Finite Group Databasebeta

← #16937413

Subgroups — #16937413

Order 4352.

454 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 97 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#14537162C211yesmaximal, characteristic, complemented
2#14529612C211yesmaximal, characteristic
3#14527345C212yesmaximal
4#14526643C212yesmaximal, complemented
5C17×#1276C211yesmaximal, characteristic, complemented
6#11763359C2211yescharacteristic
7C2×#10587555—21no—
8#11762833C2211yescharacteristic
9C2×#10587476—21no—
10C2×#10587476—21no—
11#11762715—24no—
12#11762675—21no—
13#11762675C2212yes—
14#11762544—24no—
15C17×#431C2212yes—
16C43×C17C2211yescharacteristic
17C2×#63252—21no—
18C22×Dic34—22no—
19C2×C42×C17—22no—
20C2×C42×C17D411yescharacteristic
21C2×C42×C17D411yescharacteristic
22C2×C42×C17C2311yescharacteristic
23#10587616—22no—
24C2×C4⋊Dic17—22no—
25C2×C4⋊Dic17—22no—
26C2×#63223—21no—
27#10587561—21no—
28#10587560Q811yescharacteristic, complemented
29C4×D68—22no—
30#10587555—42no—
31#10587555—42no—
32#10587555Q811yescharacteristic, complemented
33C17×C4⋊C8—22no—
34C17×C22⋊C8—22no—
35C4×C8×C17—22no—
36#10587502—24no—
37#10587502—22no—
38#10587502D414yes—
39#10587476—42no—
40#10587476—42no—
41C4⋊#3622—22no—
42C22×C4×C17—21no—
43C22×C4×C17—21no—
44C22×C4×C17C2×D411yescharacteristic
45C22×C4×C17C2×D412yes—
46C2×C22⋊D17—41no—
47C22×Dic17—24no—
48#63252—41no—
49#63252—41no—
50#63252—41no—
51#63252—41no—
52C2×D68—21no—
53C2×D68—22no—
54C2×C4×D17—41no—
55C2×Dic34—21no—
56C2×Dic34—48no—
57C2×Dic34—44no—
58C2×C8×C17—42no—
59C42×C17—21no—
60C42×C17—21no—
61C42×C17—44no—
62C42×C17D812yes—
63C42×C17SD1612yes—
64C42×C17C2×Q811yescharacteristic
65C42×C17#4112yes—
66C42×C17#4111yescharacteristic
67#63227—42no—
68C4⋊Dic17—22no—
69C4⋊Dic17—44no—
70C4⋊Dic17—42no—
71C4⋊Dic17—42no—
72#63223—41no—
73#63223—41no—
74C2×#3622—41no—
75#10362—171nomaximal, Sylow
76C23×C17#12011yescharacteristic
77C22×D17—41no—
78C2×C4×C17—24no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—42no—
82C2×C4×C17—42no—
83C2×C4×C17C22⋊Q812yes—
84C2×C4×C17C22⋊Q811yescharacteristic
85C2×C4×C17#12411yescharacteristic
86C2×C4×C17#12412yes—
87C2×C4×C17C2×D811yescharacteristic
88C2×C4×C17C2×SD1611yescharacteristic
89C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C17#13711yescharacteristic
91C22⋊D17—81no—
92C22⋊D17—81no—
93C2×Dic17—48no—
94C2×Dic17—41no—
95C2×Dic17—44no—
96D68—41no—
97D68—41no—
98D68—82no—
99C4×D17—81no—
100C4×D17—81no—
101Dic34—41no—
102Dic34—41no—
103Dic34—88no—
104C8×C17—82no—
105#3622—81no—
106#2862—171no—
107#1962—171no—
108#1517—172no—
109#1286—172no—
110#1276D1711yescharacteristic, complemented
111C22×C17—22no—
112C22×C17#39311yescharacteristic
113C22×C17#44711yescharacteristic
114C22×C17#44811yescharacteristic
115C2×D17—81no—
116C2×D17—82no—
117C4×C17—24no—
118C4×C17—22no—
119C4×C17—22no—
120C4×C17—42no—
121C4×C17#47411yescharacteristic
122C4×C17#47611yescharacteristic
123C4×C17#48611yescharacteristic
124C4×C17#48911yescharacteristic
125Dic17—84no—
126Dic17—81no—
127#532—171no—
128C2×C4⋊Q8—341no—
129#431C2×D1712yes—
130#423—171no—
131C2×#104—341no—
132C2×#104—341no—
133#391—344no—
134#382—172no—
135#382—341no—
136C43C2×D1711yescharacteristic
137#327—344no—
138C2×C17—21no—
139C2×C17—21no—
140C2×C17#202311yescharacteristic
141C2×C17#203112yes—
142D17—161no—
143C2×#41—341no—
144C22×Q8—342no—
145C2×M16—341no—
146C4⋊Q8—171no—
147C4⋊Q8—682no—
148C4⋊Q8—682no—
149#127—171no—
150#124—341no—
151#121—342no—
152C4×D4—342no—
153C2×C4⋊C4—342no—
154C2×C4⋊C4—342no—
155C2×C42—22no—
156C2×C42D6811yescharacteristic
157C2×C42D6811yescharacteristic
158C2×C42C22×D1711yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—342no—
161#104—682no—
162#104—682no—
163C22⋊C8—22no—
164C4×C8—22no—
165#95—174no—
166#95—344no—
167#95—342no—
168C17#1036211yescharacteristic, complemented, Sylow
169#41—681no—
170#41—681no—
171#41—681no—
172#41—681no—
173C2×Q8—341no—
174C2×Q8—684no—
175C2×Q8—688no—
176C2×D4—342no—
177C2×D4—341no—
178C2×D4—681no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—344no—
182C22×C4—681no—
183C22×C4C2×D6811yescharacteristic
184C22×C4D4×D1712yes—
185M16—681no—
186M16—681no—
187C2×C8—42no—
188C2×C8—681no—
189C4⋊C4—342no—
190C4⋊C4—684no—
191C4⋊C4—682no—
192C4⋊C4—682no—
193C22⋊C4—682no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#6321912yes—
198C42D13612yes—
199C42#6325212yes—
200C42#6325411yescharacteristic
201C42Q8×D1711yescharacteristic
202C23—681no—
203C23#1058756811yescharacteristic
204Q8—681no—
205Q8—681no—
206Q8—1368no—
207D4—681no—
208D4—681no—
209D4—1362no—
210D4—1361no—
211D4—1361no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—688no—
218C2×C4—684no—
219C2×C4—681no—
220C2×C4—1361no—
221C2×C4—1361no—
222C2×C4#1058756111yescharacteristic
223C2×C4#1058757112yes—
224C2×C4#1058758212yes—
225C2×C4#1058758311yescharacteristic
226C2×C4C2×#6321911yescharacteristic
227C2×C4C2×D13611yescharacteristic
228C2×C4#1058759411yescharacteristic
229C2×C4#1058759511yescharacteristic
230C8—82no—
231C8—1361no—
232C22—22no—
233C22—1361no—
234C22—1362no—
235C22#1176272211yescharacteristic
236C22#1176294511yescharacteristic
237C22#1176295011yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—1364no—
243C4—1361no—
244C4#1176309011yescharacteristic
245C4#1176310011yescharacteristic
246C4#1176315611yescharacteristic
247C4#1176316911yescharacteristic
248C2—21no—
249C2—21no—
250C2—2721no—
251C2#1452992111yescharacteristic
252C2#1452995712yes—
subgroupG/H
centerC22#11762722
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1276C2
socleC22×C17#393
prime radical#11763359C22
hypercenter#1276D17
hypocenterC17#10362
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #14537162C2semidirect
4. #14526643C2semidirect
5. C17×#1276C2semidirect
28. #10587560Q8semidirect
32. #10587555Q8semidirect
110. #1276D17semidirect
168. C17#10362semidirect

Maximal subgroups

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

subgroupindex
#145371622
#145296122
#145273452
#145266432
C17×#12762
#1036217