Finite Group Databasebeta

← C22×#17988827 #33953081

Subgroups — C22×#17988827 #33953081

Order 18688.

3006 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924018C211yesmaximal, characteristic, complemented
2C23×C8×D73C211yesmaximal, characteristic
3C22×#14556571C211yesmaximal, characteristic
4C22×#14556567C211yesmaximal, characteristic
5C2×#17988827C2124yesmaximal, direct product factor
6C22×#14556524C211yesmaximal, characteristic, complemented
7C22×#14556516C211yesmaximal, characteristic, complemented
8C22×#103×C73C211yesmaximal, characteristic, complemented
9C23×Dic146C2211yescharacteristic
10C23×C4×D73C2211yescharacteristic
11C22×#11924018—28no—
12C22×#11924018C22112yescomplemented
13C23×#10596831C2211yescharacteristic
14C22×C8×D73—28no—
15C22×C8×D73C22112yes—
16C2×#14556571—26no—
17C2×#14556571C22112yes—
18C2×#14556567—26no—
19C2×#14556567C22112yes—
20C22×C4⋊Dic73—21no—
21C22×C4⋊Dic73C2211yescharacteristic
22C22×#11924007—21no—
23C22×#11924005—21no—
24C22×#11924005—21no—
25#17988827C22164yesdirect product factor
26C2×#14556524C22112yescomplemented
27C2×#14556516C22112yescomplemented
28C23×Q8×C73C2211yescharacteristic
29C23×C8×C73C2211yescharacteristic
30C22×C4⋊C4×C73C2211yescharacteristic
31C2×#103×C73C22112yescomplemented
32C24×D73D411yescharacteristic
33C22×Dic146—23no—
34C22×Dic146—24no—
35C22×Dic146—24no—
36C22×Dic146C2316yes—
37C23×Dic73—21no—
38C23×Dic73—21no—
39C23×Dic73D411yescharacteristic
40C22×C4×D73—24no—
41C22×C4×D73—24no—
42C22×C4×D73D416yes—
43C22×C4×D73C2316yes—
44C2×#11924018—248no—
45C2×#11924018C23116yescomplemented
46C22×#10596831D418yes—
47C22×#10596831C2316yes—
48C2×C8×D73—248no—
49C2×C8×D73C23116yes—
50#14556571—224no—
51#14556571C23116yes—
52#14556567—224no—
53#14556567C23116yes—
54C2×C4⋊Dic73—212no—
55C2×C4⋊Dic73—23no—
56C2×C4⋊Dic73C2316yes—
57C2×#11924007—212no—
58C2×#11924005—212no—
59C2×#11924005—212no—
60#14556524C23116yescomplemented
61#14556516C23116yescomplemented
62C22×Q8×C73—23no—
63C22×Q8×C73—24no—
64C22×Q8×C73—24no—
65C22×Q8×C73C2316yes—
66C23×C4×C73—21no—
67C23×C4×C73—21no—
68C23×C4×C73C2311yescharacteristic
69C22×C8×C73—24no—
70C22×C8×C73C2316yes—
71C2×C4⋊C4×C73—23no—
72C2×C4⋊C4×C73C2316yes—
73#103×C73C23116yescomplemented
74C23×D73—24no—
75C23×D73—24no—
76C23×D73Q1618yescomplemented
77C23×D73C2×D416yes—
78C2×Dic146—212no—
79C2×Dic146—212no—
80C2×Dic146—26no—
81C2×Dic146—412no—
82C2×Dic146C2414yes—
83C22×Dic73—24no—
84C22×Dic73—26no—
85C22×Dic73—26no—
86C22×Dic73—44no—
87C22×Dic73—44no—
88C22×Dic73C2×D416yes—
89C2×C4×D73—224no—
90C2×C4×D73—212no—
91C2×C4×D73—212no—
92C2×C4×D73C2×D4112yes—
93C2×C4×D73C2414yes—
94#11924018—232no—
95C2×#10596831C2×D4124yes—
96C2×#10596831C2414yes—
97C8×D73—232no—
98C4⋊Dic73—26no—
99C4⋊Dic73—216no—
100C4⋊Dic73C2414yes—
101#11924007—216no—
102#11924005—216no—
103#11924005—216no—
104C24×C73C2×D411yescharacteristic
105C2×Q8×C73—26no—
106C2×Q8×C73—212no—
107C2×Q8×C73—212no—
108C2×Q8×C73—412no—
109C2×Q8×C73C2414yes—
110C22×C4×C73—26no—
111C22×C4×C73—26no—
112C22×C4×C73—44no—
113C22×C4×C73—44no—
114C22×C4×C73C2×D413yes—
115C22×C4×C73C2×D414yes—
116C22×C4×C73#4114yes—
117C22×C4×C73C2413yes—
118C2×C8×C73—212no—
119C2×C8×C73C2414yes—
120C4⋊C4×C73—26no—
121C4⋊C4×C73C2414yes—
122C23×C73#12114yes—
123C23×C73C2×Q1614yes—
124C23×C73#13514yes—
125C23×C73C22×D413yes—
126C22×D73—212no—
127C22×D73—212no—
128C22×D73—232no—
129C22×D73C2×Q16124yescomplemented
130C22×D73C22×D414yes—
131Q8×C73—24no—
132Q8×C73—24no—
133Q8×C73—412no—
134C2×C4×C73—24no—
135C2×C4×C73—24no—
136C2×C4×C73—412no—
137C2×C4×C73—412no—
138C2×C4×C73#121112yes—
139C2×C4×C73C22×D413yes—
140C2×C4×C73C22×D416yes—
141C2×C4×C73C2×#4116yes—
142C2×C4×C73C2511yescharacteristic
143C2×Dic73—212no—
144C2×Dic73—24no—
145C2×Dic73—24no—
146C2×Dic73—412no—
147C2×Dic73—412no—
148C2×Dic73C22×D414yes—
149C4×D73—224no—
150C4×D73—24no—
151C4×D73—24no—
152Dic146—24no—
153Dic146—24no—
154Dic146—412no—
155C8×C73—24no—
156#10596831C22×D418yes—
157C22×C73#467116yescomplemented
158C22×C73C2×#12116yes—
159C22×C73C22×Q1616yes—
160C22×C73C2×#13516yes—
161C22×C73C23×D411yescharacteristic
162C2×D73—24no—
163C2×D73—248no—
164C2×D73—24no—
165C2×D73C22×Q1618yescomplemented
166C4×C73—44no—
167C4×C73—44no—
168C4×C73C2×#12116yes—
169C4×C73C23×D411yescharacteristic
170C4×C73C22×#4111yescharacteristic
171Dic73—24no—
172Dic73—44no—
173Dic73—44no—
174C22×#467—731nomaximal, Sylow
175C2×C73C2×#467112yescomplemented
176C2×C73C22×#12111yescharacteristic
177C2×C73C23×Q1611yescharacteristic
178C2×C73C22×#13511yescharacteristic
179D73—28no—
180C23×Q16—731no—
181C24×C8—731no—
182C22×C22⋊Q8—731no—
183C22×C22⋊Q8—731no—
184C2×#467—7324no—
185C22×#107—731no—
186C22×#103—731no—
187C22×#103D7311yescharacteristic, complemented
188C73C22×#46711yescharacteristic, complemented, Sylow
189C23×Q8—731no—
190C23×Q8C2×D7311yescharacteristic
191C24×C4—731no—
192C22×Q16—7312no—
193C22×Q16—1468no—
194C23×C8—731no—
195C23×C8—7312no—
196C23×C8—1468no—
197C23×C8C2×D7311yescharacteristic
198C2×C22⋊Q8—7312no—
199C2×C22⋊Q8—7312no—
200C2×C22⋊Q8—1466no—
201C2×C22⋊Q8—1466no—
202C22×C4⋊C4—731no—
203C22×C4⋊C4—1461no—
204C22×C4⋊C4—1461no—
205C22×C4⋊C4C2×D7311yescharacteristic
206C22×C22⋊C4—1461no—
207C22×C22⋊C4—1461no—
208#467—7364no—
209C2×#107—7312no—
210C2×#103—7312no—
211C2×#103C2×D73112yescomplemented
212C25—731no—
213C22×Q8—23no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—736no—
217C22×Q8—1464no—
218C22×Q8—1464no—
219C22×Q8—1463no—
220C22×Q8C22×D7316yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—731no—
224C23×C4—736no—
225C23×C4—736no—
226C23×C4—1461no—
227C23×C4—1461no—
228C23×C4—1464no—
229C23×C4—1464no—
230C23×C4C22×D7311yescharacteristic
231C2×Q16—7316no—
232C2×Q16—14648no—
233C22×C8—24no—
234C22×C8—738no—
235C22×C8—736no—
236C22×C8—7316no—
237C22×C8—14648no—
238C22×C8C22×D7316yes—
239C22⋊Q8—7316no—
240C22⋊Q8—7316no—
241C22⋊Q8—14624no—
242C22⋊Q8—14624no—
243C2×C4⋊C4—23no—
244C2×C4⋊C4—736no—
245C2×C4⋊C4—14612no—
246C2×C4⋊C4—14612no—
247C2×C4⋊C4—1463no—
248C2×C4⋊C4C22×D7316yes—
249C2×C22⋊C4—14612no—
250C2×C22⋊C4—14612no—
251#107—7316no—
252#103—7316no—
253#103C22×D73116yescomplemented
254C24—738no—
255C24—736no—
256C24—1464no—
257C24—1464no—
258C24D4×D7311yescharacteristic
259C2×Q8—26no—
260C2×Q8—212no—
261C2×Q8—212no—
262C2×Q8—412no—
263C2×Q8—29212no—
264C2×Q8—734no—
265C2×Q8—14612no—
266C2×Q8—14612no—
267C2×Q8—1466no—
268C2×Q8C23×D7314yes—
269C22×C4—26no—
270C22×C4—26no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—2924no—
274C22×C4—2924no—
275C22×C4—736no—
276C22×C4—7312no—
277C22×C4—734no—
278C22×C4—1466no—
279C22×C4—1466no—
280C22×C4—1464no—
281C22×C4—14624no—
282C22×C4—14612no—
283C22×C4—14612no—
284C22×C4D4×D7313yes—
285C22×C4D4×D7314yes—
286C22×C4#1192402814yes—
287C22×C4C23×D7313yes—
288Q16—14632no—
289C2×C8—212no—
290C2×C8—7324no—
291C2×C8—734no—
292C2×C8—14632no—
293C2×C8C23×D7314yes—
294C4⋊C4—26no—
295C4⋊C4—734no—
296C4⋊C4—14616no—
297C4⋊C4—14616no—
298C4⋊C4—1466no—
299C4⋊C4C23×D7314yes—
300C22⋊C4—14616no—
301C22⋊C4—14616no—
302C23—734no—
303C23—7324no—
304C23—14612no—
305C23—14632no—
306C23—14612no—
307C23#1455656414yes—
308C23Q16×D7314yes—
309C23#1455661214yes—
310C23C2×D4×D7313yes—
311Q8—24no—
312Q8—24no—
313Q8—412no—
314Q8—29212no—
315Q8—1464no—
316Q8—1464no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—412no—
320C2×C4—412no—
321C2×C4—29212no—
322C2×C4—29212no—
323C2×C4—734no—
324C2×C4—1464no—
325C2×C4—1464no—
326C2×C4—14612no—
327C2×C4—1464no—
328C2×C4—1464no—
329C2×C4—14624no—
330C2×C4#14556564112yes—
331C2×C4C2×D4×D7316yes—
332C2×C4C2×D4×D7313yes—
333C2×C4C2×#1192402816yes—
334C2×C4C24×D7311yescharacteristic
335C8—24no—
336C8—738no—
337C22—738no—
338C22—14648no—
339C22—1464no—
340C22—1464no—
341C22#17988827116yesdirect product factor
342C22C2×#1455656416yes—
343C22C2×Q16×D7316yes—
344C22C2×#1455661216yes—
345C22C22×D4×D7311yescharacteristic
346C4—44no—
347C4—44no—
348C4—2924no—
349C4—2924no—
350C4—1464no—
351C4C2×#1455656416yes—
352C4C22×D4×D7311yescharacteristic
353C4C22×#1192402811yescharacteristic
354C2—1468no—
355C2C2×#17988827112yesdirect product factor
356C2C22×#1455656411yescharacteristic
357C2C22×Q16×D7311yescharacteristic
358C2C22×#1455661211yescharacteristic
subgroupG/H
centerC24D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC22×#103×C73C2
socleC24×C73C2×D4
prime radicalC24×D73D4
hypercenterC22×#103D73
hypocenterC73C22×#467
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C23×#11924018C2semidirect
5. C2×#17988827C2direct (N × H)
6. C22×#14556524C2semidirect
7. C22×#14556516C2semidirect
8. C22×#103×C73C2semidirect
12. C22×#11924018C22semidirect
25. #17988827C22direct (N × H)
26. C2×#14556524C22semidirect
27. C2×#14556516C22semidirect
31. C2×#103×C73C22semidirect
45. C2×#11924018C23semidirect
60. #14556524C23semidirect
61. #14556516C23semidirect
73. #103×C73C23semidirect
76. C23×D73Q16semidirect
129. C22×D73C2×Q16semidirect
157. C22×C73#467semidirect
165. C2×D73C22×Q16semidirect
175. C2×C73C2×#467semidirect
187. C22×#103D73semidirect
188. C73C22×#467semidirect
211. C2×#103C2×D73semidirect
253. #103C22×D73semidirect
341. C22#17988827direct (N × H)
355. C2C2×#17988827direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240182
C23×C8×D732
C22×#145565712
C22×#145565672
C2×#179888272
C22×#145565242
C22×#145565162
C22×#103×C732
C22×#46773