Finite Group Databasebeta

← C22×#11763712 #17978212

Subgroups — C22×#11763712 #17978212

Order 4352.

1486 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1486 conjugacy classes of subgroups — too many to draw. Only its 240 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×#10587553C211yesmaximal, characteristic
2C22×#10587551C211yesmaximal, characteristic
3C2×#11763712C2112yesmaximal, direct product factor
4C22×#10587572C211yesmaximal, characteristic
5C22×#63247—21no—
6C22×C22⋊#595—21no—
7C2×#10587553—23no—
8C2×#10587553C2216yes—
9C2×#10587551—23no—
10C2×#10587551C2216yes—
11#11763712C22116yesdirect product factor
12C23×C22⋊D17C2211yescharacteristic
13C24×Dic17C411yescharacteristic, complemented
14C2×#10587572—23no—
15C2×#10587572—23no—
16C2×#10587572C416yescomplemented
17C2×#10587572C2216yes—
18C22×C4⋊Dic17—21no—
19C22×#63227—21no—
20C22×C22⋊C4×C17C411yescharacteristic, complemented
21C25×C17C2×C411yescharacteristic
22C24×D17D411yescharacteristic
23C23×#595—41no—
24C23×C4×C17—21no—
25C22×C22⋊D17—41no—
26C22×C22⋊D17—43no—
27C22×C22⋊D17C2×C413yes—
28C22×C22⋊D17D413yes—
29C22×C22⋊D17D413yes—
30C22×C22⋊D17C2313yes—
31C23×Dic17—21no—
32C23×Dic17—23no—
33C23×Dic17—43no—
34C23×Dic17—41no—
35C23×Dic17C2×C416yescomplemented
36C23×Dic17D411yescharacteristic
37C2×C22⋊#595—26no—
38C2×C22⋊#595—43no—
39C2×#63247—26no—
40C2×#63247—43no—
41C22×#3624—41no—
42C2×C22⋊C4×C17—23no—
43C2×C22⋊C4×C17C2×C416yescomplemented
44C2×#63227—26no—
45C2×#63227—26no—
46C2×C4⋊Dic17—26no—
47C2×C4⋊Dic17—26no—
48#10587572—212no—
49#10587572—26no—
50#10587572—26no—
51#10587572C2×C4112yescomplemented
52#10587572C2314yes—
53#10587553—26no—
54#10587553C2314yes—
55#10587551—26no—
56#10587551C2314yes—
57C24×C17—23no—
58C24×C17—21no—
59C24×C17—44no—
60C24×C17C22⋊C413yes—
61C24×C17C22⋊C411yescharacteristic
62C24×C17C22×C413yes—
63C23×D17—21no—
64C23×D17—23no—
65C23×D17—44no—
66C23×D17C22⋊C413yes—
67C23×D17C2×D413yes—
68C22×#595—46no—
69C22×#595—84no—
70C22×C4×C17—26no—
71C22×C4×C17—44no—
72C2×C22⋊D17—43no—
73C2×C22⋊D17—43no—
74C2×C22⋊D17—43no—
75C2×C22⋊D17—43no—
76C2×C22⋊D17—43no—
77C2×C22⋊D17—43no—
78C2×C22⋊D17—43no—
79C2×C22⋊D17—43no—
80C2×C22⋊D17C22⋊C416yes—
81C2×C22⋊D17C22×C413yes—
82C2×C22⋊D17C2×D413yes—
83C2×C22⋊D17C2×D413yes—
84C2×C22⋊D17C2411yescharacteristic
85C22×Dic17—21no—
86C22×Dic17—23no—
87C22×Dic17—26no—
88C22×Dic17—26no—
89C22×Dic17—46no—
90C22×Dic17—46no—
91C22×Dic17—43no—
92C22×Dic17—41no—
93C22×Dic17—46no—
94C22×Dic17—46no—
95C22×Dic17C22⋊C413yes—
96C22×Dic17C22×C414yescomplemented
97C22×Dic17C2×D413yes—
98C22⋊#595—24no—
99C22⋊#595—46no—
100#63247—24no—
101#63247—46no—
102C2×#3624—46no—
103C22⋊C4×C17—26no—
104C22⋊C4×C17C22×C414yescomplemented
105#63227—212no—
106#63227—24no—
107C4⋊Dic17—212no—
108C4⋊Dic17—24no—
109C22×#352—171nomaximal, Sylow
110C23×C17—24no—
111C23×C17—23no—
112C23×C17—23no—
113C23×C17—23no—
114C23×C17—23no—
115C23×C17—46no—
116C23×C17—46no—
117C23×C17—412no—
118C23×C17—44no—
119C23×C17#9913yes—
120C23×C17#9911yescharacteristic
121C23×C17C2×C22⋊C413yes—
122C23×C17C2×C22⋊C413yes—
123C23×C17C23×C411yescharacteristic
124C22×D17—23no—
125C22×D17—23no—
126C22×D17—23no—
127C22×D17—23no—
128C22×D17—44no—
129C22×D17—46no—
130C22×D17—46no—
131C22×D17—86no—
132C22×D17C2×C22⋊C413yes—
133C22×D17C22×D411yescharacteristic
134C2×#595—44no—
135C2×#595—812no—
136C2×C4×C17—24no—
137C2×C4×C17—412no—
138C22⋊D17—46no—
139C22⋊D17—43no—
140C22⋊D17—43no—
141C22⋊D17—43no—
142C22⋊D17—41no—
143C2×Dic17—23no—
144C2×Dic17—23no—
145C2×Dic17—23no—
146C2×Dic17—23no—
147C2×Dic17—24no—
148C2×Dic17—412no—
149C2×Dic17—46no—
150C2×Dic17—46no—
151C2×Dic17—44no—
152C2×Dic17C2×C22⋊C413yes—
153C2×Dic17C22×D411yescharacteristic
154#3624—44no—
155C22×#123—171no—
156C22×#100—171no—
157C22×#99—171no—
158C2×#352—1712no—
159C22×C17—26no—
160C22×C17—26no—
161C22×C17—23no—
162C22×C17—21no—
163C22×C17—43no—
164C22×C17—41no—
165C22×C17—46no—
166C22×C17—46no—
167C22×C17—46no—
168C22×C17—46no—
169C22×C17#35214yescomplemented
170C22×C17C2×#9913yes—
171C22×C17C2×#9913yes—
172C22×C17C22×C22⋊C411yescharacteristic
173C2×D17—21no—
174C2×D17—23no—
175C2×D17—46no—
176C2×D17—46no—
177C2×D17—43no—
178C2×D17—41no—
179C2×D17—83no—
180C2×D17—83no—
181#595—84no—
182C4×C17—44no—
183Dic17—21no—
184Dic17—23no—
185Dic17—44no—
186C23×D4—171no—
187C24×C4—171no—
188C22×M16—341no—
189C2×#123—176no—
190C2×#123—176no—
191C2×#123—343no—
192C2×#123—343no—
193C22×C4⋊C4—341no—
194C22×C22⋊C4—341no—
195C22×C22⋊C4—341no—
196C22×C22⋊C4#59511yescharacteristic, complemented
197C2×#100—176no—
198C2×#100—343no—
199C2×#99—176no—
200C2×#99—343no—
201#352—1716no—
202C2×C17—21no—
203C2×C17—23no—
204C2×C17—43no—
205C2×C17—41no—
206C2×C17C2×#35216yescomplemented
207C2×C17C22×#9911yescharacteristic
208D17—41no—
209D17—43no—
210C25—171no—
211C25C2×#59511yescharacteristic
212C22×D4—173no—
213C22×D4—173no—
214C22×D4—173no—
215C22×D4—173no—
216C22×D4—683no—
217C22×D4—681no—
218C23×C4—21no—
219C23×C4—171no—
220C23×C4—176no—
221C23×C4—343no—
222C23×C4—341no—
223C23×C4—683no—
224C23×C4—681no—
225C23×C4—681no—
226C2×M16—346no—
227C2×M16—683no—
228C22×C8—681no—
229#123—174no—
230#123—1712no—
231#123—3412no—
232#123—346no—
233#123—346no—
234C2×C4⋊C4—346no—
235C2×C4⋊C4—346no—
236C2×C22⋊C4—23no—
237C2×C22⋊C4—346no—
238C2×C22⋊C4—346no—
239C2×C22⋊C4—346no—
240C2×C22⋊C4—683no—
241C2×C22⋊C4C2×#59516yescomplemented
242#100—174no—
243#100—346no—
244#99—174no—
245#99—346no—
246C17C22×#35211yescharacteristic, complemented, Sylow
247C24—23no—
248C24—21no—
249C24—44no—
250C24—173no—
251C24—173no—
252C24—341no—
253C24—343no—
254C24—684no—
255C24C22⋊#59513yes—
256C24C22⋊#59511yescharacteristic
257C24C22×#59513yes—
258C2×D4—176no—
259C2×D4—173no—
260C2×D4—173no—
261C2×D4—173no—
262C2×D4—171no—
263C2×D4—683no—
264C2×D4—683no—
265C2×D4—683no—
266C2×D4—683no—
267C2×D4—683no—
268C2×D4—683no—
269C2×D4—683no—
270C2×D4—683no—
271C22×C4—26no—
272C22×C4—44no—
273C22×C4—173no—
274C22×C4—173no—
275C22×C4—174no—
276C22×C4—343no—
277C22×C4—341no—
278C22×C4—346no—
279C22×C4—346no—
280C22×C4—686no—
281C22×C4—686no—
282C22×C4—683no—
283C22×C4—681no—
284C22×C4—686no—
285C22×C4—686no—
286C22×C4—686no—
287C22×C4—1364no—
288M16—344no—
289M16—686no—
290C2×C8—686no—
291C4⋊C4—344no—
292C4⋊C4—3412no—
293C22⋊C4—26no—
294C22⋊C4—344no—
295C22⋊C4—3412no—
296C22⋊C4—344no—
297C22⋊C4—686no—
298C22⋊C4C22×#59514yescomplemented
299C23—24no—
300C23—23no—
301C23—23no—
302C23—23no—
303C23—23no—
304C23—46no—
305C23—46no—
306C23—412no—
307C23—44no—
308C23—171no—
309C23—173no—
310C23—343no—
311C23—343no—
312C23—343no—
313C23—343no—
314C23—686no—
315C23—686no—
316C23—684no—
317C23—1366no—
318C23#1058753913yes—
319C23#1058753911yescharacteristic
320C23C2×C22⋊#59513yes—
321C23C2×C22⋊#59513yes—
322C23C23×#59511yescharacteristic
323D4—683no—
324D4—681no—
325D4—683no—
326D4—686no—
327D4—683no—
328C2×C4—24no—
329C2×C4—412no—
330C2×C4—171no—
331C2×C4—173no—
332C2×C4—343no—
333C2×C4—343no—
334C2×C4—343no—
335C2×C4—343no—
336C2×C4—344no—
337C2×C4—6812no—
338C2×C4—686no—
339C2×C4—686no—
340C2×C4—684no—
341C2×C4—684no—
342C2×C4—13612no—
343C8—684no—
344C22—26no—
345C22—26no—
346C22—23no—
347C22—21no—
348C22—43no—
349C22—41no—
350C22—46no—
351C22—46no—
352C22—46no—
353C22—46no—
354C22—341no—
355C22—343no—
356C22—681no—
357C22—683no—
358C22—686no—
359C22—686no—
360C22—1363no—
361C22—1363no—
362C22#1176371214yesdirect product factor
363C22C2×#1058753913yes—
364C22C2×#1058753913yes—
365C22C22×C22⋊#59511yescharacteristic
366C4—44no—
367C4—341no—
368C4—343no—
369C4—684no—
370C4—1364no—
371C2—21no—
372C2—23no—
373C2—43no—
374C2—41no—
375C2—681no—
376C2—683no—
377C2C2×#1176371216yesdirect product factor
378C2C22×#1058753911yescharacteristic
subgroupG/H
centerC23#10587539
commutatorC23×C17C23×C4
FrattiniC23C23×#595
FittingC22×C22⋊C4×C17C4
socleC23×C17#99
prime radicalC23×C22⋊D17C22
cosocleC2×C22⋊D17C24
hypercenterC22×C22⋊C4#595
hypocenterC17C22×#352
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#11763712C2direct (N × H)
11. #11763712C22direct (N × H)
13. C24×Dic17C4semidirect
16. C2×#10587572C4semidirect
20. C22×C22⋊C4×C17C4semidirect
35. C23×Dic17C2×C4semidirect
43. C2×C22⋊C4×C17C2×C4semidirect
51. #10587572C2×C4semidirect
96. C22×Dic17C22×C4semidirect
104. C22⋊C4×C17C22×C4semidirect
169. C22×C17#352semidirect
196. C22×C22⋊C4#595semidirect
206. C2×C17C2×#352semidirect
241. C2×C22⋊C4C2×#595semidirect
246. C17C22×#352semidirect
298. C22⋊C4C22×#595semidirect
362. C22#11763712direct (N × H)
377. C2C2×#11763712direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875532
C22×#105875512
C2×#117637122
C22×#105875722
C22×#35217