Finite Group Databasebeta

← C73×#8629 #32918033

Subgroups — C73×#8629 #32918033

Order 18688.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2929C211yesmaximal, characteristic
2C73×#1810C211yesmaximal, characteristic, complemented
3C73×#1807C211yesmaximal, characteristic, complemented
4C73×#1513C212yesmaximal
5C73×#1305C212yesmaximal, complemented
6C2×#131×C73C411yescharacteristic, complemented
7C23×C8×C73C2211yescharacteristic
8#525×C73C2211yescharacteristic
9#435×C73—21no—
10#435×C73—21no—
11C8×D4×C73—21no—
12C8×D4×C73—21no—
13C2×#108×C73—21no—
14C2×#106×C73—21no—
15#424×C73C411yescharacteristic, complemented
16#408×C73C411yescharacteristic, complemented
17#407×C73C2212yes—
18#407×C73C2211yescharacteristic
19#405×C73C411yescharacteristic, complemented
20#380×C73C2212yes—
21#341×C73—21no—
22#340×C73—21no—
23#328×C73—22no—
24#327×C73—22no—
25C2×#41×C73C2×C411yescharacteristic
26C23×C4×C73C2311yescharacteristic
27#131×C73—42no—
28C2×M16×C73—21no—
29C2×M16×C73—22no—
30C2×M16×C73C2×C411yescharacteristic
31C2×M16×C73C2×C411yescharacteristic
32C22×C8×C73—21no—
33C22×C8×C73—42no—
34C22×C8×C73C2×C411yescharacteristic
35C22×C8×C73C2×C411yescharacteristic
36C22×C8×C73D411yescharacteristic
37C22×C8×C73D411yescharacteristic
38C22×C8×C73D411yescharacteristic
39C22×C8×C73D411yescharacteristic
40#123×C73—21no—
41C22⋊Q8×C73D411yescharacteristic
42C22⋊Q8×C73Q811yescharacteristic
43#121×C73D411yescharacteristic
44#121×C73Q811yescharacteristic
45#120×C73—21no—
46C4×D4×C73—21no—
47C4×D4×C73—21no—
48#117×C73C2×C411yescharacteristic
49C2×C4⋊C4×C73—22no—
50C2×C22⋊C4×C73—22no—
51#108×C73—42no—
52#106×C73—42no—
53C4⋊C8×C73—21no—
54C4⋊C8×C73—21no—
55C22⋊C8×C73—21no—
56C22⋊C8×C73—21no—
57C22⋊C8×C73—21no—
58C22⋊C8×C73—21no—
59C22⋊C8×C73—22no—
60C22⋊C8×C73—22no—
61#97×C73—21no—
62C4×C8×C73—21no—
63#95×C73—22no—
64C24×C73#4111yescharacteristic
65#41×C73—41no—
66#41×C73—41no—
67C2×Q8×C73C4⋊C411yescharacteristic
68C2×D4×C73—21no—
69C2×D4×C73—21no—
70C2×D4×C73—21no—
71C2×D4×C73C4⋊C411yescharacteristic
72C22×C4×C73—21no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73—42no—
76C22×C4×C73C22×C411yescharacteristic
77C22×C4×C73C2×D411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73C2×D411yescharacteristic
80C22×C4×C73C2×Q811yescharacteristic
81C22×C4×C73#4111yescharacteristic
82M16×C73—42no—
83M16×C73—41no—
84M16×C73—41no—
85M16×C73—42no—
86M16×C73—42no—
87C2×C8×C73—22no—
88C2×C8×C73—22no—
89C2×C8×C73—22no—
90C2×C8×C73—21no—
91C2×C8×C73—21no—
92C2×C8×C73—41no—
93C2×C8×C73—41no—
94C2×C8×C73—41no—
95C2×C8×C73—42no—
96C2×C8×C73—42no—
97C2×C8×C73C22⋊C411yescharacteristic
98C2×C8×C73C22⋊C411yescharacteristic
99C2×C8×C73C22⋊C411yescharacteristic
100C2×C8×C73C22⋊C411yescharacteristic
101C4⋊C4×C73—21no—
102C4⋊C4×C73—21no—
103C4⋊C4×C73—42no—
104C4⋊C4×C73—42no—
105C4⋊C4×C73C4⋊C411yescharacteristic
106C4⋊C4×C73C4⋊C411yescharacteristic
107C22⋊C4×C73—21no—
108C22⋊C4×C73—21no—
109C22⋊C4×C73—21no—
110C22⋊C4×C73—21no—
111C22⋊C4×C73—21no—
112C22⋊C4×C73—82no—
113C42×C73—21no—
114C23×C73—22no—
115C23×C73—21no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73#11711yescharacteristic
119C23×C73#12111yescharacteristic
120C23×C73C22⋊Q811yescharacteristic
121Q8×C73—41no—
122D4×C73—41no—
123D4×C73—42no—
124D4×C73—42no—
125D4×C73—42no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—82no—
141C2×C4×C73—82no—
142C2×C4×C73—82no—
143C2×C4×C73C2×C22⋊C411yescharacteristic
144C2×C4×C73C2×C4⋊C411yescharacteristic
145C2×C4×C73#12111yescharacteristic
146C2×C4×C73C22⋊Q811yescharacteristic
147C8×C73—21no—
148C8×C73—21no—
149C8×C73—21no—
150C8×C73—41no—
151C8×C73—41no—
152C8×C73—82no—
153C8×C73#9912yes—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—44no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73#38011yescharacteristic
167C22×C73#47112yes—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—41no—
172C4×C73—41no—
173C4×C73—41no—
174C4×C73—162no—
175C4×C73C2×#9911yescharacteristic
176C4×C73#41011yescharacteristic
177#8629C7311yesmaximal, characteristic, direct product factor, Sylow
178C2×C73—22no—
179C2×C73—41no—
180C2×C73—41no—
181C2×C73—81no—
182C2×C73#179911yescharacteristic
183C2×C73#181411yescharacteristic
184C2×C73#181511yescharacteristic
185#2929C2×C7311yescharacteristic
186#1810C2×C7311yescharacteristic, complemented
187#1807C2×C7311yescharacteristic, complemented
188#1513C2×C7312yes—
189#1305C2×C7312yescomplemented
190C73#862911yescharacteristic, direct product factor, Sylow
191C2×#131C4×C7311yescharacteristic, complemented
192C23×C8C22×C7311yescharacteristic
193#525C22×C7311yescharacteristic
194#435—21no—
195#435—21no—
196C8×D4—21no—
197C8×D4—21no—
198C2×#108—21no—
199C2×#106—21no—
200#424C4×C7311yescharacteristic, complemented
201#408C4×C7311yescharacteristic, complemented
202#407C22×C7312yes—
203#407C22×C7311yescharacteristic
204#405C4×C7311yescharacteristic, complemented
205#380C22×C7312yes—
206#341—21no—
207#340—21no—
208#328—22no—
209#327—22no—
210C2×#41C2×C4×C7311yescharacteristic
211C23×C4C23×C7311yescharacteristic
212#131—42no—
213C2×M16—21no—
214C2×M16—22no—
215C2×M16C2×C4×C7311yescharacteristic
216C2×M16C2×C4×C7311yescharacteristic
217C22×C8—21no—
218C22×C8—42no—
219C22×C8C2×C4×C7311yescharacteristic
220C22×C8C2×C4×C7311yescharacteristic
221C22×C8D4×C7311yescharacteristic
222C22×C8D4×C7311yescharacteristic
223C22×C8D4×C7311yescharacteristic
224C22×C8D4×C7311yescharacteristic
225#123—21no—
226C22⋊Q8D4×C7311yescharacteristic
227C22⋊Q8Q8×C7311yescharacteristic
228#121D4×C7311yescharacteristic
229#121Q8×C7311yescharacteristic
230#120—21no—
231C4×D4—21no—
232C4×D4—21no—
233#117C2×C4×C7311yescharacteristic
234C2×C4⋊C4—22no—
235C2×C22⋊C4—22no—
236#108—42no—
237#106—42no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#97—21no—
247C4×C8—21no—
248#95—22no—
249C24#41×C7311yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊C4×C7311yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊C4×C7311yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4C22×C4×C7311yescharacteristic
262C22×C4C2×D4×C7311yescharacteristic
263C22×C4C2×D4×C7311yescharacteristic
264C22×C4C2×D4×C7311yescharacteristic
265C22×C4C2×Q8×C7311yescharacteristic
266C22×C4#41×C7311yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—42no—
271M16—42no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8C22⋊C4×C7311yescharacteristic
283C2×C8C22⋊C4×C7311yescharacteristic
284C2×C8C22⋊C4×C7311yescharacteristic
285C2×C8C22⋊C4×C7311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4C4⋊C4×C7311yescharacteristic
291C4⋊C4C4⋊C4×C7311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—82no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#117×C7311yescharacteristic
304C23#121×C7311yescharacteristic
305C23C22⋊Q8×C7311yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4C2×C22⋊C4×C7311yescharacteristic
329C2×C4C2×C4⋊C4×C7311yescharacteristic
330C2×C4#121×C7311yescharacteristic
331C2×C4C22⋊Q8×C7311yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—82no—
338C8#99×C7312yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#380×C7311yescharacteristic
352C22#471×C7312yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—162no—
360C4C2×#99×C7311yescharacteristic
361C4#410×C7311yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2C73×#179911yescharacteristic
367C2C73×#181411yescharacteristic
368C2C73×#181511yescharacteristic
subgroupG/H
center, socleC22×C73#380
commutatorC22×C4C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radical#525×C73C22
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×#1810C2semidirect
3. C73×#1807C2semidirect
5. C73×#1305C2semidirect
6. C2×#131×C73C4semidirect
15. #424×C73C4semidirect
16. #408×C73C4semidirect
19. #405×C73C4semidirect
177. #8629C73direct (N × H)
186. #1810C2×C73semidirect
187. #1807C2×C73semidirect
189. #1305C2×C73semidirect
190. C73#8629direct (N × H)
191. C2×#131C4×C73semidirect
200. #424C4×C73semidirect
201. #408C4×C73semidirect
204. #405C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#29292
C73×#18102
C73×#18072
C73×#15132
C73×#13052
#862973