Finite Group Databasebeta

← #16922986

Subgroups — #16922986

Order 4352.

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

normal subgroups only

This group has 400 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#14528927C211yesmaximal, characteristic, complemented
2#14528927C211yesmaximal, characteristic, complemented
3#14526607C212yesmaximal, complemented
4#14526605C212yesmaximal
5C17×#2860C211yesmaximal, characteristic
6#11762564—21no—
7#11762564—21no—
8#11762564—21no—
9#11762564—21no—
10#11762564C2212yes—
11#11762564C2212yes—
12C2×C17×#126—21no—
13C17×#390C411yescharacteristic
14C17×#390C411yescharacteristic
15C17×#390C2211yescharacteristic
16C17×#389C411yescharacteristic
17C17×#389C411yescharacteristic
18C17×#389C2211yescharacteristic
19C17×#388—21no—
20C17×#388—21no—
21C17×#388—21no—
22C43×C17C2211yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C17×#126—44no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C42×C17C2×C411yescharacteristic
38C2×C42×C17C2×C411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C42×C17C2311yescharacteristic
44C22×#3622—22no—
45C22×#3622—22no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C24×C17—21no—
50C2×D4×C17—22no—
51C2×D4×C17—41no—
52C2×D4×C17—41no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C4⋊C4×C17—22no—
64C4⋊C4×C17—41no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—42no—
67C4⋊C4×C17—42no—
68C4⋊C4×C17—42no—
69C22⋊C4×C17—42no—
70C22⋊C4×C17—42no—
71C22⋊C4×C17—42no—
72C22⋊C4×C17—42no—
73C22⋊C4×C17—42no—
74C22⋊C4×C17—42no—
75C42×C17—22no—
76C42×C17—22no—
77C42×C17—22no—
78C42×C17—41no—
79C42×C17—41no—
80C42×C17—41no—
81C42×C17—41no—
82C2×#3622—42no—
83C2×#3622—42no—
84C2×#3622—42no—
85C2×#3622—42no—
86C2×#3622—42no—
87C2×#3622—42no—
88#7726—171nomaximal, Sylow
89C23×C17—41no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17C2×C22⋊C411yescharacteristic
97D4×C17—82no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
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—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17#13112yes—
126C2×C4×C17#13512yes—
127C2×C4×C17AGL(1,ℤ/8)12yes—
128#3622—82no—
129#3622—82no—
130#2860D1711yescharacteristic
131#1839—171no—
132#1839—171no—
133#1271—172no—
134#1270—172no—
135C22×C17—81no—
136C22×C17—81no—
137C22×C17—81no—
138C22×C17—81no—
139C22×C17—81no—
140C22×C17—81no—
141C22×C17—81no—
142C22×C17#40811yescharacteristic
143C22×C17#41011yescharacteristic
144C22×C17#41311yescharacteristic
145C22×C17#41611yescharacteristic
146C22×C17#41811yescharacteristic
147C22×C17#42111yescharacteristic
148C22×C17#42111yescharacteristic
149C4×C17—22no—
150C4×C17—22no—
151C4×C17—22no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—81no—
157C4×C17—81no—
158C4×C17—81no—
159C2×#126—21no—
160#390Dic1711yescharacteristic
161#390Dic1711yescharacteristic
162#390C2×D1711yescharacteristic
163#389Dic1711yescharacteristic
164#389Dic1711yescharacteristic
165#389C2×D1711yescharacteristic
166#388—21no—
167#388—21no—
168#388—21no—
169C43C2×D1711yescharacteristic
170#336—172no—
171#336—172no—
172#336—341no—
173#336—341no—
174#336—341no—
175#336—341no—
176C2×C17—161no—
177C2×C17#147311yescharacteristic
178C2×C17#148211yescharacteristic
179C2×C17#149611yescharacteristic
180C2×C17#151411yescharacteristic
181C2×C17#152111yescharacteristic
182C2×C17#153511yescharacteristic
183C2×C17#155311yescharacteristic
184C22×D4C2×Dic1711yescharacteristic
185C22×C8—342no—
186C22×C8—342no—
187#126—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic1711yescharacteristic
192C2×C4⋊C4C2×Dic1711yescharacteristic
193C2×C4⋊C4C2×Dic1711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42C2×Dic1711yescharacteristic
201C2×C42C2×Dic1711yescharacteristic
202C2×C42C22⋊D1711yescharacteristic
203C2×C42C22⋊D1711yescharacteristic
204C2×C42C22⋊D1711yescharacteristic
205C2×C42C22⋊D1711yescharacteristic
206C2×C42C22×D1711yescharacteristic
207#95—21no—
208#95—21no—
209#95—21no—
210C17#772611yescharacteristic, complemented, Sylow
211C24—21no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4C22⋊Dic1711yescharacteristic
219C22×C4C22⋊Dic1711yescharacteristic
220C22×C4C22⋊Dic1711yescharacteristic
221C22×C4C22⋊Dic1711yescharacteristic
222C22×C4C22×Dic1711yescharacteristic
223C22×C4C2×C22⋊D1711yescharacteristic
224C22×C4C2×C22⋊D1711yescharacteristic
225C2×C8—682no—
226C2×C8—682no—
227C2×C8—682no—
228C2×C8—682no—
229C2×C8—682no—
230C2×C8—682no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊Dic1711yescharacteristic
258D4—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
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—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4#1058761812yes—
287C2×C4#1058763412yes—
288C2×C4#1058763612yes—
289C8—1362no—
290C8—1362no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1176277611yescharacteristic
299C22#1176278311yescharacteristic
300C22#1176279311yescharacteristic
301C22#1176280511yescharacteristic
302C22#1176281411yescharacteristic
303C22#1176282611yescharacteristic
304C22#1176282611yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C2—161no—
316C2#1452715311yescharacteristic
317C2#1452719211yescharacteristic
318C2#1452725311yescharacteristic
319C2#1452733111yescharacteristic
320C2#1452736111yescharacteristic
321C2#1452741511yescharacteristic
322C2#1452748911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2860C2
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#2860D17
hypocenterC17#7726
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #14528927C2semidirect
2. #14528927C2semidirect
3. #14526607C2semidirect
210. C17#7726semidirect

Maximal subgroups

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

subgroupindex
#145289272
#145266072
#145266052
C17×#28602
#772617