Finite Group Databasebeta

← #16944912

Subgroups — #16944912

Order 4352.

464 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533092C211yesmaximal, characteristic
2C4×#10587504C211yesmaximal, characteristic
3#14526791C214yesmaximal, complemented
4C17×#1465C211yesmaximal, characteristic
5C2×#10587621—21no—
6C22×C4⋊Dic17C411yescharacteristic
7C2×#10587504—21no—
8C4×#63223—21no—
9#11762739C2212yes—
10#11762734C412yes—
11#11762707—21no—
12#11762670—21no—
13#11762669—21no—
14#11762669C411yescharacteristic
15#11762578C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#407—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C2×M16×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—21no—
33#10587621—44no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—41no—
37C2×C4⋊Dic17—41no—
38C2×C4⋊Dic17C2×C412yes—
39C2×#63225—22no—
40C2×#63223—22no—
41C17×C22⋊C8—22no—
42C17×C22⋊C8—41no—
43#10587504—24no—
44#10587502—22no—
45#10587502—22no—
46#10587502C2×C412yes—
47#10587474—41no—
48C4×#3622—41no—
49C24×C17C22⋊C411yescharacteristic
50C24×C17C22⋊C411yescharacteristic
51C2×D4×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—42no—
68C22×Dic17—42no—
69C22×Dic17—42no—
70M16×C17—82no—
71C2×C8×C17—42no—
72C4⋊C4×C17—41no—
73C22⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C42×C17—21no—
76C42×C17—21no—
77C22⋊Dic17—42no—
78C22⋊Dic17—44no—
79C4⋊Dic17—42no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82C4⋊Dic17—42no—
83#63225—44no—
84#63223—24no—
85#63223—42no—
86C2×#3622—42no—
87#11780—171nomaximal, Sylow
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17#10412yes—
98C23×C17C2×C22⋊C411yescharacteristic
99D4×C17—44no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—81no—
113C2×C4×C17#9911yescharacteristic
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17AGL(1,ℤ/8)12yes—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—84no—
123C8×C17—82no—
124#3622—44no—
125#2486—171no—
126C4×#100—171no—
127#1465D1711yescharacteristic
128#1346—174no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—22no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—81no—
141C22×C17C2×#9911yescharacteristic
142C22×C17#41811yescharacteristic
143C22×C17C2×#10411yescharacteristic
144C4×C17—21no—
145C4×C17—22no—
146C4×C17—41no—
147C4×C17—41no—
148C4×C17—41no—
149C4×C17#35212yes—
150Dic17—82no—
151Dic17—82no—
152C2×#123—341no—
153C2×C4×D4C2×D1711yescharacteristic
154C22×C4⋊C4—171no—
155C2×#100—341no—
156#407—21no—
157C4×M16—341no—
158#398—172no—
159#397—172no—
160#389—341no—
161#381—341no—
162#380—171no—
163#380—341no—
164#343—172no—
165#324C2×D1712yes—
166#323—21no—
167C2×C17—21no—
168C2×C17—21no—
169C2×C17—42no—
170C2×C17—81no—
171C2×C17#150711yescharacteristic
172C2×C17C2×#35211yescharacteristic
173C2×C17#213511yescharacteristic
174C22×D4D6811yescharacteristic
175C23×C4—341no—
176C23×C4C4×D1711yescharacteristic
177C23×C4C22⋊D1711yescharacteristic
178C2×M16—41no—
179C2×M16—342no—
180#123—684no—
181C4×D4—41no—
182C4×D4—41no—
183C2×C4⋊C4—172no—
184C2×C4⋊C4—342no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4—341no—
187C2×C4⋊C4—681no—
188C2×C4⋊C4—681no—
189C2×C4⋊C4C22×D1711yescharacteristic
190C2×C22⋊C4—341no—
191C2×C22⋊C4—342no—
192C2×C22⋊C4C4×D1711yescharacteristic
193C2×C22⋊C4C22⋊D1711yescharacteristic
194C2×C42D6811yescharacteristic
195#100—344no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198#97—681no—
199C4×C8—681no—
200#95—172no—
201#95—342no—
202#95—342no—
203C17#1178011yescharacteristic, complemented, Sylow
204C24#6322711yescharacteristic
205C24#6322711yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—342no—
216C22×C4—342no—
217C22×C4—682no—
218C22×C4—682no—
219C22×C4—682no—
220C22×C4#6322711yescharacteristic
221C22×C4#6322711yescharacteristic
222C22×C4C2×C4×D1711yescharacteristic
223C22×C4C2×D6811yescharacteristic
224C22×C4C2×C22⋊D1711yescharacteristic
225M16—82no—
226M16—344no—
227M16—682no—
228C2×C8—42no—
229C2×C8—682no—
230C4⋊C4—41no—
231C4⋊C4—682no—
232C4⋊C4—682no—
233C4⋊C4—681no—
234C4⋊C4—681no—
235C4⋊C4—684no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—682no—
239C22⋊C4—684no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1058749612yes—
252C23C2×#6322711yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—682no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4—682no—
272C2×C4—682no—
273C2×C4—1364no—
274C2×C4#1058747711yescharacteristic
275C2×C4#1058747711yescharacteristic
276C2×C4#1058759412yes—
277C8—82no—
278C8—684no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#1058747711yescharacteristic
292C22#1176281511yescharacteristic
293C22C2×#1058749611yescharacteristic
294C4—22no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—1362no—
300C4—1362no—
301C4#1176259812yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#1452730111yescharacteristic
307C2C2×#1176259811yescharacteristic
308C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11780
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526791C2semidirect
203. C17#11780semidirect

Maximal subgroups

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

subgroupindex
#145330922
C4×#105875042
#145267912
C17×#14652
#1178017