Finite Group Databasebeta

← #16946900

Subgroups — #16946900

Order 4352.

372 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 63 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#14530008C211yesmaximal, characteristic
2#14527820C212yesmaximal
3#14526811C212yesmaximal, complemented
4#14526730C211yesmaximal, characteristic, complemented
5C17×#2571C211yesmaximal, characteristic, complemented
6#11763390C2211yescharacteristic
7#11763127—22no—
8C2×#10587476—21no—
9#11762808—21no—
10#11762578C2211yescharacteristic
11#11762542—22no—
12#11762535C2212yes—
13C17×#400—21no—
14C17×#400C2212yes—
15C17×#389—22no—
16C17×#384C2211yescharacteristic
17C17×#382—22no—
18C4×C4⋊C4×C17—21no—
19C2×#63252—21no—
20C2×C4⋊C4×C17—21no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C4⋊C4×C17D412yes—
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17—22no—
28C2×C42×C17D411yescharacteristic
29#10587616—22no—
30C2×#63223—21no—
31#10587583—41no—
32#10587581—41no—
33C17×#95—21no—
34C17×#95—22no—
35C17×#95—22no—
36C17×#95—22no—
37C17×#95D412yes—
38C22⋊#3622—22no—
39#10587481—41no—
40#10587480—41no—
41#10587480—42no—
42#10587479—42no—
43#10587476—42no—
44C22×C4×C17—21no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—21no—
48C22×C4×C17C2×D411yescharacteristic
49C22×C4×C17C2×D412yes—
50C2×C22⋊D17—41no—
51#63252—41no—
52#63252—41no—
53#63252—41no—
54#63252—41no—
55C2×D68—21no—
56C2×D68—22no—
57C2×C4×D17—41no—
58C2×Dic34—21no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—44no—
64C4⋊C4×C17—42no—
65C4⋊C4×C17—81no—
66C42×C17—21no—
67C42×C17—44no—
68C42×C17SD1612yes—
69#63227—42no—
70C4⋊Dic17—22no—
71#63223—41no—
72#63223—41no—
73C2×#3622—42no—
74C2×#3622—41no—
75#12181—171nomaximal, Sylow
76C23×C17#12011yescharacteristic
77C22×D17—41no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17C2×SD1611yescharacteristic
94C2×C4×C17#13711yescharacteristic
95C22⋊D17—81no—
96C22⋊D17—81no—
97C2×Dic17—41no—
98D68—41no—
99D68—41no—
100D68—82no—
101C4×D17—81no—
102C4×D17—81no—
103Dic34—41no—
104Dic34—41no—
105#3622—81no—
106#3622—82no—
107#2571D1711yescharacteristic, complemented
108#2039—171no—
109#1616—172no—
110#1353—172no—
111#1325—171no—
112C22×C17—21no—
113C22×C17—21no—
114C22×C17#44811yescharacteristic
115C22×C17#45411yescharacteristic
116C22×C17B(4,𝔽2)11yescharacteristic
117C2×D17—81no—
118C2×D17—81no—
119C2×D17—81no—
120C4×C17—24no—
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—42no—
124C4×C17—82no—
125C4×C17—81no—
126C4×C17—82no—
127Dic17—81no—
128#537—171no—
129#481—342no—
130C2×#104—341no—
131#417—341no—
132#400—21no—
133#400C2×D1712yes—
134#389—22no—
135#384C2×D1711yescharacteristic
136#382—22no—
137C4×C4⋊C4—21no—
138#343—171no—
139#327—342no—
140#324—172no—
141C2×C17—21no—
142C2×C17—21no—
143C2×C17#159911yescharacteristic
144C2×C17#219811yescharacteristic
145C2×C17#220411yescharacteristic
146D17—161no—
147C2×#41—341no—
148C2×M16—341no—
149C22⋊Q8—681no—
150#121—342no—
151#121—681no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C22⋊D1711yescharacteristic
157C2×C4⋊C4C22⋊D1712yes—
158C2×C4⋊C4C22×D1711yescharacteristic
159C2×C42—22no—
160C2×C42C22⋊D1711yescharacteristic
161#106—682no—
162#104—682no—
163#103—681no—
164#102—681no—
165#102—682no—
166C22⋊C8—342no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95C22⋊D1712yes—
172C17#1218111yescharacteristic, complemented, Sylow
173#41—681no—
174#41—681no—
175#41—681no—
176#41—681no—
177C2×Q8—341no—
178C2×D4—341no—
179C2×D4—342no—
180C2×D4—681no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—681no—
186C22×C4C2×C22⋊D1711yescharacteristic
187C22×C4C2×C22⋊D1712yes—
188M16—681no—
189M16—681no—
190C2×C8—682no—
191C2×C8—681no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—342no—
200C22⋊C4—682no—
201C42—21no—
202C42—44no—
203C42#6323012yes—
204C23—681no—
205C23#1058763911yescharacteristic
206Q8—681no—
207Q8—681no—
208D4—681no—
209D4—681no—
210D4—1362no—
211D4—1361no—
212D4—1361no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—681no—
229C2×C4—1361no—
230C2×C4—1361no—
231C2×C4C2×#6323011yescharacteristic
232C2×C4#1058762811yescharacteristic
233C8—1361no—
234C8—1362no—
235C22—21no—
236C22—21no—
237C22—1361no—
238C22—1361no—
239C22—1361no—
240C22#1176294811yescharacteristic
241C22#1176298211yescharacteristic
242C22#1176299611yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4—1361no—
251C2—21no—
252C2—21no—
253C2—2721no—
254C2#1452772411yescharacteristic
255C2#1453078011yescharacteristic
256C2#1453081011yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2571C2
socleC22×C17B(4,𝔽2)
prime radical#11763390C22
hypercenter#2571D17
hypocenterC17#12181
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #14526811C2semidirect
4. #14526730C2semidirect
5. C17×#2571C2semidirect
107. #2571D17semidirect
172. C17#12181semidirect

Maximal subgroups

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

subgroupindex
#145300082
#145278202
#145268112
#145267302
C17×#25712
#1218117