Finite Group Databasebeta

← C22×#11763662 #17965207

Subgroups — C22×#11763662 #17965207

Order 4352.

1846 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 1846 conjugacy classes of subgroups — too many to draw. Only its 360 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
1C22×#10587543C211yesmaximal, characteristic
2C22×(C2×#595)⋊C4C211yesmaximal, characteristic
3C2×#11763662C2112yesmaximal, direct product factor
4C22×#10587583C211yesmaximal, characteristic
5C24×#595—21no—
6C23×#3624—21no—
7C2×#10587543—23no—
8C2×#10587543C2216yes—
9C2×(C2×#595)⋊C4—23no—
10C2×(C2×#595)⋊C4C2216yes—
11#11763662C22116yesdirect product factor
12C23×Dic34C411yescharacteristic, complemented
13C23×C4×D17C2211yescharacteristic
14C2×#10587583—23no—
15C2×#10587583—23no—
16C2×#10587583C416yescomplemented
17C2×#10587583C2216yes—
18C22×C4⋊Dic17—21no—
19C22×#63227—21no—
20C22×C4⋊C4×C17C411yescharacteristic, complemented
21C24×D17D411yescharacteristic
22C23×#595—28no—
23C23×#595—26no—
24C23×#595—44no—
25C23×#595—44no—
26C23×C4×C17—21no—
27C23×C4×C17C2×C411yescharacteristic
28C23×Dic17—21no—
29C23×Dic17D411yescharacteristic
30C22×C4×D17—44no—
31C22×C4×D17C2×C413yes—
32C22×C4×D17D413yes—
33C22×C4×D17D413yes—
34C22×C4×D17C2313yes—
35C22×Dic34—23no—
36C22×Dic34—44no—
37C22×Dic34C2×C416yescomplemented
38C22×#3624—26no—
39C22×#3624—44no—
40C2×C4⋊C4×C17—23no—
41C2×C4⋊C4×C17C2×C416yescomplemented
42C2×#63227—26no—
43C2×#63227—26no—
44C2×C4⋊Dic17—26no—
45C2×C4⋊Dic17—26no—
46#10587583—212no—
47#10587583—26no—
48#10587583—26no—
49#10587583C2×C4112yescomplemented
50#10587583C2314yes—
51(C2×#595)⋊C4—26no—
52(C2×#595)⋊C4C2314yes—
53#10587543—26no—
54#10587543C2314yes—
55C24×C17C22⋊C411yescharacteristic
56C23×D17—24no—
57C23×D17—24no—
58C23×D17C22⋊C413yes—
59C23×D17SD1614yes—
60C23×D17Q1614yes—
61C23×D17C2×D413yes—
62C22×#595—224no—
63C22×#595—24no—
64C22×#595—432no—
65C22×#595—412no—
66C22×#595—412no—
67C22×C4×C17—24no—
68C22×C4×C17—26no—
69C22×C4×C17—44no—
70C22×C4×C17C22⋊C413yes—
71C22×C4×C17C22×C413yes—
72C22×Dic17—24no—
73C22×Dic17—26no—
74C22×Dic17—44no—
75C22×Dic17C22⋊C413yes—
76C22×Dic17C2×D413yes—
77C2×C4×D17—46no—
78C2×C4×D17—46no—
79C2×C4×D17—46no—
80C2×C4×D17—46no—
81C2×C4×D17C22⋊C416yes—
82C2×C4×D17C22×C413yes—
83C2×C4×D17C2×D413yes—
84C2×C4×D17C2×D413yes—
85C2×C4×D17C2411yescharacteristic
86C2×Dic34—26no—
87C2×Dic34—412no—
88C2×Dic34—412no—
89C2×Dic34C22×C414yescomplemented
90C2×#3624—24no—
91C2×#3624—412no—
92C4⋊C4×C17—26no—
93C4⋊C4×C17C22×C414yescomplemented
94#63227—212no—
95#63227—24no—
96C4⋊Dic17—212no—
97C4⋊Dic17—24no—
98C22×#328—171nomaximal, Sylow
99C23×C17#9914yes—
100C23×C17#10314yes—
101C23×C17#10414yes—
102C23×C17C2×C22⋊C413yes—
103C22×D17—216no—
104C22×D17—216no—
105C22×D17—26no—
106C22×D17—26no—
107C22×D17—26no—
108C22×D17—26no—
109C22×D17#103112yes—
110C22×D17C2×C22⋊C413yes—
111C22×D17C2×SD1616yes—
112C22×D17C2×Q1616yes—
113C22×D17C22×D411yescharacteristic
114C2×#595—28no—
115C2×#595—44no—
116C2×#595—448no—
117C2×#595—44no—
118C2×C4×C17—26no—
119C2×C4×C17—26no—
120C2×C4×C17—24no—
121C2×C4×C17—412no—
122C2×C4×C17C2×C22⋊C413yes—
123C2×C4×C17C23×C411yescharacteristic
124C2×Dic17—26no—
125C2×Dic17—26no—
126C2×Dic17—24no—
127C2×Dic17—412no—
128C2×Dic17C2×C22⋊C413yes—
129C2×Dic17C22×D411yescharacteristic
130C4×D17—46no—
131C4×D17—43no—
132C4×D17—43no—
133C4×D17—43no—
134C4×D17—41no—
135Dic34—412no—
136Dic34—44no—
137#3624—44no—
138C22×C22⋊Q8—171no—
139C22×C22⋊C8—171no—
140C22×#95—171no—
141C2×#328—1712no—
142C22×C17#328116yescomplemented
143C22×C17C2×#9916yes—
144C22×C17C2×#10316yes—
145C22×C17C2×#10416yes—
146C22×C17C22×C22⋊C411yescharacteristic
147C2×D17—21no—
148C2×D17—23no—
149C2×D17—21no—
150C2×D17—23no—
151C2×D17—212no—
152C2×D17—224no—
153C2×D17—212no—
154C2×D17C2×#10316yes—
155C2×D17C22×SD1611yescharacteristic
156C2×D17C22×Q1611yescharacteristic
157#595—48no—
158C4×C17—23no—
159C4×C17—21no—
160C4×C17—44no—
161Dic17—21no—
162Dic17—23no—
163Dic17—44no—
164C23×Q8—171no—
165C24×C4—171no—
166C24×C4—341no—
167C23×C8—341no—
168C2×C22⋊Q8—176no—
169C2×C22⋊Q8—176no—
170C2×C22⋊Q8—343no—
171C2×C22⋊Q8—343no—
172C22×C4⋊C4—341no—
173C22×C4⋊C4#59511yescharacteristic, complemented
174C22×C22⋊C4—341no—
175C2×C22⋊C8—176no—
176C2×C22⋊C8—343no—
177C2×#95—176no—
178C2×#95—343no—
179#328—1716no—
180C2×C17C2×#328112yescomplemented
181C2×C17C22×#9911yescharacteristic
182C2×C17C22×#10311yescharacteristic
183C2×C17C22×#10411yescharacteristic
184D17—21no—
185D17—21no—
186D17—26no—
187C25—171no—
188C22×Q8—176no—
189C22×Q8—343no—
190C22×Q8—684no—
191C23×C4—21no—
192C23×C4—171no—
193C23×C4—173no—
194C23×C4—173no—
195C23×C4—173no—
196C23×C4—173no—
197C23×C4—346no—
198C23×C4—348no—
199C23×C4—341no—
200C23×C4—684no—
201C23×C4—684no—
202C23×C4—684no—
203C23×C4C2×#59511yescharacteristic
204C22×C8—346no—
205C22×C8—684no—
206C22⋊Q8—174no—
207C22⋊Q8—1712no—
208C22⋊Q8—3412no—
209C22⋊Q8—346no—
210C22⋊Q8—346no—
211C2×C4⋊C4—23no—
212C2×C4⋊C4—346no—
213C2×C4⋊C4—346no—
214C2×C4⋊C4C2×#59516yescomplemented
215C2×C22⋊C4—346no—
216C2×C22⋊C4—346no—
217C22⋊C8—174no—
218C22⋊C8—346no—
219#95—174no—
220#95—346no—
221C17C22×#32811yescharacteristic, complemented, Sylow
222C24—173no—
223C24—173no—
224C24—174no—
225C24—174no—
226C24—344no—
227C24—344no—
228C24C22⋊#59511yescharacteristic
229C2×Q8—174no—
230C2×Q8—346no—
231C2×Q8—6812no—
232C2×Q8—6812no—
233C22×C4—24no—
234C22×C4—26no—
235C22×C4—44no—
236C22×C4—173no—
237C22×C4—173no—
238C22×C4—176no—
239C22×C4—173no—
240C22×C4—173no—
241C22×C4—173no—
242C22×C4—171no—
243C22×C4—344no—
244C22×C4—344no—
245C22×C4—3424no—
246C22×C4—346no—
247C22×C4—686no—
248C22×C4—686no—
249C22×C4—686no—
250C22×C4—686no—
251C22×C4—6812no—
252C22×C4—6812no—
253C22×C4—6832no—
254C22×C4—684no—
255C22×C4C22⋊#59513yes—
256C22×C4C22×#59513yes—
257C2×C8—344no—
258C2×C8—6812no—
259C4⋊C4—26no—
260C4⋊C4—344no—
261C4⋊C4—3412no—
262C4⋊C4C22×#59514yescomplemented
263C22⋊C4—344no—
264C22⋊C4—3412no—
265C23—176no—
266C23—1712no—
267C23—176no—
268C23—171no—
269C23—173no—
270C23—346no—
271C23—346no—
272C23—346no—
273C23—346no—
274C23—3416no—
275C23—3416no—
276C23#1058753914yes—
277C23#1058754814yes—
278C23#1058754914yes—
279C23C2×C22⋊#59513yes—
280Q8—684no—
281Q8—6812no—
282C2×C4—26no—
283C2×C4—26no—
284C2×C4—24no—
285C2×C4—412no—
286C2×C4—171no—
287C2×C4—173no—
288C2×C4—346no—
289C2×C4—346no—
290C2×C4—348no—
291C2×C4—344no—
292C2×C4—683no—
293C2×C4—683no—
294C2×C4—686no—
295C2×C4—681no—
296C2×C4—683no—
297C2×C4—684no—
298C2×C4—6848no—
299C2×C4—684no—
300C2×C4—6812no—
301C2×C4C2×C22⋊#59513yes—
302C2×C4C23×#59511yescharacteristic
303C8—684no—
304C22—171no—
305C22—171no—
306C22—176no—
307C22—3412no—
308C22—3424no—
309C22—3412no—
310C22—341no—
311C22—343no—
312C22—341no—
313C22—343no—
314C22#11763662116yesdirect product factor
315C22C2×#1058753916yes—
316C22C2×#1058754916yes—
317C22C2×#1058754816yes—
318C22C22×C22⋊#59511yescharacteristic
319C4—23no—
320C4—21no—
321C4—44no—
322C4—341no—
323C4—343no—
324C4—688no—
325C4—684no—
326C2—341no—
327C2—341no—
328C2—346no—
329C2C2×#11763662112yesdirect product factor
330C2C22×#1058753911yescharacteristic
331C2C22×#1058754911yescharacteristic
332C2C22×#1058754811yescharacteristic
subgroupG/H
centerC24C22⋊#595
commutatorC2×C4×C17C23×C4
FrattiniC2×C4C23×#595
FittingC22×C4⋊C4×C17C4
socleC24×C17C22⋊C4
prime radicalC24×D17D4
cosocleC2×C4×D17C24
hypercenterC22×C4⋊C4#595
hypocenterC17C22×#328
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#11763662C2direct (N × H)
11. #11763662C22direct (N × H)
12. C23×Dic34C4semidirect
16. C2×#10587583C4semidirect
20. C22×C4⋊C4×C17C4semidirect
37. C22×Dic34C2×C4semidirect
41. C2×C4⋊C4×C17C2×C4semidirect
49. #10587583C2×C4semidirect
89. C2×Dic34C22×C4semidirect
93. C4⋊C4×C17C22×C4semidirect
142. C22×C17#328semidirect
173. C22×C4⋊C4#595semidirect
180. C2×C17C2×#328semidirect
214. C2×C4⋊C4C2×#595semidirect
221. C17C22×#328semidirect
262. C4⋊C4C22×#595semidirect
314. C22#11763662direct (N × H)
329. C2C2×#11763662direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875432
C22×(C2×#595)⋊C42
C2×#117636622
C22×#105875832
C22×#32817