Finite Group Databasebeta

← C22×#17988810 #33952793

Subgroups — C22×#17988810 #33952793

Order 18688.

2918 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×Dic292C211yesmaximal, characteristic, complemented
2C23×#11924017C211yesmaximal, characteristic
3C22×#14556579C211yesmaximal, characteristic
4C22×C4×Dic146C211yesmaximal, characteristic, complemented
5C2×#17988810C2124yesmaximal, direct product factor
6C22×#14556516C211yesmaximal, characteristic, complemented
7C22×#14556513C211yesmaximal, characteristic
8C22×C4⋊C8×C73C211yesmaximal, characteristic, complemented
9C23×Dic146C2211yescharacteristic
10C23×Dic146C2211yescharacteristic
11C23×D292C2211yescharacteristic
12C22×Dic292—28no—
13C22×Dic292C22112yescomplemented
14C22×#11924017—28no—
15C22×#11924017C22112yes—
16C2×#14556579—26no—
17C2×#14556579C22112yes—
18C2×C4×Dic146—26no—
19C2×C4×Dic146C22112yescomplemented
20C22×C4⋊Dic73C2211yescharacteristic
21C22×#11924007—21no—
22C22×#11924005—21no—
23C22×#11924005—21no—
24C22×C4×Dic73—21no—
25#17988810C22164yesdirect product factor
26C2×#14556516C22112yescomplemented
27C2×#14556513C22112yes—
28C23×C8×C73C2211yescharacteristic
29C23×C8×C73C2211yescharacteristic
30C22×C42×C73C2211yescharacteristic
31C2×C4⋊C8×C73C22112yescomplemented
32C24×D73—21no—
33C22×Dic146—23no—
34C22×Dic146—23no—
35C22×Dic146—24no—
36C22×Dic146—24no—
37C22×Dic146—24no—
38C22×Dic146D418yes—
39C22×Dic146C2316yes—
40C22×Dic146C2316yes—
41C22×D292—24no—
42C22×D292—23no—
43C22×D292—24no—
44C22×D292C2316yes—
45C23×Dic73—21no—
46C23×Dic73—21no—
47C23×Dic73—21no—
48C2×Dic292—248no—
49C2×Dic292C23116yescomplemented
50C2×#11924017—248no—
51C2×#11924017C23116yes—
52#14556579—224no—
53#14556579C23116yes—
54C4×Dic146—224no—
55C4×Dic146C23116yescomplemented
56C2×C4⋊Dic73—23no—
57C2×C4⋊Dic73C2316yes—
58C2×#11924007—212no—
59C2×#11924005—212no—
60C2×#11924005—212no—
61C2×C4×Dic73—212no—
62#14556516C23116yescomplemented
63#14556513C23116yes—
64C23×C4×C73D411yescharacteristic
65C23×C4×C73D411yescharacteristic
66C23×C4×C73C2311yescharacteristic
67C22×C8×C73—24no—
68C22×C8×C73—24no—
69C22×C8×C73C2316yes—
70C22×C8×C73C2316yes—
71C2×C42×C73D416yes—
72C2×C42×C73C2316yes—
73C4⋊C8×C73C23116yescomplemented
74C23×D73—26no—
75C23×D73—44no—
76C23×D73—44no—
77C23×D73—44no—
78C2×Dic146—224no—
79C2×Dic146—26no—
80C2×Dic146—212no—
81C2×Dic146—212no—
82C2×Dic146—212no—
83C2×Dic146—26no—
84C2×Dic146—412no—
85C2×Dic146C2×D4124yes—
86C2×Dic146C2414yes—
87C2×Dic146C2414yes—
88C2×D292—212no—
89C2×D292—212no—
90C2×D292—26no—
91C2×D292—412no—
92C2×D292C2414yes—
93C22×Dic73—28no—
94C22×Dic73—26no—
95C22×Dic73—26no—
96C22×Dic73—26no—
97C22×Dic73—44no—
98C22×Dic73—44no—
99Dic292—232no—
100#11924017—232no—
101C4⋊Dic73—26no—
102C4⋊Dic73C2414yes—
103#11924007—216no—
104#11924005—216no—
105#11924005—216no—
106C4×Dic73—216no—
107C24×C73C2×D411yescharacteristic
108C22×C4×C73—24no—
109C22×C4×C73—24no—
110C22×C4×C73C2×D414yes—
111C22×C4×C73C2×D416yes—
112C22×C4×C73C2×D416yes—
113C22×C4×C73C2×D413yes—
114C22×C4×C73#4114yes—
115C22×C4×C73C2413yes—
116C2×C8×C73—212no—
117C2×C8×C73—212no—
118C2×C8×C73C2414yes—
119C2×C8×C73C2414yes—
120C42×C73C2×D4112yes—
121C42×C73C2414yes—
122C23×C73#12114yes—
123C23×C73#13514yes—
124C23×C73#13714yes—
125C23×C73C22×D413yes—
126C22×D73—24no—
127C22×D73—412no—
128C22×D73—412no—
129C22×D73—412no—
130C22×D73—816no—
131C2×C4×C73—212no—
132C2×C4×C73—212no—
133C2×C4×C73#121112yes—
134C2×C4×C73C22×D416yes—
135C2×C4×C73C22×D414yes—
136C2×C4×C73C22×D414yes—
137C2×C4×C73C22×D413yes—
138C2×C4×C73C2×#4116yes—
139C2×C4×C73C2511yescharacteristic
140C2×Dic73—224no—
141C2×Dic73—24no—
142C2×Dic73—24no—
143C2×Dic73—24no—
144C2×Dic73—412no—
145C2×Dic73—412no—
146D292—24no—
147D292—24no—
148D292—412no—
149Dic146—224no—
150Dic146—24no—
151Dic146—24no—
152Dic146—24no—
153Dic146—412no—
154Dic146C22×D418yes—
155C8×C73—24no—
156C8×C73—24no—
157C22×C73#464116yescomplemented
158C22×C73C2×#12116yes—
159C22×C73C2×#13516yes—
160C22×C73C2×#13716yes—
161C22×C73C23×D411yescharacteristic
162C2×D73—44no—
163C2×D73—44no—
164C2×D73—44no—
165C2×D73—824no—
166C4×C73—24no—
167C4×C73—24no—
168C4×C73C2×#12116yes—
169C4×C73C23×D411yescharacteristic
170C4×C73C22×#4111yescharacteristic
171Dic73—28no—
172Dic73—44no—
173Dic73—44no—
174C22×#464—731nomaximal, Sylow
175C2×C73C2×#464112yescomplemented
176C2×C73C22×#12111yescharacteristic
177C2×C73C22×#13511yescharacteristic
178C2×C73C22×#13711yescharacteristic
179D73—84no—
180C23×Q16—731no—
181C23×SD16—731no—
182C22×#124—731no—
183C22×C4×Q8—731no—
184C2×#464—7324no—
185C22×C4⋊C8D7311yescharacteristic, complemented
186C22×#103—731no—
187C22×#102—731no—
188C73C22×#46411yescharacteristic, complemented, Sylow
189C23×Q8—731no—
190C23×Q8—731no—
191C23×D4—731no—
192C22×Q16—7312no—
193C22×Q16—1468no—
194C22×SD16—7312no—
195C22×SD16—1468no—
196C23×C8C2×D7311yescharacteristic
197C23×C8C2×D7311yescharacteristic
198C2×#124—7312no—
199C2×#124—1466no—
200C2×C4×Q8—7312no—
201C2×C4×Q8—1466no—
202C22×C4⋊C4—731no—
203C22×C4⋊C4—1461no—
204C22×C22⋊C4—1461no—
205C22×C22⋊C4—1461no—
206C22×C42—1461no—
207C22×C42C2×D7311yescharacteristic
208#464—7364no—
209C2×C4⋊C8C2×D73112yescomplemented
210C2×#103—7312no—
211C2×#102—7312no—
212C25—1461no—
213C22×Q8—738no—
214C22×Q8—736no—
215C22×Q8—736no—
216C22×Q8—1464no—
217C22×Q8—1464no—
218C22×Q8—1463no—
219C22×Q8—1463no—
220C22×Q8—1464no—
221C22×D4—736no—
222C22×D4—1464no—
223C22×D4—1463no—
224C22×D4—1464no—
225C23×C4—1461no—
226C23×C4—1461no—
227C23×C4—1461no—
228C23×C4D29211yescharacteristic
229C23×C4D29211yescharacteristic
230C23×C4C22×D7311yescharacteristic
231C2×Q16—7316no—
232C2×Q16—14648no—
233C2×SD16—7316no—
234C2×SD16—14648no—
235C22×C8—24no—
236C22×C8—24no—
237C22×C8C22×D7316yes—
238C22×C8C22×D7316yes—
239#124—7316no—
240#124—14624no—
241C4×Q8—7316no—
242C4×Q8—14624no—
243C2×C4⋊C4—736no—
244C2×C4⋊C4—1463no—
245C2×C4⋊C4—14612no—
246C2×C22⋊C4—14612no—
247C2×C22⋊C4—14612no—
248C2×C42—14612no—
249C2×C42D29216yes—
250C2×C42C22×D7316yes—
251C4⋊C8C22×D73116yescomplemented
252#103—7316no—
253#102—7316no—
254C24—2924no—
255C24—2924no—
256C24—2924no—
257C24—1466no—
258C24C2×D29211yescharacteristic
259C2×Q8—29212no—
260C2×Q8—7324no—
261C2×Q8—734no—
262C2×Q8—734no—
263C2×Q8—14612no—
264C2×Q8—14612no—
265C2×Q8—14624no—
266C2×Q8—1466no—
267C2×Q8—1466no—
268C2×Q8—14612no—
269C2×D4—29212no—
270C2×D4—734no—
271C2×D4—14612no—
272C2×D4—1466no—
273C2×D4—14612no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—2924no—
277C22×C4—2924no—
278C22×C4—1466no—
279C22×C4—1466no—
280C22×C4—1468no—
281C22×C4—1466no—
282C22×C4D4×D7314yes—
283C22×C4C2×D29216yes—
284C22×C4C2×D29216yes—
285C22×C4C2×D29213yes—
286C22×C4#1192402914yes—
287C22×C4C23×D7313yes—
288Q16—14632no—
289SD16—14632no—
290C2×C8—212no—
291C2×C8—212no—
292C2×C8C23×D7314yes—
293C2×C8C23×D7314yes—
294C4⋊C4—734no—
295C4⋊C4—14616no—
296C4⋊C4—1466no—
297C22⋊C4—14616no—
298C22⋊C4—14616no—
299C42—14616no—
300C42C2×D292112yes—
301C42C23×D7314yes—
302C23—29212no—
303C23—29212no—
304C23—29212no—
305C23—58416no—
306C23—1464no—
307C23#1455656614yes—
308C23#1455661314yes—
309C23#1455662414yes—
310C23C22×D29213yes—
311Q8—29212no—
312Q8—738no—
313Q8—1464no—
314Q8—1464no—
315Q8—1464no—
316Q8—14624no—
317D4—29212no—
318D4—1464no—
319D4—1464no—
320C2×C4—212no—
321C2×C4—212no—
322C2×C4—29212no—
323C2×C4—29212no—
324C2×C4—1464no—
325C2×C4—1464no—
326C2×C4—1464no—
327C2×C4—14624no—
328C2×C4#14556566112yes—
329C2×C4C2×D4×D7316yes—
330C2×C4C22×D29214yes—
331C2×C4C22×D29214yes—
332C2×C4C22×D29213yes—
333C2×C4C2×#1192402916yes—
334C2×C4C24×D7311yescharacteristic
335C8—24no—
336C8—24no—
337C22—2924no—
338C22—2924no—
339C22—2924no—
340C22—58424no—
341C22#17988810116yesdirect product factor
342C22C2×#1455656616yes—
343C22C2×#1455661316yes—
344C22C2×#1455662416yes—
345C22C23×D29211yescharacteristic
346C4—24no—
347C4—24no—
348C4—2924no—
349C4—2924no—
350C4—1468no—
351C4C2×#1455656616yes—
352C4C22×D4×D7311yescharacteristic
353C4C22×#1192402911yescharacteristic
354C2—5844no—
355C2C2×#17988810112yesdirect product factor
356C2C22×#1455656611yescharacteristic
357C2C22×#1455661311yescharacteristic
358C2C22×#1455662411yescharacteristic
subgroupG/H
centerC24C2×D292
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC22×C4⋊C8×C73C2
socleC24×C73C2×D4
prime radicalC23×D292C22
hypercenterC22×C4⋊C8D73
hypocenterC73C22×#464
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C23×Dic292C2semidirect
4. C22×C4×Dic146C2semidirect
5. C2×#17988810C2direct (N × H)
6. C22×#14556516C2semidirect
8. C22×C4⋊C8×C73C2semidirect
13. C22×Dic292C22semidirect
19. C2×C4×Dic146C22semidirect
25. #17988810C22direct (N × H)
26. C2×#14556516C22semidirect
31. C2×C4⋊C8×C73C22semidirect
49. C2×Dic292C23semidirect
55. C4×Dic146C23semidirect
62. #14556516C23semidirect
73. C4⋊C8×C73C23semidirect
157. C22×C73#464semidirect
175. C2×C73C2×#464semidirect
185. C22×C4⋊C8D73semidirect
188. C73C22×#464semidirect
209. C2×C4⋊C8C2×D73semidirect
251. C4⋊C8C22×D73semidirect
341. C22#17988810direct (N × H)
355. C2C2×#17988810direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Dic2922
C23×#119240172
C22×#145565792
C22×C4×Dic1462
C2×#179888102
C22×#145565162
C22×#145565132
C22×C4⋊C8×C732
C22×#46473