Finite Group Databasebeta

← #17959356

Subgroups — #17959356

Order 4352.

446 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C4×#10587545C211yesmaximal, characteristic, complemented
2#14544046C211yesmaximal, characteristic, complemented
3#14543768C212yesmaximal, complemented
4#14543766C212yesmaximal
5#14533693C211yesmaximal, characteristic
6#11763830—21no—
7C2×#10587545—21no—
8C4×#63247—21no—
9#11763689C2212yes—
10#11763655C2212yes—
11#11763654—21no—
12C2×#10587610—21no—
13C2×C4×Dic34C2211yescharacteristic
14#11762740C411yescharacteristic
15#11762740C2211yescharacteristic
16#11762723C411yescharacteristic
17#11762723C2211yescharacteristic
18#11762697—21no—
19#11762697—21no—
20#11762675C411yescharacteristic
21#11762669—21no—
22C2×C4×D4×C17C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17—21no—
25C22×Dic34D411yescharacteristic
26C2×#63247—22no—
27C4×D4×C17—41no—
28C4×D4×C17—41no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17—21no—
31C2×C42×C17C2×C411yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17D411yescharacteristic
38#10587610—44no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225C2×C411yescharacteristic
42C2×#63225C2311yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C4×Dic34—41no—
46C4×Dic34—41no—
47C22⋊#3624—22no—
48#10587545—24no—
49#10587542—41no—
50#10587542—41no—
51C4⋊#3624—41no—
52C4×#3624—41no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic
57#10587502C2×C411yescharacteristic
58C24×C17—21no—
59C2×D4×C17—41no—
60C2×D4×C17—42no—
61C22×C4×C17—21no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C2×Dic34—22no—
75C2×Dic34—22no—
76C2×Dic34—41no—
77#63247—24no—
78#63247—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C42×C17—21no—
85C42×C17—21no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C4⋊Dic17—41no—
91C4⋊Dic17—42no—
92#63225—41no—
93#63225—41no—
94#63225—42no—
95#63225—42no—
96C4×Dic17—21no—
97C4×Dic17—21no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100#11778—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17C2×C22⋊C411yescharacteristic
109D4×C17—42no—
110D4×C17—42no—
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—41no—
118C2×C4×C17—41no—
119C2×C4×C17—84no—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#10412yes—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17AGL(1,ℤ/8)12yes—
139Dic34—44no—
140#3624—44no—
141#3624—82no—
142#2536—171no—
143C4×#101—171no—
144#1467—171no—
145#1346—172no—
146#1345—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—81no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17—81no—
156C22×C17C2×#9911yescharacteristic
157C22×C17#41811yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—41no—
163C4×C17—81no—
164Dic17—22no—
165Dic17—41no—
166Dic17—41no—
167Dic17—41no—
168Dic17—81no—
169Dic17—81no—
170C2×C22⋊Q8—341no—
171C2×C4×Q8—171no—
172C2×C4×D4#59511yescharacteristic
173#432—341no—
174C2×#101—341no—
175C4×M16—341no—
176#398—171no—
177#398—171no—
178#394—171no—
179#394—171no—
180#388—341no—
181#388—341no—
182#382—171no—
183#380—341no—
184#343—172no—
185#324—172no—
186#324—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—82no—
190C2×C17#150711yescharacteristic
191C2×C17#212011yescharacteristic
192C2×C17#213311yescharacteristic
193C22×Q8—171no—
194C22×D4C2×#59511yescharacteristic
195C23×C4—21no—
196C2×M16—342no—
197C22⋊Q8—684no—
198C4×Q8—681no—
199C4×Q8—681no—
200C4×D4—41no—
201C4×D4—41no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4C2×#59511yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42C2×#59511yescharacteristic
217C4⋊C8—681no—
218#101—344no—
219C22⋊C8—342no—
220#97—681no—
221#97—681no—
222C4×C8—681no—
223#95—171no—
224#95—171no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#1177811yescharacteristic, complemented, Sylow
229C24—21no—
230C2×Q8—342no—
231C2×Q8—342no—
232C2×Q8—681no—
233C2×D4—41no—
234C2×D4—42no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4C22⋊#59511yescharacteristic
246C22×C4C22⋊#59511yescharacteristic
247C22×C4C22×#59511yescharacteristic
248M16—344no—
249M16—682no—
250C2×C8—682no—
251C2×C8—682no—
252C4⋊C4—41no—
253C4⋊C4—681no—
254C4⋊C4—681no—
255C4⋊C4—681no—
256C4⋊C4—682no—
257C4⋊C4—682no—
258C4⋊C4—682no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C22⋊C4—682no—
265C42—21no—
266C42—21no—
267C42—341no—
268C42—341no—
269C42—341no—
270C42—341no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊#59511yescharacteristic
279Q8—684no—
280D4—42no—
281D4—42no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—84no—
291C2×C4—172no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4#1058754812yes—
308C2×C4#1058755111yescharacteristic
309C2×C4#1058755111yescharacteristic
310C8—684no—
311C8—1362no—
312C22—21no—
313C22—21no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×#1058755111yescharacteristic
322C22#1176380111yescharacteristic
323C22C2×#1058754811yescharacteristic
324C4—22no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—81no—
329C4—342no—
330C4—681no—
331C4—681no—
332C4—681no—
333C4—1361no—
334C4—1361no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2#1454418011yescharacteristic
339C2#1454503511yescharacteristic
340C2#1454507911yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×D4×C17C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×#63225C23
hypercenterC2×C4×D4#595
hypocenterC17#11778
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×#10587545C2semidirect
2. #14544046C2semidirect
3. #14543768C2semidirect
228. C17#11778semidirect

Maximal subgroups

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

subgroupindex
C4×#105875452
#145440462
#145437682
#145437662
#145336932
#1177817