Finite Group Databasebeta

← #17953367

Subgroups — #17953367

Order 4352.

388 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 84 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#14544346C212yesmaximal, complemented
2#14544272C212yesmaximal
3#14543690C212yesmaximal, complemented
4#14537492C211yesmaximal, characteristic
5#11763830C2214yes—
6#11763672—22no—
7#11763671—22no—
8#11763665—22no—
9#11763664—22no—
10C2×#10587593C411yescharacteristic
11C2×#10587634C411yescharacteristic
12#11763355C2211yescharacteristic
13#11762884—22no—
14#11762872—22no—
15#11762832C2211yescharacteristic
16#11762775C411yescharacteristic
17C2×C8×Dic17C2211yescharacteristic
18C17×#408C411yescharacteristic
19C2×#41×C17C2×C411yescharacteristic
20C2×#63252C2×C411yescharacteristic
21C2×M16×C17C2×C411yescharacteristic
22C22×C8×C17C2×C411yescharacteristic
23#10587634—42no—
24#10587633D411yescharacteristic
25#10587630Q811yescharacteristic
26#10587626Q811yescharacteristic
27#10587625—22no—
28#10587622D411yescharacteristic
29C4×C22⋊D17—22no—
30C2×C4×Dic17C2311yescharacteristic
31C2×#63223—21no—
32C2×#63223C2×C411yescharacteristic
33C22×#3622—21no—
34C22×#3622C2×C411yescharacteristic
35#10587593—42no—
36C2×#63218—21no—
37C2×C8×D17—21no—
38C22⋊#3624—24no—
39#10587542—24no—
40C4⋊#3624—24no—
41C17×C22⋊C8—22no—
42#10587491—22no—
43#10587485—22no—
44C8×Dic17D412yes—
45C8×Dic17D412yes—
46#41×C17—41no—
47#41×C17—41no—
48C2×Q8×C17C4⋊C411yescharacteristic
49C2×D4×C17—21no—
50C2×D4×C17C4⋊C411yescharacteristic
51C22×C4×C17—21no—
52C22×C4×C17C22×C411yescharacteristic
53C2×C22⋊D17—21no—
54C2×C22⋊D17—22no—
55C22×Dic17#4112yes—
56#63252—41no—
57#63252—41no—
58C2×D68C4⋊C411yescharacteristic
59C2×C4×D17—21no—
60C2×Dic34C4⋊C411yescharacteristic
61C2×#3624—44no—
62M16×C17—41no—
63C2×C8×C17—21no—
64C2×C8×C17—41no—
65C2×C8×C17C22⋊C412yes—
66C22⋊Dic17—22no—
67#63227—22no—
68#63225—22no—
69C4×Dic17C2×D411yescharacteristic
70C4×Dic17C2×D412yes—
71C4×Dic17C2×Q811yescharacteristic
72#63223—42no—
73#63223—41no—
74C2×#3622—22no—
75C2×#3622—21no—
76C2×#3622—42no—
77C2×#3622C22⋊C411yescharacteristic
78C2×#3622C22⋊C411yescharacteristic
79#63218—42no—
80C8×D17—42no—
81#8630—171nomaximal, Sylow
82C23×C17—21no—
83C23×C17#11711yescharacteristic
84C22×D17—21no—
85Q8×C17—41no—
86D4×C17—41no—
87D4×C17—42no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17C2×C22⋊C411yescharacteristic
94C2×C4×C17C2×C4⋊C411yescharacteristic
95C22⋊D17—44no—
96C22⋊D17—42no—
97C2×Dic17—21no—
98C2×Dic17—42no—
99C2×Dic17#12112yes—
100C2×Dic17C22⋊Q812yes—
101D68—41no—
102C4×D17—41no—
103C4×D17—41no—
104Dic34—41no—
105#3624—84no—
106C8×C17—22no—
107C8×C17—41no—
108#3622—21no—
109#3622—41no—
110#3622#10012yes—
111#2931—171no—
112#1568—172no—
113#1536—172no—
114#1307—172no—
115C22×C17—21no—
116C22×C17—22no—
117C22×C17—41no—
118C22×C17—42no—
119C22×C17#38011yescharacteristic
120C2×D17—41no—
121C2×D17—42no—
122C4×C17—21no—
123C4×C17—41no—
124C4×C17C2×#10011yescharacteristic
125C4×C17#41311yescharacteristic
126Dic17—22no—
127Dic17—22no—
128Dic17—41no—
129C2×#131—171no—
130C2×#131—171no—
131#532—171no—
132#436—342no—
133C8×D4—342no—
134#432—174no—
135#423—171no—
136#408—171no—
137#408#59511yescharacteristic
138C2×C4×C8—171no—
139#335—342no—
140#334—342no—
141#329—342no—
142#329—342no—
143C2×C17—41no—
144C2×C17—81no—
145C2×C17#180011yescharacteristic
146C2×C17#181412yes—
147D17—81no—
148C2×#41—171no—
149C2×#41C2×#59511yescharacteristic
150#131—682no—
151#131—682no—
152C2×M16—171no—
153C2×M16—341no—
154C2×M16—341no—
155C2×M16C2×#59511yescharacteristic
156C22×C8—171no—
157C22×C8—341no—
158C22×C8—341no—
159C22×C8C2×#59511yescharacteristic
160C4⋊Q8—171no—
161#127—171no—
162#124—171no—
163#124—171no—
164#121—342no—
165C4×D4—342no—
166C2×C42—171no—
167C4⋊C8—342no—
168C4⋊C8—344no—
169C22⋊C8—22no—
170C22⋊C8—342no—
171C22⋊C8—344no—
172#97—344no—
173C4×C8—172no—
174C4×C8—172no—
175C17#863011yescharacteristic, complemented, Sylow
176#41—41no—
177#41—41no—
178#41—681no—
179#41—681no—
180C2×Q8—171no—
181C2×Q8C4⋊#59511yescharacteristic
182C2×D4—21no—
183C2×D4—171no—
184C2×D4—342no—
185C2×D4—341no—
186C2×D4C4⋊#59511yescharacteristic
187C22×C4—21no—
188C22×C4—172no—
189C22×C4—341no—
190C22×C4C22×#59511yescharacteristic
191M16—41no—
192M16—682no—
193M16—682no—
194M16—681no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—171no—
198C2×C8—171no—
199C2×C8—342no—
200C2×C8—341no—
201C2×C8—682no—
202C2×C8—682no—
203C2×C8—684no—
204C2×C8C22⋊#59512yes—
205C4⋊C4—342no—
206C22⋊C4—342no—
207C22⋊C4—342no—
208C42—171no—
209C42—171no—
210C42—172no—
211C23—21no—
212C23—341no—
213C23#1058767711yescharacteristic
214Q8—41no—
215Q8—681no—
216D4—41no—
217D4—42no—
218D4—681no—
219D4—684no—
220D4—682no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—172no—
227C2×C4—172no—
228C2×C4—341no—
229C2×C4—682no—
230C2×C4—681no—
231C2×C4—681no—
232C2×C4C2×C4⋊#59511yescharacteristic
233C2×C4C2×C22⋊#59511yescharacteristic
234C8—22no—
235C8—41no—
236C8—172no—
237C8—341no—
238C8—681no—
239C8—1364no—
240C22—21no—
241C22—22no—
242C22—41no—
243C22—42no—
244C22—681no—
245C22—682no—
246C22#1176375111yescharacteristic
247C4—21no—
248C4—41no—
249C4—342no—
250C4—342no—
251C4—681no—
252C4D17⋊#10011yescharacteristic
253C4#1176379611yescharacteristic
254C2—41no—
255C2—81no—
256C2—1361no—
257C2#1454464911yescharacteristic
258C2#1454467112yes—
subgroupG/H
centerC22#11763751
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#408C4
socleC22×C17#380
prime radical#11763355C22
cosocleC2×C4×Dic17C23
hypercenter#408#595
hypocenterC17#8630
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #14544346C2semidirect
3. #14543690C2semidirect
175. C17#8630semidirect

Maximal subgroups

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

subgroupindex
#145443462
#145442722
#145436902
#145374922
#863017