Finite Group Databasebeta

← #16945469

Subgroups — #16945469

Order 4352.

440 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528776C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526811C214yesmaximal, complemented
4C17×#2487C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587504—21no—
7C2×C22⋊#3622—21no—
8#11762578C2212yes—
9#11762535C2212yes—
10#11762533—21no—
11C2×C4×D4×C17C2211yescharacteristic
12C22×C4⋊C4×C17C411yescharacteristic
13C17×#400C2212yes—
14C17×#399C412yes—
15C17×#389—21no—
16C17×#386—21no—
17C17×#385—21no—
18C17×#380—21no—
19C17×#380C411yescharacteristic
20C22×D4×C17D411yescharacteristic
21C23×C4×C17—21no—
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—41no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×#63223—22no—
39C22×#3622—41no—
40C17×#95—22no—
41C17×#95—22no—
42C17×#95C2×C412yes—
43#10587504—42no—
44C22⋊#3622—22no—
45C22⋊#3622—41no—
46C4⋊#3622—41no—
47C4⋊#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C22⋊C411yescharacteristic
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17SD1612yes—
53C2×D4×C17Q1612yes—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17—82no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C4⋊C4×C17—41no—
70C4⋊C4×C17—42no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—82no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—82no—
79C42×C17—21no—
80C42×C17—21no—
81#63223—42no—
82#63223—42no—
83C2×#3622—42no—
84C2×#3622—42no—
85C2×#3622—82no—
86#11903—171nomaximal, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17#10312yes—
96C23×C17#10312yes—
97C23×C17C2×C22⋊C411yescharacteristic
98D4×C17—42no—
99D4×C17—42no—
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—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—81no—
119C2×C4×C17—84no—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17C2×SD1611yescharacteristic
123C2×C4×C17C2×Q1611yescharacteristic
124#3622—82no—
125#3622—82no—
126#2487D1711yescharacteristic
127#1800—171no—
128#1459—171no—
129#1353—174no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—22no—
133C22×C17—22no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—42no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—81no—
142C22×C17C2×#9911yescharacteristic
143C22×C17C2×#10311yescharacteristic
144C22×C17#42111yescharacteristic
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—41no—
149C4×C17—41no—
150C4×C17—41no—
151C4×C17—82no—
152C4×C17—82no—
153C4×C17#35212yes—
154C2×C4×D4C2×D1711yescharacteristic
155C22×C4⋊C4Dic1711yescharacteristic
156#423—341no—
157C2×#100—341no—
158C2×C22⋊C8—341no—
159#400C2×D1712yes—
160#399Dic1712yes—
161#389—21no—
162#386—21no—
163#385—21no—
164#380—21no—
165#380Dic1711yescharacteristic
166#343—172no—
167#324—172no—
168#323—341no—
169C2×C17—21no—
170C2×C17—21no—
171C2×C17—42no—
172C2×C17—42no—
173C2×C17#152011yescharacteristic
174C2×C17C2×#35211yescharacteristic
175C2×C17#213411yescharacteristic
176C22×D4C22⋊D1711yescharacteristic
177C23×C4—21no—
178C23×C4C2×Dic1711yescharacteristic
179C23×C4C22⋊D1711yescharacteristic
180C2×M16—342no—
181C22×C8—681no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×Dic1712yes—
190C2×C4⋊C4C22×D1711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C2×Dic1711yescharacteristic
194C2×C22⋊C4C22⋊D1711yescharacteristic
195C2×C42C22⋊D1711yescharacteristic
196C4⋊C8—681no—
197C4⋊C8—681no—
198#100—682no—
199C22⋊C8—342no—
200C22⋊C8—681no—
201#95—22no—
202#95—22no—
203#95C2×Dic1712yes—
204C17#1190311yescharacteristic, complemented, Sylow
205C24C22⋊Dic1711yescharacteristic
206C24C22⋊Dic1711yescharacteristic
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#6323012yes—
210C2×D4#6323112yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4C22⋊Dic1711yescharacteristic
222C22×C4C22⋊Dic1711yescharacteristic
223C22×C4C22×Dic1711yescharacteristic
224C22×C4C2×C22⋊D1711yescharacteristic
225C22×C4C2×C22⋊D1711yescharacteristic
226M16—682no—
227M16—682no—
228C2×C8—682no—
229C2×C8—682no—
230C2×C8—1362no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—82no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C42—21no—
242C42—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1058750612yes—
252C23#1058750612yes—
253C23C2×C22⋊Dic1711yescharacteristic
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—84no—
276C2×C4#1058750511yescharacteristic
277C2×C4#1058750511yescharacteristic
278C2×C4C2×#6323011yescharacteristic
279C2×C4C2×#6323111yescharacteristic
280C8—1362no—
281C8—1362no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#1058750511yescharacteristic
295C22C2×#1058750611yescharacteristic
296C22#1176282611yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—82no—
304C4—82no—
305C4#1176259912yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#1452735611yescharacteristic
311C2C2×#1176259911yescharacteristic
312C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2487C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#2487D17
hypocenterC17#11903
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #14526811C2semidirect
204. C17#11903semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145271022
#145268112
C17×#24872
#1190317