Finite Group Databasebeta

← #16946585

Subgroups — #16946585

Order 4352.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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#14536772C211yesmaximal, characteristic, complemented
2#14527821C212yesmaximal
3#14526810C212yesmaximal, complemented
4#14526718C211yesmaximal, characteristic, complemented
5C17×#2007C211yesmaximal, characteristic
6#11763125—22no—
7#11762748C2212yes—
8#11762741—22no—
9#11762716—21no—
10#11762707—21no—
11#11762696—22no—
12#11762543—22no—
13#11762535C2212yes—
14C2×C17×#137—21no—
15C17×#537C2211yescharacteristic
16C2×C17×#104—21no—
17C17×#391C2211yescharacteristic
18C17×#343C2211yescharacteristic
19C2×#41×C17—21no—
20C22×Q8×C17D411yescharacteristic
21C17×#137—42no—
22C17×#137—42no—
23C2×Q16×C17—41no—
24C2×SD16×C17—41no—
25C2×M16×C17—21no—
26C17×C22⋊Q8—41no—
27C17×#121—22no—
28C17×#121—41no—
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17D412yes—
35C2×#63225—22no—
36C2×C4×Dic17—21no—
37C17×#104—42no—
38C17×#95—22no—
39#10587503—42no—
40#10587502—22no—
41#10587502—22no—
42#10587502—22no—
43#10587502D412yes—
44C22⋊#3622—22no—
45#10587479—42no—
46#41×C17—41no—
47#41×C17—41no—
48#41×C17—41no—
49#41×C17—41no—
50C2×Q8×C17—21no—
51C2×Q8×C17—21no—
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17SD1612yes—
55C2×D4×C17—21no—
56C2×D4×C17—22no—
57C2×D4×C17—41no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17C2×D412yes—
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65Q16×C17—44no—
66SD16×C17—44no—
67M16×C17—22no—
68M16×C17—41no—
69C2×C8×C17—41no—
70C4⋊C4×C17—22no—
71C4⋊C4×C17—41no—
72C22⋊C4×C17—42no—
73C42×C17—41no—
74C4⋊Dic17—42no—
75C4⋊Dic17—42no—
76C4⋊Dic17—81no—
77C4⋊Dic17—81no—
78#63225—44no—
79C4×Dic17—81no—
80C2×#3622—42no—
81#12131—171nomaximal, Sylow
82C23×C17—41no—
83C23×C17#12011yescharacteristic
84Q8×C17—42no—
85Q8×C17—41no—
86Q8×C17—41no—
87Q8×C17—81no—
88D4×C17—41no—
89D4×C17—41no—
90D4×C17—82no—
91D4×C17—81no—
92D4×C17—81no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—81no—
105C2×C4×C17—81no—
106C2×C4×C17C2×SD1611yescharacteristic
107C2×C4×C17#13711yescharacteristic
108C2×Dic17—42no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112C2×Dic17—42no—
113C2×Dic17—42no—
114C8×C17—42no—
115#3622—82no—
116#2809—171no—
117#2007D1711yescharacteristic
118#1616—172no—
119#1353—172no—
120#1319—171no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17—81no—
124C22×C17—81no—
125C22×C17—81no—
126C22×C17#44811yescharacteristic
127C22×C17#45311yescharacteristic
128C22×C17B(4,𝔽2)11yescharacteristic
129C4×C17—21no—
130C4×C17—21no—
131C4×C17—42no—
132C4×C17—42no—
133C4×C17—42no—
134C4×C17—81no—
135Dic17—82no—
136Dic17—82no—
137C2×#137—21no—
138#537C2×D1711yescharacteristic
139#481—342no—
140C2×#104—21no—
141#400—172no—
142#398—342no—
143#391—341no—
144#391C2×D1711yescharacteristic
145#389—341no—
146#387—342no—
147#343C2×D1711yescharacteristic
148#327—342no—
149#324—172no—
150C2×C17—21no—
151C2×C17—21no—
152C2×C17—161no—
153C2×C17#159611yescharacteristic
154C2×C17#220111yescharacteristic
155C2×C17#220411yescharacteristic
156C2×#41—21no—
157C22×Q8C22⋊D1711yescharacteristic
158#137—42no—
159#137—42no—
160C2×Q16—41no—
161C2×SD16—41no—
162C2×M16—21no—
163C22⋊Q8—41no—
164#121—22no—
165#121—41no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4—342no—
168C2×C4⋊C4—341no—
169C2×C4⋊C4—341no—
170C2×C4⋊C4—341no—
171C2×C4⋊C4C22×D1711yescharacteristic
172C2×C42—341no—
173C2×C42C22⋊D1711yescharacteristic
174#106—682no—
175#104—42no—
176#102—682no—
177C22⋊C8—342no—
178#95—22no—
179#95—172no—
180#95—342no—
181#95—342no—
182#95—342no—
183C17#1213111yescharacteristic, complemented, Sylow
184#41—41no—
185#41—41no—
186#41—41no—
187#41—41no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8#6322912yes—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—41no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—342no—
200C22×C4—342no—
201C22×C4D4×D1712yes—
202C22×C4C2×C22⋊D1711yescharacteristic
203Q16—44no—
204SD16—44no—
205M16—22no—
206M16—41no—
207C2×C8—41no—
208C2×C8—682no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—682no—
212C4⋊C4—684no—
213C4⋊C4—682no—
214C4⋊C4—1361no—
215C4⋊C4—1361no—
216C22⋊C4—42no—
217C42—41no—
218C42—1361no—
219C23—41no—
220C23#1058762311yescharacteristic
221Q8—42no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225D4—41no—
226D4—41no—
227D4—82no—
228D4—81no—
229D4—81no—
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—41no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—682no—
244C2×C4—682no—
245C2×C4—682no—
246C2×C4—682no—
247C2×C4—682no—
248C2×C4—682no—
249C2×C4C2×#6322911yescharacteristic
250C2×C4#1058763711yescharacteristic
251C8—42no—
252C8—1362no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#1176294711yescharacteristic
259C22#1176297611yescharacteristic
260C22#1176299511yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—81no—
267C4—1362no—
268C4—1362no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2#1452770611yescharacteristic
273C2#1453079611yescharacteristic
274C2#1453080911yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2007C2
socleC22×C17B(4,𝔽2)
prime radicalC17×#537C22
hypercenter#2007D17
hypocenterC17#12131
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #14536772C2semidirect
3. #14526810C2semidirect
4. #14526718C2semidirect
183. C17#12131semidirect

Maximal subgroups

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

subgroupindex
#145367722
#145278212
#145268102
#145267182
C17×#20072
#1213117