Finite Group Databasebeta

← #16954011

Subgroups — #16954011

Order 4352.

384 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 81 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#14528998C212yesmaximal
2#14528998C211yesmaximal, characteristic
3#14526911C211yesmaximal, characteristic
4#14526911C212yesmaximal
5C17×#2809C211yesmaximal, characteristic
6#11763287C2211yescharacteristic
7#11762849—22no—
8#11762849—21no—
9C2×#10587507C2212yes—
10C2×#10587507C2211yescharacteristic
11#11762569—22no—
12#11762569—21no—
13C17×#400—21no—
14C17×#398—22no—
15C17×#391C2211yescharacteristic
16C17×#391C2212yes—
17C17×#389—21no—
18C17×#387—22no—
19C22×Q8×C17C2311yescharacteristic
20C2×C4⋊C4×C17—22no—
21C2×C4⋊C4×C17—21no—
22C2×C4⋊C4×C17—21no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17D412yes—
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C42×C17D412yes—
27C2×C42×C17D411yescharacteristic
28Q8×Dic17—41no—
29Q8×Dic17—41no—
30#10587612—22no—
31#10587612—21no—
32C2×#63223—22no—
33C2×#63223—21no—
34C17×#95—21no—
35C17×#95—22no—
36C17×#95—22no—
37C17×#95—22no—
38C17×#95—22no—
39#10587507—22no—
40#10587507—21no—
41#10587507Q814yes—
42#10587507Q812yes—
43#10587479—42no—
44#10587479—41no—
45#10587478—42no—
46#10587478—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17—41no—
49C2×Q8×C17C2×Q812yes—
50C2×Q8×C17C2×Q811yescharacteristic
51C2×Q8×C17#4111yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17C2×D412yes—
57C22×C4×C17C2×D411yescharacteristic
58C4⋊C4×C17—42no—
59C4⋊C4×C17—41no—
60C4⋊C4×C17—44no—
61C4⋊C4×C17—42no—
62C4⋊C4×C17—81no—
63C4⋊C4×C17—81no—
64C42×C17—41no—
65C42×C17—42no—
66C22⋊Dic17—41no—
67C4⋊Dic17—21no—
68C4⋊Dic17—22no—
69C4⋊Dic17—22no—
70C4⋊Dic17—21no—
71C4×Dic17—42no—
72C4×Dic17—41no—
73#63223—24no—
74#63223—22no—
75#63223—41no—
76#63223—42no—
77C2×#3622—42no—
78C2×#3622—41no—
79#13465—171nomaximal, Sylow
80C23×C17#12011yescharacteristic
81Q8×C17—42no—
82Q8×C17—41no—
83Q8×C17—41no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
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—42no—
93C2×C4×C17—42no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—41no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17C22⋊Q812yes—
100C2×C4×C17C22⋊Q811yescharacteristic
101C2×C4×C17C4⋊Q812yes—
102C2×C4×C17C4⋊Q811yescharacteristic
103C2×Dic17—41no—
104C2×Dic17—41no—
105#3622—44no—
106#3622—42no—
107#2809D1711yescharacteristic
108#1857—172no—
109#1857—171no—
110#1394—171no—
111#1394—172no—
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—82no—
123C4×C17—81no—
124C4×C17—82no—
125Dic17—81no—
126Dic17—81no—
127#519—171no—
128#428—342no—
129#428—341no—
130C2×#101—172no—
131C2×#101—171no—
132#400—21no—
133#398—22no—
134#391C2×D1711yescharacteristic
135#391C2×D1712yes—
136#389—21no—
137#387—22no—
138#339—342no—
139#339—341no—
140C2×C17—21no—
141C2×C17—21no—
142C2×C17#206211yescharacteristic
143C2×C17#220311yescharacteristic
144C2×C17#220411yescharacteristic
145C22×Q8C22×D1711yescharacteristic
146C2×M16—342no—
147C2×M16—341no—
148C4×Q8—681no—
149C4×Q8—681no—
150#117—342no—
151#117—341no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C22⋊D1712yes—
157C2×C4⋊C4C22⋊D1711yescharacteristic
158C2×C42C22⋊D1712yes—
159C2×C42C22⋊D1711yescharacteristic
160#107—682no—
161#107—681no—
162#106—682no—
163#106—681no—
164#101—174no—
165#101—172no—
166#101—342no—
167#101—341no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173C17#1346511yescharacteristic, complemented, Sylow
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8Q8×D1711yescharacteristic
177C2×Q8Q8×D1712yes—
178C2×Q8#6325611yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C2×C22⋊D1712yes—
184C22×C4C2×C22⋊D1711yescharacteristic
185M16—344no—
186M16—342no—
187M16—681no—
188M16—682no—
189C2×C8—682no—
190C2×C8—681no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—342no—
198C4⋊C4—341no—
199C4⋊C4—341no—
200C4⋊C4—342no—
201C22⋊C4—681no—
202C42—41no—
203C42—42no—
204C42—682no—
205C42—681no—
206C23#1058763911yescharacteristic
207Q8—42no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—681no—
226C2×C4—681no—
227C2×C4#1058763012yes—
228C2×C4#1058763011yescharacteristic
229C2×C4#1058763212yes—
230C2×C4#1058763211yescharacteristic
231C8—684no—
232C8—682no—
233C22—21no—
234C22—21no—
235C22#1176272811yescharacteristic
236C22#1176299611yescharacteristic
237C22#1176299911yescharacteristic
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—81no—
245C4—82no—
246C4—1361no—
247C4—1361no—
248C2—21no—
249C2—21no—
250C2#1453013611yescharacteristic
251C2#1453080511yescharacteristic
252C2#1453081011yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC22×Q8×C17C23
FrattiniC22×Q8C22×D17
FittingC17×#2809C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2809D17
hypocenterC17#13465
LCSG ▷ C22×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

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
#145289982
#145269112
C17×#28092
#1346517