Finite Group Databasebeta

← C73×#25609 #32935013

Subgroups — C73×#25609 #32935013

Order 18688.

1052 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#541×C73C211yesmaximal, characteristic
2C2×#494×C73C211yesmaximal, characteristic, complemented
3C2×#491×C73C211yesmaximal, characteristic, complemented
4C73×#2621C211yesmaximal, characteristic
5C73×#2001C214yesmaximal
6C73×#1991C211yesmaximal, characteristic, complemented
7C73×#1960C211yesmaximal, characteristic, complemented
8C73×#1878C214yesmaximal, complemented
9C73×#1750C211yesmaximal, characteristic
10C22×Q16×C73C2214yes—
11#541×C73—24no—
12#541×C73—24no—
13C2×C4⋊Q8×C73C2211yescharacteristic
14C2×C4⋊Q8×C73C2211yescharacteristic
15C2×#126×C73—22no—
16C2×#125×C73—22no—
17C2×#125×C73C2211yescharacteristic
18C2×C22⋊Q8×C73C2212yes—
19C2×C22⋊Q8×C73C2212yes—
20C2×C4×Q8×C73C2212yes—
21C2×#117×C73C2211yescharacteristic
22#494×C73—24no—
23#491×C73—24no—
24C2×#103×C73C2214yes—
25C2×#103×C73C2214yes—
26C2×C22⋊C8×C73C2212yes—
27C2×#97×C73C2211yescharacteristic
28C2×C4×C8×C73C2211yescharacteristic
29#398×C73—22no—
30#397×C73—22no—
31#391×C73C2212yes—
32#382×C73C2211yescharacteristic
33#340×C73C2214yes—
34#336×C73C2212yes—
35C22×Q8×C73C2312yes—
36C22×Q8×C73C2312yes—
37C23×C4×C73C2311yescharacteristic
38C2×Q16×C73—28no—
39C2×Q16×C73—28no—
40C2×Q16×C73—44no—
41C22×C8×C73C2312yes—
42C22×C8×C73C2312yes—
43C4⋊Q8×C73—22no—
44C4⋊Q8×C73—22no—
45C4⋊Q8×C73—22no—
46C4⋊Q8×C73D414yes—
47#126×C73—216no—
48#125×C73—216no—
49#125×C73—22no—
50#125×C73D414yes—
51C22⋊Q8×C73—24no—
52C22⋊Q8×C73—24no—
53C22⋊Q8×C73—44no—
54C4×Q8×C73—24no—
55C4×Q8×C73—24no—
56#117×C73—22no—
57#117×C73—22no—
58C2×C4⋊C4×C73—22no—
59C2×C4⋊C4×C73—22no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73—22no—
62C2×C4⋊C4×C73—22no—
63C2×C4⋊C4×C73—22no—
64C2×C4⋊C4×C73D412yes—
65C2×C4⋊C4×C73C2312yes—
66C2×C4⋊C4×C73C2312yes—
67C2×C22⋊C4×C73—22no—
68C2×C22⋊C4×C73—22no—
69C2×C22⋊C4×C73D412yes—
70C2×C42×C73—22no—
71C2×C42×C73C2311yescharacteristic
72C2×C42×C73C2311yescharacteristic
73#103×C73—28no—
74#103×C73—44no—
75C22⋊C8×C73—24no—
76#97×C73—22no—
77C4×C8×C73—22no—
78#95×C73—22no—
79#95×C73—24no—
80C24×C73C2×D411yescharacteristic
81C2×Q8×C73—22no—
82C2×Q8×C73—22no—
83C2×Q8×C73—22no—
84C2×Q8×C73—24no—
85C2×Q8×C73—24no—
86C2×Q8×C73—24no—
87C2×Q8×C73—24no—
88C2×Q8×C73C2×D414yes—
89C22×C4×C73—22no—
90C22×C4×C73—22no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—22no—
94C22×C4×C73—22no—
95C22×C4×C73—22no—
96C22×C4×C73—22no—
97C22×C4×C73C2×D412yes—
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2×D412yes—
100C22×C4×C73C2411yescharacteristic
101Q16×C73—416no—
102C2×C8×C73—22no—
103C2×C8×C73—22no—
104C2×C8×C73—22no—
105C2×C8×C73—22no—
106C2×C8×C73—22no—
107C2×C8×C73#4114yes—
108C4⋊C4×C73—28no—
109C4⋊C4×C73—28no—
110C4⋊C4×C73—28no—
111C4⋊C4×C73—28no—
112C4⋊C4×C73—24no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—24no—
115C4⋊C4×C73—44no—
116C4⋊C4×C73—44no—
117C4⋊C4×C73C2×D414yes—
118C22⋊C4×C73—28no—
119C22⋊C4×C73—24no—
120C22⋊C4×C73—44no—
121C42×C73—28no—
122C42×C73—21no—
123C42×C73—21no—
124C42×C73C2×D412yes—
125C42×C73C2×D412yes—
126C23×C73—21no—
127C23×C73—21no—
128C23×C73—21no—
129C23×C73—24no—
130C23×C73C22×D411yescharacteristic
131Q8×C73—28no—
132Q8×C73—44no—
133Q8×C73—44no—
134Q8×C73—44no—
135C2×C4×C73—24no—
136C2×C4×C73—22no—
137C2×C4×C73—24no—
138C2×C4×C73—24no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—24no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—44no—
146C2×C4×C73—42no—
147C2×C4×C73—42no—
148C2×C4×C73—42no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—44no—
153C2×C4×C73#12014yes—
154C2×C4×C73C22×D411yescharacteristic
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C2×#4112yes—
157C2×C4×C73H(5,𝔽2)12yes—
158C8×C73—24no—
159C8×C73—42no—
160C22×C73—42no—
161C22×C73—41no—
162C22×C73—42no—
163C22×C73—42no—
164C22×C73C2×#12011yescharacteristic
165C22×C73#53211yescharacteristic
166C22×C73#53511yescharacteristic
167C22×C73#57814yes—
168C4×C73—24no—
169C4×C73—41no—
170C4×C73—44no—
171C4×C73—42no—
172C4×C73—44no—
173C4×C73—82no—
174C4×C73—82no—
175C4×C73#54612yes—
176C4×C73#55012yes—
177#25609C7311yesmaximal, characteristic, direct product factor, Sylow
178C2×C73—81no—
179C2×C73#243111yescharacteristic
180C2×C73#301112yes—
181C2×C73#317712yes—
182C2×C73#322312yes—
183C2×#541C2×C7311yescharacteristic
184C2×#494C2×C7311yescharacteristic, complemented
185C2×#491C2×C7311yescharacteristic, complemented
186#2621C2×C7311yescharacteristic
187#2001C2×C7314yes—
188#1991C2×C7311yescharacteristic, complemented
189#1960C2×C7311yescharacteristic, complemented
190#1878C2×C7314yescomplemented
191#1750C2×C7311yescharacteristic
192C73#2560911yescharacteristic, direct product factor, Sylow
193C22×Q16C22×C7314yes—
194#541—24no—
195#541—24no—
196C2×C4⋊Q8C22×C7311yescharacteristic
197C2×C4⋊Q8C22×C7311yescharacteristic
198C2×#126—22no—
199C2×#125—22no—
200C2×#125C22×C7311yescharacteristic
201C2×C22⋊Q8C22×C7312yes—
202C2×C22⋊Q8C22×C7312yes—
203C2×C4×Q8C22×C7312yes—
204C2×#117C22×C7311yescharacteristic
205#494—24no—
206#491—24no—
207C2×#103C22×C7314yes—
208C2×#103C22×C7314yes—
209C2×C22⋊C8C22×C7312yes—
210C2×#97C22×C7311yescharacteristic
211C2×C4×C8C22×C7311yescharacteristic
212#398—22no—
213#397—22no—
214#391C22×C7312yes—
215#382C22×C7311yescharacteristic
216#340C22×C7314yes—
217#336C22×C7312yes—
218C22×Q8C23×C7312yes—
219C22×Q8C23×C7312yes—
220C23×C4C23×C7311yescharacteristic
221C2×Q16—28no—
222C2×Q16—28no—
223C2×Q16—44no—
224C22×C8C23×C7312yes—
225C22×C8C23×C7312yes—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8D4×C7314yes—
230#126—216no—
231#125—216no—
232#125—22no—
233#125D4×C7314yes—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—44no—
237C4×Q8—24no—
238C4×Q8—24no—
239#117—22no—
240#117—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4D4×C7312yes—
248C2×C4⋊C4C23×C7312yes—
249C2×C4⋊C4C23×C7312yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4D4×C7312yes—
253C2×C42—22no—
254C2×C42C23×C7311yescharacteristic
255C2×C42C23×C7311yescharacteristic
256#103—28no—
257#103—44no—
258C22⋊C8—24no—
259#97—22no—
260C4×C8—22no—
261#95—22no—
262#95—24no—
263C24C2×D4×C7311yescharacteristic
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8C2×D4×C7314yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4C2×D4×C7312yes—
281C22×C4C2×D4×C7311yescharacteristic
282C22×C4C2×D4×C7312yes—
283C22×C4C24×C7311yescharacteristic
284Q16—416no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8#41×C7314yes—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4C2×D4×C7314yes—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—44no—
304C42—28no—
305C42—21no—
306C42—21no—
307C42C2×D4×C7312yes—
308C42C2×D4×C7312yes—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—24no—
313C23C22×D4×C7311yescharacteristic
314Q8—28no—
315Q8—44no—
316Q8—44no—
317Q8—44no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4#120×C7314yes—
337C2×C4C22×D4×C7311yescharacteristic
338C2×C4C22×D4×C7311yescharacteristic
339C2×C4C2×#41×C7312yes—
340C2×C4H(5,𝔽2)×C7312yes—
341C8—24no—
342C8—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22C2×#120×C7311yescharacteristic
348C22#532×C7311yescharacteristic
349C22#535×C7311yescharacteristic
350C22#578×C7314yes—
351C4—24no—
352C4—42no—
353C4—44no—
354C4—41no—
355C4—44no—
356C4—82no—
357C4—82no—
358C4#546×C7312yes—
359C4#550×C7312yes—
360C2—81no—
361C2C73×#243111yescharacteristic
362C2C73×#301112yes—
363C2C73×#317712yes—
364C2C73×#322312yes—
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC24×C73C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C2×#117×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#494×C73C2semidirect
3. C2×#491×C73C2semidirect
6. C73×#1991C2semidirect
7. C73×#1960C2semidirect
8. C73×#1878C2semidirect
177. #25609C73direct (N × H)
184. C2×#494C2×C73semidirect
185. C2×#491C2×C73semidirect
188. #1991C2×C73semidirect
189. #1960C2×C73semidirect
190. #1878C2×C73semidirect
192. C73#25609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#541×C732
C2×#494×C732
C2×#491×C732
C73×#26212
C73×#20012
C73×#19912
C73×#19602
C73×#18782
C73×#17502
#2560973