Finite Group Databasebeta

← #17950255

Subgroups — #17950255

Order 4352.

420 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 94 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#14544700C211yesmaximal, characteristic, complemented
2#14544699C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543601C212yesmaximal
5#14537152C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×C4⋊#3624—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587584—21no—
13#11762711C411yescharacteristic
14#11762710C411yescharacteristic
15#11762710C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762705C411yescharacteristic
18#11762701—21no—
19#11762699—21no—
20#11762698—21no—
21C42×Dic17C2211yescharacteristic
22C17×#389C411yescharacteristic
23C22×D68C2×C411yescharacteristic
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×#63227—21no—
31C2×#63227—21no—
32C2×#63227—21no—
33C2×#63227—21no—
34C2×#63227—21no—
35C2×#63227—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587584—42no—
46#10587584—42no—
47#10587544—42no—
48C4⋊#3624—42no—
49C17×#95—21no—
50#10587502—21no—
51#10587502—21no—
52C23×D17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22×C411yescharacteristic
58C22×Dic17—21no—
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C2×D411yescharacteristic
62C22×Dic17C2×D411yescharacteristic
63C2×D68—22no—
64C2×D68—41no—
65C2×D68—41no—
66C2×#3624—42no—
67C2×#3624—42no—
68C2×#3624—42no—
69C2×#3624—42no—
70C2×#3624—42no—
71C2×#3624—42no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—42no—
75C42×C17—22no—
76#63227—42no—
77#63227—42no—
78#63227—42no—
79#63227—42no—
80#63227—41no—
81#63227—41no—
82#63227—41no—
83#63227—41no—
84C4⋊Dic17—41no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87#63225—42no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93C4×Dic17—41no—
94C4×Dic17D812yes—
95C4×Dic17SD1612yes—
96#7732—171nomaximal, Sylow
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—41no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17—41no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17—41no—
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—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17#10212yes—
119C2×C4×C17#13112yes—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17#10212yes—
133C2×Dic17C2×D811yescharacteristic
134C2×Dic17C2×SD1611yescharacteristic
135C2×Dic17AGL(1,ℤ/8)11yescharacteristic
136C2×Dic17#13711yescharacteristic
137D68—82no—
138#3624—82no—
139#3624—82no—
140#2860—171no—
141#1831—171no—
142#1831—171no—
143#1271—172no—
144#1270—172no—
145C22×C17#40811yescharacteristic
146C22×C17#41011yescharacteristic
147C22×C17#41311yescharacteristic
148C22×C17C2×#10211yescharacteristic
149C22×C17#41711yescharacteristic
150C22×C17#42111yescharacteristic
151C22×C17#42111yescharacteristic
152C2×D17—81no—
153C2×D17—81no—
154C2×D17—81no—
155C2×D17—81no—
156C2×D17—81no—
157C2×D17—81no—
158C2×D17—81no—
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—41no—
162C4×C17—81no—
163C4×C17—81no—
164Dic17—22no—
165Dic17—21no—
166Dic17—21no—
167Dic17—21no—
168Dic17—21no—
169Dic17—41no—
170Dic17—81no—
171C2×#126—341no—
172C2×C4⋊C8—341no—
173C2×C4⋊C8—341no—
174#390—171no—
175#390—171no—
176#390—171no—
177#389—171no—
178#389—171no—
179#389#59511yescharacteristic
180#388—341no—
181#388—341no—
182#388—341no—
183C43—171no—
184#336—172no—
185#336—172no—
186#336—341no—
187#336—341no—
188C2×C17#147311yescharacteristic
189C2×C17#148011yescharacteristic
190C2×C17#149511yescharacteristic
191C2×C17#151211yescharacteristic
192C2×C17#151711yescharacteristic
193C2×C17#153511yescharacteristic
194C2×C17#154711yescharacteristic
195D17—161no—
196C22×D4—171no—
197C22×C8—342no—
198C22×C8—342no—
199#126—682no—
200#126—682no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4C2×#59511yescharacteristic
206C2×C4⋊C4C2×#59511yescharacteristic
207C2×C22⋊C4—341no—
208C2×C22⋊C4—341no—
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C42—171no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42—171no—
217C2×C42—171no—
218C2×C42—171no—
219C2×C42C2×#59511yescharacteristic
220C4⋊C8—682no—
221C4⋊C8—682no—
222#95—21no—
223#95—341no—
224#95—341no—
225C17#773211yescharacteristic, complemented, Sylow
226C24—341no—
227C2×D4—342no—
228C2×D4—681no—
229C2×D4—681no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—341no—
237C22×C4C22⋊#59511yescharacteristic
238C22×C4C22⋊#59511yescharacteristic
239C22×C4C22×#59511yescharacteristic
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—682no—
243C2×C8—682no—
244C2×C8—682no—
245C2×C8—682no—
246C4⋊C4—22no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—681no—
250C4⋊C4—682no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C22⋊C4—682no—
254C22⋊C4—681no—
255C22⋊C4—682no—
256C22⋊C4—681no—
257C22⋊C4—681no—
258C22⋊C4—681no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C42—22no—
262C42—172no—
263C42—172no—
264C42—341no—
265C42—341no—
266C42—681no—
267C42—681no—
268C42—681no—
269C42—681no—
270C23—681no—
271C23—681no—
272C23—681no—
273C23—681no—
274C23—681no—
275C23—681no—
276C23—681no—
277C23C2×C22⋊#59511yescharacteristic
278D4—1362no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—171no—
293C2×C4—172no—
294C2×C4—171no—
295C2×C4—171no—
296C2×C4—171no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4#1058754712yes—
310C2×C4#1058767812yes—
311C8—1362no—
312C8—1362no—
313C22—1361no—
314C22—1361no—
315C22—1361no—
316C22—1361no—
317C22—1361no—
318C22—1361no—
319C22—1361no—
320C22#1176378211yescharacteristic
321C22#1176379011yescharacteristic
322C22#1176379511yescharacteristic
323C22C2×#1058754711yescharacteristic
324C22#1176380011yescharacteristic
325C22#1176380611yescharacteristic
326C22#1176380811yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—341no—
333C4—341no—
334C4—341no—
335C4—341no—
336C4—342no—
337C4—681no—
338C4—1361no—
339C2—2721no—
340C2#1454406811yescharacteristic
341C2#1454408911yescharacteristic
342C2#1454414111yescharacteristic
343C2#1454419811yescharacteristic
344C2#1454421511yescharacteristic
345C2#1454427011yescharacteristic
346C2#1454429511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7732
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #14544700C2semidirect
2. #14544699C2semidirect
3. #14543602C2semidirect
225. C17#7732semidirect

Maximal subgroups

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

subgroupindex
#145447002
#145446992
#145436022
#145436012
#145371522
#773217