Finite Group Databasebeta

← #16922643

Subgroups — #16922643

Order 4352.

624 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 119 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#14531348C211yesmaximal, characteristic
2#14528899C212yesmaximal, complemented
3#14526605C211yesmaximal, characteristic, complemented
4#14526604C212yesmaximal, complemented
5C17×#1270C211yesmaximal, characteristic, complemented
6C2×#10587626C411yescharacteristic
7C2×D4×Dic17C411yescharacteristic
8C2×#10587579C411yescharacteristic
9C2×#10587567—22no—
10C2×#10587485—22no—
11#11762711C2211yescharacteristic
12#11762709—22no—
13#11762678C411yescharacteristic
14C42×Dic17C2211yescharacteristic
15#11762564C2212yes—
16#11762563—22no—
17C17×#390C2211yescharacteristic
18C17×#336C2212yes—
19C22×D4×C17C2×C411yescharacteristic
20C22×C22⋊D17—22no—
21C23×Dic17—21no—
22C22×D68C2×C411yescharacteristic
23C22×C4×D17—21no—
24C22×C8×C17—22no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C22⋊C4×C17—22no—
27C2×C42×C17C2311yescharacteristic
28C2×C22⋊Dic17—21no—
29#10587626—22no—
30#10587626—22no—
31D4×Dic17—42no—
32C2×#63227—22no—
33C2×#63227—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—22no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D412yes—
39C2×C4×Dic17D412yes—
40C22×#3622—22no—
41#10587579—42no—
42#10587567—44no—
43#10587567—44no—
44#10587485—44no—
45C24×C17—21no—
46C23×D17—21no—
47C2×D4×C17—21no—
48C2×D4×C17—41no—
49C2×D4×C17—41no—
50C2×D4×C17M1612yes—
51C22×C4×C17—21no—
52C22×C4×C17C22×C411yescharacteristic
53C22×C4×C17C2×D412yes—
54C2×C22⋊D17—28no—
55C2×C22⋊D17—44no—
56C2×C22⋊D17—44no—
57C22×Dic17—21no—
58C22×Dic17—42no—
59C22×Dic17—44no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C22⋊C412yes—
62C22×Dic17C22⋊C411yescharacteristic
63C2×D68—21no—
64C2×D68—41no—
65C2×D68—41no—
66C2×D68M1612yes—
67C2×C4×D17—42no—
68C2×C4×D17—44no—
69C2×C8×C17—42no—
70C2×C8×C17—42no—
71C2×C8×C17—42no—
72C4⋊C4×C17—41no—
73C22⋊C4×C17—42no—
74C22⋊C4×C17—42no—
75C42×C17—41no—
76C22⋊Dic17—42no—
77#63227—42no—
78#63227—42no—
79#63227—42no—
80C4⋊Dic17—41no—
81#63225—44no—
82C4×Dic17—21no—
83C4×Dic17—21no—
84C4×Dic17—22no—
85C4×Dic17—42no—
86C4×Dic17D814yes—
87C4×Dic17SD1614yes—
88C2×#3622—42no—
89C2×#3622—42no—
90C2×#3622—42no—
91#7651—171nomaximal, Sylow
92C23×C17—22no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—22no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17—42no—
102C22×D17—42no—
103D4×C17—42no—
104D4×C17—42no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17C2×M1611yescharacteristic
112C2×C4×C17C2×M1611yescharacteristic
113C2×C4×C17C2×D812yes—
114C2×C4×C17C2×SD1612yes—
115C22⋊D17—416no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—21no—
119C2×Dic17—22no—
120C2×Dic17—41no—
121C2×Dic17—42no—
122C2×Dic17—84no—
123C2×Dic17#10012yes—
124C2×Dic17#10214yes—
125C2×Dic17#10214yes—
126D68—42no—
127D68—42no—
128C4×D17—84no—
129C8×C17—82no—
130#3622—82no—
131C4×#127—171no—
132#1831—172no—
133#1270—171no—
134#1270—172no—
135#1270D1711yescharacteristic, complemented
136C22×C17—42no—
137C22×C17—44no—
138C22×C17—81no—
139C22×C17—81no—
140C22×C17—82no—
141C22×C17#40711yescharacteristic
142C22×C17#41011yescharacteristic
143C22×C17C2×#10011yescharacteristic
144C22×C17C2×#10212yes—
145C22×C17#42112yes—
146C2×D17—42no—
147C2×D17—44no—
148C2×D17—81no—
149C2×D17—81no—
150C2×D17—82no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—22no—
155C4×C17—81no—
156Dic17—24no—
157Dic17—41no—
158Dic17—81no—
159Dic17#33114yes—
160C2×#127—171no—
161C2×C4×D4—171no—
162C2×C4×D4—171no—
163C2×C4×D4—342no—
164C2×C4⋊C8—342no—
165#390—171no—
166#390—342no—
167#390C2×D1711yescharacteristic
168#384—171no—
169C43—171no—
170#336—172no—
171#336—342no—
172#336C2×D1712yes—
173C2×C17—82no—
174C2×C17#146611yescharacteristic
175C2×C17#148612yes—
176C2×C17#151712yes—
177C2×C17#153611yescharacteristic
178C2×C17C2×#33111yescharacteristic
179D17—82no—
180C22×D4—171no—
181C22×D4—342no—
182C22×D4C4×D1711yescharacteristic
183C23×C4—341no—
184C23×C4—341no—
185C22×C8—22no—
186C22×C8—342no—
187#127—342no—
188#127—342no—
189C4×D4—684no—
190C4×D4—684no—
191C4×D4—682no—
192C4×D4—682no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4C4×D1711yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—341no—
198C2×C22⋊C4—341no—
199C2×C22⋊C4—342no—
200C2×C42—172no—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42—172no—
204C2×C42C22×D1711yescharacteristic
205C4⋊C8—684no—
206C17#765111yescharacteristic, complemented, Sylow
207C24—21no—
208C24—341no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—172no—
213C2×D4—348no—
214C2×D4—341no—
215C2×D4—681no—
216C2×D4—681no—
217C2×D4—684no—
218C2×D4—684no—
219C2×D4#6321812yes—
220C22×C4—21no—
221C22×C4—171no—
222C22×C4—172no—
223C22×C4—171no—
224C22×C4—341no—
225C22×C4—684no—
226C22×C4—682no—
227C22×C4—684no—
228C22×C4—682no—
229C22×C4C2×C4×D1711yescharacteristic
230C22×C4D4×D1712yes—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—682no—
235C2×C8—682no—
236C2×C8—682no—
237C4⋊C4—41no—
238C4⋊C4—681no—
239C4⋊C4—684no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—682no—
243C22⋊C4—682no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C42—41no—
247C42—174no—
248C42—174no—
249C42—342no—
250C42—341no—
251C42—341no—
252C42—682no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—342no—
259C23—681no—
260C23—681no—
261C23—682no—
262C23—682no—
263C23C22⋊C4×D1711yescharacteristic
264D4—42no—
265D4—42no—
266D4—682no—
267D4—682no—
268D4—6816no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—172no—
276C2×C4—174no—
277C2×C4—174no—
278C2×C4—342no—
279C2×C4—342no—
280C2×C4—341no—
281C2×C4—342no—
282C2×C4—681no—
283C2×C4—682no—
284C2×C4—1364no—
285C2×C4—1364no—
286C2×C4C2×#6321811yescharacteristic
287C2×C4M16×D1711yescharacteristic
288C2×C4D8×D1712yes—
289C2×C4SD16×D1712yes—
290C8—82no—
291C8—1362no—
292C22—42no—
293C22—44no—
294C22—82no—
295C22—81no—
296C22—81no—
297C22—682no—
298C22—684no—
299C22—1362no—
300C22—1361no—
301C22—1361no—
302C22#1176277111yescharacteristic
303C22#1176278111yescharacteristic
304C22#100×D1711yescharacteristic
305C22#102×D1712yes—
306C22#1176282412yes—
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—174no—
313C4—344no—
314C4—681no—
315C4—1361no—
316C2—82no—
317C2—1362no—
318C2#1452712311yescharacteristic
319C2#1452720712yes—
320C2#1452734212yes—
321C2#1452741711yescharacteristic
322C2D17×#33111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1270C2
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587626C4
cosocleC2×C42×C17C23
hypercenter#1270D17
hypocenterC17#7651
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #14528899C2semidirect
3. #14526605C2semidirect
4. #14526604C2semidirect
5. C17×#1270C2semidirect
135. #1270D17semidirect
206. C17#7651semidirect

Maximal subgroups

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

subgroupindex
#145313482
#145288992
#145266052
#145266042
C17×#12702
#765117