Finite Group Databasebeta

← #16954014

Subgroups — #16954014

Order 4352.

384 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 79 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
2#14528999C212yesmaximal
3#14526911C211yesmaximal, characteristic
4#14526910C212yesmaximal
5C17×#1857C211yesmaximal, characteristic
6#11762852—22no—
7C2×#10587507C2212yes—
8#11762748—21no—
9#11762741—22no—
10#11762716C2212yes—
11#11762707—21no—
12#11762696—22no—
13#11762568—22no—
14C17×#519C2211yescharacteristic
15C17×#428—21no—
16C2×C17×#101C2211yescharacteristic
17C17×#391C2211yescharacteristic
18C17×#339—21no—
19C22×Q8×C17C2311yescharacteristic
20C2×M16×C17—21no—
21C4×Q8×C17—41no—
22C4×Q8×C17—41no—
23C17×#117—21no—
24C17×#117—22no—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C42×C17D411yescharacteristic
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17D412yes—
31C2×#63225—22no—
32C2×C4×Dic17D412yes—
33C2×#63223—22no—
34C17×#107—41no—
35C17×#106—41no—
36C17×#101—21no—
37C17×#101Q812yes—
38C17×#95—22no—
39#10587507—22no—
40#10587507Q814yes—
41#10587502—21no—
42#10587502—22no—
43#10587502—22no—
44#10587502—22no—
45#10587479—42no—
46#10587478—42no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C2×Q8×C17C2×Q811yescharacteristic
50C2×Q8×C17C2×Q812yes—
51C2×Q8×C17#4111yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C2×D412yes—
55C22×C4×C17C2×D411yescharacteristic
56C22×Dic17—22no—
57C22×Dic17—22no—
58M16×C17—22no—
59M16×C17—41no—
60C2×C8×C17—41no—
61C4⋊C4×C17—22no—
62C4⋊C4×C17—22no—
63C4⋊C4×C17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17—41no—
66C22⋊C4×C17—41no—
67C42×C17—41no—
68C42×C17—41no—
69C42×C17—42no—
70C4⋊Dic17—42no—
71C4⋊Dic17—42no—
72C4⋊Dic17—81no—
73C4⋊Dic17—81no—
74#63225—44no—
75C4×Dic17—42no—
76#63223—24no—
77#63223—42no—
78C2×#3622—42no—
79#13465—171nomaximal, Sylow
80C23×C17#12011yescharacteristic
81Q8×C17—41no—
82Q8×C17—42no—
83Q8×C17—41no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—42no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17C22⋊Q812yes—
96C2×C4×C17C22⋊Q811yescharacteristic
97C2×C4×C17C4⋊Q812yes—
98C2×C4×C17C4⋊Q811yescharacteristic
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C8×C17—42no—
106#3622—44no—
107#2809—171no—
108#1857—172no—
109#1857D1711yescharacteristic
110#1394—172no—
111#1394—171no—
112C22×C17—21no—
113C22×C17—21no—
114C22×C17#39511yescharacteristic
115C22×C17B(4,𝔽2)11yescharacteristic
116C22×C17#45811yescharacteristic
117C4×C17—22no—
118C4×C17—21no—
119C4×C17—22no—
120C4×C17—21no—
121C4×C17—42no—
122C4×C17—81no—
123C4×C17—81no—
124C4×C17—81no—
125Dic17—82no—
126Dic17—82no—
127#519C2×D1711yescharacteristic
128#428—21no—
129#428—342no—
130C2×#101—172no—
131C2×#101C2×D1711yescharacteristic
132#400—341no—
133#398—342no—
134#391—172no—
135#391C2×D1711yescharacteristic
136#389—341no—
137#387—342no—
138#339—21no—
139#339—342no—
140C2×C17—21no—
141C2×C17—21no—
142C2×C17#206211yescharacteristic
143C2×C17#220311yescharacteristic
144C2×C17#220411yescharacteristic
145C22×Q8C22×D1711yescharacteristic
146C2×M16—21no—
147C2×M16—342no—
148C4×Q8—41no—
149C4×Q8—41no—
150#117—22no—
151#117—21no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—342no—
154C2×C4⋊C4—341no—
155C2×C4⋊C4—341no—
156C2×C4⋊C4—341no—
157C2×C4⋊C4C22⋊D1711yescharacteristic
158C2×C42—172no—
159C2×C42C22⋊D1711yescharacteristic
160#107—41no—
161#107—682no—
162#106—41no—
163#106—682no—
164#101—21no—
165#101—174no—
166#101—342no—
167#101Dic3412yes—
168#95—22no—
169#95—341no—
170#95—342no—
171#95—342no—
172#95—342no—
173C17#1346511yescharacteristic, complemented, Sylow
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C2×Dic3411yescharacteristic
177C2×Q8#6325211yescharacteristic
178C2×Q8Q8×D1712yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—342no—
182C22×C4—342no—
183C22×C4D4×D1712yes—
184C22×C4C2×C22⋊D1711yescharacteristic
185M16—22no—
186M16—41no—
187M16—344no—
188M16—682no—
189C2×C8—41no—
190C2×C8—682no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—682no—
197C4⋊C4—684no—
198C4⋊C4—682no—
199C4⋊C4—1361no—
200C4⋊C4—1361no—
201C22⋊C4—41no—
202C42—41no—
203C42—41no—
204C42—42no—
205C42—682no—
206C23#1058762311yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—42no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—682no—
222C2×C4—682no—
223C2×C4—682no—
224C2×C4—682no—
225C2×C4—682no—
226C2×C4—682no—
227C2×C4#1058757412yes—
228C2×C4#1058758212yes—
229C2×C4#1058761011yescharacteristic
230C2×C4#1058763011yescharacteristic
231C8—42no—
232C8—684no—
233C22—21no—
234C22—21no—
235C22#1176272911yescharacteristic
236C22#1176299511yescharacteristic
237C22#1176299811yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4—1362no—
247C4—1362no—
248C2—21no—
249C2—21no—
250C2#1453013711yescharacteristic
251C2#1453080411yescharacteristic
252C2#1453080911yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC22×Q8×C17C23
FrattiniC22×Q8C22×D17
FittingC17×#1857C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#1857D17
hypocenterC17#13465
LCSG ▷ C22×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
173. C17#13465semidirect

Maximal subgroups

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

subgroupindex
#145367722
#145289992
#145269112
#145269102
C17×#18572
#1346517