Finite Group Databasebeta

← C22×#16863444 #31719523

Subgroups — C22×#16863444 #31719523

Order 17152.

2718 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11761612C211yesmaximal, characteristic
2C23×#11761612C211yesmaximal, characteristic
3C2×#16863444C2124yesmaximal, direct product factor
4C22×#14523978C214yesmaximal, complemented
5C23×M16×C67C211yesmaximal, characteristic
6C22×#11761612—21no—
7C22×#11761612—26no—
8C22×#11761612C412yes—
9C22×#11761612C412yes—
10C22×#11761612C4112yes—
11C22×#11761612C2212yes—
12C22×#11761612C22112yes—
13C22×#11761612C22112yes—
14C23×C67⋊C8—21no—
15C23×C67⋊C8C2212yes—
16#16863444C22164yesdirect product factor
17C2×#14523978C22148yescomplemented
18C24×C4×C67C2211yescharacteristic
19C22×M16×C67—21no—
20C22×M16×C67—26no—
21C22×M16×C67C2212yes—
22C22×M16×C67C22112yes—
23C23×C8×C67—21no—
24C2×#11761612—212no—
25C2×#11761612—26no—
26C2×#11761612—224no—
27C2×#11761612C2×C4124yes—
28C2×#11761612C2×C4124yes—
29C2×#11761612C2×C4148yes—
30C2×#11761612C23112yes—
31C2×#11761612C23116yes—
32C2×#11761612C23116yes—
33C22×C67⋊C8—212no—
34C22×C67⋊C8—22no—
35C22×C67⋊C8C2×C4112yes—
36C22×C67⋊C8C2×C414yes—
37C22×C67⋊C8C23112yes—
38#14523978C23164yescomplemented
39C25×C67Q811yescharacteristic
40C23×C4×C67C2×C411yescharacteristic
41C23×C4×C67C2×C411yescharacteristic
42C23×C4×C67C2×C416yes—
43C23×C4×C67D411yescharacteristic
44C23×C4×C67D411yescharacteristic
45C23×C4×C67D416yes—
46C23×C4×C67Q811yescharacteristic
47C23×C4×C67Q816yes—
48C23×C4×C67C2311yescharacteristic
49C23×C4×C67C2316yes—
50C2×M16×C67—212no—
51C2×M16×C67—26no—
52C2×M16×C67—224no—
53C2×M16×C67C23112yes—
54C2×M16×C67C23116yes—
55C22×C8×C67—212no—
56C22×C8×C67—22no—
57#11761612—216no—
58#11761612—212no—
59#11761612C22×C4132yes—
60#11761612C22×C4132yes—
61#11761612C2418yes—
62C2×C67⋊C8—212no—
63C2×C67⋊C8—216no—
64C2×C67⋊C8C22×C4124yes—
65C2×C67⋊C8C22×C4124yes—
66C2×C67⋊C8C2418yes—
67C24×C67—44no—
68C24×C67C4⋊C411yescharacteristic
69C24×C67C4⋊C411yescharacteristic
70C24×C67C4⋊C416yes—
71C24×C67C2×Q811yescharacteristic
72C24×C67C2×Q816yes—
73C22×C4×C67C4⋊C416yes—
74C22×C4×C67C4⋊C416yes—
75C22×C4×C67C4⋊C416yes—
76C22×C4×C67C4⋊C416yes—
77C22×C4×C67C4⋊C416yes—
78C22×C4×C67C4⋊C4124yes—
79C22×C4×C67C4⋊C411yescharacteristic
80C22×C4×C67C4⋊C411yescharacteristic
81C22×C4×C67C22×C413yes—
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C22×C416yes—
84C22×C4×C67C22×C416yes—
85C22×C4×C67C22×C4112yes—
86C22×C4×C67C2×D416yes—
87C22×C4×C67C2×D416yes—
88C22×C4×C67C2×D413yes—
89C22×C4×C67C2×D4112yes—
90C22×C4×C67C2×D411yescharacteristic
91C22×C4×C67C2×Q813yes—
92C22×C4×C67C2×Q816yes—
93C22×C4×C67C2×Q8112yes—
94C22×C4×C67C2413yes—
95C22×C4×C67C2414yes—
96M16×C67—216no—
97M16×C67—212no—
98M16×C67C2418yes—
99C2×C8×C67—212no—
100C2×C8×C67—216no—
101C23×C67—24no—
102C23×C67—24no—
103C23×C67—24no—
104C23×C67—412no—
105C23×C67—412no—
106C23×C67C2×C4⋊C413yes—
107C23×C67C2×C4⋊C416yes—
108C23×C67C2×C4⋊C416yes—
109C23×C67C2×C4⋊C4112yes—
110C23×C67C2×C4⋊C411yescharacteristic
111C23×C67C22×Q813yes—
112C23×C67C22×Q814yes—
113C2×C4×C67C2×C4⋊C416yes—
114C2×C4×C67C2×C4⋊C4112yes—
115C2×C4×C67C2×C4⋊C4112yes—
116C2×C4×C67C2×C4⋊C4112yes—
117C2×C4×C67C2×C4⋊C4112yes—
118C2×C4×C67C2×C4⋊C4112yes—
119C2×C4×C67C2×C4⋊C413yes—
120C2×C4×C67C2×C4⋊C416yes—
121C2×C4×C67C2×C4⋊C416yes—
122C2×C4×C67C2×C4⋊C413yes—
123C2×C4×C67C23×C413yes—
124C2×C4×C67C23×C413yes—
125C2×C4×C67C23×C414yes—
126C2×C4×C67C23×C414yes—
127C2×C4×C67C22×D413yes—
128C2×C4×C67C22×D414yes—
129C2×C4×C67C22×D414yes—
130C2×C4×C67C22×D413yes—
131C2×C4×C67C22×Q813yes—
132C2×C4×C67C22×Q814yes—
133C2×C4×C67C2511yescharacteristic
134C8×C67—28no—
135C67⋊C8—28no—
136C67⋊C8C23×C4116yes—
137C22×C67—212no—
138C22×C67—212no—
139C22×C67—26no—
140C22×C67—26no—
141C22×C67—412no—
142C22×C67—44no—
143C22×C67#43014yescomplemented
144C22×C67C22×C4⋊C413yes—
145C22×C67C22×C4⋊C414yes—
146C22×C67C22×C4⋊C414yes—
147C22×C67C22×C4⋊C413yes—
148C22×C67C23×Q811yescharacteristic
149C4×C67C22×C4⋊C413yes—
150C4×C67C22×C4⋊C413yes—
151C4×C67C22×C4⋊C414yes—
152C4×C67C22×C4⋊C414yes—
153C4×C67C24×C411yescharacteristic
154C4×C67C23×D411yescharacteristic
155C22×#430—671nomaximal, Sylow
156C2×C67—24no—
157C2×C67—24no—
158C2×C67—23no—
159C2×C67—21no—
160C2×C67C2×#43016yescomplemented
161C2×C67C23×C4⋊C411yescharacteristic
162C23×M16—671no—
163C23×M16—671no—
164C23×M16D6711yescharacteristic
165C2×#430—6724no—
166C22×#108—674no—
167C67C22×#43011yescharacteristic, complemented, Sylow
168C24×C4C2×D6711yescharacteristic
169C22×M16—26no—
170C22×M16—21no—
171C22×M16—672no—
172C22×M16—672no—
173C22×M16—6712no—
174C22×M16—6712no—
175C22×M16—672no—
176C22×M16—6712no—
177C22×M16—1346no—
178C22×M16—1341no—
179C22×M16C2×D6712yes—
180C22×M16C2×D67112yes—
181C23×C8—21no—
182C23×C8—672no—
183C23×C8—1341no—
184#430—6764no—
185C2×#108—6748no—
186C25Dic13411yescharacteristic
187C23×C4Dic13411yescharacteristic
188C23×C4Dic13416yes—
189C23×C4C4×D6711yescharacteristic
190C23×C4C4×D6711yescharacteristic
191C23×C4C4×D6716yes—
192C23×C4C22⋊D6716yes—
193C23×C4C22⋊D6711yescharacteristic
194C23×C4C22⋊D6711yescharacteristic
195C23×C4C22×D6716yes—
196C23×C4C22×D6711yescharacteristic
197C2×M16—212no—
198C2×M16—26no—
199C2×M16—224no—
200C2×M16—6724no—
201C2×M16—6724no—
202C2×M16—6748no—
203C2×M16—6716no—
204C2×M16—6712no—
205C2×M16—6716no—
206C2×M16—13412no—
207C2×M16—1346no—
208C2×M16—13424no—
209C2×M16C22×D67112yes—
210C2×M16C22×D67116yes—
211C22×C8—212no—
212C22×C8—22no—
213C22×C8—674no—
214C22×C8—6712no—
215C22×C8—6712no—
216C22×C8—13412no—
217C22×C8—1342no—
218#108—6764no—
219C24—44no—
220C24Dic67⋊C411yescharacteristic
221C24Dic67⋊C411yescharacteristic
222C24Dic67⋊C416yes—
223C24C2×Dic13411yescharacteristic
224C24C2×Dic13416yes—
225C22×C4Dic67⋊C416yes—
226C22×C4Dic67⋊C416yes—
227C22×C4Dic67⋊C416yes—
228C22×C4Dic67⋊C4124yes—
229C22×C4Dic67⋊C411yescharacteristic
230C22×C4Dic67⋊C411yescharacteristic
231C22×C4Dic67⋊C416yes—
232C22×C4Dic67⋊C416yes—
233C22×C4C2×C4×D6711yescharacteristic
234C22×C4C2×C4×D6713yes—
235C22×C4C2×C4×D6716yes—
236C22×C4C2×C4×D6716yes—
237C22×C4C2×C4×D67112yes—
238C22×C4C2×C22⋊D6716yes—
239C22×C4C2×C22⋊D6716yes—
240C22×C4C2×C22⋊D6713yes—
241C22×C4C2×C22⋊D67112yes—
242C22×C4C2×C22⋊D6711yescharacteristic
243C22×C4C2×Dic13416yes—
244C22×C4C2×Dic134112yes—
245C22×C4C2×Dic13413yes—
246C22×C4C23×D6713yes—
247C22×C4C23×D6714yes—
248M16—216no—
249M16—212no—
250M16—6732no—
251M16—6732no—
252M16—678no—
253M16—13416no—
254M16—13412no—
255M16C23×D6718yes—
256C2×C8—212no—
257C2×C8—216no—
258C2×C8—6724no—
259C2×C8—6724no—
260C2×C8—678no—
261C2×C8—13412no—
262C2×C8—13416no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—412no—
267C23—412no—
268C23C2×Dic67⋊C413yes—
269C23C2×Dic67⋊C416yes—
270C23C2×Dic67⋊C416yes—
271C23C2×Dic67⋊C4112yes—
272C23C2×Dic67⋊C411yescharacteristic
273C23C22×Dic13413yes—
274C23C22×Dic13414yes—
275C2×C4C2×Dic67⋊C416yes—
276C2×C4C2×Dic67⋊C4112yes—
277C2×C4C2×Dic67⋊C4112yes—
278C2×C4C2×Dic67⋊C4112yes—
279C2×C4C2×Dic67⋊C4112yes—
280C2×C4C2×Dic67⋊C4112yes—
281C2×C4C2×Dic67⋊C413yes—
282C2×C4C2×Dic67⋊C416yes—
283C2×C4C2×Dic67⋊C416yes—
284C2×C4C2×Dic67⋊C413yes—
285C2×C4C22×C4×D6713yes—
286C2×C4C22×C4×D6713yes—
287C2×C4C22×C4×D6714yes—
288C2×C4C22×C4×D6714yes—
289C2×C4C22×C22⋊D6713yes—
290C2×C4C22×C22⋊D6714yes—
291C2×C4C22×C22⋊D6714yes—
292C2×C4C22×C22⋊D6713yes—
293C2×C4C22×Dic13413yes—
294C2×C4C22×Dic13414yes—
295C2×C4C24×D6711yescharacteristic
296C8—28no—
297C8—6716no—
298C8—1348no—
299C22—212no—
300C22—212no—
301C22—26no—
302C22—26no—
303C22—412no—
304C22—44no—
305C22#1686344414yesdirect product factor
306C22C22×Dic67⋊C413yes—
307C22C22×Dic67⋊C414yes—
308C22C22×Dic67⋊C414yes—
309C22C22×Dic67⋊C413yes—
310C22C23×Dic13411yescharacteristic
311C4C22×Dic67⋊C413yes—
312C4C22×Dic67⋊C414yes—
313C4C22×Dic67⋊C414yes—
314C4C22×Dic67⋊C413yes—
315C4C23×C4×D6711yescharacteristic
316C4C23×C22⋊D6711yescharacteristic
317C2—24no—
318C2—24no—
319C2—23no—
320C2—21no—
321C2C2×#1686344416yesdirect product factor
322C2C23×Dic67⋊C411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D67
commutatorC4×C67C24×C4
FrattiniC2×C4C24×D67
FittingC23×M16×C67C2
socleC23×C67C2×C4⋊C4
prime radicalC25×C67Q8
cosocleC2×C4×C67C25
hypercenterC23×M16D67
hypocenterC67C22×#430
LCSG ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16863444C2direct (N × H)
4. C22×#14523978C2semidirect
16. #16863444C22direct (N × H)
17. C2×#14523978C22semidirect
38. #14523978C23semidirect
143. C22×C67#430semidirect
160. C2×C67C2×#430semidirect
167. C67C22×#430semidirect
305. C22#16863444direct (N × H)
321. C2C2×#16863444direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616122
C2×#168634442
C22×#145239782
C23×M16×C672
C22×#43067