Finite Group Databasebeta

← #16946736

Subgroups — #16946736

Order 4352.

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

normal subgroups only

This group has 424 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#14527786C212yesmaximal
3#14526790C212yesmaximal, complemented
4#14526730C211yesmaximal, characteristic, complemented
5C17×#2582C211yesmaximal, characteristic, complemented
6#11763390C2211yescharacteristic
7#11763122—22no—
8C2×#10587476—21no—
9#11762808—21no—
10#11762578C2211yescharacteristic
11#11762542—22no—
12#11762535C2212yes—
13C2×C4×Q8×C17—21no—
14C17×#398—21no—
15C17×#398C2212yes—
16C17×#391—22no—
17C17×#389—22no—
18C17×#384C2211yescharacteristic
19C22×Q8×C17—21no—
20C2×#63252—21no—
21C4×Q8×C17—42no—
22C4×Q8×C17—42no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17D412yes—
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C42×C17—22no—
31C2×C42×C17D411yescharacteristic
32#10587616—22no—
33C2×#63223—21no—
34#10587583—41no—
35#10587581—41no—
36C17×#95—22no—
37C17×#95—21no—
38C17×#95—22no—
39C17×#95D412yes—
40C22⋊#3622—22no—
41#10587481—41no—
42#10587480—41no—
43#10587480—42no—
44#10587478—42no—
45#10587476—42no—
46C2×Q8×C17—24no—
47C2×Q8×C17—42no—
48C2×Q8×C17—42no—
49C22×C4×C17—21no—
50C22×C4×C17—22no—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
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—41no—
65C4⋊C4×C17—42no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—44no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—42no—
71C4⋊C4×C17—81no—
72C42×C17—21no—
73C42×C17—44no—
74C42×C17D812yes—
75#63227—42no—
76C4⋊Dic17—22no—
77#63223—41no—
78#63223—41no—
79C2×#3622—42no—
80C2×#3622—41no—
81#12153—171nomaximal, Sylow
82C23×C17#12011yescharacteristic
83C22×D17—41no—
84Q8×C17—48no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—24no—
91C2×C4×C17—22no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—41no—
100C2×C4×C17C2×D811yescharacteristic
101C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
102C22⋊D17—81no—
103C22⋊D17—81no—
104C2×Dic17—41no—
105D68—41no—
106D68—41no—
107D68—82no—
108C4×D17—81no—
109C4×D17—81no—
110Dic34—41no—
111Dic34—41no—
112#3622—81no—
113#3622—82no—
114#2582D1711yescharacteristic, complemented
115#2039—171no—
116#1610—172no—
117#1346—172no—
118#1325—171no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17#44711yescharacteristic
122C22×C17#45411yescharacteristic
123C22×C17B(4,𝔽2)11yescharacteristic
124C2×D17—81no—
125C2×D17—81no—
126C2×D17—81no—
127C4×C17—24no—
128C4×C17—21no—
129C4×C17—21no—
130C4×C17—44no—
131C4×C17—42no—
132C4×C17—82no—
133C4×C17—81no—
134Dic17—81no—
135#537—171no—
136C2×C4×Q8—21no—
137#480—342no—
138C2×#104—341no—
139#417—341no—
140#398—21no—
141#398C2×D1712yes—
142#391—22no—
143#389—22no—
144#384C2×D1711yescharacteristic
145#343—171no—
146#327—342no—
147#324—172no—
148C2×C17—21no—
149C2×C17—21no—
150C2×C17#159811yescharacteristic
151C2×C17#219811yescharacteristic
152C2×C17#220511yescharacteristic
153D17—161no—
154C2×#41—341no—
155C22×Q8—21no—
156C2×M16—341no—
157C22⋊Q8—681no—
158#121—342no—
159#121—681no—
160C4×Q8—42no—
161C4×Q8—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C22⋊D1711yescharacteristic
167C2×C4⋊C4C22⋊D1712yes—
168C2×C4⋊C4C22×D1711yescharacteristic
169C2×C42—22no—
170C2×C42C22⋊D1711yescharacteristic
171#107—682no—
172#104—682no—
173#103—681no—
174#102—681no—
175#102—682no—
176C22⋊C8—342no—
177#95—22no—
178#95—21no—
179#95—22no—
180#95C22⋊D1712yes—
181C17#1215311yescharacteristic, complemented, Sylow
182#41—681no—
183#41—681no—
184#41—681no—
185#41—681no—
186C2×Q8—24no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—341no—
190C2×D4—341no—
191C2×D4—342no—
192C2×D4—681no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—681no—
198C22×C4C2×C22⋊D1711yescharacteristic
199C22×C4C2×C22⋊D1712yes—
200M16—681no—
201M16—681no—
202C2×C8—682no—
203C2×C8—681no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—342no—
213C22⋊C4—682no—
214C42—21no—
215C42—44no—
216C42#6322812yes—
217C23—681no—
218C23#1058763911yescharacteristic
219Q8—48no—
220Q8—681no—
221Q8—681no—
222D4—681no—
223D4—681no—
224D4—1362no—
225D4—1361no—
226D4—1361no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—681no—
243C2×C4—1361no—
244C2×C4—1361no—
245C2×C4C2×#6322811yescharacteristic
246C2×C4#1058761811yescharacteristic
247C8—1361no—
248C8—1362no—
249C22—21no—
250C22—21no—
251C22—1361no—
252C22—1361no—
253C22—1361no—
254C22#1176294311yescharacteristic
255C22#1176298211yescharacteristic
256C22#1176299611yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—1361no—
265C2—21no—
266C2—21no—
267C2—2721no—
268C2#1452771911yescharacteristic
269C2#1453078011yescharacteristic
270C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2582C2
socleC22×C17B(4,𝔽2)
prime radical#11763390C22
hypercenter#2582D17
hypocenterC17#12153
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

Complements

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

NH ≅ G/Ntype
3. #14526790C2semidirect
4. #14526730C2semidirect
5. C17×#2582C2semidirect
114. #2582D17semidirect
181. C17#12153semidirect

Maximal subgroups

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

subgroupindex
#145300082
#145277862
#145267902
#145267302
C17×#25822
#1215317