Finite Group Databasebeta

← #16946586

Subgroups — #16946586

Order 4352.

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

normal subgroups only

This group has 398 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#14529830C211yesmaximal, characteristic
2#14527820C212yesmaximal
3#14526811C212yesmaximal, complemented
4#14526718C211yesmaximal, characteristic, complemented
5C17×#2809C211yesmaximal, characteristic, complemented
6C2×#10587628—21no—
7#11763390C2211yescharacteristic
8#11763127—22no—
9C2×#10587476—21no—
10#11762578C2211yescharacteristic
11#11762542—22no—
12#11762535C2212yes—
13C17×#400C2212yes—
14C17×#398—22no—
15C17×#391—21no—
16C17×#391C2211yescharacteristic
17C17×#389—21no—
18C17×#387—22no—
19C22×Q8×C17D411yescharacteristic
20C2×#63252—21no—
21C2×C4⋊C4×C17—22no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17—21no—
25C2×C4⋊C4×C17D412yes—
26C2×C4⋊C4×C17C2311yescharacteristic
27C2×C42×C17—21no—
28C2×C42×C17D411yescharacteristic
29C2×#63231—41no—
30#10587628—42no—
31#10587628—42no—
32C2×#63230—41no—
33#10587616—22no—
34C2×#63223—21no—
35#10587583—41no—
36#10587581—41no—
37C17×#95—22no—
38C17×#95—22no—
39C17×#95—22no—
40C17×#95—22no—
41C17×#95D412yes—
42C22⋊#3622—22no—
43#10587480—42no—
44#10587479—42no—
45#10587476—42no—
46C2×Q8×C17—21no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C2×Q8×C17SD1612yes—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17C2×D411yescharacteristic
55C22×C4×C17C2×D412yes—
56C2×C22⋊D17—41no—
57#63252—41no—
58#63252—41no—
59#63252—41no—
60#63252—41no—
61C2×D68—21no—
62C2×D68—22no—
63C2×C4×D17—41no—
64C2×Dic34—21no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—42no—
67C4⋊C4×C17—44no—
68C4⋊C4×C17—42no—
69C4⋊C4×C17—81no—
70C4⋊C4×C17—81no—
71C42×C17—41no—
72C42×C17—81no—
73#63231—44no—
74#63230—44no—
75#63227—42no—
76C4⋊Dic17—22no—
77#63223—22no—
78#63223—41no—
79C2×#3622—41no—
80C2×#3622—42no—
81#12131—171nomaximal, Sylow
82C23×C17#12011yescharacteristic
83C22×D17—41no—
84Q8×C17—42no—
85Q8×C17—81no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
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—42no—
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—42no—
115#3622—82no—
116#2809D1711yescharacteristic, complemented
117#2007—171no—
118#1616—172no—
119#1353—172no—
120#1319—171no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17#44811yescharacteristic
124C22×C17#45311yescharacteristic
125C22×C17B(4,𝔽2)11yescharacteristic
126C2×D17—81no—
127C2×D17—81no—
128C2×D17—81no—
129C4×C17—21no—
130C4×C17—21no—
131C4×C17—42no—
132C4×C17—42no—
133C4×C17—42no—
134C4×C17—82no—
135C4×C17—82no—
136Dic17—81no—
137C2×#137—341no—
138#537—171no—
139#481—342no—
140C2×#104—341no—
141#400C2×D1712yes—
142#398—22no—
143#391—21no—
144#391C2×D1711yescharacteristic
145#389—21no—
146#387—22no—
147#343—171no—
148#327—342no—
149#324—172no—
150C2×C17—21no—
151C2×C17—21no—
152C2×C17#159611yescharacteristic
153C2×C17#220111yescharacteristic
154C2×C17#220411yescharacteristic
155D17—161no—
156C2×#41—341no—
157C22×Q8C22⋊D1711yescharacteristic
158#137—682no—
159#137—682no—
160C2×Q16—681no—
161C2×SD16—681no—
162C2×M16—341no—
163C22⋊Q8—681no—
164#121—342no—
165#121—681no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C22⋊D1712yes—
171C2×C4⋊C4C22×D1711yescharacteristic
172C2×C42—21no—
173C2×C42C22⋊D1711yescharacteristic
174#106—682no—
175#104—682no—
176#102—682no—
177C22⋊C8—342no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95C22⋊D1712yes—
183C17#1213111yescharacteristic, complemented, Sylow
184#41—681no—
185#41—681no—
186#41—681no—
187#41—681no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—341no—
192C2×Q8#6323012yes—
193C2×D4—341no—
194C2×D4—342no—
195C2×D4—681no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—681no—
201C22×C4C2×C22⋊D1711yescharacteristic
202C22×C4C2×C22⋊D1712yes—
203Q16—684no—
204SD16—684no—
205M16—342no—
206M16—681no—
207C2×C8—681no—
208C2×C8—682no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—342no—
216C22⋊C4—682no—
217C42—41no—
218C42—81no—
219C23—681no—
220C23#1058763911yescharacteristic
221Q8—42no—
222Q8—81no—
223Q8—681no—
224Q8—681no—
225D4—681no—
226D4—681no—
227D4—1362no—
228D4—1361no—
229D4—1361no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—681no—
247C2×C4—1361no—
248C2×C4—1361no—
249C2×C4C2×#6323011yescharacteristic
250C2×C4#1058762811yescharacteristic
251C8—682no—
252C8—1362no—
253C22—21no—
254C22—21no—
255C22—1361no—
256C22—1361no—
257C22—1361no—
258C22#1176294811yescharacteristic
259C22#1176297711yescharacteristic
260C22#1176299611yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4—82no—
268C4—1361no—
269C2—21no—
270C2—21no—
271C2—2721no—
272C2#1452770711yescharacteristic
273C2#1453079711yescharacteristic
274C2#1453081011yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2809C2
socleC22×C17B(4,𝔽2)
prime radical#11763390C22
hypercenter#2809D17
hypocenterC17#12131
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
3. #14526811C2semidirect
4. #14526718C2semidirect
5. C17×#2809C2semidirect
116. #2809D17semidirect
183. C17#12131semidirect

Maximal subgroups

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

subgroupindex
#145298302
#145278202
#145268112
#145267182
C17×#28092
#1213117