Finite Group Databasebeta

← #17949779

Subgroups — #17949779

Order 4352.

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

normal subgroups only

This group has 514 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#14544638C212yesmaximal
2#14543603C214yesmaximal, complemented
3#14531266C211yesmaximal, characteristic
4C2×C4⋊#3624—22no—
5C2×C22⋊#3624—22no—
6#11763675C2214yes—
7C2×#10587612C411yescharacteristic
8C22×C4×Dic17C2211yescharacteristic
9#11762705C412yes—
10#11762705C2212yes—
11#11762681—22no—
12C4×#63225—22no—
13C2×C17×#117C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C23×Dic17D412yes—
16C22×#3624—24no—
17C17×#117—42no—
18C2×C4⋊C4×C17C2×C412yes—
19C2×C22⋊C4×C17—21no—
20C2×C42×C17—21no—
21C2×C22⋊Dic17—21no—
22#10587612—22no—
23#10587612C814yescomplemented
24C2×C4⋊Dic17C2×C412yes—
25C2×#63225—22no—
26C2×#63225—22no—
27C2×C4×Dic17—21no—
28C2×C4×Dic17—42no—
29C2×C4×Dic17C2×C411yescharacteristic
30C2×C4×Dic17D412yes—
31C2×C4×Dic17C2311yescharacteristic
32C22⋊#3624—44no—
33C4⋊#3624—44no—
34#10587502—22no—
35#10587502—22no—
36C24×C17C22⋊C411yescharacteristic
37C22×C4×C17—22no—
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17C22⋊C411yescharacteristic
41C22×C4×C17C2×C812yes—
42C22×C4×C17M1612yes—
43C22×C4×C17C22×C411yescharacteristic
44C22×Dic17—22no—
45C22×Dic17—22no—
46C22×Dic17—22no—
47C22×Dic17—42no—
48C22×Dic17—42no—
49C22×Dic17C22⋊C412yes—
50C22×Dic17C2×D412yes—
51C2×#3624—44no—
52C2×#3624—44no—
53C2×#3624—44no—
54C4⋊C4×C17—42no—
55C22⋊C4×C17—42no—
56C42×C17—42no—
57C22⋊Dic17—24no—
58C4⋊Dic17—22no—
59C4⋊Dic17C2×C814yes—
60#63225—44no—
61#63225—42no—
62#63225—42no—
63C4×Dic17—24no—
64C4×Dic17—24no—
65C4×Dic17—41no—
66C4×Dic17—41no—
67#7598—171nomaximal, Sylow
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17C22⋊C812yes—
73C23×C17C2×C22⋊C411yescharacteristic
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—24no—
77C2×C4×C17—24no—
78C2×C4×C17—42no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17C22⋊C812yes—
82C2×C4×C17#9912yes—
83C2×C4×C17#10112yes—
84C2×C4×C17C22×C811yescharacteristic
85C2×C4×C17C2×M1611yescharacteristic
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—24no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—44no—
93C2×Dic17—44no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17AGL(1,ℤ/8)12yes—
97C2×Dic17#13712yes—
98#3624—84no—
99#2296—171no—
100#1797—172no—
101#1271—174no—
102C22×C17—22no—
103C22×C17—24no—
104C22×C17—41no—
105C22×C17—41no—
106C22×C17—42no—
107C22×C17C2×C22⋊C811yescharacteristic
108C22×C17C2×#9911yescharacteristic
109C22×C17C2×#10111yescharacteristic
110C22×C17#41712yes—
111C22×C17#42112yes—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—82no—
115C4×C17#32414yes—
116Dic17—22no—
117Dic17—22no—
118Dic17—44no—
119Dic17—42no—
120C2×#117—171no—
121C2×#117#59511yescharacteristic
122C22×C42—171no—
123C2×C4⋊C8—342no—
124C2×C22⋊C8—342no—
125#389—172no—
126#389—172no—
127#385—342no—
128C4×C4⋊C4—342no—
129#336—174no—
130C2×C17—42no—
131C2×C17C2×#32411yescharacteristic
132C2×C17#147712yes—
133C2×C17#151312yes—
134C2×C17#153711yescharacteristic
135C2×C17#156111yescharacteristic
136C23×C4—172no—
137C23×C4C2×#59511yescharacteristic
138C22×C8—344no—
139#117—42no—
140#117—174no—
141#117—342no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—342no—
144C2×C4⋊C4—342no—
145C2×C4⋊C4C2×#59512yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—341no—
148C2×C42—21no—
149C2×C42—172no—
150C2×C42—171no—
151C2×C42—171no—
152C2×C42—341no—
153C2×C42—682no—
154C4⋊C8—684no—
155C22⋊C8—684no—
156#95—342no—
157#95—342no—
158C17#759811yescharacteristic, complemented, Sylow
159C24C22⋊#59511yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—172no—
164C22×C4—172no—
165C22×C4—342no—
166C22×C4—342no—
167C22×C4—342no—
168C22×C4—682no—
169C22×C4—682no—
170C22×C4#6324212yes—
171C22×C4#6324312yes—
172C22×C4C22⋊#59511yescharacteristic
173C22×C4C22×#59511yescharacteristic
174C2×C8—684no—
175C2×C8—684no—
176C2×C8—684no—
177C4⋊C4—42no—
178C4⋊C4—174no—
179C4⋊C4—342no—
180C4⋊C4—682no—
181C4⋊C4—682no—
182C4⋊C4—684no—
183C22⋊C4—42no—
184C22⋊C4—344no—
185C42—42no—
186C42—344no—
187C42—344no—
188C42—681no—
189C42—681no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23#1058754312yes—
195C23C2×C22⋊#59511yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—172no—
204C2×C4—172no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—344no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—684no—
212C2×C4—684no—
213C2×C4—682no—
214C2×C4—682no—
215C2×C4#1058754312yes—
216C2×C4#1058754512yes—
217C2×C4#1058755112yes—
218C2×C4C2×#6324211yescharacteristic
219C2×C4C2×#6324311yescharacteristic
220C8—1364no—
221C22—22no—
222C22—24no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22C2×#1058754311yescharacteristic
227C22C2×#1058755111yescharacteristic
228C22C2×#1058754511yescharacteristic
229C22#1176380012yes—
230C22#1176380612yes—
231C4—21no—
232C4—21no—
233C4—82no—
234C4—342no—
235C4—342no—
236C4—682no—
237C4—684no—
238C4#1176365514yes—
239C2—42no—
240C2C2×#1176365511yescharacteristic
241C2#1454408112yes—
242C2#1454419912yes—
243C2#1454427511yescharacteristic
244C2#1454432311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#117#595
hypocenterC17#7598
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #14543603C2semidirect
23. #10587612C8semidirect
158. C17#7598semidirect

Maximal subgroups

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

subgroupindex
#145446382
#145436032
#145312662
#759817