Finite Group Databasebeta

← #17949669

Subgroups — #17949669

Order 4352.

528 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544498C212yesmaximal, 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—28no—
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—42no—
55C22×Dic17—44no—
56C22×Dic17C22⋊C411yescharacteristic
57C22×Dic17C22⋊C411yescharacteristic
58C22×Dic17C2×D412yes—
59C2×#3624—28no—
60C2×#3624—44no—
61C2×#3624—44no—
62C22⋊C4×C17—28no—
63C42×C17C2×C812yes—
64C42×C17M1612yes—
65C22⋊Dic17—42no—
66C22⋊Dic17—44no—
67C4⋊Dic17—42no—
68C4⋊Dic17—41no—
69C4⋊Dic17—41no—
70C4×Dic17—22no—
71C4×Dic17—24no—
72C4×Dic17—24no—
73C4×Dic17—41no—
74C4×Dic17—42no—
75#7567—171nomaximal, Sylow
76C23×C17—22no—
77C23×C17—41no—
78C23×C17—41no—
79C23×C17—42no—
80C23×C17—42no—
81C23×C17C2×C22⋊C411yescharacteristic
82Q8×C17—42no—
83Q8×C17—42no—
84D4×C17—42no—
85D4×C17—42no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17C22⋊C814yes—
94C2×C4×C17C22×C811yescharacteristic
95C2×C4×C17C2×M1611yescharacteristic
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—21no—
99C2×Dic17—22no—
100C2×Dic17—41no—
101C2×Dic17—41no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—84no—
105C2×Dic17#10012yes—
106C2×Dic17#10414yes—
107C2×Dic17#13514yes—
108#3624—48no—
109C4×#124—171no—
110#1744—172no—
111#1272—172no—
112#1270—172no—
113C22×C17—42no—
114C22×C17—44no—
115C22×C17—81no—
116C22×C17—82no—
117C22×C17—81no—
118C22×C17C2×C22⋊C811yescharacteristic
119C22×C17#41011yescharacteristic
120C22×C17C2×#10011yescharacteristic
121C22×C17#41612yes—
122C22×C17C2×#10412yes—
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—24no—
126C4×C17—42no—
127Dic17—24no—
128Dic17—41no—
129Dic17—81no—
130Dic17—81no—
131Dic17#32914yes—
132C2×#124#59511yescharacteristic
133C2×C4×Q8—171no—
134C2×C4×D4—171no—
135C2×C4×C8—342no—
136#391—171no—
137#390—171no—
138#388—342no—
139#382—171no—
140C4×C22⋊C4—342no—
141C43—171no—
142#336—174no—
143#336—342no—
144C2×C17—82no—
145C2×C17#146011yescharacteristic
146C2×C17#147612yes—
147C2×C17#150012yes—
148C2×C17C2×#32911yescharacteristic
149C2×C17#154311yescharacteristic
150C22×Q8C2×#59511yescharacteristic
151C22×D4C2×#59511yescharacteristic
152C23×C4—341no—
153C22×C8—344no—
154#124—22no—
155#124#362414yescomplemented
156C4×Q8—682no—
157C4×D4—682no—
158C2×C4⋊C4—171no—
159C2×C4⋊C4—171no—
160C2×C4⋊C4—341no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—341no—
163C2×C22⋊C4—342no—
164C2×C42—172no—
165C2×C42—172no—
166C2×C42—171no—
167C2×C42—171no—
168C2×C42—341no—
169C2×C42C2×#59511yescharacteristic
170C4×C8—348no—
171#95—342no—
172#95—342no—
173C17#756711yescharacteristic, complemented, Sylow
174C24—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×#362412yes—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4C2×#362412yes—
183C22×C4—21no—
184C22×C4—171no—
185C22×C4—172no—
186C22×C4—171no—
187C22×C4—341no—
188C22×C4—341no—
189C22×C4—682no—
190C22×C4—684no—
191C22×C4C22⋊#59512yes—
192C22×C4C22×#59511yescharacteristic
193C2×C8—348no—
194C2×C8—684no—
195C2×C8—684no—
196C4⋊C4—681no—
197C4⋊C4—681no—
198C4⋊C4—682no—
199C22⋊C4—28no—
200C22⋊C4—684no—
201C22⋊C4—682no—
202C42—344no—
203C42—342no—
204C42—344no—
205C42—681no—
206C42—682no—
207C42C2×#362412yes—
208C42#6324712yes—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊#59511yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—172no—
227C2×C4—174no—
228C2×C4—174no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—681no—
234C2×C4—681no—
235C2×C4—682no—
236C2×C4—682no—
237C2×C4—1364no—
238C2×C4C22⋊#362414yes—
239C2×C4C22×#362411yescharacteristic
240C2×C4C2×#6324711yescharacteristic
241C8—688no—
242C22—42no—
243C22—44no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22C2×C22⋊#362411yescharacteristic
248C22#1176378911yescharacteristic
249C22D17⋊#10011yescharacteristic
250C22#1176379912yes—
251C22D17⋊#10412yes—
252C4—21no—
253C4—21no—
254C4—24no—
255C4—42no—
256C4—174no—
257C4—344no—
258C4—681no—
259C4—1361no—
260C4—1361no—
261C2—82no—
262C2#1454402411yescharacteristic
263C2#1454407512yes—
264C2#1454415612yes—
265C2D17⋊#32911yescharacteristic
266C2#1454428811yescharacteristic
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#7567
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. #14544498C2semidirect
2. #14543604C2semidirect
22. C17×#124C8semidirect
155. #124#3624semidirect
173. C17#7567semidirect

Maximal subgroups

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

subgroupindex
#145444982
#145436042
#145436002
#124×Dic172
#756717