Finite Group Databasebeta

← #16946924

Subgroups — #16946924

Order 4352.

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

normal subgroups only

This group has 360 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#14530013C211yesmaximal, characteristic
2#14527820C212yesmaximal
3#14526811C212yesmaximal, complemented
4#14526728C211yesmaximal, characteristic, complemented
5C17×#2844C211yesmaximal, characteristic, complemented
6#11763390C2211yescharacteristic
7#11763127—22no—
8C2×#10587476—21no—
9#11762808—21no—
10#11762578C2211yescharacteristic
11#11762542—22no—
12#11762535C2212yes—
13C17×#400—22no—
14C17×#400C2212yes—
15C17×#395—22no—
16C17×#395—21no—
17C17×#382—21no—
18C17×#382C2211yescharacteristic
19C22×Q8×C17—21no—
20C2×#63252—21no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—22no—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C4⋊C4×C17D412yes—
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17D411yescharacteristic
28#10587616—22no—
29C2×#63223—21no—
30#10587583—41no—
31#10587581—41no—
32C17×#95—22no—
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—
44C2×Q8×C17—82no—
45C2×Q8×C17—81no—
46C22×C4×C17—21no—
47C22×C4×C17—22no—
48C22×C4×C17—21no—
49C22×C4×C17—22no—
50C22×C4×C17C2×D411yescharacteristic
51C22×C4×C17C2×D412yes—
52C2×C22⋊D17—41no—
53#63252—41no—
54#63252—41no—
55#63252—41no—
56#63252—41no—
57C2×D68—21no—
58C2×D68—22no—
59C2×C4×D17—41no—
60C2×Dic34—21no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—42no—
63C4⋊C4×C17—41no—
64C4⋊C4×C17—81no—
65C4⋊C4×C17—81no—
66C4⋊C4×C17—82no—
67C42×C17—21no—
68C42×C17SD1612yes—
69#63227—42no—
70C4⋊Dic17—22no—
71#63223—41no—
72#63223—41no—
73C2×#3622—42no—
74C2×#3622—41no—
75#12185—171nomaximal, Sylow
76C23×C17#12011yescharacteristic
77C22×D17—41no—
78Q8×C17—84no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
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—42no—
92C2×C4×C17—42no—
93C2×C4×C17—41no—
94C2×C4×C17C2×SD1611yescharacteristic
95C2×C4×C17#13711yescharacteristic
96C22⋊D17—81no—
97C22⋊D17—81no—
98C2×Dic17—41no—
99D68—41no—
100D68—41no—
101D68—82no—
102C4×D17—81no—
103C4×D17—81no—
104Dic34—41no—
105Dic34—41no—
106#3622—81no—
107#3622—82no—
108#2844D1711yescharacteristic, complemented
109#2040—171no—
110#1616—172no—
111#1353—172no—
112#1324—171no—
113C22×C17—21no—
114C22×C17—21no—
115C22×C17#44811yescharacteristic
116C22×C17#45411yescharacteristic
117C22×C17B(4,𝔽2)11yescharacteristic
118C2×D17—81no—
119C2×D17—81no—
120C2×D17—81no—
121C4×C17—21no—
122C4×C17—21no—
123C4×C17—42no—
124C4×C17—42no—
125C4×C17—82no—
126C4×C17—82no—
127C4×C17—81no—
128Dic17—81no—
129#537—171no—
130#481—342no—
131C2×#104—341no—
132#417—341no—
133#400—22no—
134#400C2×D1712yes—
135#395—22no—
136#395—21no—
137#382—21no—
138#382C2×D1711yescharacteristic
139#343—171no—
140#327—342no—
141#324—172no—
142C2×C17—21no—
143C2×C17—21no—
144C2×C17#159911yescharacteristic
145C2×C17#220111yescharacteristic
146C2×C17#220211yescharacteristic
147D17—161no—
148C2×#41—341no—
149C22×Q8—21no—
150C2×M16—341no—
151C22⋊Q8—681no—
152#121—342no—
153#121—681no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊D1711yescharacteristic
158C2×C4⋊C4C22⋊D1712yes—
159C2×C4⋊C4C22×D1711yescharacteristic
160C2×C42C22⋊D1711yescharacteristic
161#106—682no—
162#104—682no—
163#103—681no—
164#102—681no—
165#102—682no—
166C22⋊C8—342no—
167#95—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95C22⋊D1712yes—
173C17#1218511yescharacteristic, complemented, Sylow
174#41—681no—
175#41—681no—
176#41—681no—
177#41—681no—
178C2×Q8—82no—
179C2×Q8—81no—
180C2×Q8—341no—
181C2×D4—341no—
182C2×D4—342no—
183C2×D4—681no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—681no—
189C22×C4C2×C22⋊D1711yescharacteristic
190C22×C4C2×C22⋊D1712yes—
191M16—681no—
192M16—681no—
193C2×C8—682no—
194C2×C8—681no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—82no—
201C4⋊C4—342no—
202C22⋊C4—682no—
203C42—21no—
204C42#6323012yes—
205C23—681no—
206C23#1058763911yescharacteristic
207Q8—84no—
208Q8—681no—
209Q8—681no—
210D4—681no—
211D4—681no—
212D4—1362no—
213D4—1361no—
214D4—1361no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—681no—
231C2×C4—1361no—
232C2×C4—1361no—
233C2×C4C2×#6323011yescharacteristic
234C2×C4#1058762811yescharacteristic
235C8—1361no—
236C8—1362no—
237C22—21no—
238C22—21no—
239C22—1361no—
240C22—1361no—
241C22—1361no—
242C22#1176294811yescharacteristic
243C22#1176298211yescharacteristic
244C22#1176299611yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—1361no—
253C2—21no—
254C2—21no—
255C2—2721no—
256C2#1452772411yescharacteristic
257C2#1453079711yescharacteristic
258C2#1453080111yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2844C2
socleC22×C17B(4,𝔽2)
prime radical#11763390C22
hypercenter#2844D17
hypocenterC17#12185
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
3. #14526811C2semidirect
4. #14526728C2semidirect
5. C17×#2844C2semidirect
108. #2844D17semidirect
173. C17#12185semidirect

Maximal subgroups

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

subgroupindex
#145300132
#145278202
#145268112
#145267282
C17×#28442
#1218517