Finite Group Databasebeta

← C22×#17988907 #33953738

Subgroups — C22×#17988907 #33953738

Order 18688.

3006 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556566C211yesmaximal, characteristic
2C22×C4⋊C4×D73C211yesmaximal, characteristic, complemented
3C2×#17988907C2124yesmaximal, direct product factor
4C22×#14556522C211yesmaximal, characteristic, complemented
5C22×#14556513C211yesmaximal, characteristic, complemented
6C22×#14556511C211yesmaximal, characteristic, complemented
7C22×#14556503C211yesmaximal, characteristic, complemented
8C22×#106×C73C211yesmaximal, characteristic, complemented
9C23×D292—21no—
10C23×D292C2211yescharacteristic
11C23×C4×D73—21no—
12C23×C4×D73C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14C2×#14556566—26no—
15C2×#14556566C22112yes—
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007—21no—
18C2×C4⋊C4×D73—26no—
19C2×C4⋊C4×D73C22112yescomplemented
20C22×#11924005—21no—
21#17988907C22164yesdirect product factor
22C2×#14556522C22112yescomplemented
23C2×#14556513C22112yescomplemented
24C2×#14556511C22112yescomplemented
25C2×#14556503C22112yescomplemented
26C23×C8×C73C2211yescharacteristic
27C22×C4⋊C4×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2211yescharacteristic
29C2×#106×C73C22112yescomplemented
30C24×D73—21no—
31C24×D73D411yescharacteristic
32C22×D292—212no—
33C22×D292—23no—
34C22×D292—24no—
35C22×D292—24no—
36C22×D292—48no—
37C22×D292C2316yes—
38C23×Dic73—21no—
39C23×Dic73D411yescharacteristic
40C22×C4×D73—212no—
41C22×C4×D73—24no—
42C22×C4×D73—24no—
43C22×C4×D73—48no—
44C22×C4×D73D416yes—
45C22×C4×D73C2316yes—
46C22×#10596831—24no—
47C22×#10596831C2316yes—
48#14556566—224no—
49#14556566C23116yes—
50C2×C4⋊Dic73—23no—
51C2×C4⋊Dic73C2316yes—
52C2×#11924007—212no—
53C4⋊C4×D73—224no—
54C4⋊C4×D73C23116yescomplemented
55C2×#11924005—212no—
56#14556522C23116yescomplemented
57#14556513C23116yescomplemented
58#14556511C23116yescomplemented
59#14556503C23116yescomplemented
60C23×C4×C73—21no—
61C23×C4×C73—21no—
62C23×C4×C73C2311yescharacteristic
63C22×C8×C73—24no—
64C22×C8×C73C2316yes—
65C2×C4⋊C4×C73—23no—
66C2×C4⋊C4×C73—23no—
67C2×C4⋊C4×C73C2316yes—
68C2×C4⋊C4×C73C2316yes—
69#106×C73C23116yescomplemented
70C23×D73—24no—
71C23×D73—24no—
72C23×D73—26no—
73C23×D73—44no—
74C23×D73—44no—
75C23×D73—44no—
76C23×D73SD1618yes—
77C23×D73C2×D416yes—
78C2×D292—212no—
79C2×D292—212no—
80C2×D292—216no—
81C2×D292—26no—
82C2×D292—448no—
83C2×D292—412no—
84C2×D292C2414yes—
85C22×Dic73—24no—
86C22×Dic73—26no—
87C22×Dic73—44no—
88C22×Dic73C2×D416yes—
89C2×C4×D73—224no—
90C2×C4×D73—212no—
91C2×C4×D73—212no—
92C2×C4×D73—216no—
93C2×C4×D73—448no—
94C2×C4×D73C2×D4112yes—
95C2×C4×D73C2414yes—
96C2×#10596831—212no—
97C2×#10596831C2414yes—
98C4⋊Dic73—26no—
99C4⋊Dic73C2414yes—
100#11924007—216no—
101#11924005—216no—
102C24×C73C2×D411yescharacteristic
103C22×C4×C73—26no—
104C22×C4×C73—26no—
105C22×C4×C73—44no—
106C22×C4×C73—44no—
107C22×C4×C73C2×D413yes—
108C22×C4×C73#4114yes—
109C22×C4×C73#4114yes—
110C22×C4×C73C2413yes—
111C2×C8×C73—212no—
112C2×C8×C73C2414yes—
113C4⋊C4×C73—26no—
114C4⋊C4×C73—26no—
115C4⋊C4×C73C2414yes—
116C4⋊C4×C73C2414yes—
117C23×C73#12314yes—
118C23×C73C2×SD1614yes—
119C23×C73AGL(1,ℤ/8)14yes—
120C23×C73C22×D413yes—
121C22×D73—212no—
122C22×D73—212no—
123C22×D73—232no—
124C22×D73—24no—
125C22×D73—412no—
126C22×D73—412no—
127C22×D73—412no—
128C22×D73—816no—
129C22×D73C2×SD16124yes—
130C22×D73C22×D414yes—
131C2×C4×C73—24no—
132C2×C4×C73—24no—
133C2×C4×C73—412no—
134C2×C4×C73—412no—
135C2×C4×C73#123112yes—
136C2×C4×C73C22×D413yes—
137C2×C4×C73C2×#4116yes—
138C2×C4×C73C2×#4116yes—
139C2×C4×C73C2511yescharacteristic
140C2×Dic73—212no—
141C2×Dic73—24no—
142C2×Dic73—412no—
143C2×Dic73C22×D414yes—
144D292—24no—
145D292—24no—
146D292—432no—
147D292—412no—
148C4×D73—224no—
149C4×D73—24no—
150C4×D73—24no—
151C4×D73—432no—
152C8×C73—24no—
153#10596831—24no—
154C22×C73#481116yescomplemented
155C22×C73C2×#12316yes—
156C22×C73C22×SD1616yes—
157C22×C73C2×AGL(1,ℤ/8)16yes—
158C22×C73C23×D411yescharacteristic
159C2×D73—248no—
160C2×D73—24no—
161C2×D73—24no—
162C2×D73—44no—
163C2×D73—44no—
164C2×D73—44no—
165C2×D73—824no—
166C2×D73C22×SD1618yes—
167C4×C73—44no—
168C4×C73—44no—
169C4×C73C2×#12316yes—
170C4×C73C22×#4111yescharacteristic
171C4×C73C22×#4111yescharacteristic
172Dic73—24no—
173Dic73—44no—
174C22×#481—731nomaximal, Sylow
175C2×C73C2×#481112yescomplemented
176C2×C73C22×#12311yescharacteristic
177C2×C73C23×SD1611yescharacteristic
178C2×C73C22×AGL(1,ℤ/8)11yescharacteristic
179D73—28no—
180D73—84no—
181C22×#121—731no—
182C23×C4⋊C4—731no—
183C2×#481—7324no—
184C22×#106—731no—
185C22×#106D7311yescharacteristic, complemented
186C22×#102—731no—
187C22×#102—731no—
188C22×C22⋊C8—731no—
189C73C22×#48111yescharacteristic, complemented, Sylow
190C23×D4—731no—
191C23×D4—1461no—
192C24×C4—731no—
193C24×C4—1461no—
194C23×C8—731no—
195C23×C8C2×D7311yescharacteristic
196C2×#121—7312no—
197C2×#121—1466no—
198C22×C4⋊C4—731no—
199C22×C4⋊C4—7312no—
200C22×C4⋊C4—1466no—
201C22×C4⋊C4—1461no—
202C22×C4⋊C4C2×D7311yescharacteristic
203C22×C4⋊C4C2×D7311yescharacteristic
204C22×C22⋊C4—1461no—
205#481—7364no—
206C2×#106—7312no—
207C2×#106C2×D73112yescomplemented
208C2×#102—7312no—
209C2×#102—7312no—
210C2×C22⋊C8—7312no—
211C25—731no—
212C25—1461no—
213C22×D4—2928no—
214C22×D4—736no—
215C22×D4—14612no—
216C22×D4—1463no—
217C22×D4—1464no—
218C22×D4—1464no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—2928no—
222C23×C4—731no—
223C23×C4—736no—
224C23×C4—736no—
225C23×C4—14612no—
226C23×C4—1461no—
227C23×C4—1464no—
228C23×C4—1464no—
229C23×C4C22×D7311yescharacteristic
230C22×C8—24no—
231C22×C8—736no—
232C22×C8—1464no—
233C22×C8C22×D7316yes—
234#121—7316no—
235#121—14624no—
236C2×C4⋊C4—23no—
237C2×C4⋊C4—23no—
238C2×C4⋊C4—736no—
239C2×C4⋊C4—7316no—
240C2×C4⋊C4—14624no—
241C2×C4⋊C4—14612no—
242C2×C4⋊C4—1463no—
243C2×C4⋊C4C22×D7316yes—
244C2×C4⋊C4C22×D7316yes—
245C2×C22⋊C4—14612no—
246#106—7316no—
247#106C22×D73116yescomplemented
248#102—7316no—
249#102—7316no—
250C22⋊C8—7316no—
251C24—2924no—
252C24—2924no—
253C24—2924no—
254C24—738no—
255C24—736no—
256C24—1466no—
257C24—1464no—
258C24—1464no—
259C24D4×D7311yescharacteristic
260C2×D4—29248no—
261C2×D4—29212no—
262C2×D4—734no—
263C2×D4—14616no—
264C2×D4—1466no—
265C2×D4—14612no—
266C2×D4—14612no—
267C22×C4—26no—
268C22×C4—26no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—29248no—
272C22×C4—2924no—
273C22×C4—736no—
274C22×C4—7312no—
275C22×C4—734no—
276C22×C4—14616no—
277C22×C4—1466no—
278C22×C4—1464no—
279C22×C4—14624no—
280C22×C4—14612no—
281C22×C4—14612no—
282C22×C4D4×D7313yes—
283C22×C4#1192402814yes—
284C22×C4#1192402914yes—
285C22×C4C23×D7313yes—
286C2×C8—212no—
287C2×C8—734no—
288C2×C8—14612no—
289C2×C8C23×D7314yes—
290C4⋊C4—26no—
291C4⋊C4—26no—
292C4⋊C4—734no—
293C4⋊C4—14616no—
294C4⋊C4—1466no—
295C4⋊C4C23×D7314yes—
296C4⋊C4C23×D7314yes—
297C22⋊C4—14616no—
298C23—29212no—
299C23—29212no—
300C23—29212no—
301C23—58416no—
302C23—734no—
303C23—7324no—
304C23—1464no—
305C23—14632no—
306C23—14612no—
307C23—14612no—
308C23#1455657614yes—
309C23SD16×D7314yes—
310C23#1455661814yes—
311C23C2×D4×D7313yes—
312D4—29232no—
313D4—29212no—
314D4—1464no—
315D4—1464no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—412no—
319C2×C4—412no—
320C2×C4—29232no—
321C2×C4—29212no—
322C2×C4—734no—
323C2×C4—1464no—
324C2×C4—14612no—
325C2×C4—1464no—
326C2×C4—1464no—
327C2×C4—14624no—
328C2×C4#14556576112yes—
329C2×C4C2×D4×D7313yes—
330C2×C4C2×#1192402816yes—
331C2×C4C2×#1192402916yes—
332C2×C4C24×D7311yescharacteristic
333C8—24no—
334C8—1464no—
335C22—2924no—
336C22—2924no—
337C22—2924no—
338C22—58424no—
339C22—738no—
340C22—1464no—
341C22—14648no—
342C22—1464no—
343C22#17988907116yesdirect product factor
344C22C2×#1455657616yes—
345C22C2×SD16×D7316yes—
346C22C2×#1455661816yes—
347C22C22×D4×D7311yescharacteristic
348C4—44no—
349C4—44no—
350C4—2924no—
351C4—1464no—
352C4C2×#1455657616yes—
353C4C22×#1192402811yescharacteristic
354C4C22×#1192402911yescharacteristic
355C2—5844no—
356C2—1468no—
357C2C2×#17988907112yesdirect product factor
358C2C22×#1455657611yescharacteristic
359C2C22×SD16×D7311yescharacteristic
360C2C22×#1455661811yescharacteristic
subgroupG/H
centerC24D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC22×#106×C73C2
socleC24×C73C2×D4
prime radicalC22×#14556566C2
hypercenterC22×#106D73
hypocenterC73C22×#481
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D73C2semidirect
3. C2×#17988907C2direct (N × H)
4. C22×#14556522C2semidirect
5. C22×#14556513C2semidirect
6. C22×#14556511C2semidirect
7. C22×#14556503C2semidirect
8. C22×#106×C73C2semidirect
19. C2×C4⋊C4×D73C22semidirect
21. #17988907C22direct (N × H)
22. C2×#14556522C22semidirect
23. C2×#14556513C22semidirect
24. C2×#14556511C22semidirect
25. C2×#14556503C22semidirect
29. C2×#106×C73C22semidirect
54. C4⋊C4×D73C23semidirect
56. #14556522C23semidirect
57. #14556513C23semidirect
58. #14556511C23semidirect
59. #14556503C23semidirect
69. #106×C73C23semidirect
154. C22×C73#481semidirect
175. C2×C73C2×#481semidirect
185. C22×#106D73semidirect
189. C73C22×#481semidirect
207. C2×#106C2×D73semidirect
247. #106C22×D73semidirect
343. C22#17988907direct (N × H)
357. C2C2×#17988907direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565662
C22×C4⋊C4×D732
C2×#179889072
C22×#145565222
C22×#145565132
C22×#145565112
C22×#145565032
C22×#106×C732
C22×#48173