Finite Group Databasebeta

← #16922638

Subgroups — #16922638

Order 4352.

468 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 99 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#14531327C211yesmaximal, characteristic
2#14528926C212yesmaximal, complemented
3#14526610C211yesmaximal, characteristic, complemented
4#14526604C212yesmaximal
5C17×#1272C211yesmaximal, characteristic, complemented
6C2×#10587633C411yescharacteristic
7C2×Q8×Dic17C411yescharacteristic
8C2×#10587579C411yescharacteristic
9#11762711C2211yescharacteristic
10#11762701—22no—
11#11762674C411yescharacteristic
12#11762656—22no—
13C42×Dic17C2211yescharacteristic
14#11762564C2212yes—
15#11762563—22no—
16#11762563—22no—
17C17×#391C2211yescharacteristic
18C17×#336C2212yes—
19C22×Q8×C17C2×C411yescharacteristic
20C22×D68C2×C411yescharacteristic
21C22×C4×D17—21no—
22C22×C8×C17—22no—
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C42×C17C2311yescharacteristic
25#10587633—22no—
26#10587633—22no—
27Q8×Dic17—42no—
28C2×#63227—22no—
29C2×#63227—22no—
30C2×#63227—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17C2×C411yescharacteristic
36C2×C4×Dic17D412yes—
37C2×C4×Dic17D412yes—
38C22×#3622—22no—
39#10587579—42no—
40C17×#95—22no—
41#10587502—22no—
42C23×D17—21no—
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×Dic17C22⋊C412yes—
53C22×Dic17C22⋊C411yescharacteristic
54C22×Dic17C22⋊C411yescharacteristic
55C2×D68—21no—
56C2×D68—41no—
57C2×D68—41no—
58C2×D68M1612yes—
59C2×C4×D17—44no—
60C2×C4×D17—42no—
61C2×C8×C17—42no—
62C2×C8×C17—42no—
63C2×C8×C17—42no—
64C4⋊C4×C17—41no—
65C42×C17—41no—
66#63227—28no—
67#63227—42no—
68#63227—42no—
69#63227—42no—
70C4⋊Dic17—42no—
71C4⋊Dic17—41no—
72C4×Dic17—21no—
73C4×Dic17—21no—
74C4×Dic17—22no—
75C4×Dic17—22no—
76C4×Dic17—22no—
77C4×Dic17—42no—
78C4×Dic17—42no—
79C2×#3622—42no—
80C2×#3622—42no—
81C2×#3622—42no—
82#7649—171nomaximal, Sylow
83C23×C17C2×C22⋊C411yescharacteristic
84C22×D17—22no—
85C22×D17—41no—
86C22×D17—41no—
87C22×D17—42no—
88C22×D17—42no—
89Q8×C17—42no—
90Q8×C17—42no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—42no—
100C2×C4×C17C2×M1611yescharacteristic
101C2×C4×C17C2×M1611yescharacteristic
102C2×C4×C17#13512yes—
103C2×C4×C17#13512yes—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—41no—
111C2×Dic17—42no—
112C2×Dic17#10012yes—
113D68—42no—
114D68—42no—
115C4×D17—84no—
116C8×C17—82no—
117#3622—82no—
118C4×#124—171no—
119#1839—172no—
120#1272—171no—
121#1272D1711yescharacteristic, complemented
122#1270—172no—
123C22×C17#40711yescharacteristic
124C22×C17#41011yescharacteristic
125C22×C17C2×#10011yescharacteristic
126C22×C17#41612yes—
127C22×C17#42112yes—
128C2×D17—42no—
129C2×D17—44no—
130C2×D17—81no—
131C2×D17—82no—
132C2×D17—81no—
133C4×C17—21no—
134C4×C17—21no—
135C4×C17—22no—
136C4×C17—22no—
137C4×C17—42no—
138C4×C17—81no—
139Dic17—22no—
140Dic17—41no—
141Dic17—42no—
142Dic17—81no—
143C2×#124—171no—
144C2×C4×Q8—171no—
145C2×C4×D4—171no—
146#391C2×D1711yescharacteristic
147#390—171no—
148#388—342no—
149#382—171no—
150C4×C22⋊C4—342no—
151C43—171no—
152#336—172no—
153#336—342no—
154#336—342no—
155#336C2×D1712yes—
156C2×C17#146611yescharacteristic
157C2×C17#148812yes—
158C2×C17#152112yes—
159C2×C17#153311yescharacteristic
160C2×C17#154311yescharacteristic
161D17—82no—
162C22×Q8C4×D1711yescharacteristic
163C22×D4—171no—
164C23×C4—341no—
165C22×C8—22no—
166C22×C8—342no—
167#124—342no—
168#124—342no—
169C4×Q8—682no—
170C4×D4—682no—
171C2×C4⋊C4—171no—
172C2×C4⋊C4—341no—
173C2×C4⋊C4C4×D1711yescharacteristic
174C2×C22⋊C4—342no—
175C2×C22⋊C4—341no—
176C2×C22⋊C4—342no—
177C2×C42—171no—
178C2×C42—171no—
179C2×C42—172no—
180C2×C42—172no—
181C2×C42—341no—
182C2×C42C22×D1711yescharacteristic
183#95—22no—
184#95—342no—
185C17#764911yescharacteristic, complemented, Sylow
186C24—341no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8#6321812yes—
191C2×D4—172no—
192C2×D4—341no—
193C2×D4—681no—
194C2×D4—681no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—171no—
198C22×C4—172no—
199C22×C4—171no—
200C22×C4—341no—
201C22×C4—682no—
202C22×C4—684no—
203C22×C4C2×C4×D1711yescharacteristic
204C22×C4D4×D1712yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—682no—
209C2×C8—682no—
210C2×C8—682no—
211C4⋊C4—41no—
212C4⋊C4—681no—
213C4⋊C4—682no—
214C22⋊C4—348no—
215C22⋊C4—682no—
216C22⋊C4—682no—
217C22⋊C4—682no—
218C42—41no—
219C42—342no—
220C42—341no—
221C42—341no—
222C42—342no—
223C42—342no—
224C42—682no—
225C42—682no—
226C23—342no—
227C23—681no—
228C23—681no—
229C23—682no—
230C23—682no—
231C23C22⋊C4×D1711yescharacteristic
232Q8—42no—
233Q8—42no—
234D4—682no—
235D4—682no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—172no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—681no—
253C2×C4—682no—
254C2×C4—1364no—
255C2×C4C2×#6321811yescharacteristic
256C2×C4M16×D1711yescharacteristic
257C2×C4#1058760212yes—
258C2×C4#1058760512yes—
259C8—82no—
260C8—1362no—
261C22—682no—
262C22—684no—
263C22—1362no—
264C22—1361no—
265C22—1361no—
266C22#1176277111yescharacteristic
267C22#1176278111yescharacteristic
268C22#100×D1711yescharacteristic
269C22#1176280412yes—
270C22#1176282412yes—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—342no—
278C4—681no—
279C4—682no—
280C4—1361no—
281C2—1362no—
282C2#1452712311yescharacteristic
283C2#1452721712yes—
284C2#1452736012yes—
285C2#1452740611yescharacteristic
286C2#1452744511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#1272D17
hypocenterC17#7649
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
2. #14528926C2semidirect
3. #14526610C2semidirect
5. C17×#1272C2semidirect
121. #1272D17semidirect
185. C17#7649semidirect

Maximal subgroups

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

subgroupindex
#145313272
#145289262
#145266102
#145266042
C17×#12722
#764917