Finite Group Databasebeta

← #16946895

Subgroups — #16946895

Order 4352.

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

normal subgroups only

This group has 472 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#14527752C212yesmaximal
3#14526814C212yesmaximal, complemented
4#14526730C211yesmaximal, characteristic, complemented
5C17×#2551C211yesmaximal, characteristic, complemented
6#11763390C2211yescharacteristic
7#11762948—22no—
8C2×#10587476—21no—
9#11762808—21no—
10#11762578C2211yescharacteristic
11#11762542—22no—
12#11762535C2212yes—
13C2×C17×C4⋊Q8—22no—
14C17×#395—21no—
15C17×#395C2212yes—
16C17×#389—22no—
17C17×#384C2211yescharacteristic
18C4×C4⋊C4×C17—21no—
19C22×Q8×C17—21no—
20C22×Q8×C17D412yes—
21C2×#63252—21no—
22C17×C4⋊Q8—48no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—21no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C42×C17—22no—
31C2×C42×C17D411yescharacteristic
32C2×#63230—42no—
33#10587616—22no—
34C2×#63223—21no—
35#10587583—41no—
36#10587581—41no—
37C17×#95—21no—
38C17×#95—22no—
39C17×#95D412yes—
40#10587506—42no—
41C22⋊#3622—22no—
42#10587481—41no—
43#10587480—41no—
44#10587476—42no—
45C2×Q8×C17—44no—
46C2×Q8×C17—44no—
47C2×Q8×C17—42no—
48C2×Q8×C17—81no—
49C22×C4×C17—21no—
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—21no—
53C22×C4×C17C2×D411yescharacteristic
54C22×C4×C17C2×D412yes—
55C2×C22⋊D17—41no—
56#63252—41no—
57#63252—41no—
58#63252—41no—
59#63252—41no—
60C2×D68—21no—
61C2×D68—22no—
62C2×C4×D17—41no—
63C2×Dic34—21no—
64C4⋊C4×C17—24no—
65C4⋊C4×C17—24no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—44no—
69C4⋊C4×C17—44no—
70C4⋊C4×C17—44no—
71C42×C17—21no—
72C42×C17—44no—
73C42×C17SD1612yescomplemented
74#63230—84no—
75#63227—42no—
76C4⋊Dic17—22no—
77#63223—41no—
78#63223—41no—
79C2×#3622—41no—
80C2×#3622—42no—
81#12180—171nomaximal, Sylow
82C23×C17#12011yescharacteristic
83C22×D17—41no—
84Q8×C17—44no—
85Q8×C17—82no—
86Q8×C17—84no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—24no—
93C2×C4×C17—24no—
94C2×C4×C17—22no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—41no—
102C2×C4×C17C2×SD1611yescharacteristic
103C2×C4×C17#13711yescharacteristic
104C22⋊D17—81no—
105C22⋊D17—81no—
106C2×Dic17—41no—
107D68—41no—
108D68—41no—
109D68—82no—
110C4×D17—81no—
111C4×D17—81no—
112Dic34—41no—
113Dic34—41no—
114#3622—81no—
115#3622—82no—
116#2551D1711yescharacteristic, complemented
117#2039—171no—
118#1604—172no—
119#1354—172no—
120#1325—171no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17#44811yescharacteristic
124C22×C17#45411yescharacteristic
125C22×C17B(4,𝔽2)11yescharacteristic
126C2×D17—81no—
127C2×D17—81no—
128C2×D17—81no—
129C4×C17—24no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—44no—
133C4×C17—42no—
134C4×C17—44no—
135C4×C17—81no—
136Dic17—81no—
137#537—171no—
138C2×C4⋊Q8—22no—
139#448—342no—
140C2×#104—341no—
141#417—341no—
142#395—21no—
143#395C2×D1712yes—
144#389—22no—
145#384C2×D1711yescharacteristic
146C4×C4⋊C4—21no—
147#343—171no—
148#327—342no—
149#324—172no—
150C2×C17—21no—
151C2×C17—21no—
152C2×C17#159911yescharacteristic
153C2×C17#219611yescharacteristic
154C2×C17#220511yescharacteristic
155D17—161no—
156C2×#41—341no—
157C22×Q8—21no—
158C22×Q8C22⋊D1712yes—
159C2×SD16—682no—
160C2×M16—341no—
161C4⋊Q8—48no—
162C22⋊Q8—681no—
163#121—342no—
164#121—681no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22⋊D1711yescharacteristic
171C2×C4⋊C4C22×D1711yescharacteristic
172C2×C42—22no—
173C2×C42C22⋊D1711yescharacteristic
174#104—682no—
175#103—681no—
176#103—682no—
177#102—681no—
178C22⋊C8—342no—
179#95—21no—
180#95—22no—
181#95C22⋊D1712yes—
182C17#1218011yescharacteristic, complemented, Sylow
183#41—681no—
184#41—681no—
185#41—681no—
186#41—681no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—81no—
191C2×Q8—341no—
192C2×D4—341no—
193C2×D4—342no—
194C2×D4—681no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—681no—
200C22×C4C2×C22⋊D1711yescharacteristic
201C22×C4C2×C22⋊D1712yes—
202SD16—1364no—
203M16—681no—
204M16—681no—
205C2×C8—682no—
206C2×C8—681no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—342no—
215C22⋊C4—682no—
216C42—21no—
217C42—44no—
218C42#6323012yescomplemented
219C23—681no—
220C23#1058763911yescharacteristic
221Q8—44no—
222Q8—82no—
223Q8—84no—
224Q8—681no—
225Q8—681no—
226D4—681no—
227D4—681no—
228D4—1362no—
229D4—1361no—
230D4—1361no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—681no—
247C2×C4—1361no—
248C2×C4—1361no—
249C2×C4C2×#6323011yescharacteristic
250C2×C4#1058762811yescharacteristic
251C8—1361no—
252C8—1362no—
253C22—21no—
254C22—21no—
255C22—1361no—
256C22—1361no—
257C22—1361no—
258C22#1176294811yescharacteristic
259C22#1176298211yescharacteristic
260C22#1176299611yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—1361no—
269C2—21no—
270C2—21no—
271C2—2721no—
272C2#1452772411yescharacteristic
273C2#1453076811yescharacteristic
274C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2551C2
socleC22×C17B(4,𝔽2)
prime radical#11763390C22
hypercenter#2551D17
hypocenterC17#12180
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
4. #14526730C2semidirect
5. C17×#2551C2semidirect
73. C42×C17SD16semidirect
116. #2551D17semidirect
182. C17#12180semidirect
218. C42#63230semidirect

Maximal subgroups

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

subgroupindex
#145300082
#145277522
#145268142
#145267302
C17×#25512
#1218017