Finite Group Databasebeta

← #17950108

Subgroups — #17950108

Order 4352.

444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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#14544501C211yesmaximal, characteristic, complemented
2#14544499C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543600C212yesmaximal
5#14537148C211yesmaximal, characteristic
6C2×#10587542—21no—
7C2×C4×#3624—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#10587542—24no—
47C4×#3624—24no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51C24×C17—21no—
52C2×D4×C17—22no—
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C22×C4×C17—21no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C22×Dic17—21no—
60C22×Dic17—21no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C2×D411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C2×#3624—24no—
66C2×#3624—24no—
67C2×#3624—42no—
68C2×#3624—42no—
69C2×#3624—42no—
70C2×#3624—42no—
71C4⋊C4×C17—22no—
72C22⋊C4×C17—42no—
73C22⋊C4×C17—42no—
74C42×C17—41no—
75C22⋊Dic17—42no—
76C22⋊Dic17—42no—
77C22⋊Dic17—42no—
78C22⋊Dic17—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82#63225—42no—
83#63225—42no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—22no—
87C4×Dic17—22no—
88C4×Dic17—22no—
89C4×Dic17—41no—
90#7692—171nomaximal, Sylow
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17C2×C22⋊C411yescharacteristic
99D4×C17—82no—
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—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17#10412yes—
111C2×C4×C17#13112yes—
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—21no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17#10412yes—
128C2×Dic17#13512yes—
129C2×Dic17AGL(1,ℤ/8)12yes—
130#3624—44no—
131#3624—44no—
132#2860—171no—
133#1745—171no—
134#1744—171no—
135#1271—172no—
136#1270—172no—
137C22×C17—81no—
138C22×C17—81no—
139C22×C17—81no—
140C22×C17—81no—
141C22×C17—81no—
142C22×C17—81no—
143C22×C17—81no—
144C22×C17#40811yescharacteristic
145C22×C17#41011yescharacteristic
146C22×C17#41311yescharacteristic
147C22×C17#41611yescharacteristic
148C22×C17#41811yescharacteristic
149C22×C17C2×#10411yescharacteristic
150C22×C17C2×#10411yescharacteristic
151C4×C17—22no—
152C4×C17—22no—
153C4×C17—41no—
154C4×C17—81no—
155Dic17—22no—
156Dic17—22no—
157Dic17—22no—
158Dic17—41no—
159Dic17—81no—
160Dic17—81no—
161C2×#126—341no—
162C2×#97—341no—
163C2×C4×C8—341no—
164#390—171no—
165#390—171no—
166#390#59511yescharacteristic
167#389—171no—
168#389—171no—
169#389—171no—
170#388—341no—
171#388—341no—
172#388—341no—
173C43—171no—
174#336—172no—
175#336—172no—
176#336—341no—
177#336—341no—
178C2×C17—161no—
179C2×C17#147311yescharacteristic
180C2×C17#148411yescharacteristic
181C2×C17#149411yescharacteristic
182C2×C17#150011yescharacteristic
183C2×C17#150511yescharacteristic
184C2×C17#152411yescharacteristic
185C2×C17#155311yescharacteristic
186C22×D4C2×#59511yescharacteristic
187C22×C8—342no—
188C22×C8—342no—
189#126—684no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4C2×#59511yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—341no—
199C2×C22⋊C4—341no—
200C2×C22⋊C4—341no—
201C2×C22⋊C4—341no—
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42—171no—
207C2×C42—171no—
208C2×C42C2×#59511yescharacteristic
209#97—344no—
210C4×C8—344no—
211#95—341no—
212#95—341no—
213#95—341no—
214C17#769211yescharacteristic, complemented, Sylow
215C24—21no—
216C2×D4—22no—
217C2×D4—41no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—171no—
223C22×C4—171no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4C22⋊#59511yescharacteristic
227C22×C4C22⋊#59511yescharacteristic
228C22×C4C22×#59511yescharacteristic
229C2×C8—344no—
230C2×C8—344no—
231C2×C8—682no—
232C2×C8—682no—
233C2×C8—682no—
234C2×C8—682no—
235C4⋊C4—22no—
236C4⋊C4—681no—
237C4⋊C4—681no—
238C4⋊C4—682no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—682no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C22⋊C4—682no—
247C42—41no—
248C42—342no—
249C42—342no—
250C42—342no—
251C42—342no—
252C42—342no—
253C42—681no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×C22⋊#59511yescharacteristic
262D4—82no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—172no—
274C2×C4—172no—
275C2×C4—172no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4#1058754812yes—
292C2×C4#1058767812yes—
293C8—684no—
294C8—684no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22#1176378311yescharacteristic
303C22#1176378811yescharacteristic
304C22#1176379611yescharacteristic
305C22#1176379911yescharacteristic
306C22#1176380111yescharacteristic
307C22C2×#1058754811yescharacteristic
308C22D17⋊#10411yescharacteristic
309C4—22no—
310C4—22no—
311C4—41no—
312C4—81no—
313C4—342no—
314C4—342no—
315C4—342no—
316C4—681no—
317C4—1361no—
318C4—1361no—
319C2—161no—
320C2#1454406511yescharacteristic
321C2#1454410411yescharacteristic
322C2#1454413611yescharacteristic
323C2#1454415511yescharacteristic
324C2#1454417411yescharacteristic
325C2#1454423511yescharacteristic
326C2#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#7692
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. #14544501C2semidirect
2. #14544499C2semidirect
3. #14543602C2semidirect
214. C17#7692semidirect

Maximal subgroups

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

subgroupindex
#145445012
#145444992
#145436022
#145436002
#145371482
#769217