Finite Group Databasebeta

← #17949713

Subgroups — #17949713

Order 4352.

500 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 106 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#14544499C211yesmaximal, characteristic, complemented
3#14543603C212yesmaximal, complemented
4#14543600C212yesmaximal
5#14531246C211yesmaximal, characteristic
6C2×C4×#3624—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×D4×Dic17C411yescharacteristic
13C2×#10587563—21no—
14#11762712C2211yescharacteristic
15#11762707C411yescharacteristic
16#11762705C2211yescharacteristic
17#11762703—21no—
18#11762697—21no—
19C4⋊C4×Dic17C411yescharacteristic
20C22⋊C4×Dic17—21no—
21C42×Dic17C2211yescharacteristic
22C17×#390C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×Dic17—21no—
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C22⋊C4×C17—21no—
29C2×C22⋊C4×C17—21no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34D4×Dic17—22no—
35D4×Dic17C814yescomplemented
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17C2311yescharacteristic
47#10587563—44no—
48C4×#3624—24no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52C24×C17—21no—
53C2×D4×C17—21no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17C2×C812yes—
57C22×C4×C17—21no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—21no—
62C22×Dic17—24no—
63C22×Dic17—21no—
64C22×Dic17—42no—
65C22×Dic17—42no—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C2×D411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C2×#3624—24no—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C42×C17—41no—
80C22⋊Dic17—24no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C4⋊Dic17—21no—
84C4⋊Dic17—41no—
85C4⋊Dic17C2×C812yes—
86#63225—42no—
87#63225—42no—
88C4×Dic17—22no—
89C4×Dic17—22no—
90C4×Dic17—22no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93C4×Dic17—42no—
94C4×Dic17C2×C812yes—
95C4×Dic17M1612yes—
96#7580—171nomaximal, Sylow
97C23×C17—22no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17C2×C22⋊C411yescharacteristic
105D4×C17—42no—
106D4×C17—42no—
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—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17#10412yes—
118C2×C4×C17C22×C811yescharacteristic
119C2×C4×C17C2×M1611yescharacteristic
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—22no—
129C2×Dic17—48no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17C22⋊C812yes—
136C2×Dic17C22⋊C812yes—
137C2×Dic17#13512yes—
138C2×Dic17AGL(1,ℤ/8)12yes—
139#3624—44no—
140#3624—82no—
141#2293—171no—
142#1839—171no—
143#1744—171no—
144#1271—172no—
145#1270—172no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—81no—
150C22×C17—81no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17C2×C22⋊C811yescharacteristic
154C22×C17#41011yescharacteristic
155C22×C17#41311yescharacteristic
156C22×C17#41611yescharacteristic
157C22×C17#41811yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C22×C17#42111yescharacteristic
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—22no—
163C4×C17—41no—
164C4×C17—81no—
165Dic17—22no—
166Dic17—22no—
167Dic17—22no—
168Dic17—22no—
169Dic17—42no—
170Dic17—81no—
171C2×C4×D4—171no—
172C2×#117—341no—
173C2×C4×C8—341no—
174#390—171no—
175#390#59511yescharacteristic
176#389—171no—
177#389—171no—
178#388—341no—
179#388—341no—
180C4×C4⋊C4—171no—
181C4×C22⋊C4—341no—
182C43—171no—
183#336—172no—
184#336—172no—
185#336—341no—
186#336—341no—
187#336—341no—
188C2×C17—82no—
189C2×C17#145811yescharacteristic
190C2×C17#147611yescharacteristic
191C2×C17#147811yescharacteristic
192C2×C17#150511yescharacteristic
193C2×C17#152111yescharacteristic
194C2×C17#153211yescharacteristic
195C2×C17#155311yescharacteristic
196C22×D4C2×#59511yescharacteristic
197C23×C4—341no—
198C22×C8—342no—
199C22×C8—342no—
200C4×D4—174no—
201C4×D4—342no—
202#117—684no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4C2×#59511yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
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×C42—341no—
220C2×C42C2×#59511yescharacteristic
221C4×C8—344no—
222#95—341no—
223#95—341no—
224#95—341no—
225C17#758011yescharacteristic, complemented, Sylow
226C24—21no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4#6324212yes—
231C22×C4—21no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—344no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—682no—
240C22×C4—682no—
241C22×C4C22⋊#59511yescharacteristic
242C22×C4C22⋊#59511yescharacteristic
243C22×C4C22×#59511yescharacteristic
244C2×C8—344no—
245C2×C8—682no—
246C2×C8—682no—
247C2×C8—682no—
248C2×C8—682no—
249C2×C8—682no—
250C4⋊C4—41no—
251C4⋊C4—172no—
252C4⋊C4—341no—
253C4⋊C4—681no—
254C4⋊C4—682no—
255C4⋊C4—682no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—344no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C42—41no—
262C42—172no—
263C42—172no—
264C42—342no—
265C42—342no—
266C42—342no—
267C42—681no—
268C42—681no—
269C42—682no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊#59511yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—172no—
291C2×C4—172no—
292C2×C4—172no—
293C2×C4—172no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—342no—
303C2×C4—688no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4#1058754812yes—
310C2×C4C2×#6324211yescharacteristic
311C2×C4D17⋊M1611yescharacteristic
312C8—684no—
313C8—1362no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#1176377711yescharacteristic
322C22#1176378911yescharacteristic
323C22#1176379611yescharacteristic
324C22#1176379911yescharacteristic
325C22#1176380111yescharacteristic
326C22C2×#1058754811yescharacteristic
327C22#1176380611yescharacteristic
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—81no—
333C4—342no—
334C4—342no—
335C4—342no—
336C4—342no—
337C4—682no—
338C4—1361no—
339C2—82no—
340C2#1454401611yescharacteristic
341C2#1454407611yescharacteristic
342C2#1454408411yescharacteristic
343C2#1454417311yescharacteristic
344C2#1454422511yescharacteristic
345C2#1454426011yescharacteristic
346C2#1454430811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#7580
LCSG ▷ C2×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. #14544499C2semidirect
3. #14543603C2semidirect
35. D4×Dic17C8semidirect
225. C17#7580semidirect

Maximal subgroups

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

subgroupindex
#145447122
#145444992
#145436032
#145436002
#145312462
#758017