Finite Group Databasebeta

← #16945025

Subgroups — #16945025

Order 4352.

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

normal subgroups only

This group has 488 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#14534534C211yesmaximal, characteristic
2#14528867C211yesmaximal, characteristic, complemented
3#14526788C214yesmaximal, complemented
4C17×#1464C211yesmaximal, characteristic, complemented
5C2×#10587560C411yescharacteristic
6C2×#10587616—21no—
7#11762773—21no—
8#11762726—21no—
9#11762726C2212yes—
10#11762718—21no—
11#11762718C412yes—
12#11762711—21no—
13#11762675C411yescharacteristic
14#11762578—21no—
15#11762578C2212yes—
16C2×C17×#117C2211yescharacteristic
17C17×#432—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C23×C4×C17D411yescharacteristic
21C22×C22⋊D17—22no—
22C22×D68—21no—
23C22×D68—21no—
24C22×D68C2×C412yes—
25C17×#117—41no—
26C17×#117—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17D411yescharacteristic
33#10587616—44no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×C4⋊Dic17—21no—
38C2×#63223—22no—
39#10587560—41no—
40#10587560—41no—
41C17×C4⋊C8—41no—
42C17×C22⋊C8—22no—
43C17×#97—41no—
44#10587502—21no—
45#10587502C2×C412yes—
46C22⋊#3622—41no—
47C22⋊#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C23×D17—22no—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—22no—
53C22×C4×C17—41no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17C2×D411yescharacteristic
60C2×C22⋊D17—44no—
61C2×C22⋊D17—82no—
62C2×C22⋊D17—82no—
63C22×Dic17—22no—
64C2×D68—42no—
65C2×D68—42no—
66C2×D68—42no—
67C2×D68—42no—
68C2×D68—41no—
69C2×D68—41no—
70C2×D68—82no—
71C2×D68—82no—
72C2×C8×C17—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—21no—
81#63227—44no—
82#63227—82no—
83#63227—82no—
84C4⋊Dic17—42no—
85#63223—42no—
86#63223—42no—
87C2×#3622—42no—
88#11806—171nomaximal, Sylow
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17C2×C22⋊C411yescharacteristic
95C22×D17—42no—
96C22×D17—42no—
97C22×D17—42no—
98C22×D17—42no—
99C22×D17—42no—
100C22×D17—42no—
101C22×D17—42no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17AGL(1,ℤ/8)12yes—
119C22⋊D17—88no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123D68—82no—
124D68—82no—
125D68—84no—
126D68—84no—
127C8×C17—82no—
128#3622—82no—
129#2619—171no—
130#1822—171no—
131#1464D1711yescharacteristic, complemented
132#1345—174no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—42no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17—41no—
139C22×C17—81no—
140C22×C17C2×#9911yescharacteristic
141C22×C17#41811yescharacteristic
142C22×C17#42111yescharacteristic
143C2×D17—82no—
144C2×D17—82no—
145C2×D17—82no—
146C2×D17—82no—
147C2×D17—82no—
148C2×D17—82no—
149C2×D17—82no—
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157Dic17—82no—
158C2×#127—171no—
159C2×#121—341no—
160C2×#117C2×D1711yescharacteristic
161#432—21no—
162#407—341no—
163#394—172no—
164#394—341no—
165#392—172no—
166#392—341no—
167#390—341no—
168#382—171no—
169#343—172no—
170#343—341no—
171#324C2×D1712yes—
172#323—21no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—81no—
176C2×C17#151011yescharacteristic
177C2×C17#212211yescharacteristic
178C2×C17#213411yescharacteristic
179D17—162no—
180C22×D4—172no—
181C22×D4—342no—
182C22×D4—341no—
183C22×D4—341no—
184C23×C4D6811yescharacteristic
185C2×M16—342no—
186#127—681no—
187#127—681no—
188#121—684no—
189#117—41no—
190#117—41no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4D6811yescharacteristic
193C2×C4⋊C4C22×D1711yescharacteristic
194C2×C22⋊C4—342no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4—342no—
197C2×C22⋊C4C4×D1711yescharacteristic
198C2×C22⋊C4C22⋊D1711yescharacteristic
199C2×C42C4×D1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—41no—
202C22⋊C8—22no—
203C22⋊C8—681no—
204C22⋊C8—681no—
205#97—41no—
206#95—172no—
207#95—341no—
208C17#1180611yescharacteristic, complemented, Sylow
209C24—342no—
210C24#6322711yescharacteristic
211C2×D4—682no—
212C2×D4—682no—
213C2×D4—682no—
214C2×D4—682no—
215C2×D4—684no—
216C2×D4—681no—
217C2×D4—681no—
218C2×D4—1362no—
219C2×D4—1362no—
220C2×D4—1362no—
221C2×D4—1362no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—342no—
227C22×C4#6322711yescharacteristic
228C22×C4#6322711yescharacteristic
229C22×C4#6322711yescharacteristic
230C22×C4C2×C4×D1711yescharacteristic
231C22×C4C2×D6811yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233M16—682no—
234M16—682no—
235C2×C8—42no—
236C2×C8—682no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—682no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—684no—
243C22⋊C4—1362no—
244C22⋊C4—1362no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—682no—
255C23—682no—
256C23—682no—
257C23—682no—
258C23—682no—
259C23—682no—
260C23—682no—
261C23C2×#6322711yescharacteristic
262D4—1362no—
263D4—1362no—
264D4—1364no—
265D4—1364no—
266D4—1368no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4#1058747711yescharacteristic
285C2×C4#1058747711yescharacteristic
286C2×C4#1058759412yes—
287C8—82no—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—1362no—
297C22—1362no—
298C22—1362no—
299C22—1362no—
300C22—1362no—
301C22—1362no—
302C22—1362no—
303C22C2×#1058747711yescharacteristic
304C22#1176281511yescharacteristic
305C22#1176282511yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—1362no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—2722no—
318C2#1452731411yescharacteristic
319C2#1453043311yescharacteristic
320C2#1453047511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1464C2
socleC22×C17C2×#99
prime radical#14534534C2
cosocleC2×C4⋊C4×C17C23
hypercenter#1464D17
hypocenterC17#11806
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
2. #14528867C2semidirect
3. #14526788C2semidirect
4. C17×#1464C2semidirect
131. #1464D17semidirect
208. C17#11806semidirect

Maximal subgroups

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

subgroupindex
#145345342
#145288672
#145267882
C17×#14642
#1180617