Finite Group Databasebeta

← C73×#9203 #32918607

Subgroups — C73×#9203 #32918607

Order 18688.

762 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2617C211yesmaximal, characteristic
2C73×#2315C211yesmaximal, characteristic, complemented
3C73×#1519C212yesmaximal, complemented
4C2×#323×C73C211yesmaximal, characteristic, complemented
5C73×#1289C212yesmaximal, complemented
6C2×#121×C73C2211yescharacteristic
7C2×#120×C73—21no—
8C22×C22⋊C4×C73—21no—
9C22×C22⋊C4×C73C2211yescharacteristic
10C2×C22⋊C8×C73C2212yes—
11#397×C73C412yescomplemented
12#394×C73—22no—
13#392×C73C412yescomplemented
14#386×C73—21no—
15#385×C73—24no—
16#380×C73—21no—
17#380×C73C2211yescharacteristic
18#380×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
28C22×C8×C73—22no—
29#121×C73—22no—
30#121×C73—22no—
31#120×C73—42no—
32#120×C73—42no—
33C2×C4⋊C4×C73—22no—
34C2×C4⋊C4×C73—22no—
35C2×C4⋊C4×C73—21no—
36C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
37C2×C22⋊C4×C73—22no—
38C2×C22⋊C4×C73—22no—
39C2×C22⋊C4×C73—22no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73—42no—
46C2×C22⋊C4×C73—42no—
47C2×C22⋊C4×C73C2×C412yes—
48C2×C22⋊C4×C73D412yes—
49C22⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51#95×C73—22no—
52#95×C73—22no—
53#95×C73C2×C412yescomplemented
54#95×C73D414yescomplemented
55C24×C73—21no—
56C24×C73—21no—
57C24×C73—21no—
58C24×C73—21no—
59C24×C73—21no—
60C24×C73—81no—
61C24×C73—81no—
62C24×C73C22⋊C412yes—
63C24×C73D812yes—
64C24×C73SD1612yes—
65C24×C73C2×D411yescharacteristic
66C2×D4×C73—22no—
67C2×D4×C73—44no—
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73—41no—
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C2×D4×C73—82no—
75C2×D4×C73—82no—
76C2×D4×C73—81no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—22no—
81C22×C4×C73—24no—
82C22×C4×C73—44no—
83C22×C4×C73—81no—
84C22×C4×C73—81no—
85C22×C4×C73—82no—
86C22×C4×C73—82no—
87C22×C4×C73C22⋊C412yes—
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D412yes—
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73#4112yes—
92C2×C8×C73—42no—
93C2×C8×C73—82no—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—82no—
96C4⋊C4×C73—81no—
97C4⋊C4×C73—82no—
98C22⋊C4×C73—24no—
99C22⋊C4×C73—24no—
100C22⋊C4×C73—44no—
101C22⋊C4×C73—44no—
102C22⋊C4×C73—42no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—42no—
107C22⋊C4×C73—82no—
108C23×C73—22no—
109C23×C73—22no—
110C23×C73—22no—
111C23×C73—22no—
112C23×C73—22no—
113C23×C73—22no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—42no—
117C23×C73—42no—
118C23×C73—42no—
119C23×C73—42no—
120C23×C73—42no—
121C23×C73—42no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—81no—
126C23×C73—81no—
127C23×C73—82no—
128C23×C73—82no—
129C23×C73—82no—
130C23×C73—82no—
131C23×C73#10214yes—
132C23×C73C2×C22⋊C411yescharacteristic
133C23×C73#12011yescharacteristic
134C23×C73#12311yescharacteristic
135C23×C73C2×D811yescharacteristic
136C23×C73C2×SD1611yescharacteristic
137C23×C73AGL(1,ℤ/8)11yescharacteristic
138C23×C73#13711yescharacteristic
139D4×C73—82no—
140D4×C73—88no—
141D4×C73—81no—
142D4×C73—81no—
143D4×C73—81no—
144D4×C73—81no—
145C2×C4×C73—22no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—42no—
149C2×C4×C73—41no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—44no—
154C2×C4×C73—44no—
155C2×C4×C73—42no—
156C2×C4×C73—41no—
157C2×C4×C73—42no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—44no—
161C2×C4×C73—81no—
162C2×C4×C73—88no—
163C2×C4×C73C4×D412yes—
164C2×C4×C73#12112yes—
165C8×C73—82no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—24no—
173C22×C73—41no—
174C22×C73—41no—
175C22×C73—41no—
176C22×C73—41no—
177C22×C73—44no—
178C22×C73—44no—
179C22×C73—44no—
180C22×C73—42no—
181C22×C73—42no—
182C22×C73—82no—
183C22×C73—82no—
184C22×C73—82no—
185C22×C73—82no—
186C22×C73—82no—
187C22×C73—81no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73#38611yescharacteristic
191C22×C73C2×#10211yescharacteristic
192C22×C73#41711yescharacteristic
193C22×C73#44711yescharacteristic
194C22×C73#44811yescharacteristic
195C22×C73#48011yescharacteristic
196C22×C73#48111yescharacteristic
197C4×C73—22no—
198C4×C73—22no—
199C4×C73—42no—
200C4×C73—84no—
201C4×C73—82no—
202C4×C73—81no—
203#9203C7311yesmaximal, characteristic, direct product factor, Sylow
204C2×C73—21no—
205C2×C73—21no—
206C2×C73—21no—
207C2×C73—21no—
208C2×C73—21no—
209C2×C73—21no—
210C2×C73—44no—
211C2×C73—161no—
212C2×C73#187411yescharacteristic
213C2×C73#191111yescharacteristic
214C2×C73#191311yescharacteristic
215#2617C2×C7311yescharacteristic
216#2315C2×C7311yescharacteristic, complemented
217#1519C2×C7312yescomplemented
218C2×#323C2×C7311yescharacteristic, complemented
219#1289C2×C7312yescomplemented
220C73#920311yescharacteristic, direct product factor, Sylow
221C2×#121C22×C7311yescharacteristic
222C2×#120—21no—
223C22×C22⋊C4—21no—
224C22×C22⋊C4C22×C7311yescharacteristic
225C2×C22⋊C8C22×C7312yes—
226#397C4×C7312yescomplemented
227#394—22no—
228#392C4×C7312yescomplemented
229#386—21no—
230#385—24no—
231#380—21no—
232#380C22×C7311yescharacteristic
233#380C22×C7312yes—
234#327—24no—
235#323—22no—
236C25D4×C7311yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2×C4×C7311yescharacteristic, complemented
240C23×C4—22no—
241C23×C4D4×C7311yescharacteristic
242C23×C4C23×C7311yescharacteristic
243C22×C8—22no—
244#121—22no—
245#121—22no—
246#120—42no—
247#120—42no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×C4×C7311yescharacteristic, complemented
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4C2×C4×C7312yes—
263C2×C22⋊C4D4×C7312yes—
264C22⋊C8—22no—
265C22⋊C8—22no—
266#95—22no—
267#95—22no—
268#95C2×C4×C7312yescomplemented
269#95D4×C7314yescomplemented
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—81no—
276C24—81no—
277C24C22⋊C4×C7312yes—
278C24D8×C7312yes—
279C24SD16×C7312yes—
280C24C2×D4×C7311yescharacteristic
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—81no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—44no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C22⋊C4×C7312yes—
303C22×C4C22×C4×C7311yescharacteristic
304C22×C4C2×D4×C7312yes—
305C22×C4C2×D4×C7311yescharacteristic
306C22×C4#41×C7312yes—
307C2×C8—42no—
308C2×C8—82no—
309C4⋊C4—22no—
310C4⋊C4—82no—
311C4⋊C4—81no—
312C4⋊C4—82no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—82no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—81no—
341C23—81no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23#102×C7314yes—
347C23C2×C22⋊C4×C7311yescharacteristic
348C23#120×C7311yescharacteristic
349C23#123×C7311yescharacteristic
350C23C2×D8×C7311yescharacteristic
351C23C2×SD16×C7311yescharacteristic
352C23AGL(1,ℤ/8)×C7311yescharacteristic
353C23#137×C7311yescharacteristic
354D4—82no—
355D4—88no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—81no—
377C2×C4—88no—
378C2×C4C4×D4×C7312yes—
379C2×C4#121×C7312yes—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#386×C7311yescharacteristic
406C22C2×#102×C7311yescharacteristic
407C22#417×C7311yescharacteristic
408C22#447×C7311yescharacteristic
409C22#448×C7311yescharacteristic
410C22#480×C7311yescharacteristic
411C22#481×C7311yescharacteristic
412C4—22no—
413C4—22no—
414C4—42no—
415C4—84no—
416C4—82no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—44no—
425C2—161no—
426C2C73×#187411yescharacteristic
427C2C73×#191111yescharacteristic
428C2C73×#191311yescharacteristic
subgroupG/H
center, socleC22×C73#386
commutatorC22×C4C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radicalC73×#2617C2
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×#2315C2semidirect
3. C73×#1519C2semidirect
4. C2×#323×C73C2semidirect
5. C73×#1289C2semidirect
11. #397×C73C4semidirect
13. #392×C73C4semidirect
24. C22×D4×C73C2×C4semidirect
36. C2×C4⋊C4×C73C2×C4semidirect
53. #95×C73C2×C4semidirect
54. #95×C73D4semidirect
203. #9203C73direct (N × H)
216. #2315C2×C73semidirect
217. #1519C2×C73semidirect
218. C2×#323C2×C73semidirect
219. #1289C2×C73semidirect
220. C73#9203direct (N × H)
226. #397C4×C73semidirect
228. #392C4×C73semidirect
239. C22×D4C2×C4×C73semidirect
251. C2×C4⋊C4C2×C4×C73semidirect
268. #95C2×C4×C73semidirect
269. #95D4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#26172
C73×#23152
C73×#15192
C2×#323×C732
C73×#12892
#920373