Finite Group Databasebeta

← #17953215

Subgroups — #17953215

Order 4352.

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

normal subgroups only

This group has 540 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#14544570C212yesmaximal, complemented
2C2×#11763665C212yesmaximal
3#14543660C212yesmaximal, complemented
4C4×#10587633C211yesmaximal, characteristic
5C2×C4⋊#3624—22no—
6C2×#10587542C2214yes—
7#11763675—22no—
8#11763665—24no—
9C2×#10587633C2211yescharacteristic
10C2×C4×D68C411yescharacteristic
11#11762716C411yescharacteristic
12#11762710C411yescharacteristic
13#11762699—22no—
14#11762674C2211yescharacteristic
15C4×#63227—22no—
16C42×Dic17C2211yescharacteristic
17C2×C4×Q8×C17C411yescharacteristic
18C22×Q8×C17C2×C411yescharacteristic
19C22×D68C2×C411yescharacteristic
20C22×C4×D17—21no—
21C22×#3624—24no—
22C4×Q8×C17—42no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C42×C17—21no—
26C2×C42×C17C2×C411yescharacteristic
27#10587633—22no—
28#10587633D412yes—
29#10587633Q812yes—
30C2×#63227—21no—
31C2×#63227—22no—
32C2×#63227—22no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D412yes—
36C2×C4×Dic17D412yes—
37C2×C4×Dic17C2311yescharacteristic
38C4×D68—42no—
39#10587542—24no—
40#10587542—24no—
41C4⋊#3624—28no—
42#10587502—22no—
43#10587502—22no—
44C23×D17—21no—
45C2×Q8×C17—21no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17C4⋊C412yes—
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22×C411yescharacteristic
54C22×Dic17—21no—
55C22×Dic17C22⋊C412yes—
56C22×Dic17C2×D412yes—
57C2×D68—21no—
58C2×D68—41no—
59C2×D68—41no—
60C2×D68C4⋊C412yes—
61C2×C4×D17—42no—
62C2×C4×D17—44no—
63C2×#3624—28no—
64C2×#3624—44no—
65C2×#3624—44no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—42no—
68C42×C17—22no—
69C42×C17—42no—
70#63227—24no—
71#63227—24no—
72#63227—44no—
73#63227—42no—
74C4⋊Dic17—41no—
75C4×Dic17—24no—
76C4×Dic17—41no—
77C4×Dic17—42no—
78C4×Dic17C2×D411yescharacteristic
79C4×Dic17C2×Q811yescharacteristic
80C4×Dic17#4112yes—
81#8579—171nomaximal, Sylow
82C23×C17C2×C22⋊C411yescharacteristic
83C22×D17—22no—
84C22×D17—41no—
85C22×D17—41no—
86C22×D17—42no—
87C22×D17—42no—
88Q8×C17—42no—
89Q8×C17—42no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17—42no—
99C2×C4×C17#10012yes—
100C2×C4×C17#10414yes—
101C2×C4×C17C2×C4⋊C411yescharacteristic
102C2×C4×C17#11711yescharacteristic
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—41no—
109C2×Dic17—42no—
110C2×Dic17#12112yes—
111C2×Dic17C22⋊Q812yes—
112D68—42no—
113D68—42no—
114C4×D17—84no—
115#3624—48no—
116C4×#124—171no—
117#1767—172no—
118C2×#329—172no—
119#1295—172no—
120C22×C17#38011yescharacteristic
121C22×C17C2×#10011yescharacteristic
122C22×C17#41311yescharacteristic
123C22×C17C2×#10412yes—
124C22×C17#42112yes—
125C2×D17—42no—
126C2×D17—44no—
127C2×D17—81no—
128C2×D17—81no—
129C2×D17—82no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—41no—
133C4×C17—42no—
134C4×C17—81no—
135C4×C17#32914yescomplemented
136Dic17—22no—
137Dic17—22no—
138Dic17—42no—
139Dic17—81no—
140C2×#124—171no—
141C2×C4×Q8#59511yescharacteristic
142C2×C4×D4—171no—
143C2×C4⋊C8—342no—
144C2×#97—174no—
145#391—171no—
146#390—171no—
147#388—342no—
148#382—171no—
149C4×C22⋊C4—342no—
150C43—171no—
151#336—342no—
152#329—344no—
153C2×C17C2×#32911yescharacteristic
154C2×C17#152912yes—
155C2×C17#153611yescharacteristic
156C2×C17#180011yescharacteristic
157C2×C17#180612yes—
158D17—82no—
159C22×Q8C2×#59511yescharacteristic
160C22×D4—171no—
161C23×C4—341no—
162C22×C8—344no—
163#124—172no—
164#124—172no—
165#124—342no—
166C4×Q8—42no—
167C4×D4—682no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4C2×#59511yescharacteristic
171C2×C22⋊C4—342no—
172C2×C22⋊C4—342no—
173C2×C22⋊C4—341no—
174C2×C42—21no—
175C2×C42—171no—
176C2×C42—172no—
177C2×C42—172no—
178C2×C42—171no—
179C2×C42C2×#59511yescharacteristic
180C4⋊C8—348no—
181#97—344no—
182#97—344no—
183#95—342no—
184#95—342no—
185C17#857911yescharacteristic, complemented, Sylow
186C24—341no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C4⋊#59512yes—
191C2×D4—172no—
192C2×D4—341no—
193C2×D4—681no—
194C2×D4—681no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—172no—
198C22×C4—172no—
199C22×C4—341no—
200C22×C4—684no—
201C22×C4—682no—
202C22×C4C22⋊#59511yescharacteristic
203C22×C4C22⋊#59511yescharacteristic
204C22×C4C22×#59511yescharacteristic
205C2×C8—348no—
206C2×C8—684no—
207C2×C8—684no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—681no—
211C22⋊C4—344no—
212C22⋊C4—344no—
213C22⋊C4—682no—
214C22⋊C4—684no—
215C42—22no—
216C42—42no—
217C42—171no—
218C42—171no—
219C42—172no—
220C42—344no—
221C42—681no—
222C42—682no—
223C23—342no—
224C23—681no—
225C23—681no—
226C23—682no—
227C23—682no—
228C23C2×C22⋊#59511yescharacteristic
229Q8—42no—
230Q8—42no—
231D4—682no—
232D4—682no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—172no—
243C2×C4—172no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—681no—
250C2×C4—682no—
251C2×C4—1364no—
252C2×C4#1058755014yes—
253C2×C4#1058755312yes—
254C2×C4C2×C4⋊#59511yescharacteristic
255C2×C4#1058767711yescharacteristic
256C8—688no—
257C22—682no—
258C22—684no—
259C22—1361no—
260C22—1362no—
261C22—1361no—
262C22#1176375111yescharacteristic
263C22C2×#1058755311yescharacteristic
264C22#1176379511yescharacteristic
265C22C2×#1058755012yes—
266C22#1176380612yes—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—81no—
272C4—342no—
273C4—342no—
274C4—682no—
275C4—1361no—
276C4#1176366514yescomplemented
277C2—1362no—
278C2C2×#1176366511yescharacteristic
279C2#1454425112yes—
280C2#1454427211yescharacteristic
281C2#1454464811yescharacteristic
282C2#1454465712yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×Q8×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4×Q8#595
hypocenterC17#8579
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #14544570C2semidirect
3. #14543660C2semidirect
135. C4×C17#329semidirect
185. C17#8579semidirect
276. C4#11763665semidirect

Maximal subgroups

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

subgroupindex
#145445702
C2×#117636652
#145436602
C4×#105876332
#857917