Finite Group Databasebeta

← C22×#11763084 #17901711

Subgroups — C22×#11763084 #17901711

Order 4352.

2630 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587637C212yesmaximal
2C23×Dic68C211yesmaximal, characteristic, complemented
3C2×#11763084C2124yesmaximal, direct product factor
4C22×#10587490C211yesmaximal, characteristic, complemented
5C22×#10587495C212yesmaximal, complemented
6C22×C17×#131C211yesmaximal, characteristic
7C23×Dic34C2212yes—
8C2×#10587637—212no—
9C2×#10587637C22124yes—
10C22×Dic68—21no—
11C22×Dic68—21no—
12C22×Dic68—26no—
13C22×Dic68C22112yescomplemented
14C22×#63231—22no—
15C22×#63229—22no—
16C22×#63223C2212yes—
17#11763084C22164yesdirect product factor
18C2×#10587490C22112yescomplemented
19C2×#10587495C22124yescomplemented
20C22×#41×C17C2211yescharacteristic
21C2×C17×#131—26no—
22C2×C17×#131C22112yes—
23C22×M16×C17—21no—
24C22×M16×C17C2211yescharacteristic
25C23×C8×C17—21no—
26C23×C8×C17C2211yescharacteristic
27C2×#41×C17—23no—
28C2×#41×C17—23no—
29C2×#41×C17D416yes—
30C2×#41×C17C2316yes—
31C22×Q8×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17D411yescharacteristic
34C23×C4×C17—21no—
35C23×C4×C17C2311yescharacteristic
36C23×Dic17—22no—
37C22×Dic34—22no—
38C22×Dic34—22no—
39C22×Dic34—26no—
40C22×Dic34—26no—
41C22×Dic34—26no—
42C22×Dic34C23112yes—
43C17×#131—224no—
44C17×#131C23116yes—
45C2×M16×C17—23no—
46C2×M16×C17—212no—
47C2×M16×C17C2316yes—
48C22×C8×C17—21no—
49C22×C8×C17—212no—
50C22×C8×C17D411yescharacteristic
51C22×C8×C17D411yescharacteristic
52C22×C8×C17D416yes—
53C22×C8×C17C2316yes—
54#10587637—248no—
55#10587637C23132yes—
56C2×#63231—224no—
57C2×#63229—224no—
58C2×#63223—26no—
59C2×#63223C23112yes—
60C22×#3622—22no—
61C2×Dic68—212no—
62C2×Dic68—212no—
63C2×Dic68—224no—
64C2×Dic68C23116yescomplemented
65#10587495C23132yescomplemented
66#10587490C23116yescomplemented
67C24×C17—21no—
68C24×C17#4111yescharacteristic
69#41×C17—212no—
70#41×C17—26no—
71#41×C17—26no—
72#41×C17C2×D4112yes—
73#41×C17C2414yes—
74C2×Q8×C17—23no—
75C2×Q8×C17C2×D416yes—
76C2×D4×C17—26no—
77C2×D4×C17—26no—
78C2×D4×C17—23no—
79C2×D4×C17C2×D416yes—
80C22×C4×C17—21no—
81C22×C4×C17—26no—
82C22×C4×C17—26no—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D413yes—
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D413yes—
87C22×C4×C17#4113yes—
88C22×C4×C17C2413yes—
89C22×Dic17—212no—
90C22×Dic17—46no—
91C22×Dic17—42no—
92C2×Dic34—212no—
93C2×Dic34—212no—
94C2×Dic34—212no—
95C2×Dic34—212no—
96C2×Dic34—212no—
97C2×Dic34—46no—
98C2×Dic34—46no—
99C2×Dic34—412no—
100C2×Dic34C2418yes—
101M16×C17—26no—
102M16×C17—216no—
103M16×C17C2414yes—
104C2×C8×C17—26no—
105C2×C8×C17—216no—
106C2×C8×C17C2×D416yes—
107C2×C8×C17C2×D416yes—
108C2×C8×C17C2×D4112yes—
109C2×C8×C17C2414yes—
110#63231—232no—
111#63229—232no—
112#63223—212no—
113#63223C2418yes—
114C2×#3622—212no—
115Dic68—216no—
116Dic68—216no—
117C22×#473—171nomaximal, Sylow
118C23×C17—24no—
119C23×C17—26no—
120C23×C17—44no—
121C23×C17#12113yes—
122C23×C17#12111yescharacteristic
123C23×C17C2×#4113yes—
124Q8×C17—26no—
125Q8×C17C22×D414yes—
126D4×C17—212no—
127D4×C17—24no—
128D4×C17—26no—
129D4×C17C22×D414yes—
130C2×C4×C17—26no—
131C2×C4×C17—212no—
132C2×C4×C17—24no—
133C2×C4×C17#12113yes—
134C2×C4×C17#12113yes—
135C2×C4×C17#12116yes—
136C2×C4×C17C22×D413yes—
137C2×C4×C17C22×D413yes—
138C2×C4×C17C22×D413yes—
139C2×C4×C17C22×D413yes—
140C2×C4×C17C2×#4113yes—
141C2×C4×C17C2511yescharacteristic
142C2×Dic17—28no—
143C2×Dic17—412no—
144C2×Dic17—412no—
145Dic34—28no—
146Dic34—28no—
147Dic34—412no—
148Dic34—412no—
149C8×C17—24no—
150C8×C17C22×D414yes—
151C8×C17C22×D414yes—
152#3622—28no—
153C22×#137—172no—
154C23×Q16—171no—
155C22×#131D1711yescharacteristic
156C2×#473—1724no—
157C22×#108—171no—
158C22×#101—172no—
159C22×C17—26no—
160C22×C17—26no—
161C22×C17—24no—
162C22×C17—412no—
163C22×C17#47314yescomplemented
164C22×C17C2×#12113yes—
165C22×C17C2×#12113yes—
166C22×C17C22×#4111yescharacteristic
167C4×C17—24no—
168C4×C17C2×#12113yes—
169C4×C17C2×#12113yes—
170C4×C17C23×D411yescharacteristic
171C4×C17C23×D411yescharacteristic
172Dic17—48no—
173C22×#41C2×D1711yescharacteristic
174C23×Q8—172no—
175C2×#137—1724no—
176C2×#137—3412no—
177C22×Q16—1712no—
178C22×Q16—342no—
179C22×Q16—341no—
180C22×Q16—341no—
181C22×Q16—346no—
182C22×SD16—342no—
183C2×#131—26no—
184C2×#131C2×D17112yes—
185C22×M16—21no—
186C22×M16—172no—
187C22×M16C2×D1711yescharacteristic
188C23×C8—21no—
189C23×C8C2×D1711yescharacteristic
190#473—1764no—
191C2×#108—1712no—
192C2×#101—1724no—
193C2×C17—23no—
194C2×C17—21no—
195C2×C17—44no—
196C2×C17C2×#47316yescomplemented
197C2×C17C22×#12111yescharacteristic
198C2×#41—23no—
199C2×#41—23no—
200C2×#41D6816yes—
201C2×#41C22×D1716yes—
202C22×Q8—1712no—
203C22×Q8—346no—
204C22×Q8—342no—
205C22×Q8—346no—
206C22×Q8—346no—
207C22×Q8—342no—
208C22×Q8D6811yescharacteristic
209C22×D4—21no—
210C22×D4D6811yescharacteristic
211C23×C4—21no—
212C23×C4—342no—
213C23×C4C22×D1711yescharacteristic
214#137—1732no—
215#137—3448no—
216C2×Q16—1716no—
217C2×Q16—3424no—
218C2×Q16—3412no—
219C2×Q16—3412no—
220C2×Q16—3424no—
221C2×SD16—3424no—
222#131—224no—
223#131C22×D17116yes—
224C2×M16—23no—
225C2×M16—212no—
226C2×M16—1712no—
227C2×M16—346no—
228C2×M16C22×D1716yes—
229C22×C8—21no—
230C22×C8—212no—
231C22×C8—342no—
232C22×C8C22⋊D1711yescharacteristic
233C22×C8C22⋊D1711yescharacteristic
234C22×C8C22⋊D1716yes—
235C22×C8C22×D1716yes—
236#108—1716no—
237#101—1732no—
238C17C22×#47311yescharacteristic, complemented, Sylow
239C24—21no—
240C24#6325211yescharacteristic
241#41—26no—
242#41—26no—
243#41—212no—
244#41C2×D68112yes—
245#41C23×D1714yes—
246C2×Q8—23no—
247C2×Q8—178no—
248C2×Q8—3412no—
249C2×Q8—3412no—
250C2×Q8—3412no—
251C2×Q8—3412no—
252C2×Q8—3412no—
253C2×Q8—686no—
254C2×Q8—686no—
255C2×Q8—6812no—
256C2×Q8C2×D6816yes—
257C2×D4—26no—
258C2×D4—26no—
259C2×D4—23no—
260C2×D4C2×D6816yes—
261C22×C4—21no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—3412no—
265C22×C4—686no—
266C22×C4—682no—
267C22×C4C2×D6813yes—
268C22×C4C2×D6811yescharacteristic
269C22×C4#6325213yes—
270C22×C4C2×C22⋊D1713yes—
271C22×C4C2×C22⋊D1711yescharacteristic
272C22×C4C23×D1713yes—
273Q16—3432no—
274Q16—3416no—
275Q16—3416no—
276SD16—3432no—
277M16—26no—
278M16—216no—
279M16—178no—
280M16—3412no—
281M16C23×D1714yes—
282C2×C8—26no—
283C2×C8—216no—
284C2×C8—3412no—
285C2×C8C2×C22⋊D1716yes—
286C2×C8C2×C22⋊D1716yes—
287C2×C8C2×C22⋊D17112yes—
288C2×C8C23×D1714yes—
289C23—24no—
290C23—26no—
291C23—44no—
292C23#1058761613yes—
293C23#1058761611yescharacteristic
294C23C2×#6325213yes—
295Q8—26no—
296Q8—348no—
297Q8—348no—
298Q8—6812no—
299Q8—6812no—
300Q8C22×D6814yes—
301D4—212no—
302D4—24no—
303D4—26no—
304D4C22×D6814yes—
305C2×C4—26no—
306C2×C4—212no—
307C2×C4—24no—
308C2×C4—348no—
309C2×C4—6812no—
310C2×C4—6812no—
311C2×C4#1058761613yes—
312C2×C4#1058761613yes—
313C2×C4#1058761616yes—
314C2×C4C22×D6813yes—
315C2×C4C22×D6813yes—
316C2×C4C2×#6325213yes—
317C2×C4C22×C22⋊D1713yes—
318C2×C4C22×C22⋊D1713yes—
319C2×C4C24×D1711yescharacteristic
320C8—24no—
321C8—348no—
322C8C22×C22⋊D1714yes—
323C8C22×C22⋊D1714yes—
324C22—26no—
325C22—26no—
326C22—24no—
327C22—412no—
328C22#1176308414yesdirect product factor
329C22C2×#1058761613yes—
330C22C2×#1058761613yes—
331C22C22×#6325211yescharacteristic
332C4—24no—
333C4—688no—
334C4C2×#1058761613yes—
335C4C2×#1058761613yes—
336C4C23×C22⋊D1711yescharacteristic
337C4C23×D6811yescharacteristic
338C2—23no—
339C2—21no—
340C2—44no—
341C2C2×#1176308416yesdirect product factor
342C2C22×#1058761611yescharacteristic
subgroupG/H
centerC23#10587616
commutator, cosocleC2×C4×C17C25
FrattiniC2×C4C24×D17
FittingC22×C17×#131C2
socleC23×C17#121
prime radicalC22×#41×C17C22
hypercenterC22×#131D17
hypocenterC17C22×#473
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#131

Complements

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

NH ≅ G/Ntype
2. C23×Dic68C2semidirect
3. C2×#11763084C2direct (N × H)
4. C22×#10587490C2semidirect
5. C22×#10587495C2semidirect
13. C22×Dic68C22semidirect
17. #11763084C22direct (N × H)
18. C2×#10587490C22semidirect
19. C2×#10587495C22semidirect
64. C2×Dic68C23semidirect
65. #10587495C23semidirect
66. #10587490C23semidirect
163. C22×C17#473semidirect
196. C2×C17C2×#473semidirect
238. C17C22×#473semidirect
328. C22#11763084direct (N × H)
341. C2C2×#11763084direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105876372
C23×Dic682
C2×#117630842
C22×#105874902
C22×#105874952
C22×C17×#1312
C22×#47317