Finite Group Databasebeta

← C73×#9409 #32918813

Subgroups — C73×#9409 #32918813

Order 18688.

840 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2539C211yesmaximal, characteristic
2C73×#2273C211yesmaximal, characteristic, complemented
3C73×#1507C212yesmaximal, complemented
4C73×#1462C211yesmaximal, characteristic, complemented
5C73×#1297C212yesmaximal, complemented
6C2×#123×C73—21no—
7C2×#121×C73C2211yescharacteristic
8C2×#120×C73—21no—
9C22×C22⋊C4×C73—21no—
10C22×C22⋊C4×C73C2211yescharacteristic
11#407×C73C2212yes—
12#399×C73C412yescomplemented
13#392×C73C412yescomplemented
14#390×C73—22no—
15C4×C22⋊C4×C73—21no—
16C4×C22⋊C4×C73—24no—
17C4×C22⋊C4×C73C2211yescharacteristic
18C4×C22⋊C4×C73C2212yes—
19#327×C73—24no—
20#323×C73—22no—
21C25×C73D411yescharacteristic
22C22×D4×C73—21no—
23C22×D4×C73—22no—
24C22×D4×C73C2×C411yescharacteristic, complemented
25C23×C4×C73—22no—
26C23×C4×C73D411yescharacteristic
27C23×C4×C73C2311yescharacteristic
28C2×M16×C73—22no—
29#123×C73—44no—
30#121×C73—22no—
31#121×C73—22no—
32#120×C73—44no—
33C2×C4⋊C4×C73—22no—
34C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
35C2×C22⋊C4×C73—22no—
36C2×C22⋊C4×C73—28no—
37C2×C22⋊C4×C73—22no—
38C2×C22⋊C4×C73—21no—
39C2×C22⋊C4×C73—22no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—22no—
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73C2×C412yes—
45C2×C22⋊C4×C73D412yes—
46C2×C42×C73—22no—
47C2×C42×C73—22no—
48C2×C42×C73—21no—
49C22⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51#95×C73—22no—
52#95×C73C2×C412yescomplemented
53#95×C73D414yescomplemented
54C24×C73—21no—
55C24×C73—24no—
56C24×C73—21no—
57C24×C73—21no—
58C24×C73—42no—
59C24×C73—81no—
60C24×C73C22⋊C412yes—
61C24×C73C2×D411yescharacteristic
62C2×D4×C73—22no—
63C2×D4×C73—44no—
64C2×D4×C73—42no—
65C2×D4×C73—42no—
66C2×D4×C73—44no—
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C2×D4×C73—82no—
70C2×D4×C73—81no—
71C22×C4×C73—22no—
72C22×C4×C73—21no—
73C22×C4×C73—24no—
74C22×C4×C73—28no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—42no—
78C22×C4×C73—81no—
79C22×C4×C73—82no—
80C22×C4×C73—82no—
81C22×C4×C73C22⋊C412yes—
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D412yes—
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73#4112yes—
86M16×C73—82no—
87C2×C8×C73—42no—
88C4⋊C4×C73—22no—
89C4⋊C4×C73—44no—
90C22⋊C4×C73—28no—
91C22⋊C4×C73—28no—
92C22⋊C4×C73—24no—
93C22⋊C4×C73—24no—
94C22⋊C4×C73—24no—
95C22⋊C4×C73—44no—
96C22⋊C4×C73—44no—
97C22⋊C4×C73—42no—
98C42×C73—82no—
99C42×C73—81no—
100C42×C73—82no—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73—24no—
104C23×C73—22no—
105C23×C73—22no—
106C23×C73—22no—
107C23×C73—22no—
108C23×C73—41no—
109C23×C73—42no—
110C23×C73—42no—
111C23×C73—42no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—42no—
115C23×C73—42no—
116C23×C73—41no—
117C23×C73—42no—
118C23×C73—81no—
119C23×C73—81no—
120C23×C73—88no—
121C23×C73#10414yes—
122C23×C73C2×C22⋊C411yescharacteristic
123C23×C73#12011yescharacteristic
124C23×C73#12311yescharacteristic
125D4×C73—82no—
126D4×C73—88no—
127D4×C73—82no—
128D4×C73—82no—
129C2×C4×C73—28no—
130C2×C4×C73—24no—
131C2×C4×C73—22no—
132C2×C4×C73—41no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73—41no—
136C2×C4×C73—44no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73—44no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—81no—
146C2×C4×C73—88no—
147C2×C4×C73C4×D412yes—
148C2×C4×C73#12112yes—
149C8×C73—82no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—24no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—44no—
159C22×C73—44no—
160C22×C73—44no—
161C22×C73—44no—
162C22×C73—42no—
163C22×C73—42no—
164C22×C73—82no—
165C22×C73—82no—
166C22×C73—82no—
167C22×C73—84no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73#38611yescharacteristic
172C22×C73C2×#10412yes—
173C22×C73#45312yes—
174C4×C73—22no—
175C4×C73—22no—
176C4×C73—48no—
177C4×C73—44no—
178C4×C73—42no—
179C4×C73—81no—
180#9409C7311yesmaximal, characteristic, direct product factor, Sylow
181C2×C73—22no—
182C2×C73—22no—
183C2×C73—41no—
184C2×C73—44no—
185C2×C73—161no—
186C2×C73#188711yescharacteristic
187C2×C73#189611yescharacteristic
188C2×C73#191311yescharacteristic
189#2539C2×C7311yescharacteristic
190#2273C2×C7311yescharacteristic, complemented
191#1507C2×C7312yescomplemented
192#1462C2×C7311yescharacteristic, complemented
193#1297C2×C7312yescomplemented
194C73#940911yescharacteristic, direct product factor, Sylow
195C2×#123—21no—
196C2×#121C22×C7311yescharacteristic
197C2×#120—21no—
198C22×C22⋊C4—21no—
199C22×C22⋊C4C22×C7311yescharacteristic
200#407C22×C7312yes—
201#399C4×C7312yescomplemented
202#392C4×C7312yescomplemented
203#390—22no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4C22×C7311yescharacteristic
207C4×C22⋊C4C22×C7312yes—
208#327—24no—
209#323—22no—
210C25D4×C7311yescharacteristic
211C22×D4—21no—
212C22×D4—22no—
213C22×D4C2×C4×C7311yescharacteristic, complemented
214C23×C4—22no—
215C23×C4D4×C7311yescharacteristic
216C23×C4C23×C7311yescharacteristic
217C2×M16—22no—
218#123—44no—
219#121—22no—
220#121—22no—
221#120—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×C4×C7311yescharacteristic, complemented
224C2×C22⋊C4—28no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4C2×C4×C7312yes—
234C2×C22⋊C4D4×C7312yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240#95—22no—
241#95C2×C4×C7312yescomplemented
242#95D4×C7314yescomplemented
243C24—21no—
244C24—24no—
245C24—21no—
246C24—21no—
247C24—42no—
248C24—81no—
249C24C22⋊C4×C7312yes—
250C24C2×D4×C7311yescharacteristic
251C2×D4—22no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—82no—
259C2×D4—81no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—81no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C22⋊C4×C7312yes—
271C22×C4C22×C4×C7311yescharacteristic
272C22×C4C2×D4×C7312yes—
273C22×C4C2×D4×C7311yescharacteristic
274C22×C4#41×C7312yes—
275M16—82no—
276C2×C8—42no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C42—82no—
288C42—81no—
289C42—82no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23—88no—
310C23#104×C7314yes—
311C23C2×C22⋊C4×C7311yescharacteristic
312C23#120×C7311yescharacteristic
313C23#123×C7311yescharacteristic
314D4—82no—
315D4—88no—
316D4—82no—
317D4—82no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—88no—
336C2×C4C4×D4×C7312yes—
337C2×C4#121×C7312yes—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#386×C7311yescharacteristic
361C22C2×#104×C7312yes—
362C22#453×C7312yes—
363C4—22no—
364C4—22no—
365C4—48no—
366C4—44no—
367C4—42no—
368C4—81no—
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—161no—
374C2C73×#188711yescharacteristic
375C2C73×#189611yescharacteristic
376C2C73×#191311yescharacteristic
subgroupG/H
center, socleC22×C73#386
commutatorC22×C4C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radicalC73×#2539C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2273C2semidirect
3. C73×#1507C2semidirect
4. C73×#1462C2semidirect
5. C73×#1297C2semidirect
12. #399×C73C4semidirect
13. #392×C73C4semidirect
24. C22×D4×C73C2×C4semidirect
34. C2×C4⋊C4×C73C2×C4semidirect
52. #95×C73C2×C4semidirect
53. #95×C73D4semidirect
180. #9409C73direct (N × H)
190. #2273C2×C73semidirect
191. #1507C2×C73semidirect
192. #1462C2×C73semidirect
193. #1297C2×C73semidirect
194. C73#9409direct (N × H)
201. #399C4×C73semidirect
202. #392C4×C73semidirect
213. C22×D4C2×C4×C73semidirect
223. C2×C4⋊C4C2×C4×C73semidirect
241. #95C2×C4×C73semidirect
242. #95D4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#25392
C73×#22732
C73×#15072
C73×#14622
C73×#12972
#940973