Finite Group Databasebeta

← #16943813

Subgroups — #16943813

Order 4352.

440 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 75 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#14531498C211yesmaximal, characteristic
2#14527121C212yesmaximal
3#14526729C212yesmaximal, complemented
4#14526718C211yesmaximal, characteristic, complemented
5C17×#1319C211yesmaximal, characteristic, complemented
6C2×#10587633C411yescharacteristic
7C2×Q8×Dic17C411yescharacteristic
8C2×#10587579C2211yescharacteristic
9#11762771—22no—
10#11762711C411yescharacteristic
11#11762698—22no—
12#11762683—22no—
13#11762678C2211yescharacteristic
14#11762655C411yescharacteristic
15#11762535C2212yes—
16#11762532—22no—
17C17×#391C2211yescharacteristic
18C17×#324C2212yes—
19C22×Q8×C17C2×C411yescharacteristic
20C22×D68C2×C411yescharacteristic
21C22×C4×D17D412yes—
22C2×C4⋊C4×C17C2311yescharacteristic
23C2×C42×C17C2×C411yescharacteristic
24#10587633—22no—
25#10587633—22no—
26Q8×Dic17—42no—
27C2×#63227—22no—
28C2×#63227—22no—
29C2×#63227D412yes—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×#63225—22no—
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17C2×C411yescharacteristic
36C2×#63218—42no—
37#10587579—42no—
38C17×C22⋊C8—22no—
39C17×#95—22no—
40#10587502—21no—
41#10587502—21no—
42C22⋊#3622—22no—
43#10587491—42no—
44C23×D17C22⋊C412yes—
45C2×Q8×C17—21no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C2×Q8×C17M1612yescomplemented
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17C22×C411yescharacteristic
52C22×C4×C17C2×D412yes—
53C22×Dic17—21no—
54C22×Dic17—21no—
55C22×Dic17C22⋊C411yescharacteristic
56C22×Dic17C22⋊C411yescharacteristic
57C2×D68—21no—
58C2×D68—21no—
59C2×D68—41no—
60C2×D68—41no—
61C2×C4×D17—22no—
62C2×C4×D17—22no—
63C2×C4×D17—82no—
64C2×C8×C17—42no—
65C4⋊C4×C17—41no—
66C42×C17—41no—
67#63227—28no—
68#63227—42no—
69#63227—82no—
70C4⋊Dic17—42no—
71C4⋊Dic17—41no—
72#63225—82no—
73C4×Dic17—21no—
74C4×Dic17—22no—
75C4×Dic17—42no—
76C4×Dic17M1612yescomplemented
77C2×#3622—42no—
78#63218—84no—
79#11535—171nomaximal, Sylow
80C23×C17C2×C22⋊C411yescharacteristic
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—42no—
85C22×D17—42no—
86Q8×C17—42no—
87Q8×C17—42no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—41no—
93C2×C4×C17—41no—
94C2×C4×C17—42no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17C2×M1611yescharacteristic
99C2×C4×C17C2×M1611yescharacteristic
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—42no—
107C2×Dic17—41no—
108C2×Dic17#9912yes—
109D68—42no—
110D68—42no—
111C4×D17—44no—
112C4×D17—82no—
113C8×C17—82no—
114#3622—82no—
115#2326—171no—
116#1465—172no—
117#1325—172no—
118#1319—171no—
119#1319D1711yescharacteristic, complemented
120C22×C17—21no—
121C22×C17—21no—
122C22×C17#40711yescharacteristic
123C22×C17C2×#9911yescharacteristic
124C22×C17#41311yescharacteristic
125C2×D17—44no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—82no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—42no—
133C4×C17—42no—
134C4×C17—81no—
135Dic17—22no—
136Dic17—42no—
137Dic17—41no—
138Dic17—81no—
139C2×#124—171no—
140C2×C4×Q8—171no—
141C2×C4×D4—171no—
142#407—342no—
143#391C2×D1711yescharacteristic
144#390—171no—
145#388—342no—
146#385—342no—
147#384—171no—
148#376—171no—
149#324—172no—
150#324C2×D1712yes—
151#323—342no—
152C2×C17—21no—
153C2×C17—21no—
154C2×C17#146511yescharacteristic
155C2×C17#212711yescharacteristic
156C2×C17#213511yescharacteristic
157D17—82no—
158C22×Q8C4×D1711yescharacteristic
159C22×D4—171no—
160C23×C4—172no—
161C2×M16—682no—
162#124—342no—
163#124—342no—
164C4×Q8—682no—
165C4×D4—682no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4—341no—
168C2×C4⋊C4—342no—
169C2×C4⋊C4C22×D1711yescharacteristic
170C2×C22⋊C4—172no—
171C2×C22⋊C4—342no—
172C2×C22⋊C4—342no—
173C2×C42—171no—
174C2×C42—171no—
175C2×C42—341no—
176C2×C42C4×D1711yescharacteristic
177C22⋊C8—22no—
178C22⋊C8—342no—
179C22⋊C8—682no—
180#95—22no—
181#95—341no—
182#95—341no—
183C17#1153511yescharacteristic, complemented, Sylow
184C24—172no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8#6321812yescomplemented
189C2×D4—341no—
190C2×D4—341no—
191C2×D4—681no—
192C2×D4—681no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—342no—
198C22×C4—342no—
199C22×C4—341no—
200C22×C4—341no—
201C22×C4—1362no—
202C22×C4C2×C4×D1711yescharacteristic
203C22×C4D4×D1712yes—
204M16—1364no—
205C2×C8—42no—
206C2×C8—682no—
207C4⋊C4—41no—
208C4⋊C4—681no—
209C4⋊C4—682no—
210C4⋊C4—1362no—
211C22⋊C4—348no—
212C22⋊C4—682no—
213C22⋊C4—1362no—
214C42—41no—
215C42—172no—
216C42—342no—
217C42—341no—
218C42—682no—
219C23—342no—
220C23—342no—
221C23—342no—
222C23—682no—
223C23—682no—
224C23C22⋊C4×D1711yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—682no—
228D4—682no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—172no—
240C2×C4—342no—
241C2×C4—341no—
242C2×C4—684no—
243C2×C4—681no—
244C2×C4—681no—
245C2×C4—681no—
246C2×C4—681no—
247C2×C4—682no—
248C2×C4—681no—
249C2×C4—1362no—
250C2×C4C2×#6321811yescharacteristic
251C2×C4M16×D1711yescharacteristic
252C8—82no—
253C8—1362no—
254C22—21no—
255C22—21no—
256C22—684no—
257C22—682no—
258C22—682no—
259C22—682no—
260C22—1362no—
261C22#1176277111yescharacteristic
262C22#99×D1711yescharacteristic
263C22#1176279011yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—342no—
270C4—682no—
271C4—681no—
272C4—1361no—
273C2—21no—
274C2—21no—
275C2—1362no—
276C2#1452711911yescharacteristic
277C2#1453045211yescharacteristic
278C2#1453047711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1319C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1319D17
hypocenterC17#11535
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #14526729C2semidirect
4. #14526718C2semidirect
5. C17×#1319C2semidirect
48. C2×Q8×C17M16semidirect
76. C4×Dic17M16semidirect
119. #1319D17semidirect
183. C17#11535semidirect
188. C2×Q8#63218semidirect

Maximal subgroups

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

subgroupindex
#145314982
#145271212
#145267292
#145267182
C17×#13192
#1153517