Finite Group Databasebeta

← #16932169

Subgroups — #16932169

Order 4352.

474 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 135 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#14537865C211yesmaximal, characteristic
2#14528780C211yesmaximal, characteristic
3#14527213C212yesmaximal
4#14526642C212yesmaximal
5C17×#2581C211yesmaximal, characteristic
6Q8×#3622—24no—
7#11762862—22no—
8#11762862C2212yes—
9#11762833C2212yes—
10C2×#10587507—21no—
11#11762578—21no—
12Q8⋊#3622—24no—
13C2×C4×Q8×C17C2211yescharacteristic
14C2×C4×Q8×C17C2211yescharacteristic
15C17×#400C412yes—
16C17×#395C412yes—
17C17×#391—21no—
18C17×#389—21no—
19C17×#387—22no—
20C17×#382C2211yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C22×Q8×C17D411yescharacteristic
23C4×Q8×C17—22no—
24C4×Q8×C17—22no—
25C4×Q8×C17—22no—
26C4×Q8×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C42×C17—21no—
34C2×C42×C17D412yes—
35C2×C42×C17C2311yescharacteristic
36C2×#63223—22no—
37C17×#95—22no—
38C17×#95—22no—
39C17×#95—22no—
40C17×#95—21no—
41C17×#95C2×C412yes—
42#10587507—42no—
43C22⋊#3622—22no—
44C4⋊#3622—24no—
45C4⋊#3622—22no—
46C4⋊#3622—22no—
47C4⋊#3622—22no—
48C4×#3622—24no—
49C4×#3622D414yes—
50C2×Q8×C17—22no—
51C2×Q8×C17—22no—
52C2×Q8×C17—22no—
53C2×Q8×C17—42no—
54C2×Q8×C17SD1612yes—
55C2×Q8×C17Q1612yes—
56C22×C4×C17—21no—
57C22×C4×C17—22no—
58C22×C4×C17—21no—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C4⋊C4×C17—24no—
65C4⋊C4×C17—22no—
66C4⋊C4×C17—22no—
67C4⋊C4×C17—44no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—42no—
71C42×C17—22no—
72C42×C17—22no—
73C42×C17—42no—
74C42×C17C2×D412yes—
75C42×C17#4112yes—
76#63223—42no—
77#63223—42no—
78C2×#3622—24no—
79C2×#3622—42no—
80#9484—171nomaximal, Sylow
81C23×C17C2×C22⋊C411yescharacteristic
82Q8×C17—24no—
83Q8×C17—24no—
84Q8×C17—24no—
85Q8×C17—42no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—41no—
99C2×C4×C17—42no—
100C2×C4×C17#10314yes—
101C2×C4×C17C4×D412yes—
102C2×C4×C17#12011yescharacteristic
103C2×C4×C17#12112yes—
104C2×C4×C17#12311yescharacteristic
105C2×C4×C17C2×SD1611yescharacteristic
106C2×C4×C17C2×Q1611yescharacteristic
107C2×C4×C17#13512yes—
108#3622—44no—
109#3622—82no—
110#2982—171no—
111#2581D1711yescharacteristic
112#1801—171no—
113#1487—172no—
114#1286—172no—
115C22×C17—21no—
116C22×C17—21no—
117C22×C17#38611yescharacteristic
118C22×C17C2×#10311yescharacteristic
119C22×C17#41611yescharacteristic
120C4×C17—22no—
121C4×C17—24no—
122C4×C17—21no—
123C4×C17—21no—
124C4×C17—42no—
125C4×C17—42no—
126C4×C17—82no—
127C4×C17—81no—
128C4×C17#46512yes—
129C4×C17#46712yes—
130C2×C4×Q8C2×D1711yescharacteristic
131C2×C4×Q8C2×D1711yescharacteristic
132C8×Q8—344no—
133#431—172no—
134#431—342no—
135#423—172no—
136C2×#101—341no—
137#400Dic1712yes—
138#395Dic1712yes—
139#391—21no—
140#389—21no—
141#387—22no—
142#382C2×D1711yescharacteristic
143#343—341no—
144Q8⋊C8—344no—
145C2×C17—21no—
146C2×C17—21no—
147C2×C17#189111yescharacteristic
148C2×C17#189211yescharacteristic
149C2×C17#191211yescharacteristic
150C22×Q8C2×Dic1711yescharacteristic
151C22×Q8C22⋊D1711yescharacteristic
152C2×M16—342no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×Dic1711yescharacteristic
161C2×C4⋊C4C2×Dic1712yes—
162C2×C4⋊C4C22⋊D1711yescharacteristic
163C2×C42—21no—
164C2×C42C22⋊D1712yes—
165C2×C42C22×D1711yescharacteristic
166C4⋊C8—344no—
167C4⋊C8—342no—
168C4⋊C8—342no—
169C4⋊C8—342no—
170#101—682no—
171C22⋊C8—342no—
172C4×C8—174no—
173C4×C8—344no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—21no—
178#95C2×Dic1712yes—
179C17#948411yescharacteristic, complemented, Sylow
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8#6323012yes—
185C2×Q8#6323112yes—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4C22⋊Dic1712yes—
190C22×C4C22⋊Dic1712yes—
191C22×C4C22×Dic1711yescharacteristic
192C22×C4C2×C22⋊D1711yescharacteristic
193C22×C4C2×C22⋊D1711yescharacteristic
194M16—682no—
195M16—682no—
196C2×C8—344no—
197C2×C8—682no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C42—22no—
206C42—22no—
207C42—42no—
208C42D4×D1712yes—
209C42#6325412yes—
210C23C2×C22⋊Dic1711yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—24no—
214Q8—42no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4#1058750614yes—
230C2×C4D4×Dic1712yes—
231C2×C4#1058762111yescharacteristic
232C2×C4#1058762311yescharacteristic
233C2×C4#1058762512yes—
234C2×C4C2×#6323011yescharacteristic
235C2×C4C2×#6323111yescharacteristic
236C2×C4#1058763612yes—
237C8—684no—
238C8—1362no—
239C22—21no—
240C22—21no—
241C22#1176268911yescharacteristic
242C22C2×#1058750611yescharacteristic
243C22#1176280511yescharacteristic
244C4—21no—
245C4—21no—
246C4—22no—
247C4—24no—
248C4—42no—
249C4—42no—
250C4—82no—
251C4—81no—
252C4#1176304012yes—
253C4#1176304812yes—
254C2—21no—
255C2—21no—
256C2#1452920311yescharacteristic
257C2#1452920811yescharacteristic
258C2#1452932211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2581C2
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2581D17
hypocenterC17#9484
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
179. C17#9484semidirect

Maximal subgroups

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

subgroupindex
#145378652
#145287802
#145272132
#145266422
C17×#25812
#948417