Finite Group Databasebeta

← C22×#17988612 #33949598

Subgroups — C22×#17988612 #33949598

Order 18688.

3006 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924011C211yesmaximal, characteristic
2C23×C8×D73C211yesmaximal, characteristic
3C22×#14556548C211yesmaximal, characteristic
4C2×#17988612C2124yesmaximal, direct product factor
5C22×C4⋊#10596831C211yesmaximal, characteristic, complemented
6C22×#14556520C211yesmaximal, characteristic, complemented
7C22×#14556520C211yesmaximal, characteristic, complemented
8C22×C4⋊C8×C73C211yesmaximal, characteristic, complemented
9C23×C4×D73C2211yescharacteristic
10C22×#11924011—28no—
11C22×#11924011C22112yes—
12C23×#10596831C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14C22×C8×D73—28no—
15C22×C8×D73C22112yes—
16C2×#14556548C4112yes—
17C2×#14556548C22112yes—
18C22×#11924007C411yescharacteristic
19C22×#11924007C411yescharacteristic
20C22×#11924005C411yescharacteristic
21C22×#11924005C411yescharacteristic
22C22×C4×Dic73C2211yescharacteristic
23#17988612C22164yesdirect product factor
24C2×C4⋊#10596831C22112yescomplemented
25C2×#14556520C22112yescomplemented
26C2×#14556520C22112yescomplemented
27C23×C8×C73C2211yescharacteristic
28C23×C8×C73C2211yescharacteristic
29C22×C42×C73C2211yescharacteristic
30C2×C4⋊C8×C73C22112yescomplemented
31C24×D73C2×C411yescharacteristic
32C23×Dic73C2×C411yescharacteristic
33C23×Dic73C2×C411yescharacteristic
34C23×Dic73C2×C411yescharacteristic
35C22×C4×D73C2×C416yes—
36C22×C4×D73D418yes—
37C22×C4×D73Q818yes—
38C22×C4×D73C2316yes—
39C2×#11924011—248no—
40C2×#11924011C23116yes—
41C22×#10596831—24no—
42C22×#10596831—24no—
43C22×#10596831C2316yes—
44C22×#10596831C2316yes—
45C2×C8×D73—248no—
46C2×C8×D73C23116yes—
47#14556548C2×C4148yes—
48#14556548C23116yes—
49C2×#11924007C2×C4112yes—
50C2×#11924007C2×C4112yes—
51C2×#11924005C2×C4112yes—
52C2×#11924005C2×C4112yes—
53C2×C4×Dic73C2×C416yes—
54C2×C4×Dic73C2316yes—
55C4⋊#10596831C23116yescomplemented
56#14556520C23116yescomplemented
57#14556520C23116yescomplemented
58C23×C4×C73C2×C411yescharacteristic
59C23×C4×C73C2×C411yescharacteristic
60C23×C4×C73C2311yescharacteristic
61C22×C8×C73—24no—
62C22×C8×C73—24no—
63C22×C8×C73C2316yes—
64C22×C8×C73C2316yes—
65C2×C42×C73C2×C416yes—
66C2×C42×C73C2316yes—
67C4⋊C8×C73C23116yescomplemented
68C23×D73—24no—
69C23×D73—24no—
70C23×D73C4⋊C418yes—
71C23×D73C22×C416yes—
72C22×Dic73—24no—
73C22×Dic73—24no—
74C22×Dic73C4⋊C418yes—
75C22×Dic73C22×C416yes—
76C22×Dic73C22×C416yes—
77C22×Dic73C22×C416yes—
78C2×C4×D73C4⋊C4148yes—
79C2×C4×D73C22×C4112yes—
80C2×C4×D73C2×D4124yes—
81C2×C4×D73C2×Q8124yes—
82C2×C4×D73C2414yes—
83#11924011—232no—
84C2×#10596831—212no—
85C2×#10596831—212no—
86C2×#10596831C2414yes—
87C2×#10596831C2414yes—
88C8×D73—232no—
89#11924007C22×C4116yes—
90#11924007C22×C4116yes—
91#11924005C22×C4116yes—
92#11924005C22×C4116yes—
93C4×Dic73C22×C4112yes—
94C4×Dic73C2414yes—
95C24×C73C22×C411yescharacteristic
96C22×C4×C73—24no—
97C22×C4×C73—24no—
98C22×C4×C73C22×C416yes—
99C22×C4×C73C22×C416yes—
100C22×C4×C73C22×C413yes—
101C22×C4×C73C2×D414yes—
102C22×C4×C73C2×Q814yes—
103C22×C4×C73C2413yes—
104C2×C8×C73—212no—
105C2×C8×C73—212no—
106C2×C8×C73C2414yes—
107C2×C8×C73C2414yes—
108C42×C73C22×C4112yes—
109C42×C73C2414yes—
110C23×C73C2×C4⋊C414yes—
111C23×C73#13114yes—
112C23×C73#13114yes—
113C23×C73C23×C413yes—
114C22×D73—212no—
115C22×D73—212no—
116C22×D73—232no—
117C22×D73C2×C4⋊C4124yes—
118C22×D73C23×C414yes—
119C2×C4×C73—212no—
120C2×C4×C73—212no—
121C2×C4×C73C2×C4⋊C4112yes—
122C2×C4×C73C23×C414yes—
123C2×C4×C73C23×C414yes—
124C2×C4×C73C23×C413yes—
125C2×C4×C73C22×D416yes—
126C2×C4×C73C22×Q816yes—
127C2×C4×C73C2511yescharacteristic
128C2×Dic73—212no—
129C2×Dic73—212no—
130C2×Dic73C2×C4⋊C4124yes—
131C2×Dic73C23×C414yes—
132C2×Dic73C23×C414yes—
133C2×Dic73C23×C414yes—
134C4×D73C2×C4⋊C4148yes—
135C4×D73C22×D418yes—
136C4×D73C22×Q818yes—
137C8×C73—24no—
138C8×C73—24no—
139#10596831—24no—
140#10596831—24no—
141C22×C73#424116yescomplemented
142C22×C73C22×C4⋊C416yes—
143C22×C73C2×#13116yes—
144C22×C73C2×#13116yes—
145C22×C73C24×C411yescharacteristic
146C2×D73—248no—
147C2×D73—24no—
148C2×D73—24no—
149C2×D73C22×C4⋊C418yes—
150C4×C73—24no—
151C4×C73—24no—
152C4×C73C22×C4⋊C416yes—
153C4×C73C23×D411yescharacteristic
154C4×C73C23×Q811yescharacteristic
155Dic73—24no—
156Dic73—24no—
157Dic73C22×C4⋊C418yes—
158C22×#424—731nomaximal, Sylow
159C2×C73C2×#424112yescomplemented
160C2×C73C23×C4⋊C411yescharacteristic
161C2×C73C22×#13111yescharacteristic
162C2×C73C22×#13111yescharacteristic
163D73—28no—
164C23×M16—731no—
165C24×C8—731no—
166C22×#117—731no—
167C2×#424—7324no—
168C22×C4⋊C8—731no—
169C22×C4⋊C8—731no—
170C22×C4⋊C8—731no—
171C22×C4⋊C8D7311yescharacteristic, complemented
172C73C22×#42411yescharacteristic, complemented, Sylow
173C24×C4—731no—
174C22×M16—7312no—
175C22×M16—1468no—
176C23×C8—731no—
177C23×C8—731no—
178C23×C8—7312no—
179C23×C8—1468no—
180C23×C8C2×D7311yescharacteristic
181C23×C8C2×D7311yescharacteristic
182C2×#117—7312no—
183C2×#117—7312no—
184C22×C4⋊C4—731no—
185C22×C4⋊C4—731no—
186C22×C22⋊C4—731no—
187C22×C22⋊C4—731no—
188C22×C42—731no—
189C22×C42C2×D7311yescharacteristic
190#424—7364no—
191C2×C4⋊C8—7312no—
192C2×C4⋊C8—7312no—
193C2×C4⋊C8—7312no—
194C2×C4⋊C8C2×D73112yescomplemented
195C25—731no—
196C23×C4—731no—
197C23×C4—736no—
198C23×C4—738no—
199C23×C4—738no—
200C23×C4—736no—
201C23×C4—731no—
202C23×C4—731no—
203C23×C4C4×D7311yescharacteristic
204C23×C4C4×D7311yescharacteristic
205C23×C4C22×D7311yescharacteristic
206C2×M16—7316no—
207C2×M16—14648no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—736no—
211C22×C8—736no—
212C22×C8—7316no—
213C22×C8—14648no—
214C22×C8—1464no—
215C22×C8—1464no—
216C22×C8C22×D7316yes—
217C22×C8C22×D7316yes—
218#117—7348no—
219#117—7316no—
220C2×C4⋊C4—7312no—
221C2×C4⋊C4—7312no—
222C2×C22⋊C4—7312no—
223C2×C22⋊C4—7312no—
224C2×C42—736no—
225C2×C42—736no—
226C2×C42C4×D7316yes—
227C2×C42C22×D7316yes—
228C4⋊C8—7316no—
229C4⋊C8—7316no—
230C4⋊C8—7316no—
231C4⋊C8C22×D73116yescomplemented
232C24—738no—
233C24—736no—
234C24—1464no—
235C24—1464no—
236C24C2×C4×D7311yescharacteristic
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—738no—
240C22×C4—736no—
241C22×C4—7348no—
242C22×C4—7312no—
243C22×C4—7324no—
244C22×C4—7324no—
245C22×C4—734no—
246C22×C4—736no—
247C22×C4—736no—
248C22×C4—1464no—
249C22×C4—1464no—
250C22×C4C2×C4×D7316yes—
251C22×C4C2×C4×D7316yes—
252C22×C4C2×C4×D7313yes—
253C22×C4D4×D7314yes—
254C22×C4Q8×D7314yes—
255C22×C4C23×D7313yes—
256M16—14632no—
257C2×C8—212no—
258C2×C8—212no—
259C2×C8—734no—
260C2×C8—734no—
261C2×C8—14632no—
262C2×C8—14612no—
263C2×C8—14612no—
264C2×C8C23×D7314yes—
265C2×C8C23×D7314yes—
266C4⋊C4—7316no—
267C4⋊C4—7316no—
268C22⋊C4—7316no—
269C22⋊C4—7316no—
270C42—7312no—
271C42—734no—
272C42C2×C4×D73112yes—
273C42C23×D7314yes—
274C23—734no—
275C23—7324no—
276C23—14632no—
277C23—14612no—
278C23—14612no—
279C23C4⋊C4×D7314yes—
280C23#1455659714yes—
281C23#1455659914yes—
282C23C22×C4×D7313yes—
283C2×C4—212no—
284C2×C4—212no—
285C2×C4—734no—
286C2×C4—7324no—
287C2×C4—738no—
288C2×C4—738no—
289C2×C4—7348no—
290C2×C4—734no—
291C2×C4—734no—
292C2×C4—14612no—
293C2×C4—14612no—
294C2×C4C4⋊C4×D73112yes—
295C2×C4C22×C4×D7314yes—
296C2×C4C22×C4×D7314yes—
297C2×C4C22×C4×D7313yes—
298C2×C4C2×D4×D7316yes—
299C2×C4C2×Q8×D7316yes—
300C2×C4C24×D7311yescharacteristic
301C8—24no—
302C8—24no—
303C8—1464no—
304C8—1464no—
305C22—738no—
306C22—14648no—
307C22—1464no—
308C22—1464no—
309C22#17988612116yesdirect product factor
310C22C2×C4⋊C4×D7316yes—
311C22C2×#1455659716yes—
312C22C2×#1455659916yes—
313C22C23×C4×D7311yescharacteristic
314C4—24no—
315C4—24no—
316C4—738no—
317C4—1464no—
318C4—1464no—
319C4C2×C4⋊C4×D7316yes—
320C4C22×D4×D7311yescharacteristic
321C4C22×Q8×D7311yescharacteristic
322C2—1468no—
323C2C2×#17988612112yesdirect product factor
324C2C22×C4⋊C4×D7311yescharacteristic
325C2C22×#1455659711yescharacteristic
326C2C22×#1455659911yescharacteristic
subgroupG/H
centerC23×C4C22×D73
commutatorC22×C73C24×C4
FrattiniC2×C4C24×D73
FittingC22×C4⋊C8×C73C2
socleC24×C73C22×C4
prime radicalC24×D73C2×C4
cosocleC2×C4×C73C25
hypercenterC22×C4⋊C8D73
hypocenterC73C22×#424
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#17988612C2direct (N × H)
5. C22×C4⋊#10596831C2semidirect
6. C22×#14556520C2semidirect
7. C22×#14556520C2semidirect
8. C22×C4⋊C8×C73C2semidirect
23. #17988612C22direct (N × H)
24. C2×C4⋊#10596831C22semidirect
25. C2×#14556520C22semidirect
26. C2×#14556520C22semidirect
30. C2×C4⋊C8×C73C22semidirect
55. C4⋊#10596831C23semidirect
56. #14556520C23semidirect
57. #14556520C23semidirect
67. C4⋊C8×C73C23semidirect
141. C22×C73#424semidirect
159. C2×C73C2×#424semidirect
171. C22×C4⋊C8D73semidirect
172. C73C22×#424semidirect
194. C2×C4⋊C8C2×D73semidirect
231. C4⋊C8C22×D73semidirect
309. C22#17988612direct (N × H)
323. C2C2×#17988612direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240112
C23×C8×D732
C22×#145565482
C2×#179886122
C22×C4⋊#105968312
C22×#145565202
C22×C4⋊C8×C732
C22×#42473