Finite Group Databasebeta

← #17949789

Subgroups — #17949789

Order 4352.

492 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544699C212yesmaximal, complemented
2#14543604C212yesmaximal, complemented
3#14543600C212yesmaximal
4#124×Dic17C211yesmaximal, characteristic
5C2×C4⋊#3624—22no—
6#11763675—22no—
7#11763675C2214yes—
8C2×Q8×Dic17C411yescharacteristic
9C2×D4×Dic17C411yescharacteristic
10#11762716C2211yescharacteristic
11#11762712C2211yescharacteristic
12#11762702—22no—
13#11762675C411yescharacteristic
14C22⋊C4×Dic17—22no—
15C42×Dic17C2211yescharacteristic
16C2×C17×#124C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×Dic17—21no—
20C22×#3624—24no—
21C17×#124—22no—
22C17×#124C814yescomplemented
23C2×C22⋊C4×C17—22no—
24C2×C42×C17C2×C411yescharacteristic
25C2×C22⋊Dic17—21no—
26C2×C22⋊Dic17—22no—
27Q8×Dic17—42no—
28D4×Dic17—42no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17C2×C411yescharacteristic
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4×Dic17—21no—
33C2×C4×Dic17C2×C411yescharacteristic
34C2×C4×Dic17D412yes—
35C2×C4×Dic17D412yes—
36C2×C4×Dic17C2311yescharacteristic
37C4⋊#3624—44no—
38#10587502—22no—
39#10587502—22no—
40C24×C17—21no—
41C2×Q8×C17—21no—
42C2×Q8×C17—41no—
43C2×Q8×C17—41no—
44C2×Q8×C17C2×C812yes—
45C2×D4×C17—21no—
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D4×C17C2×C812yes—
49C22×C4×C17—21no—
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22×C411yescharacteristic
52C22×Dic17—21no—
53C22×Dic17—21no—
54C22×Dic17—44no—
55C22×Dic17—42no—
56C22×Dic17C22⋊C411yescharacteristic
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C2×D412yes—
59C2×#3624—44no—
60C2×#3624—44no—
61C2×#3624—44no—
62C22⋊C4×C17—28no—
63C42×C17C2×C812yes—
64C42×C17M1612yes—
65C22⋊Dic17—42no—
66C22⋊Dic17—42no—
67C22⋊Dic17—42no—
68C4⋊Dic17—42no—
69C4⋊Dic17—41no—
70C4⋊Dic17—41no—
71C4×Dic17—22no—
72C4×Dic17—22no—
73C4×Dic17—22no—
74C4×Dic17—42no—
75C4×Dic17—41no—
76C4×Dic17—42no—
77#7601—171nomaximal, Sylow
78C23×C17—22no—
79C23×C17—41no—
80C23×C17—41no—
81C23×C17—42no—
82C23×C17—42no—
83C23×C17C2×C22⋊C411yescharacteristic
84Q8×C17—42no—
85Q8×C17—42no—
86D4×C17—42no—
87D4×C17—42no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17C22⋊C814yes—
96C2×C4×C17C22×C811yescharacteristic
97C2×C4×C17C2×M1611yescharacteristic
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—42no—
106C2×Dic17—42no—
107C2×Dic17—84no—
108C2×Dic17#10012yes—
109C2×Dic17AGL(1,ℤ/8)12yes—
110C2×Dic17#13712yes—
111#3624—84no—
112C4×#124—171no—
113#1831—172no—
114#1272—172no—
115#1270—172no—
116C22×C17—42no—
117C22×C17—44no—
118C22×C17—81no—
119C22×C17—82no—
120C22×C17—81no—
121C22×C17C2×C22⋊C811yescharacteristic
122C22×C17#41011yescharacteristic
123C22×C17C2×#10011yescharacteristic
124C22×C17#41712yes—
125C22×C17#42112yes—
126C4×C17—21no—
127C4×C17—21no—
128C4×C17—24no—
129C4×C17—42no—
130Dic17—22no—
131Dic17—22no—
132Dic17—22no—
133Dic17—41no—
134Dic17—81no—
135Dic17—81no—
136C2×#124#59511yescharacteristic
137C2×C4×Q8—171no—
138C2×C4×D4—171no—
139C2×C4⋊C8—342no—
140#391—171no—
141#390—171no—
142#388—342no—
143#382—171no—
144C4×C22⋊C4—342no—
145C43—171no—
146#336—174no—
147#336—342no—
148C2×C17—82no—
149C2×C17#146011yescharacteristic
150C2×C17#147712yes—
151C2×C17#151212yes—
152C2×C17#153311yescharacteristic
153C2×C17#156011yescharacteristic
154C22×Q8C2×#59511yescharacteristic
155C22×D4C2×#59511yescharacteristic
156C23×C4—341no—
157C22×C8—344no—
158#124—22no—
159#124#362414yescomplemented
160C4×Q8—682no—
161C4×D4—682no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—341no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—341no—
167C2×C22⋊C4—342no—
168C2×C42—172no—
169C2×C42—171no—
170C2×C42—172no—
171C2×C42—171no—
172C2×C42—341no—
173C2×C42C2×#59511yescharacteristic
174C4⋊C8—684no—
175#95—342no—
176#95—342no—
177C17#760111yescharacteristic, complemented, Sylow
178C24—21no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C2×#362412yes—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C2×#362412yes—
187C22×C4—21no—
188C22×C4—171no—
189C22×C4—172no—
190C22×C4—171no—
191C22×C4—341no—
192C22×C4—341no—
193C22×C4—682no—
194C22×C4—684no—
195C22×C4C22⋊#59512yes—
196C22×C4C22×#59511yescharacteristic
197C2×C8—684no—
198C2×C8—684no—
199C2×C8—684no—
200C4⋊C4—681no—
201C4⋊C4—681no—
202C4⋊C4—682no—
203C22⋊C4—28no—
204C22⋊C4—682no—
205C22⋊C4—682no—
206C22⋊C4—682no—
207C42—342no—
208C42—342no—
209C42—342no—
210C42—681no—
211C42—682no—
212C42—682no—
213C42C2×#362412yes—
214C42#6324712yes—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23C2×C22⋊#59511yescharacteristic
221Q8—42no—
222Q8—42no—
223D4—42no—
224D4—42no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—172no—
233C2×C4—172no—
234C2×C4—172no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—341no—
239C2×C4—342no—
240C2×C4—681no—
241C2×C4—681no—
242C2×C4—682no—
243C2×C4—682no—
244C2×C4—1364no—
245C2×C4C22⋊#362414yes—
246C2×C4C22×#362411yescharacteristic
247C2×C4C2×#6324711yescharacteristic
248C8—1364no—
249C22—42no—
250C22—44no—
251C22—82no—
252C22—81no—
253C22—81no—
254C22C2×C22⋊#362411yescharacteristic
255C22#1176378911yescharacteristic
256C22D17⋊#10011yescharacteristic
257C22#1176380012yes—
258C22#1176380612yes—
259C4—21no—
260C4—21no—
261C4—24no—
262C4—42no—
263C4—342no—
264C4—342no—
265C4—342no—
266C4—681no—
267C4—1361no—
268C4—1361no—
269C2—82no—
270C2#1454402411yescharacteristic
271C2#1454407912yes—
272C2#1454419612yes—
273C2#1454426411yescharacteristic
274C2#1454432211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenterC2×#124#595
hypocenterC17#7601
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14544699C2semidirect
2. #14543604C2semidirect
22. C17×#124C8semidirect
159. #124#3624semidirect
177. C17#7601semidirect

Maximal subgroups

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

subgroupindex
#145446992
#145436042
#145436002
#124×Dic172
#760117