Finite Group Databasebeta

← #17950230

Subgroups — #17950230

Order 4352.

400 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 84 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#14544712C211yesmaximal, characteristic, complemented
2#14544712C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543600C212yesmaximal
5#14537148C211yesmaximal, characteristic
6#11763675—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587565—21no—
13#11762712C411yescharacteristic
14#11762712C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762707C411yescharacteristic
17#11762705C411yescharacteristic
18#11762703—21no—
19#11762702—21no—
20#11762697—21no—
21C42×Dic17C2211yescharacteristic
22C17×#390C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C22⋊C4×C17—21no—
28C2×C22⋊C4×C17—21no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587565—44no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49C24×C17—21no—
50C2×D4×C17—22no—
51C2×D4×C17—41no—
52C2×D4×C17—41no—
53C22×C4×C17—21no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C22×Dic17—21no—
58C22×Dic17—21no—
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C2×D411yescharacteristic
62C22×Dic17C2×D411yescharacteristic
63C2×#3624—42no—
64C2×#3624—42no—
65C2×#3624—42no—
66C2×#3624—42no—
67C2×#3624—42no—
68C2×#3624—42no—
69C4⋊C4×C17—22no—
70C22⋊C4×C17—42no—
71C22⋊C4×C17—42no—
72C42×C17—41no—
73C22⋊Dic17—42no—
74C22⋊Dic17—42no—
75C22⋊Dic17—42no—
76C22⋊Dic17—42no—
77C4⋊Dic17—42no—
78C4⋊Dic17—41no—
79C4⋊Dic17—41no—
80#63225—42no—
81#63225—42no—
82C4×Dic17—22no—
83C4×Dic17—22no—
84C4×Dic17—22no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C4×Dic17—41no—
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—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17#13112yes—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17#13512yes—
127C2×Dic17AGL(1,ℤ/8)12yes—
128#3624—82no—
129#3624—82no—
130#2860—171no—
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—41no—
151C4×C17—41no—
152C4×C17—81no—
153Dic17—22no—
154Dic17—22no—
155Dic17—41no—
156Dic17—41no—
157Dic17—81no—
158Dic17—81no—
159C2×#126—341no—
160#390—171no—
161#390—171no—
162#390#59511yescharacteristic
163#389—171no—
164#389—171no—
165#389—171no—
166#388—341no—
167#388—341no—
168#388—341no—
169C43—171no—
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×#59511yescharacteristic
185C22×C8—342no—
186C22×C8—342no—
187#126—684no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4C2×#59511yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—341no—
197C2×C22⋊C4—341no—
198C2×C22⋊C4—341no—
199C2×C22⋊C4—341no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42C2×#59511yescharacteristic
207#95—341no—
208#95—341no—
209#95—341no—
210C17#772611yescharacteristic, complemented, Sylow
211C24—21no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4C22⋊#59511yescharacteristic
223C22×C4C22⋊#59511yescharacteristic
224C22×C4C22×#59511yescharacteristic
225C2×C8—682no—
226C2×C8—682no—
227C2×C8—682no—
228C2×C8—682no—
229C2×C8—682no—
230C2×C8—682no—
231C4⋊C4—22no—
232C4⋊C4—681no—
233C4⋊C4—681no—
234C4⋊C4—682no—
235C4⋊C4—682no—
236C4⋊C4—682no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—682no—
240C22⋊C4—682no—
241C22⋊C4—682no—
242C22⋊C4—682no—
243C42—41no—
244C42—342no—
245C42—342no—
246C42—342no—
247C42—681no—
248C42—681no—
249C42—681no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊#59511yescharacteristic
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—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—172no—
271C2×C4—172no—
272C2×C4—341no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4#1058767812yes—
289C8—1362no—
290C8—1362no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1176378311yescharacteristic
299C22#1176378811yescharacteristic
300C22#1176379611yescharacteristic
301C22#1176379911yescharacteristic
302C22#1176380111yescharacteristic
303C22#1176380611yescharacteristic
304C22#1176380711yescharacteristic
305C4—22no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—342no—
310C4—342no—
311C4—681no—
312C4—681no—
313C4—1361no—
314C4—1361no—
315C2—161no—
316C2#1454406511yescharacteristic
317C2#1454409611yescharacteristic
318C2#1454414411yescharacteristic
319C2#1454420311yescharacteristic
320C2#1454422611yescharacteristic
321C2#1454426911yescharacteristic
322C2#1454430811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#7726
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #14544712C2semidirect
2. #14544712C2semidirect
3. #14543602C2semidirect
210. C17#7726semidirect

Maximal subgroups

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

subgroupindex
#145447122
#145436022
#145436002
#145371482
#772617