Finite Group Databasebeta

← C22×#17988932 #33955167

Subgroups — C22×#17988932 #33955167

Order 18688.

3086 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556590C211yesmaximal, characteristic, complemented
2C22×#14556587C211yesmaximal, characteristic
3C2×#17988932C2124yesmaximal, direct product factor
4C22×#14556513C211yesmaximal, characteristic, complemented
5C22×#14556512C211yesmaximal, characteristic, complemented
6C22×#14556511C211yesmaximal, characteristic, complemented
7C22×C8×Dic73C211yesmaximal, characteristic, complemented
8C22×#102×C73C211yesmaximal, characteristic, complemented
9C23×Dic146—21no—
10C23×D292C2211yescharacteristic
11C23×C22⋊D73—21no—
12C23×C22⋊D73—21no—
13C23×#10596831C2211yescharacteristic
14C2×#14556590—26no—
15C2×#14556590C22112yescomplemented
16C2×#14556587—26no—
17C2×#14556587C22112yes—
18C22×C4⋊Dic73C2211yescharacteristic
19C22×#11924007—21no—
20C22×C4×Dic73C2211yescharacteristic
21#17988932C22164yesdirect product factor
22C2×#14556513C22112yescomplemented
23C2×#14556512C22112yescomplemented
24C2×#14556511C22112yescomplemented
25C2×C8×Dic73C22112yescomplemented
26C23×D4×C73C2211yescharacteristic
27C23×C8×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2211yescharacteristic
29C2×#102×C73C22112yescomplemented
30C24×D73—21no—
31C22×Dic146—212no—
32C22×Dic146—48no—
33C22×D292—23no—
34C22×D292—24no—
35C22×D292—24no—
36C22×D292C2316yes—
37C22×C22⋊D73—212no—
38C22×C22⋊D73—212no—
39C22×C22⋊D73—48no—
40C22×C22⋊D73—48no—
41C23×Dic73—21no—
42C23×Dic73D411yescharacteristic
43C23×Dic73D411yescharacteristic
44C22×#10596831—24no—
45C22×#10596831C2316yes—
46#14556590—224no—
47#14556590C23116yescomplemented
48#14556587—224no—
49#14556587C23116yes—
50C2×C4⋊Dic73—23no—
51C2×C4⋊Dic73C2316yes—
52C2×#11924007—212no—
53C2×C4×Dic73D416yes—
54C2×C4×Dic73C2316yes—
55#14556513C23116yescomplemented
56#14556512C23116yescomplemented
57#14556511C23116yescomplemented
58C8×Dic73C23116yescomplemented
59C25×C73—21no—
60C22×D4×C73—23no—
61C22×D4×C73—24no—
62C22×D4×C73—24no—
63C22×D4×C73C2316yes—
64C23×C4×C73—21no—
65C23×C4×C73C2311yescharacteristic
66C22×C8×C73—24no—
67C22×C8×C73C2316yes—
68C2×C4⋊C4×C73—23no—
69C2×C4⋊C4×C73C2316yes—
70#102×C73C23116yescomplemented
71C23×D73—26no—
72C23×D73—44no—
73C23×D73—44no—
74C23×D73—44no—
75C2×Dic146—216no—
76C2×Dic146—448no—
77C2×D292—212no—
78C2×D292—212no—
79C2×D292—26no—
80C2×D292—412no—
81C2×D292C2414yes—
82C2×C22⋊D73—216no—
83C2×C22⋊D73—216no—
84C2×C22⋊D73—448no—
85C2×C22⋊D73—448no—
86C22×Dic73—26no—
87C22×Dic73—44no—
88C22×Dic73D818yes—
89C22×Dic73SD1618yes—
90C22×Dic73C2×D416yes—
91C22×Dic73C2×D416yes—
92C2×#10596831—212no—
93C2×#10596831C2414yes—
94C4⋊Dic73—26no—
95C4⋊Dic73C2414yes—
96#11924007—216no—
97C4×Dic73C2×D4112yes—
98C4×Dic73C2414yes—
99C24×C73—26no—
100C24×C73—44no—
101C24×C73—44no—
102C24×C73—44no—
103C24×C73C2×D411yescharacteristic
104C2×D4×C73—26no—
105C2×D4×C73—212no—
106C2×D4×C73—212no—
107C2×D4×C73—412no—
108C2×D4×C73C2414yes—
109C22×C4×C73—26no—
110C22×C4×C73—44no—
111C22×C4×C73C2×D413yes—
112C22×C4×C73#4114yes—
113C22×C4×C73#4114yes—
114C22×C4×C73C2413yes—
115C2×C8×C73—212no—
116C2×C8×C73C2414yes—
117C4⋊C4×C73—26no—
118C4⋊C4×C73C2414yes—
119C23×C73—24no—
120C23×C73—412no—
121C23×C73—412no—
122C23×C73—412no—
123C23×C73—816no—
124C23×C73#12414yes—
125C23×C73C2×D814yes—
126C23×C73C2×SD1614yes—
127C23×C73C22×D413yes—
128C22×D73—24no—
129C22×D73—412no—
130C22×D73—412no—
131C22×D73—412no—
132C22×D73—816no—
133D4×C73—24no—
134D4×C73—24no—
135D4×C73—412no—
136C2×C4×C73—24no—
137C2×C4×C73—412no—
138C2×C4×C73#124112yes—
139C2×C4×C73C22×D413yes—
140C2×C4×C73C2×#4116yes—
141C2×C4×C73C2×#4116yes—
142C2×C4×C73C2511yescharacteristic
143C22⋊D73—432no—
144C22⋊D73—432no—
145C2×Dic73—24no—
146C2×Dic73—412no—
147C2×Dic73C2×D8124yes—
148C2×Dic73C2×SD16124yes—
149C2×Dic73C22×D414yes—
150C2×Dic73C22×D414yes—
151D292—24no—
152D292—24no—
153D292—412no—
154Dic146—432no—
155C8×C73—24no—
156#10596831—24no—
157C22×C73—44no—
158C22×C73—44no—
159C22×C73—44no—
160C22×C73—824no—
161C22×C73#486116yescomplemented
162C22×C73C2×#12416yes—
163C22×C73C22×D816yes—
164C22×C73C22×SD1616yes—
165C22×C73C23×D411yescharacteristic
166C2×D73—44no—
167C2×D73—44no—
168C2×D73—44no—
169C2×D73—824no—
170C4×C73—44no—
171C4×C73C2×#12416yes—
172C4×C73C22×#4111yescharacteristic
173C4×C73C22×#4111yescharacteristic
174Dic73—44no—
175Dic73C22×D818yes—
176Dic73C22×SD1618yes—
177C22×#486—731nomaximal, Sylow
178C2×C73—84no—
179C2×C73C2×#486112yescomplemented
180C2×C73C22×#12411yescharacteristic
181C2×C73C23×D811yescharacteristic
182C2×C73C23×SD1611yescharacteristic
183D73—84no—
184C22×C4⋊Q8—731no—
185C22×#127—731no—
186C2×#486—7324no—
187C22×#102—731no—
188C22×#102—731no—
189C22×#102—731no—
190C22×#102D7311yescharacteristic, complemented
191C22×C4×C8—731no—
192C73C22×#48611yescharacteristic, complemented, Sylow
193C23×Q8—1461no—
194C23×D4—731no—
195C23×D4—1461no—
196C23×D4—1461no—
197C23×D4C2×D7311yescharacteristic
198C23×C8—731no—
199C23×C8C2×D7311yescharacteristic
200C2×C4⋊Q8—7312no—
201C2×C4⋊Q8—1466no—
202C2×#127—7312no—
203C2×#127—1466no—
204C22×C4⋊C4—731no—
205C22×C4⋊C4—1461no—
206C22×C4⋊C4C2×D7311yescharacteristic
207C22×C42—731no—
208#486—7364no—
209C2×#102—7312no—
210C2×#102—7312no—
211C2×#102—7312no—
212C2×#102C2×D73112yescomplemented
213C2×C4×C8—7312no—
214C25—21no—
215C25—1461no—
216C22×Q8—2928no—
217C22×Q8—14612no—
218C22×D4—23no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—2928no—
222C22×D4—2928no—
223C22×D4—736no—
224C22×D4—14612no—
225C22×D4—14612no—
226C22×D4—1463no—
227C22×D4—1464no—
228C22×D4—1464no—
229C22×D4C22×D7316yes—
230C23×C4—21no—
231C23×C4—731no—
232C23×C4—731no—
233C23×C4—1461no—
234C23×C4C22×D7311yescharacteristic
235C22×C8—24no—
236C22×C8—736no—
237C22×C8—1464no—
238C22×C8C22×D7316yes—
239C4⋊Q8—7316no—
240C4⋊Q8—14624no—
241#127—7316no—
242#127—14624no—
243C2×C4⋊C4—23no—
244C2×C4⋊C4—736no—
245C2×C4⋊C4—14612no—
246C2×C4⋊C4—1463no—
247C2×C4⋊C4C22×D7316yes—
248C2×C42—736no—
249C2×C42—736no—
250#102—7316no—
251#102—7316no—
252#102—7316no—
253#102C22×D73116yescomplemented
254C4×C8—7316no—
255C24—26no—
256C24—44no—
257C24—44no—
258C24—44no—
259C24—2924no—
260C24—2924no—
261C24—2924no—
262C24—1466no—
263C24D4×D7311yescharacteristic
264C2×Q8—29248no—
265C2×Q8—14616no—
266C2×D4—26no—
267C2×D4—212no—
268C2×D4—212no—
269C2×D4—412no—
270C2×D4—29248no—
271C2×D4—29248no—
272C2×D4—29212no—
273C2×D4—734no—
274C2×D4—14616no—
275C2×D4—14616no—
276C2×D4—1466no—
277C2×D4—14612no—
278C2×D4—14612no—
279C2×D4C23×D7314yes—
280C22×C4—26no—
281C22×C4—44no—
282C22×C4—2924no—
283C22×C4—738no—
284C22×C4—738no—
285C22×C4—736no—
286C22×C4—736no—
287C22×C4—1466no—
288C22×C4D4×D7313yes—
289C22×C4#1192402714yes—
290C22×C4#1192402814yes—
291C22×C4C23×D7313yes—
292C2×C8—212no—
293C2×C8—734no—
294C2×C8—14612no—
295C2×C8C23×D7314yes—
296C4⋊C4—26no—
297C4⋊C4—734no—
298C4⋊C4—14616no—
299C4⋊C4—1466no—
300C4⋊C4C23×D7314yes—
301C42—7312no—
302C42—734no—
303C23—24no—
304C23—412no—
305C23—412no—
306C23—412no—
307C23—816no—
308C23—29212no—
309C23—29212no—
310C23—29212no—
311C23—58416no—
312C23—1464no—
313C23#1455657714yes—
314C23D8×D7314yes—
315C23SD16×D7314yes—
316C23C2×D4×D7313yes—
317Q8—29232no—
318D4—24no—
319D4—24no—
320D4—412no—
321D4—29232no—
322D4—29232no—
323D4—29212no—
324D4—1464no—
325D4—1464no—
326C2×C4—24no—
327C2×C4—412no—
328C2×C4—29212no—
329C2×C4—734no—
330C2×C4—734no—
331C2×C4—7324no—
332C2×C4—7324no—
333C2×C4—1464no—
334C2×C4#14556577112yes—
335C2×C4C2×D4×D7313yes—
336C2×C4C2×#1192402716yes—
337C2×C4C2×#1192402816yes—
338C2×C4C24×D7311yescharacteristic
339C8—24no—
340C8—1464no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—824no—
345C22—2924no—
346C22—2924no—
347C22—2924no—
348C22—58424no—
349C22#17988932116yesdirect product factor
350C22C2×#1455657716yes—
351C22C2×D8×D7316yes—
352C22C2×SD16×D7316yes—
353C22C22×D4×D7311yescharacteristic
354C4—44no—
355C4—2924no—
356C4—738no—
357C4—738no—
358C4C2×#1455657716yes—
359C4C22×#1192402711yescharacteristic
360C4C22×#1192402811yescharacteristic
361C2—84no—
362C2—5844no—
363C2C2×#17988932112yesdirect product factor
364C2C22×#1455657711yescharacteristic
365C2C22×D8×D7311yescharacteristic
366C2C22×SD16×D7311yescharacteristic
subgroupG/H
centerC24D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC22×#102×C73C2
socleC24×C73C2×D4
prime radicalC22×#14556587C2
hypercenterC22×#102D73
hypocenterC73C22×#486
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14556590C2semidirect
3. C2×#17988932C2direct (N × H)
4. C22×#14556513C2semidirect
5. C22×#14556512C2semidirect
6. C22×#14556511C2semidirect
7. C22×C8×Dic73C2semidirect
8. C22×#102×C73C2semidirect
15. C2×#14556590C22semidirect
21. #17988932C22direct (N × H)
22. C2×#14556513C22semidirect
23. C2×#14556512C22semidirect
24. C2×#14556511C22semidirect
25. C2×C8×Dic73C22semidirect
29. C2×#102×C73C22semidirect
47. #14556590C23semidirect
55. #14556513C23semidirect
56. #14556512C23semidirect
57. #14556511C23semidirect
58. C8×Dic73C23semidirect
70. #102×C73C23semidirect
161. C22×C73#486semidirect
179. C2×C73C2×#486semidirect
190. C22×#102D73semidirect
192. C73C22×#486semidirect
212. C2×#102C2×D73semidirect
253. #102C22×D73semidirect
349. C22#17988932direct (N × H)
363. C2C2×#17988932direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565902
C22×#145565872
C2×#179889322
C22×#145565132
C22×#145565122
C22×#145565112
C22×C8×Dic732
C22×#102×C732
C22×#48673