Finite Group Databasebeta

← C73×#20781 #32930185

Subgroups — C73×#20781 #32930185

Order 18688.

758 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#117×C73C211yesmaximal, characteristic
2C73×#1839C212yesmaximal, complemented
3C73×#1832C212yesmaximal, complemented
4C73×#1817C212yesmaximal
5C73×#1798C212yesmaximal
6C73×#1767C211yesmaximal, characteristic, complemented
7C4×C4⋊C8×C73C211yesmaximal, characteristic, complemented
8C73×#1750C212yesmaximal
9C73×#1746C212yesmaximal, complemented
10C2×#117×C73C2211yescharacteristic
11C2×#117×C73C2211yescharacteristic
12C22×C42×C73C2211yescharacteristic
13C2×C4⋊C8×C73C2212yes—
14C2×C4⋊C8×C73C2212yes—
15C2×C22⋊C8×C73C2214yes—
16C2×#97×C73C2212yes—
17C2×C4×C8×C73C2212yes—
18C4×C4⋊C4×C73C412yes—
19C4×C4⋊C4×C73C412yes—
20C4×C22⋊C4×C73C412yes—
21C4×C22⋊C4×C73C412yes—
22#376×C73C2211yescharacteristic
23#376×C73C2211yescharacteristic
24C43×C73C2211yescharacteristic
25C43×C73C2211yescharacteristic
26#336×C73C2214yes—
27#336×C73C2214yes—
28#336×C73C2214yes—
29#336×C73C2212yes—
30#336×C73C2212yes—
31C23×C4×C73C2×C412yes—
32C23×C4×C73C2311yescharacteristic
33C22×C8×C73C2314yes—
34C22×C8×C73C2314yes—
35#117×C73—22no—
36#117×C73—22no—
37#117×C73—24no—
38C2×C4⋊C4×C73C2×C412yes—
39C2×C4⋊C4×C73C2×C411yescharacteristic
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C22⋊C4×C73C2×C412yes—
42C2×C22⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73C2×C411yescharacteristic
44C2×C42×C73—24no—
45C2×C42×C73C2×C412yes—
46C2×C42×C73C2×C412yes—
47C2×C42×C73C2×C412yes—
48C2×C42×C73C2×C412yes—
49C2×C42×C73C2×C412yes—
50C2×C42×C73C2311yescharacteristic
51C2×C42×C73C2311yescharacteristic
52C2×C42×C73C2311yescharacteristic
53C2×C42×C73C2311yescharacteristic
54C2×C42×C73C2311yescharacteristic
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—24no—
57C4⋊C8×C73—24no—
58C22⋊C8×C73—28no—
59#97×C73—24no—
60C4×C8×C73—24no—
61#95×C73C2×C412yes—
62#95×C73C2×C412yes—
63#95×C73C2×C412yes—
64#95×C73C2×C412yes—
65C24×C73C22×C411yescharacteristic
66C22×C4×C73—22no—
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—24no—
70C22×C4×C73—24no—
71C22×C4×C73—42no—
72C22×C4×C73C22×C412yes—
73C22×C4×C73C22×C411yescharacteristic
74C22×C4×C73C22×C412yes—
75C22×C4×C73C22×C411yescharacteristic
76C22×C4×C73C22×C411yescharacteristic
77C22×C4×C73C22×C412yes—
78C22×C4×C73C22×C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C22×C412yes—
81C22×C4×C73#4114yes—
82C22×C4×C73C2411yescharacteristic
83C2×C8×C73—24no—
84C2×C8×C73—24no—
85C2×C8×C73—24no—
86C2×C8×C73—24no—
87C2×C8×C73—24no—
88C2×C8×C73—24no—
89C4⋊C4×C73—22no—
90C4⋊C4×C73—22no—
91C4⋊C4×C73—24no—
92C22⋊C4×C73—22no—
93C22⋊C4×C73—22no—
94C22⋊C4×C73—24no—
95C42×C73—24no—
96C42×C73—24no—
97C42×C73—24no—
98C42×C73—22no—
99C42×C73—22no—
100C42×C73—22no—
101C42×C73—21no—
102C42×C73—21no—
103C42×C73—42no—
104C42×C73—42no—
105C42×C73#4112yes—
106C42×C73#4112yes—
107C42×C73#4114yes—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—22no—
111C23×C73—22no—
112C23×C73#11712yes—
113C23×C73C23×C411yescharacteristic
114C2×C4×C73—22no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—22no—
118C2×C4×C73—22no—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—22no—
122C2×C4×C73—24no—
123C2×C4×C73—24no—
124C2×C4×C73—22no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—22no—
128C2×C4×C73—22no—
129C2×C4×C73—22no—
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—48no—
135C2×C4×C73#11712yes—
136C2×C4×C73#11712yes—
137C2×C4×C73#11712yes—
138C2×C4×C73#11714yes—
139C2×C4×C73#13114yes—
140C2×C4×C73C2×#4112yes—
141C2×C4×C73C2×#4112yes—
142C2×C4×C73C2×#4111yescharacteristic
143C2×C4×C73C2×#4111yescharacteristic
144C8×C73—44no—
145C8×C73—44no—
146C22×C73—24no—
147C22×C73—22no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—42no—
151C22×C73C2×#11711yescharacteristic
152C22×C73C2×#11711yescharacteristic
153C22×C73C2×#11711yescharacteristic
154C22×C73C2×#13112yes—
155C22×C73#56812yes—
156C4×C73—22no—
157C4×C73—22no—
158C4×C73—24no—
159C4×C73—24no—
160C4×C73—22no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—42no—
164C4×C73—42no—
165C4×C73#52912yes—
166C4×C73#52912yes—
167#20781C7311yesmaximal, characteristic, direct product factor, Sylow
168C2×C73—42no—
169C2×C73#228311yescharacteristic
170C2×C73#292211yescharacteristic
171C2×C73#292311yescharacteristic
172C2×C73#292412yes—
173C2×C73#292512yes—
174C4×#117C2×C7311yescharacteristic
175#1839C2×C7312yescomplemented
176#1832C2×C7312yescomplemented
177#1817C2×C7312yes—
178#1798C2×C7312yes—
179#1767C2×C7311yescharacteristic, complemented
180C4×C4⋊C8C2×C7311yescharacteristic, complemented
181#1750C2×C7312yes—
182#1746C2×C7312yescomplemented
183C73#2078111yescharacteristic, direct product factor, Sylow
184C2×#117C22×C7311yescharacteristic
185C2×#117C22×C7311yescharacteristic
186C22×C42C22×C7311yescharacteristic
187C2×C4⋊C8C22×C7312yes—
188C2×C4⋊C8C22×C7312yes—
189C2×C22⋊C8C22×C7314yes—
190C2×#97C22×C7312yes—
191C2×C4×C8C22×C7312yes—
192C4×C4⋊C4C4×C7312yes—
193C4×C4⋊C4C4×C7312yes—
194C4×C22⋊C4C4×C7312yes—
195C4×C22⋊C4C4×C7312yes—
196#376C22×C7311yescharacteristic
197#376C22×C7311yescharacteristic
198C43C22×C7311yescharacteristic
199C43C22×C7311yescharacteristic
200#336C22×C7314yes—
201#336C22×C7314yes—
202#336C22×C7312yes—
203#336C22×C7312yes—
204#336C22×C7314yes—
205C23×C4C2×C4×C7312yes—
206C23×C4C23×C7311yescharacteristic
207C22×C8C23×C7314yes—
208C22×C8C23×C7314yes—
209#117—22no—
210#117—22no—
211#117—24no—
212C2×C4⋊C4C2×C4×C7311yescharacteristic
213C2×C4⋊C4C2×C4×C7311yescharacteristic
214C2×C4⋊C4C2×C4×C7312yes—
215C2×C22⋊C4C2×C4×C7311yescharacteristic
216C2×C22⋊C4C2×C4×C7311yescharacteristic
217C2×C22⋊C4C2×C4×C7312yes—
218C2×C42—24no—
219C2×C42C2×C4×C7312yes—
220C2×C42C2×C4×C7312yes—
221C2×C42C2×C4×C7312yes—
222C2×C42C2×C4×C7312yes—
223C2×C42C2×C4×C7312yes—
224C2×C42C23×C7311yescharacteristic
225C2×C42C23×C7311yescharacteristic
226C2×C42C23×C7311yescharacteristic
227C2×C42C23×C7311yescharacteristic
228C2×C42C23×C7311yescharacteristic
229C2×C42C23×C7311yescharacteristic
230C4⋊C8—24no—
231C4⋊C8—24no—
232C22⋊C8—28no—
233#97—24no—
234C4×C8—24no—
235#95C2×C4×C7312yes—
236#95C2×C4×C7312yes—
237#95C2×C4×C7312yes—
238#95C2×C4×C7312yes—
239C24C22×C4×C7311yescharacteristic
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4C22×C4×C7312yes—
247C22×C4C22×C4×C7311yescharacteristic
248C22×C4C22×C4×C7312yes—
249C22×C4C22×C4×C7311yescharacteristic
250C22×C4C22×C4×C7311yescharacteristic
251C22×C4C22×C4×C7312yes—
252C22×C4C22×C4×C7311yescharacteristic
253C22×C4C22×C4×C7311yescharacteristic
254C22×C4C22×C4×C7312yes—
255C22×C4#41×C7314yes—
256C22×C4C24×C7311yescharacteristic
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—24no—
269C42—24no—
270C42—24no—
271C42—24no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—21no—
276C42—21no—
277C42—42no—
278C42—42no—
279C42#41×C7312yes—
280C42#41×C7312yes—
281C42#41×C7314yes—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23#117×C7312yes—
287C23C23×C4×C7311yescharacteristic
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—48no—
309C2×C4#117×C7312yes—
310C2×C4#117×C7312yes—
311C2×C4#117×C7312yes—
312C2×C4#117×C7314yes—
313C2×C4#131×C7314yes—
314C2×C4C2×#41×C7312yes—
315C2×C4C2×#41×C7312yes—
316C2×C4C2×#41×C7311yescharacteristic
317C2×C4C2×#41×C7311yescharacteristic
318C8—44no—
319C8—44no—
320C22—22no—
321C22—24no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22C2×#117×C7311yescharacteristic
326C22C2×#117×C7311yescharacteristic
327C22C2×#117×C7311yescharacteristic
328C22C2×#131×C7312yes—
329C22#568×C7312yes—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—24no—
334C4—24no—
335C4—41no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4#529×C7312yes—
340C4#529×C7312yes—
341C2—42no—
342C2C73×#228311yescharacteristic
343C2C73×#292211yescharacteristic
344C2C73×#292311yescharacteristic
345C2C73×#292412yes—
346C2C73×#292512yes—
subgroupG/H
centerC22×C4×C73C24
commutatorC23C23×C4×C73
Frattini, cosocleC22×C4C24×C73
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1839C2semidirect
3. C73×#1832C2semidirect
6. C73×#1767C2semidirect
7. C4×C4⋊C8×C73C2semidirect
9. C73×#1746C2semidirect
167. #20781C73direct (N × H)
175. #1839C2×C73semidirect
176. #1832C2×C73semidirect
179. #1767C2×C73semidirect
180. C4×C4⋊C8C2×C73semidirect
182. #1746C2×C73semidirect
183. C73#20781direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#117×C732
C73×#18392
C73×#18322
C73×#18172
C73×#17982
C73×#17672
C4×C4⋊C8×C732
C73×#17502
C73×#17462
#2078173