Finite Group Databasebeta

← #17949768

Subgroups — #17949768

Order 4352.

630 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 124 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#14544674C212yesmaximal
2#14543605C214yesmaximal, complemented
3C2×#11762714C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5#11763675—22no—
6#11763675C2214yes—
7C23×Dic34C411yescharacteristic
8C22×C4×Dic17C2211yescharacteristic
9#11762714—22no—
10#11762714C412yes—
11#11762714C2212yes—
12C2×#10587502—22no—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17—21no—
15C23×C4×C17C2×C411yescharacteristic
16C23×Dic17—21no—
17C23×Dic17D412yes—
18C22×Dic34—21no—
19C22×Dic34—22no—
20C22×Dic34—42no—
21C22×Dic34—42no—
22C22×Dic34C814yescomplemented
23C22×Dic34C2×C412yes—
24C22×#3624—24no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—42no—
27C2×C4⋊C4×C17C2×C412yes—
28C2×C4×Dic17—42no—
29C2×C4×Dic17C2×C411yescharacteristic
30C2×C4×Dic17D412yes—
31C2×C4×Dic17C2311yescharacteristic
32C22⋊#3624—44no—
33#10587502—24no—
34#10587502—24no—
35C24×C17C22⋊C411yescharacteristic
36C22×C4×C17—22no—
37C22×C4×C17—22no—
38C22×C4×C17—22no—
39C22×C4×C17—44no—
40C22×C4×C17—42no—
41C22×C4×C17C22⋊C411yescharacteristic
42C22×C4×C17C2×C812yes—
43C22×C4×C17M1612yes—
44C22×C4×C17C22×C411yescharacteristic
45C22×Dic17—24no—
46C22×Dic17—24no—
47C22×Dic17—22no—
48C22×Dic17—24no—
49C22×Dic17—42no—
50C22×Dic17—42no—
51C22×Dic17—42no—
52C22×Dic17C22⋊C412yes—
53C22×Dic17C2×D412yes—
54C2×Dic34—24no—
55C2×Dic34—22no—
56C2×Dic34—48no—
57C2×Dic34—42no—
58C2×Dic34—42no—
59C2×Dic34—48no—
60C2×Dic34—42no—
61C2×Dic34—42no—
62C2×Dic34C2×C814yes—
63C2×#3624—44no—
64C2×#3624—44no—
65C2×#3624—44no—
66C4⋊C4×C17—42no—
67C4⋊C4×C17—42no—
68C4×Dic17—41no—
69C4×Dic17—42no—
70C4×Dic17—41no—
71#7595—171nomaximal, Sylow
72C23×C17—22no—
73C23×C17—22no—
74C23×C17C22⋊C812yes—
75C23×C17#9912yes—
76C23×C17#10012yes—
77C23×C17C2×C22⋊C411yescharacteristic
78C2×C4×C17—24no—
79C2×C4×C17—24no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—42no—
85C2×C4×C17—44no—
86C2×C4×C17—84no—
87C2×C4×C17C22⋊C812yes—
88C2×C4×C17C22×C811yescharacteristic
89C2×C4×C17C2×M1611yescharacteristic
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—24no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—48no—
97C2×Dic17—48no—
98C2×Dic17—42no—
99C2×Dic17—42no—
100C2×Dic17#13714yes—
101Dic34—48no—
102Dic34—44no—
103Dic34—44no—
104#3624—84no—
105C2×#391—171no—
106#1817—172no—
107#1272—174no—
108C22×C17—22no—
109C22×C17—24no—
110C22×C17—24no—
111C22×C17—22no—
112C22×C17#32314yes—
113C22×C17C2×C22⋊C811yescharacteristic
114C22×C17C2×#9911yescharacteristic
115C22×C17C2×#10011yescharacteristic
116C22×C17#41912yes—
117C22×C17#42112yes—
118C4×C17—22no—
119C4×C17—21no—
120C4×C17—21no—
121C4×C17—82no—
122Dic17—24no—
123Dic17—44no—
124Dic17—42no—
125C23×Q8—171no—
126C22×C4⋊C4#59511yescharacteristic
127C22×C42—171no—
128C2×C22⋊C8—342no—
129#391—172no—
130#391—172no—
131#391—342no—
132C2×#95—342no—
133#336—174no—
134#336—342no—
135C2×C17—24no—
136C2×C17C2×#32311yescharacteristic
137C2×C17#147912yes—
138C2×C17#151112yes—
139C2×C17#153311yescharacteristic
140C2×C17#155811yescharacteristic
141C22×Q8—174no—
142C22×Q8—172no—
143C22×Q8—342no—
144C22×Q8—341no—
145C22×Q8—682no—
146C22×Q8—682no—
147C23×C4—21no—
148C23×C4—172no—
149C23×C4—341no—
150C23×C4C2×#59511yescharacteristic
151C22×C8—344no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C2×#59512yes—
155C2×C42—172no—
156C2×C42—171no—
157C2×C42—171no—
158C2×C42—682no—
159C22⋊C8—684no—
160#95—344no—
161#95—344no—
162C17#759511yescharacteristic, complemented, Sylow
163C24C22⋊#59511yescharacteristic
164C2×Q8—174no—
165C2×Q8—342no—
166C2×Q8—344no—
167C2×Q8—682no—
168C2×Q8—682no—
169C2×Q8—688no—
170C2×Q8—682no—
171C2×Q8—682no—
172C2×Q8—688no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—172no—
179C22×C4—172no—
180C22×C4—344no—
181C22×C4—342no—
182C22×C4—344no—
183C22×C4—344no—
184C22×C4—682no—
185C22×C4—682no—
186C22×C4—682no—
187C22×C4#6324212yes—
188C22×C4#6324312yes—
189C22×C4C22⋊#59511yescharacteristic
190C22×C4C22×#59511yescharacteristic
191C2×C8—684no—
192C2×C8—684no—
193C2×C8—684no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C42—682no—
197C42—681no—
198C42—681no—
199C23—22no—
200C23—22no—
201C23#1058753912yes—
202C23#1058754312yes—
203C23#1058755312yes—
204C23C2×C22⋊#59511yescharacteristic
205Q8—684no—
206Q8—684no—
207Q8—688no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—84no—
217C2×C4—174no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—344no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—688no—
225C2×C4—682no—
226C2×C4—682no—
227C2×C4—688no—
228C2×C4#1058754312yes—
229C2×C4C2×#6324211yescharacteristic
230C2×C4C2×#6324311yescharacteristic
231C8—1364no—
232C22—24no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22#1176365314yes—
237C22C2×#1058754311yescharacteristic
238C22C2×#1058753911yescharacteristic
239C22C2×#1058755311yescharacteristic
240C22#1176380212yes—
241C22#1176380712yes—
242C4—21no—
243C4—21no—
244C4—22no—
245C4—82no—
246C4—344no—
247C4—682no—
248C4—684no—
249C2—24no—
250C2C2×#1176365311yescharacteristic
251C2#1454408812yes—
252C2#1454419212yes—
253C2#1454426311yescharacteristic
254C2#1454431711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#7595
LCSG ▷ C2×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. #14543605C2semidirect
22. C22×Dic34C8semidirect
162. C17#7595semidirect

Maximal subgroups

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

subgroupindex
#145446742
#145436052
C2×#117627142
#759517