Finite Group Databasebeta

← C22×#17988346 #33083365

Subgroups — C22×#17988346 #33083365

Order 18688.

2830 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924012C211yesmaximal, characteristic
2C23×C4×Dic73C211yesmaximal, characteristic
3C2×#17988346C2112yesmaximal, direct product factor
4C22×#117×C73C211yesmaximal, characteristic
5C24×Dic73—21no—
6C22×#11924012—24no—
7C22×#11924012—23no—
8C22×#11924012C418yescomplemented
9C22×#11924012C2216yes—
10C23×#10596831—21no—
11C22×C4×Dic73—23no—
12C22×C4×Dic73—24no—
13C22×C4×Dic73C418yescomplemented
14C22×C4×Dic73C2216yes—
15#17988346C22116yesdirect product factor
16C24×C4×C73C2211yescharacteristic
17C2×#117×C73—23no—
18C2×#117×C73—23no—
19C2×#117×C73C416yescomplemented
20C2×#117×C73C2216yes—
21C22×C4⋊C4×C73C412yescomplemented
22C22×C22⋊C4×C73—21no—
23C22×C42×C73—21no—
24C23×Dic73—216no—
25C23×Dic73—28no—
26C23×Dic73—26no—
27C2×#11924012—224no—
28C2×#11924012—26no—
29C2×#11924012—212no—
30C2×#11924012C2×C4124yescomplemented
31C2×#11924012C2314yes—
32C22×#10596831—28no—
33C22×#10596831—26no—
34C2×C4×Dic73—224no—
35C2×C4×Dic73—26no—
36C2×C4×Dic73—212no—
37C2×C4×Dic73C2×C4124yescomplemented
38C2×C4×Dic73C2314yes—
39C25×C73D411yescharacteristic
40C23×C4×C73—22no—
41C23×C4×C73C2×C414yes—
42C23×C4×C73C2×C414yes—
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73C2×C413yes—
45C23×C4×C73D413yes—
46C23×C4×C73D414yes—
47C23×C4×C73D413yes—
48C23×C4×C73D411yescharacteristic
49C23×C4×C73Q814yes—
50C23×C4×C73C2313yes—
51#117×C73—212no—
52#117×C73—26no—
53#117×C73—26no—
54#117×C73C2×C4112yescomplemented
55#117×C73C2314yes—
56C2×C4⋊C4×C73—26no—
57C2×C4⋊C4×C73C2×C4112yescomplemented
58C2×C22⋊C4×C73—26no—
59C2×C22⋊C4×C73—26no—
60C2×C42×C73—26no—
61C2×C42×C73—26no—
62C22×Dic73—264no—
63C22×Dic73—248no—
64C22×Dic73—224no—
65C22×Dic73—24no—
66#11924012—224no—
67#11924012—24no—
68#11924012C22×C418yescomplemented
69C2×#10596831—224no—
70C2×#10596831—24no—
71C4×Dic73—224no—
72C4×Dic73—24no—
73C4×Dic73C22×C418yescomplemented
74C24×C73—28no—
75C24×C73C22⋊C414yes—
76C24×C73C22⋊C413yes—
77C24×C73C22⋊C411yescharacteristic
78C24×C73C4⋊C414yes—
79C24×C73C2×D413yes—
80C22×C4×C73—212no—
81C22×C4×C73—48no—
82C22×C4×C73C42112yes—
83C22×C4×C73C4214yes—
84C22×C4×C73C22⋊C4112yes—
85C22×C4×C73C22⋊C414yes—
86C22×C4×C73C22⋊C4112yes—
87C22×C4×C73C22⋊C413yes—
88C22×C4×C73C22⋊C413yes—
89C22×C4×C73C22⋊C416yes—
90C22×C4×C73C4⋊C4112yes—
91C22×C4×C73C4⋊C4112yes—
92C22×C4×C73C4⋊C414yes—
93C22×C4×C73C4⋊C4112yes—
94C22×C4×C73C4⋊C414yes—
95C22×C4×C73C22×C416yes—
96C22×C4×C73C22×C416yes—
97C22×C4×C73C22×C413yes—
98C22×C4×C73C22×C413yes—
99C22×C4×C73C2×D413yes—
100C22×C4×C73C2×D413yes—
101C22×C4×C73C2×D413yes—
102C22×C4×C73C2×D416yes—
103C22×C4×C73C2×Q816yes—
104C22×C4×C73C2411yescharacteristic
105C4⋊C4×C73—212no—
106C4⋊C4×C73C22×C418yescomplemented
107C22⋊C4×C73—212no—
108C22⋊C4×C73—24no—
109C42×C73—212no—
110C42×C73—24no—
111C23×C73—216no—
112C23×C73—216no—
113C23×C73—212no—
114C23×C73—212no—
115C23×C73#95112yes—
116C23×C73#9514yes—
117C23×C73#10418yes—
118C23×C73C2×C22⋊C416yes—
119C23×C73C2×C22⋊C413yes—
120C23×C73C2×C22⋊C413yes—
121C23×C73C2×C4⋊C416yes—
122C23×C73C22×D411yescharacteristic
123C2×C4×C73—28no—
124C2×C4×C73—424no—
125C2×C4×C73#95124yes—
126C2×C4×C73#95112yes—
127C2×C4×C73#95112yes—
128C2×C4×C73C2×C4216yes—
129C2×C4×C73C2×C4216yes—
130C2×C4×C73C2×C22⋊C416yes—
131C2×C4×C73C2×C22⋊C416yes—
132C2×C4×C73C2×C22⋊C416yes—
133C2×C4×C73C2×C22⋊C413yes—
134C2×C4×C73C2×C22⋊C413yes—
135C2×C4×C73C2×C4⋊C416yes—
136C2×C4×C73C2×C4⋊C416yes—
137C2×C4×C73C2×C4⋊C416yes—
138C2×C4×C73C2×C4⋊C416yes—
139C2×C4×C73C2×C4⋊C416yes—
140C2×C4×C73C23×C411yescharacteristic
141C2×C4×C73C23×C411yescharacteristic
142C2×C4×C73C23×C411yescharacteristic
143C2×C4×C73C22×D411yescharacteristic
144C2×C4×C73C22×D411yescharacteristic
145C2×C4×C73C22×Q811yescharacteristic
146C2×Dic73—216no—
147C2×Dic73—296no—
148C2×Dic73—28no—
149#10596831—28no—
150C22×C73—22no—
151C22×C73—26no—
152C22×C73—212no—
153C22×C73—224no—
154C22×C73—212no—
155C22×C73#339116yescomplemented
156C22×C73C2×#9516yes—
157C22×C73C2×#9516yes—
158C22×C73C2×#104112yes—
159C22×C73C22×C22⋊C411yescharacteristic
160C22×C73C22×C22⋊C411yescharacteristic
161C22×C73C22×C4⋊C411yescharacteristic
162C4×C73—48no—
163C4×C73C2×#9516yes—
164C4×C73C2×#9516yes—
165C4×C73C22×C4211yescharacteristic
166C4×C73C22×C22⋊C411yescharacteristic
167C4×C73C22×C4⋊C411yescharacteristic
168C4×C73C22×C4⋊C411yescharacteristic
169Dic73—216no—
170C22×#339—731nomaximal, Sylow
171C2×C73—21no—
172C2×C73—21no—
173C2×C73—26no—
174C2×C73C2×#339112yescomplemented
175C2×C73C22×#9511yescharacteristic
176C2×C73C22×#10412yes—
177C23×M16—731no—
178C22×#117D7311yescharacteristic
179C23×C42—731no—
180C2×#339—7312no—
181C73C22×#33911yescharacteristic, complemented, Sylow
182C24×C4—1461no—
183C24×C4C2×D7311yescharacteristic
184C22×M16—738no—
185C22×M16—736no—
186C22×M16—1464no—
187C22×M16—1463no—
188C23×C8—1461no—
189C2×#117—23no—
190C2×#117—23no—
191C2×#117Dic7316yescomplemented
192C2×#117C2×D7316yes—
193C22×C4⋊C4Dic7312yescomplemented
194C22×C22⋊C4—21no—
195C22×C42—21no—
196C22×C42—738no—
197C22×C42—736no—
198C22×C42—1464no—
199C22×C42—1463no—
200#339—7316no—
201C25C22⋊D7311yescharacteristic
202C23×C4—22no—
203C23×C4—14616no—
204C23×C4—1468no—
205C23×C4—1466no—
206C23×C4Dic14614yes—
207C23×C4C4×D7314yes—
208C23×C4C4×D7314yes—
209C23×C4D29214yes—
210C23×C4C2×Dic7311yescharacteristic
211C23×C4C2×Dic7313yes—
212C23×C4C22⋊D7313yes—
213C23×C4C22⋊D7313yes—
214C23×C4C22⋊D7311yescharacteristic
215C23×C4C22×D7313yes—
216C2×M16—7324no—
217C2×M16—734no—
218C2×M16—14624no—
219C2×M16—1466no—
220C2×M16—14612no—
221C22×C8—1468no—
222C22×C8—1466no—
223#117—212no—
224#117—26no—
225#117—26no—
226#117C2×Dic73112yescomplemented
227#117C22×D7314yes—
228C2×C4⋊C4—26no—
229C2×C4⋊C4C2×Dic73112yescomplemented
230C2×C22⋊C4—26no—
231C2×C22⋊C4—26no—
232C2×C42—26no—
233C2×C42—26no—
234C2×C42—7324no—
235C2×C42—734no—
236C2×C42—14624no—
237C2×C42—1466no—
238C2×C42—14612no—
239C24—28no—
240C24#1192400514yes—
241C24C22⋊Dic7313yes—
242C24C22⋊Dic7311yescharacteristic
243C24#1192400714yes—
244C24C2×C22⋊D7313yes—
245C22×C4—212no—
246C22×C4—48no—
247C22×C4—14648no—
248C22×C4—14664no—
249C22×C4—14624no—
250C22×C4—1464no—
251C22×C4C4×Dic73112yes—
252C22×C4C4×Dic7314yes—
253C22×C4#11924005112yes—
254C22×C4#1192400514yes—
255C22×C4#11924005112yes—
256C22×C4C22⋊Dic7313yes—
257C22×C4C22⋊Dic7313yes—
258C22×C4C22⋊Dic7316yes—
259C22×C4#11924007112yes—
260C22×C4#11924007112yes—
261C22×C4#1192400714yes—
262C22×C4C4⋊Dic73112yes—
263C22×C4C4⋊Dic7314yes—
264C22×C4C2×C4×D7316yes—
265C22×C4C2×C4×D7316yes—
266C22×C4C22×Dic7313yes—
267C22×C4C22×Dic7313yes—
268C22×C4C2×C22⋊D7313yes—
269C22×C4C2×C22⋊D7313yes—
270C22×C4C2×C22⋊D7313yes—
271C22×C4C2×D29216yes—
272C22×C4C2×Dic14616yes—
273C22×C4C23×D7311yescharacteristic
274M16—738no—
275M16—14624no—
276M16—1464no—
277C2×C8—14624no—
278C2×C8—1464no—
279C4⋊C4—212no—
280C4⋊C4C22×Dic7318yescomplemented
281C22⋊C4—212no—
282C22⋊C4—24no—
283C42—212no—
284C42—24no—
285C42—738no—
286C42—14624no—
287C42—1464no—
288C23—212no—
289C23—212no—
290C23—216no—
291C23—216no—
292C23#14556498112yes—
293C23#1455649814yes—
294C23#1455651818yes—
295C23C2×#1192400516yes—
296C23C2×C22⋊Dic7313yes—
297C23C2×C22⋊Dic7313yes—
298C23C2×#1192400716yes—
299C23C22×C22⋊D7311yescharacteristic
300C2×C4—28no—
301C2×C4—424no—
302C2×C4—14616no—
303C2×C4—14696no—
304C2×C4—1468no—
305C2×C4#14556498124yes—
306C2×C4#14556498112yes—
307C2×C4#14556498112yes—
308C2×C4C2×C4×Dic7316yes—
309C2×C4C2×C4×Dic7316yes—
310C2×C4C2×#1192400516yes—
311C2×C4C2×#1192400516yes—
312C2×C4C2×#1192400516yes—
313C2×C4C2×C22⋊Dic7313yes—
314C2×C4C2×C22⋊Dic7313yes—
315C2×C4C2×#1192400716yes—
316C2×C4C2×#1192400716yes—
317C2×C4C2×#1192400716yes—
318C2×C4C2×C4⋊Dic7316yes—
319C2×C4C2×C4⋊Dic7316yes—
320C2×C4C22×C4×D7311yescharacteristic
321C2×C4C22×C4×D7311yescharacteristic
322C2×C4C23×Dic7311yescharacteristic
323C2×C4C22×C22⋊D7311yescharacteristic
324C2×C4C22×D29211yescharacteristic
325C2×C4C22×Dic14611yescharacteristic
326C8—1468no—
327C22—22no—
328C22—26no—
329C22—212no—
330C22—224no—
331C22—212no—
332C22#17988346116yesdirect product factor
333C22C2×#1455649816yes—
334C22C2×#1455649816yes—
335C22C2×#14556518112yes—
336C22C22×#1192400511yescharacteristic
337C22C22×C22⋊Dic7311yescharacteristic
338C22C22×#1192400711yescharacteristic
339C4—48no—
340C4—14616no—
341C4C2×#1455649816yes—
342C4C2×#1455649816yes—
343C4C22×C4×Dic7311yescharacteristic
344C4C22×#1192400511yescharacteristic
345C4C22×#1192400711yescharacteristic
346C4C22×C4⋊Dic7311yescharacteristic
347C2—21no—
348C2—26no—
349C2—21no—
350C2C2×#17988346112yesdirect product factor
351C2C22×#1455649811yescharacteristic
352C2C22×#1455651812yes—
subgroupG/H
centerC23×C4C22⋊D73
commutatorC4×C73C22×C42
FrattiniC22×C4C23×D73
FittingC22×#117×C73C2
socleC24×C73C22⋊C4
prime radicalC25×C73D4
cosocleC22×C4×C73C24
hypercenterC22×#117D73
hypocenterC73C22×#339
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#17988346C2direct (N × H)
8. C22×#11924012C4semidirect
13. C22×C4×Dic73C4semidirect
15. #17988346C22direct (N × H)
19. C2×#117×C73C4semidirect
21. C22×C4⋊C4×C73C4semidirect
30. C2×#11924012C2×C4semidirect
37. C2×C4×Dic73C2×C4semidirect
54. #117×C73C2×C4semidirect
57. C2×C4⋊C4×C73C2×C4semidirect
68. #11924012C22×C4semidirect
73. C4×Dic73C22×C4semidirect
106. C4⋊C4×C73C22×C4semidirect
155. C22×C73#339semidirect
174. C2×C73C2×#339semidirect
181. C73C22×#339semidirect
191. C2×#117Dic73semidirect
193. C22×C4⋊C4Dic73semidirect
226. #117C2×Dic73semidirect
229. C2×C4⋊C4C2×Dic73semidirect
280. C4⋊C4C22×Dic73semidirect
332. C22#17988346direct (N × H)
350. C2C2×#17988346direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240122
C23×C4×Dic732
C2×#179883462
C22×#117×C732
C22×#33973