Finite Group Databasebeta

← #16922650

Subgroups — #16922650

Order 4352.

456 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 119 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#14531353C211yesmaximal, characteristic
2#14528899C212yesmaximal
3#14526610C211yesmaximal, characteristic
4#14526609C212yesmaximal
5C17×#1272C211yesmaximal, characteristic
6C2×#10587630C411yescharacteristic
7C2×Q8×Dic17C411yescharacteristic
8C2×#10587573C411yescharacteristic
9C2×#10587485—22no—
10#11762715C2211yescharacteristic
11#11762706—22no—
12#11762678C411yescharacteristic
13#11762660—22no—
14C42×Dic17C2211yescharacteristic
15#11762564C2212yes—
16#11762563—22no—
17C17×#391C2211yescharacteristic
18C17×#336C2212yes—
19C22×Q8×C17C2×C411yescharacteristic
20C22×Dic34C2×C411yescharacteristic
21C22×C8×C17—22no—
22C2×C4⋊C4×C17C2×C411yescharacteristic
23C2×C42×C17C2311yescharacteristic
24Q8×Dic17—42no—
25#10587630—22no—
26#10587630—22no—
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17C2×C411yescharacteristic
29C2×#63225—22no—
30C2×#63225—21no—
31C2×#63225—22no—
32C2×C4×Dic17—21no—
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17C2×C411yescharacteristic
36C2×C4×Dic17D412yes—
37C2×C4×Dic17D412yes—
38C22×#3622—22no—
39#10587573—42no—
40C17×#95—22no—
41#10587502—22no—
42#10587485—44no—
43C2×Q8×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17M1612yes—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C22×C411yescharacteristic
50C22×C4×C17C2×D412yes—
51C22×Dic17—21no—
52C22×Dic17—21no—
53C22×Dic17C22⋊C411yescharacteristic
54C22×Dic17C22⋊C412yes—
55C22×Dic17C22⋊C411yescharacteristic
56C2×Dic34—21no—
57C2×Dic34—41no—
58C2×Dic34—41no—
59C2×Dic34M1612yes—
60C2×C8×C17—42no—
61C2×C8×C17—42no—
62C2×C8×C17—42no—
63C4⋊C4×C17—41no—
64C42×C17—41no—
65C4⋊Dic17—42no—
66C4⋊Dic17—41no—
67#63225—28no—
68#63225—42no—
69#63225—44no—
70C4×Dic17—21no—
71C4×Dic17—21no—
72C4×Dic17—22no—
73C4×Dic17—42no—
74C4×Dic17—42no—
75C4×Dic17—42no—
76C4×Dic17SD1614yes—
77C4×Dic17Q1614yes—
78C2×#3622—42no—
79C2×#3622—42no—
80C2×#3622—42no—
81#7653—171nomaximal, Sylow
82C23×C17C2×C22⋊C411yescharacteristic
83Q8×C17—42no—
84Q8×C17—42no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—41no—
93C2×C4×C17—42no—
94C2×C4×C17C2×M1611yescharacteristic
95C2×C4×C17C2×M1611yescharacteristic
96C2×C4×C17C2×SD1612yes—
97C2×C4×C17C2×Q1612yes—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—42no—
107C2×Dic17#10012yes—
108C2×Dic17#10314yes—
109C2×Dic17#10314yes—
110Dic34—42no—
111Dic34—42no—
112C8×C17—82no—
113#3622—82no—
114C4×C4⋊Q8—171no—
115#1831—172no—
116#1272—171no—
117#1272—172no—
118#1272D1711yescharacteristic
119C22×C17#40711yescharacteristic
120C22×C17#41011yescharacteristic
121C22×C17C2×#10011yescharacteristic
122C22×C17C2×#10312yes—
123C22×C17#42112yes—
124C4×C17—21no—
125C4×C17—21no—
126C4×C17—22no—
127C4×C17—22no—
128C4×C17—42no—
129C4×C17—81no—
130Dic17—24no—
131Dic17—41no—
132Dic17—42no—
133Dic17—81no—
134Dic17#33314yes—
135C2×C4⋊Q8—171no—
136C2×C4×Q8—171no—
137C2×C4×Q8—171no—
138C2×C4⋊C8—342no—
139#391—171no—
140#391C2×D1711yescharacteristic
141#389—342no—
142#384—171no—
143C4×C4⋊C4—342no—
144C43—171no—
145#336—172no—
146#336—342no—
147#336C2×D1712yes—
148C2×C17#146611yescharacteristic
149C2×C17#148712yes—
150C2×C17#151812yes—
151C2×C17#153611yescharacteristic
152C2×C17C2×#33311yescharacteristic
153C22×Q8—171no—
154C22×Q8C4×D1711yescharacteristic
155C22×C8—22no—
156C22×C8—342no—
157C4⋊Q8—342no—
158C4⋊Q8—342no—
159C4×Q8—682no—
160C4×Q8—682no—
161C2×C4⋊C4—171no—
162C2×C4⋊C4—342no—
163C2×C4⋊C4—341no—
164C2×C4⋊C4—341no—
165C2×C4⋊C4—342no—
166C2×C4⋊C4C4×D1711yescharacteristic
167C2×C42—172no—
168C2×C42—171no—
169C2×C42—171no—
170C2×C42—172no—
171C2×C42—341no—
172C2×C42—341no—
173C2×C42C22×D1711yescharacteristic
174C4⋊C8—684no—
175#95—22no—
176#95—342no—
177C17#765311yescharacteristic, complemented, Sylow
178C2×Q8—21no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—172no—
182C2×Q8—341no—
183C2×Q8—681no—
184C2×Q8—681no—
185C2×Q8#6321812yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—171no—
189C22×C4—172no—
190C22×C4—171no—
191C22×C4—341no—
192C22×C4—341no—
193C22×C4C2×C4×D1711yescharacteristic
194C22×C4D4×D1712yes—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—682no—
199C2×C8—682no—
200C2×C8—682no—
201C4⋊C4—41no—
202C4⋊C4—348no—
203C4⋊C4—681no—
204C4⋊C4—682no—
205C4⋊C4—682no—
206C4⋊C4—684no—
207C42—41no—
208C42—174no—
209C42—174no—
210C42—342no—
211C42—341no—
212C42—341no—
213C42—682no—
214C42—682no—
215C42—682no—
216C23C22⋊C4×D1711yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—682no—
220Q8—682no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—172no—
231C2×C4—174no—
232C2×C4—174no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—341no—
237C2×C4—342no—
238C2×C4—681no—
239C2×C4—681no—
240C2×C4—681no—
241C2×C4—682no—
242C2×C4C2×#6321811yescharacteristic
243C2×C4M16×D1711yescharacteristic
244C2×C4SD16×D1712yes—
245C2×C4Q16×D1712yes—
246C8—82no—
247C8—1362no—
248C22#1176277111yescharacteristic
249C22#1176278111yescharacteristic
250C22#100×D1711yescharacteristic
251C22#103×D1712yes—
252C22#1176282412yes—
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—81no—
259C4—174no—
260C4—344no—
261C4—682no—
262C4—681no—
263C4—1361no—
264C2#1452712311yescharacteristic
265C2#1452721112yes—
266C2#1452734612yes—
267C2#1452741711yescharacteristic
268C2D17×#33311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1272D17
hypocenterC17#7653
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
177. C17#7653semidirect

Maximal subgroups

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

subgroupindex
#145313532
#145288992
#145266102
#145266092
C17×#12722
#765317