Finite Group Databasebeta

← C73×#9235 #32918639

Subgroups — C73×#9235 #32918639

Order 18688.

550 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3101C211yesmaximal, characteristic
2C73×#2293C211yesmaximal, characteristic
3C73×#1994C211yesmaximal, characteristic, complemented
4C73×#1503C211yesmaximal, characteristic, complemented
5C73×#1486C211yesmaximal, characteristic
6C73×#1303C211yesmaximal, characteristic, complemented
7C73×#1276C211yesmaximal, characteristic, complemented
8#533×C73C2211yescharacteristic
9C2×C4×D4×C73C2211yescharacteristic
10C2×#117×C73—21no—
11#483×C73—21no—
12#476×C73C412yescomplemented
13#461×C73C412yescomplemented
14#450×C73—21no—
15C4×SD16×C73—22no—
16#431×C73C2211yescharacteristic
17#431×C73C2211yescharacteristic
18#423×C73C2211yescharacteristic
19#390×C73—21no—
20#389×C73—21no—
21#388×C73—22no—
22C4×C4⋊C4×C73C2211yescharacteristic
23C4×C22⋊C4×C73—21no—
24C43×C73C2211yescharacteristic
25#339×C73—21no—
26#339×C73—21no—
27#328×C73—22no—
28D4⋊C8×C73—22no—
29C22×D4×C73D411yescharacteristic
30C23×C4×C73—21no—
31C2×SD16×C73—22no—
32C2×M16×C73—21no—
33C4⋊Q8×C73C2×C411yescharacteristic
34C4⋊Q8×C73C2×C411yescharacteristic
35#125×C73—21no—
36C22⋊Q8×C73—21no—
37C22⋊Q8×C73—21no—
38C4×Q8×C73—21no—
39C4×Q8×C73—21no—
40C4×D4×C73—21no—
41C4×D4×C73—21no—
42C4×D4×C73C2×C412yes—
43C4×D4×C73D412yes—
44#117×C73—21no—
45#117×C73—42no—
46#117×C73—42no—
47C2×C4⋊C4×C73—22no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—22no—
52C2×C22⋊C4×C73—22no—
53C2×C42×C73—21no—
54C2×C42×C73—21no—
55C2×C42×C73—21no—
56C2×C42×C73D411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C42×C73C2311yescharacteristic
59#106×C73—22no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73—21no—
62C4⋊C8×C73—21no—
63C4⋊C8×C73C2×C412yes—
64#103×C73—22no—
65#102×C73—22no—
66C22⋊C8×C73—21no—
67C22⋊C8×C73—21no—
68C4×C8×C73—21no—
69C4×C8×C73—21no—
70#95×C73—22no—
71#95×C73D412yes—
72C24×C73—21no—
73C2×Q8×C73—21no—
74C2×Q8×C73—21no—
75C2×D4×C73—21no—
76C2×D4×C73—42no—
77C2×D4×C73C22⋊C412yes—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—22no—
84C22×C4×C73—41no—
85C22×C4×C73—41no—
86C22×C4×C73—41no—
87C22×C4×C73—41no—
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91SD16×C73—44no—
92M16×C73—41no—
93M16×C73—41no—
94C2×C8×C73—22no—
95C2×C8×C73—41no—
96C2×C8×C73—41no—
97C4⋊C4×C73—21no—
98C4⋊C4×C73—21no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73—21no—
101C4⋊C4×C73—21no—
102C4⋊C4×C73—42no—
103C4⋊C4×C73—44no—
104C4⋊C4×C73—41no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73C22⋊C412yes—
107C22⋊C4×C73—22no—
108C22⋊C4×C73—22no—
109C22⋊C4×C73—44no—
110C22⋊C4×C73—44no—
111C22⋊C4×C73—41no—
112C42×C73—21no—
113C42×C73—21no—
114C42×C73—21no—
115C42×C73—42no—
116C42×C73—42no—
117C42×C73—41no—
118C42×C73—41no—
119C42×C73—41no—
120C42×C73SD1612yescomplemented
121C42×C73C22×C411yescharacteristic
122C42×C73C2×D411yescharacteristic
123C42×C73#4111yescharacteristic
124C42×C73#4111yescharacteristic
125C23×C73—22no—
126C23×C73—42no—
127C23×C73—42no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73#12011yescharacteristic
131Q8×C73—42no—
132Q8×C73—81no—
133D4×C73—22no—
134D4×C73—22no—
135D4×C73—42no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—42no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—42no—
156C2×C4×C73C2×C22⋊C411yescharacteristic
157C2×C4×C73C4×D411yescharacteristic
158C2×C4×C73C4×D411yescharacteristic
159C2×C4×C73#12111yescharacteristic
160C2×C4×C73#12111yescharacteristic
161C2×C4×C73#12311yescharacteristic
162C2×C4×C73C2×SD1611yescharacteristic
163C2×C4×C73#13511yescharacteristic
164C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C73#13711yescharacteristic
166C8×C73—42no—
167C8×C73—81no—
168C8×C73—81no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—42no—
172C22×C73—42no—
173C22×C73—42no—
174C22×C73—82no—
175C22×C73—82no—
176C22×C73#38611yescharacteristic
177C22×C73#45011yescharacteristic
178C22×C73#45211yescharacteristic
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—42no—
183C4×C73—42no—
184C4×C73—42no—
185C4×C73—82no—
186C4×C73—81no—
187C4×C73—81no—
188C4×C73#42112yes—
189C4×C73C4×SD1611yescharacteristic
190C4×C73#44011yescharacteristic
191C4×C73#46111yescharacteristic
192C4×C73#46311yescharacteristic
193#9235C7311yesmaximal, characteristic, direct product factor, Sylow
194C2×C73—21no—
195C2×C73—21no—
196C2×C73—82no—
197C2×C73#187711yescharacteristic
198C2×C73#189711yescharacteristic
199C2×C73#190611yescharacteristic
200#3101C2×C7311yescharacteristic
201#2293C2×C7311yescharacteristic
202#1994C2×C7311yescharacteristic, complemented
203#1503C2×C7311yescharacteristic, complemented
204#1486C2×C7311yescharacteristic
205#1303C2×C7311yescharacteristic, complemented
206#1276C2×C7311yescharacteristic, complemented
207C73#923511yescharacteristic, direct product factor, Sylow
208#533C22×C7311yescharacteristic
209C2×C4×D4C22×C7311yescharacteristic
210C2×#117—21no—
211#483—21no—
212#476C4×C7312yescomplemented
213#461C4×C7312yescomplemented
214#450—21no—
215C4×SD16—22no—
216#431C22×C7311yescharacteristic
217#431C22×C7311yescharacteristic
218#423C22×C7311yescharacteristic
219#390—21no—
220#389—21no—
221#388—22no—
222C4×C4⋊C4C22×C7311yescharacteristic
223C4×C22⋊C4—21no—
224C43C22×C7311yescharacteristic
225#339—21no—
226#339—21no—
227#328—22no—
228D4⋊C8—22no—
229C22×D4D4×C7311yescharacteristic
230C23×C4—21no—
231C2×SD16—22no—
232C2×M16—21no—
233C4⋊Q8C2×C4×C7311yescharacteristic
234C4⋊Q8C2×C4×C7311yescharacteristic
235#125—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C2×C4×C7312yes—
243C4×D4D4×C7312yes—
244#117—21no—
245#117—42no—
246#117—42no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4D4×C7311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42D4×C7311yescharacteristic
257C2×C42D4×C7311yescharacteristic
258C2×C42C23×C7311yescharacteristic
259#106—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8C2×C4×C7312yes—
264#103—22no—
265#102—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270#95—22no—
271#95D4×C7312yes—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4C22⋊C4×C7312yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C2×D4×C7311yescharacteristic
289C22×C4C2×D4×C7311yescharacteristic
290C22×C4C2×D4×C7311yescharacteristic
291SD16—44no—
292M16—41no—
293M16—41no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4C22⋊C4×C7312yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—41no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—42no—
316C42—42no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42SD16×C7312yescomplemented
321C42C22×C4×C7311yescharacteristic
322C42C2×D4×C7311yescharacteristic
323C42#41×C7311yescharacteristic
324C42#41×C7311yescharacteristic
325C23—22no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23#120×C7311yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—22no—
334D4—22no—
335D4—42no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4C2×C22⋊C4×C7311yescharacteristic
357C2×C4C4×D4×C7311yescharacteristic
358C2×C4C4×D4×C7311yescharacteristic
359C2×C4#121×C7311yescharacteristic
360C2×C4#121×C7311yescharacteristic
361C2×C4#123×C7311yescharacteristic
362C2×C4C2×SD16×C7311yescharacteristic
363C2×C4#135×C7311yescharacteristic
364C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
365C2×C4#137×C7311yescharacteristic
366C8—42no—
367C8—81no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—82no—
375C22—82no—
376C22#386×C7311yescharacteristic
377C22#450×C7311yescharacteristic
378C22#452×C7311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4#421×C7312yes—
389C4C4×SD16×C7311yescharacteristic
390C4#440×C7311yescharacteristic
391C4#461×C7311yescharacteristic
392C4#463×C7311yescharacteristic
393C2—21no—
394C2—21no—
395C2—82no—
396C2C73×#187711yescharacteristic
397C2C73×#189711yescharacteristic
398C2C73×#190611yescharacteristic
subgroupG/H
centerC2×C4×C73#121
commutatorC42C22×C4×C73
Frattini, cosocleC2×C42C23×C73
socleC22×C73#386
prime radicalC22×D4×C73D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#1994C2semidirect
4. C73×#1503C2semidirect
6. C73×#1303C2semidirect
7. C73×#1276C2semidirect
12. #476×C73C4semidirect
13. #461×C73C4semidirect
120. C42×C73SD16semidirect
193. #9235C73direct (N × H)
202. #1994C2×C73semidirect
203. #1503C2×C73semidirect
205. #1303C2×C73semidirect
206. #1276C2×C73semidirect
207. C73#9235direct (N × H)
212. #476C4×C73semidirect
213. #461C4×C73semidirect
320. C42SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#31012
C73×#22932
C73×#19942
C73×#15032
C73×#14862
C73×#13032
C73×#12762
#923573