no—
23C2×C4⋊Dic17—42no—
24C2×C4⋊Dic17C2×C412yes—
25C2×#63225—24no—
26C2×#63225—24no—
27C2×#63225—44no—
28C2×#63225—44no—
29C2×C4×Dic17—24no—
30C2×C4×Dic17C2×C411yescharacteristic
31C2×C4×Dic17D412yes—
32C2×C4×Dic17C2311yescharacteristic
33C22⋊#3624—44no—
34#10587544—28no—
35C24×C17C22⋊C411yescharacteristic
36C22×C4×C17—22no—
37C22×C4×C17—22no—
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17—42no—
41C22×C4×C17—44no—
42C22×C4×C17C22⋊C411yescharacteristic
43C22×C4×C17C22×C411yescharacteristic
44C22×Dic17—24no—
45C22×Dic17—24no—
46C22×Dic17—22no—
47C22×Dic17—42no—
48C22×Dic17—44no—
49C22×Dic17C22⋊C412yes—
50C22×Dic17D814yes—
51C22×Dic17SD1614yes—
52C22×Dic17C2×D412yes—
53C2×#3624—44no—
54C2×#3624—44no—
55C2×#3624—44no—
56C4⋊C4×C17—42no—
57C4⋊C4×C17—42no—
58C4⋊Dic17—42no—
59C4⋊Dic17—42no—
60#63225—48no—
61#63225—44no—
62#63225—44no—
63C4×Dic17—22no—
64C4×Dic17—24no—
65C4×Dic17#4114yes—
66#8723—171nomaximal, Sylow
67C23×C17—21no—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17#10012yes—
71C23×C17#10214yes—
72C23×C17C2×C22⋊C411yescharacteristic
73C2×C4×C17—28no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—42no—
79C2×C4×C17—44no—
80C2×C4×C17—84no—
81C2×C4×C17#10112yes—
82C2×C4×C17#11712yes—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—28no—
86C2×Dic17—28no—
87C2×Dic17—42no—
88C2×Dic17—44no—
89C2×Dic17—84no—
90C2×Dic17#10214yes—
91C2×Dic17#12314yes—
92C2×Dic17C2×D812yes—
93C2×Dic17C2×SD1612yes—
94C2×Dic17#13714yes—
95#3624—84no—
96C2×#384—171no—
97#1797—172no—
98#1296—174no—
99C22×C17—24no—
100C22×C17—22no—
101C22×C17—22no—
102C22×C17—24no—
103C22×C17#33114yes—
104C22×C17#38111yescharacteristic
105C22×C17C2×#10011yescharacteristic
106C22×C17C2×#10111yescharacteristic
107C22×C17C2×#10212yes—
108C22×C17#41912yes—
109C4×C17—22no—
110C4×C17—22no—
111C4×C17—82no—
112Dic17—24no—
113Dic17—82no—
114Dic17#48014yes—
115Dic17#48314yes—
116C22×C4⋊C4—171no—
117C22×C4⋊C4—342no—
118C22×C4⋊C4#59511yescharacteristic
119C22×C42—171no—
120C2×C4⋊C8—174no—
121C2×C22⋊C8—342no—
122#384—172no—
123#384—172no—
124#384—342no—
125#336—342no—
126C2×C17—24no—
127C2×C17C2×#33111yescharacteristic
128C2×C17#154612yes—
129C2×C17#155811yescharacteristic
130C2×C17#182111yescharacteristic
131C2×C17#182312yes—
132C23×C4—21no—
133C23×C4—172no—
134C23×C4—341no—
135C23×C4C2×#59511yescharacteristic
136C22×C8—344no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—344no—
141C2×C4⋊C4—344no—
142C2×C4⋊C4—341no—
143C2×C4⋊C4—682no—
144C2×C4⋊C4—684no—
145C2×C4⋊C4—684no—
146C2×C4⋊C4C2×#59512yes—
147C2×C42—171no—
148C2×C42—172no—
149C2×C42—171no—
150C2×C42—344no—
151C4⋊C8—348no—
152C22⋊C8—684no—
153C17#872311yescharacteristic, complemented, Sylow
154C24C22⋊#59511yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—174no—
164C22×C4—174no—
165C22×C4—344no—
166C22×C4—344no—
167C22×C4—342no—
168C22×C4—684no—
169C22×C4—682no—
170C22×C4C22⋊#59511yescharacteristic
171C22×C4C22×#59511yescharacteristic
172C2×C8—684no—
173C2×C8—684no—
174C2×C8—684no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—682no—
178C4⋊C4—682no—
179C4⋊C4—684no—
180C4⋊C4—684no—
181C4⋊C4—688no—
182C42—174no—
183C42—344no—
184C42—342no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23#1058754714yes—
189C23#1058755312yes—
190C23C2×C22⋊#59511yescharacteristic
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—84no—
199C2×C4—174no—
200C2×C4—172no—
201C2×C4—172no—
202C2×C4—174no—
203C2×C4—174no—
204C2×C4—348no—
205C2×C4—348no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—682no—
209C2×C4—684no—
210C2×C4—1364no—
211C2×C4#1058754512yes—
212C2×C4#1058767712yes—
213C8—1364no—
214C22—22no—
215C22—24no—
216C22—24no—
217C22—22no—
218C22#1176366714yes—
219C22#1176375211yescharacteristic
220C22C2×#1058755311yescharacteristic
221C22C2×#1058754511yescharacteristic
222C22C2×#1058754712yes—
223C22#1176380212yes—
224C4—22no—
225C4—22no—
226C4—82no—
227C4—174no—
228C4—174no—
229C4—344no—
230C4—1362no—
231C2—24no—
232C2C2×#1176366711yescharacteristic
233C2#1454429312yes—
234C2#1454431711yescharacteristic
235C2#1454468311yescharacteristic
236C2#1454468712yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8723
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543662C2semidirect
153. C17#8723semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145436622
C2×#117626782
#872317