Finite Group Databasebeta

← #16945003

Subgroups — #16945003

Order 4352.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#14534550C211yesmaximal, characteristic
2#14528861C211yesmaximal, characteristic
3#14526785C214yesmaximal
4C17×#1464C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587610—21no—
7#11762739—21no—
8#11762739C412yescomplemented
9#11762734—21no—
10#11762734C2212yes—
11#11762715—21no—
12#11762675C411yescharacteristic, complemented
13#11762672—21no—
14#11762576—21no—
15#11762576C2212yes—
16C2×C17×#117C2211yescharacteristic
17C17×#432—21no—
18C17×#324—21no—
19C17×#323C2212yes—
20C23×C4×C17D411yescharacteristic
21C22×Dic34—21no—
22C17×#117—41no—
23C17×#117—41no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C4⋊C4×C17D411yescharacteristic
26C2×C22⋊C4×C17D411yescharacteristic
27C2×C22⋊C4×C17C2311yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—22no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C412yes—
35#10587610—44no—
36C2×#63225—22no—
37#10587555—41no—
38#10587555—41no—
39C17×C4⋊C8—41no—
40C17×C22⋊C8—22no—
41C17×#97—41no—
42#10587502—22no—
43#10587502—21no—
44#10587502—22no—
45#10587502—41no—
46#10587502—41no—
47#10587502C2×C412yes—
48C24×C17C2×D411yescharacteristic
49C22×C4×C17—22no—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—41no—
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×C4×C17C2×D411yescharacteristic
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—42no—
62C2×Dic34—42no—
63C2×Dic34—42no—
64C2×Dic34—41no—
65C2×Dic34—41no—
66C2×C8×C17—42no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—41no—
69C22⋊C4×C17—41no—
70C22⋊C4×C17—41no—
71C42×C17—21no—
72C42×C17—21no—
73C42×C17—21no—
74C42×C17—21no—
75C22⋊Dic17—44no—
76C22⋊Dic17—42no—
77C22⋊Dic17—42no—
78C4⋊Dic17—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17—42no—
81#63225—44no—
82#11801—171nomaximal, Sylow
83C23×C17—21no—
84C23×C17—21no—
85C23×C17—41no—
86C23×C17—41no—
87C23×C17C2×C22⋊C411yescharacteristic
88C23×C17AGL(1,ℤ/8)12yes—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17#9911yescharacteristic
105C2×C4×C17#9911yescharacteristic
106C2×Dic17—42no—
107C2×Dic17—42no—
108C2×Dic17—42no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112C2×Dic17—82no—
113C2×Dic17—82no—
114C2×Dic17—82no—
115Dic34—82no—
116Dic34—82no—
117C8×C17—82no—
118#2621—171no—
119#1820—171no—
120#1464D1711yescharacteristic
121#1344—174no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—22no—
125C22×C17—42no—
126C22×C17—41no—
127C22×C17—41no—
128C22×C17—81no—
129C22×C17C2×#9911yescharacteristic
130C22×C17#41811yescharacteristic
131C22×C17#42111yescharacteristic
132C4×C17—21no—
133C4×C17—21no—
134C4×C17—41no—
135C4×C17—41no—
136C4×C17—41no—
137C4×C17—41no—
138C4×C17—41no—
139Dic17—82no—
140Dic17—82no—
141Dic17—162no—
142C2×C4⋊Q8—171no—
143C2×C22⋊Q8—341no—
144C2×#117C2×D1711yescharacteristic
145#432—21no—
146#398—172no—
147#398—341no—
148#397—172no—
149#397—341no—
150#391—341no—
151#382—171no—
152#381—341no—
153#342—172no—
154#342—341no—
155#324—21no—
156#323C2×D1712yes—
157C2×C17—21no—
158C2×C17—21no—
159C2×C17—42no—
160C2×C17#151011yescharacteristic
161C2×C17#212311yescharacteristic
162C2×C17#212911yescharacteristic
163C22×Q8—341no—
164C23×C4D6811yescharacteristic
165C4⋊Q8—681no—
166C4⋊Q8—681no—
167C22⋊Q8—684no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—172no—
171C2×C4⋊C4—342no—
172C2×C4⋊C4—341no—
173C2×C4⋊C4—341no—
174C2×C4⋊C4C4×D1711yescharacteristic
175C2×C4⋊C4C22⋊D1711yescharacteristic
176C2×C22⋊C4—342no—
177C2×C22⋊C4—342no—
178C2×C22⋊C4D6811yescharacteristic
179C2×C22⋊C4C22×D1711yescharacteristic
180C2×C42C4×D1711yescharacteristic
181C2×C42C22⋊D1711yescharacteristic
182C4⋊C8—41no—
183C22⋊C8—22no—
184#97—41no—
185#95—172no—
186#95—342no—
187#95—341no—
188#95—342no—
189#95—681no—
190#95—681no—
191C17#1180111yescharacteristic, complemented, Sylow
192C24C2×D6811yescharacteristic
193C2×Q8—682no—
194C2×Q8—682no—
195C2×Q8—681no—
196C2×Q8—681no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—342no—
202C22×C4—342no—
203C22×C4—682no—
204C22×C4#6322711yescharacteristic
205C22×C4#6322711yescharacteristic
206C22×C4#6322711yescharacteristic
207C22×C4#6322711yescharacteristic
208C22×C4C2×C4×D1711yescharacteristic
209C22×C4C2×C22⋊D1711yescharacteristic
210C2×C8—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—682no—
214C4⋊C4—684no—
215C4⋊C4—682no—
216C4⋊C4—682no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—684no—
220C22⋊C4—682no—
221C22⋊C4—682no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23#1058759412yes—
231C23C2×#6322711yescharacteristic
232Q8—1362no—
233Q8—1362no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—682no—
250C2×C4—682no—
251C2×C4—682no—
252C2×C4—682no—
253C2×C4—682no—
254C2×C4—682no—
255C2×C4—1362no—
256C2×C4—1362no—
257C2×C4—1362no—
258C2×C4#1058747711yescharacteristic
259C2×C4#1058747711yescharacteristic
260C8—82no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1058747711yescharacteristic
269C22#1176281511yescharacteristic
270C22#1176282511yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—2722no—
279C4—1362no—
280C4—1362no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#1452731411yescharacteristic
285C2#1453043911yescharacteristic
286C2#1453045811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1464C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1464D17
hypocenterC17#11801
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
5. C2×#10587555C4semidirect
8. #11762739C4semidirect
12. #11762675C4semidirect
191. C17#11801semidirect

Maximal subgroups

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

subgroupindex
#145345502
#145288612
#145267852
C17×#14642
#1180117