Finite Group Databasebeta

← #16943815

Subgroups — #16943815

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#14531492C211yesmaximal, characteristic
2#14527121C212yesmaximal
3#14526730C211yesmaximal, characteristic, complemented
4#14526717C212yesmaximal, complemented
5C17×#1325C211yesmaximal, characteristic, complemented
6C2×#10587561C411yescharacteristic
7C2×#10587573C411yescharacteristic
8C2×#10587579C2211yescharacteristic
9#11762771—22no—
10#11762714C2211yescharacteristic
11#11762710C411yescharacteristic
12#11762699—22no—
13#11762680—22no—
14#11762655C411yescharacteristic
15#11762535C2212yes—
16#11762532—22no—
17C17×#384C2211yescharacteristic
18C17×#324C2212yes—
19C22×D68C2×C411yescharacteristic
20C22×C4×D17D412yes—
21C22×Dic34C2×C411yescharacteristic
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C4⋊C4×C17C2311yescharacteristic
25C2×C42×C17C2×C411yescharacteristic
26C2×#63227—22no—
27C2×#63227—22no—
28C2×#63227D412yes—
29C2×#63225—21no—
30C2×C4×Dic17—21no—
31C2×C4×Dic17C2×C411yescharacteristic
32C2×C4×Dic17C2×C411yescharacteristic
33C2×#63218—42no—
34#10587579—42no—
35#10587573—42no—
36#10587561—22no—
37#10587561—22no—
38C17×C22⋊C8—22no—
39#10587502—22no—
40#10587502—21no—
41#10587502—21no—
42C22⋊#3622—22no—
43#10587491—42no—
44C23×D17C22⋊C412yes—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C22×C411yescharacteristic
48C22×C4×C17C2×D412yes—
49C22×Dic17—21no—
50C22×Dic17—21no—
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C22⋊C411yescharacteristic
53C2×D68—21no—
54C2×D68—21no—
55C2×D68—41no—
56C2×D68—41no—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—82no—
60C2×Dic34—21no—
61C2×Dic34—41no—
62C2×Dic34—41no—
63C2×Dic34M1612yescomplemented
64C2×C8×C17—42no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—82no—
68C42×C17—21no—
69C42×C17M1612yescomplemented
70#63227—28no—
71#63227—42no—
72#63227—82no—
73#63225—42no—
74C4×Dic17—22no—
75C4×Dic17—42no—
76C4×Dic17—41no—
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—
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—42no—
93C2×C4×C17—42no—
94C2×C4×C17—41no—
95C2×C4×C17C2×M1611yescharacteristic
96C2×C4×C17C2×M1611yescharacteristic
97C2×Dic17—22no—
98C2×Dic17—21no—
99C2×Dic17—41no—
100C2×Dic17—41no—
101C2×Dic17—41no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17#9912yes—
107D68—42no—
108D68—42no—
109C4×D17—44no—
110C4×D17—82no—
111Dic34—42no—
112Dic34—42no—
113C8×C17—82no—
114#3622—82no—
115#2326—171no—
116#1465—172no—
117#1325—171no—
118#1325D1711yescharacteristic, complemented
119#1319—172no—
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—41no—
137Dic17—42no—
138Dic17—81no—
139C2×#124—171no—
140C2×C4×Q8—171no—
141C2×C4×D4—171no—
142#407—342no—
143#391—171no—
144#390—171no—
145#388—342no—
146#385—342no—
147#384C2×D1711yescharacteristic
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×Q8—171no—
159C22×D4—171no—
160C23×C4—172no—
161C2×M16—682no—
162#124—342no—
163#124—342no—
164C4×Q8—682no—
165C4×D4—682no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—341no—
168C2×C4⋊C4C4×D1711yescharacteristic
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—341no—
181#95—341no—
182#95—342no—
183C17#1153511yescharacteristic, complemented, Sylow
184C24—172no—
185C2×Q8—172no—
186C2×Q8—341no—
187C2×Q8—681no—
188C2×Q8—681no—
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—41no—
209C4⋊C4—82no—
210C4⋊C4—682no—
211C22⋊C4—348no—
212C22⋊C4—682no—
213C22⋊C4—1362no—
214C42—21no—
215C42—342no—
216C42—681no—
217C42—682no—
218C42#6321812yescomplemented
219C23—342no—
220C23—342no—
221C23—342no—
222C23—682no—
223C23—682no—
224C23C22⋊C4×D1711yescharacteristic
225Q8—682no—
226Q8—682no—
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—42no—
237C2×C4—41no—
238C2×C4—172no—
239C2×C4—342no—
240C2×C4—341no—
241C2×C4—684no—
242C2×C4—681no—
243C2×C4—681no—
244C2×C4—681no—
245C2×C4—681no—
246C2×C4—681no—
247C2×C4—681no—
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#1176279211yescharacteristic
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#1452712111yescharacteristic
277C2#1453045011yescharacteristic
278C2#1453047711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1325C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1325D17
hypocenterC17#11535
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #14526730C2semidirect
4. #14526717C2semidirect
5. C17×#1325C2semidirect
63. C2×Dic34M16semidirect
69. C42×C17M16semidirect
118. #1325D17semidirect
183. C17#11535semidirect
218. C42#63218semidirect

Maximal subgroups

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

subgroupindex
#145314922
#145271212
#145267302
#145267172
C17×#13252
#1153517