Finite Group Databasebeta

← C73×#9350 #32918754

Subgroups — C73×#9350 #32918754

Order 18688.

472 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3040C211yesmaximal, characteristic
2C73×#2971C211yesmaximal, characteristic, complemented
3C73×#1527C211yesmaximal, characteristic, complemented
4C73×#1488C211yesmaximal, characteristic, complemented
5C73×#1469C211yesmaximal, characteristic, complemented
6C73×#1307C211yesmaximal, characteristic, complemented
7C73×#1299C211yesmaximal, characteristic, complemented
8C2×#137×C73C411yescharacteristic
9C2×AGL(1,ℤ/8)×C73C411yescharacteristic
10C2×#124×C73C2211yescharacteristic
11C4×#41×C73C2211yescharacteristic
12#464×C73—21no—
13#464×C73—21no—
14#463×C73—21no—
15#463×C73—21no—
16C8×Q8×C73—21no—
17#436×C73—21no—
18#436×C73—22no—
19#432×C73—21no—
20#432×C73C2211yescharacteristic
21#432×C73C2211yescharacteristic
22#431×C73C2211yescharacteristic
23#423×C73C2211yescharacteristic
24#419×C73C411yescharacteristic
25#418×C73C411yescharacteristic
26C4×M16×C73—21no—
27C4×M16×C73C2211yescharacteristic
28#329×C73—21no—
29#329×C73—21no—
30Q8⋊C8×C73—21no—
31D4⋊C8×C73—21no—
32#323×C73—22no—
33C2×#41×C73C2×C411yescharacteristic
34C22×Q8×C73C2×C411yescharacteristic
35C22×D4×C73C2×C411yescharacteristic
36#137×C73—42no—
37AGL(1,ℤ/8)×C73—42no—
38C2×Q16×C73—21no—
39C2×SD16×C73—21no—
40C2×SD16×C73—21no—
41C2×D8×C73—21no—
42C2×M16×C73—22no—
43C2×M16×C73—21no—
44C2×M16×C73C2×C411yescharacteristic
45C2×M16×C73C2×C411yescharacteristic
46#124×C73—21no—
47#124×C73—21no—
48#124×C73—21no—
49#124×C73—21no—
50C4×Q8×C73D411yescharacteristic
51C4×Q8×C73D411yescharacteristic
52C4×D4×C73—21no—
53C4×D4×C73—21no—
54C4×D4×C73D411yescharacteristic
55C4×D4×C73D411yescharacteristic
56#117×C73—21no—
57#117×C73C2×C411yescharacteristic
58C2×C22⋊C4×C73—21no—
59C2×C22⋊C4×C73—21no—
60C2×C42×C73—21no—
61C2×C42×C73C2311yescharacteristic
62C4⋊C8×C73—22no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73—21no—
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68#103×C73—21no—
69#103×C73—21no—
70#102×C73—21no—
71#102×C73—21no—
72C22⋊C8×C73—22no—
73C22⋊C8×C73—21no—
74C22⋊C8×C73—21no—
75C22⋊C8×C73D412yes—
76#97×C73—21no—
77#97×C73—21no—
78#97×C73D411yescharacteristic
79#97×C73D411yescharacteristic
80C4×C8×C73—22no—
81C4×C8×C73—21no—
82C4×C8×C73—21no—
83C24×C73—21no—
84#41×C73—22no—
85#41×C73—41no—
86C2×Q8×C73—21no—
87C2×Q8×C73—21no—
88C2×Q8×C73—41no—
89C2×Q8×C73—41no—
90C2×Q8×C73C22⋊C411yescharacteristic
91C2×D4×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73—21no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C2×D4×C73C22⋊C411yescharacteristic
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73C22×C411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73#4111yescharacteristic
104Q16×C73—42no—
105SD16×C73—42no—
106SD16×C73—42no—
107D8×C73—42no—
108M16×C73—42no—
109M16×C73—44no—
110M16×C73—41no—
111M16×C73—41no—
112C2×C8×C73—21no—
113C2×C8×C73—21no—
114C2×C8×C73—21no—
115C2×C8×C73—22no—
116C2×C8×C73—21no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73C22⋊C411yescharacteristic
122C4⋊C4×C73C22⋊C411yescharacteristic
123C22⋊C4×C73—21no—
124C22⋊C4×C73—21no—
125C22⋊C4×C73—21no—
126C22⋊C4×C73—41no—
127C22⋊C4×C73—41no—
128C22⋊C4×C73—41no—
129C22⋊C4×C73—41no—
130C42×C73—21no—
131C42×C73—21no—
132C42×C73—21no—
133C42×C73C2×D411yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73#4111yescharacteristic
137C23×C73—21no—
138C23×C73—42no—
139C23×C73—41no—
140C23×C73—81no—
141C23×C73—81no—
142C23×C73C4×D411yescharacteristic
143Q8×C73—21no—
144Q8×C73—41no—
145Q8×C73—41no—
146Q8×C73—81no—
147Q8×C73#10012yes—
148D4×C73—21no—
149D4×C73—21no—
150D4×C73—42no—
151D4×C73—41no—
152D4×C73—41no—
153D4×C73—81no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—42no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—81no—
165C2×C4×C73—81no—
166C2×C4×C73C2×C22⋊C411yescharacteristic
167C2×C4×C73C4×D411yescharacteristic
168C2×C4×C73#12011yescharacteristic
169C2×C4×C73#12111yescharacteristic
170C2×C4×C73#12111yescharacteristic
171C2×C4×C73#12311yescharacteristic
172C8×C73—22no—
173C8×C73—42no—
174C8×C73—41no—
175C8×C73—41no—
176C8×C73—41no—
177C8×C73—81no—
178C8×C73—81no—
179C22×C73—21no—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—41no—
183C22×C73—41no—
184C22×C73—41no—
185C22×C73—81no—
186C22×C73—81no—
187C22×C73—81no—
188C22×C73—162no—
189C22×C73#38611yescharacteristic
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—22no—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—41no—
197C4×C73—41no—
198C4×C73—81no—
199C4×C73C2×#10011yescharacteristic
200C4×C73#41311yescharacteristic
201#9350C7311yesmaximal, characteristic, direct product factor, Sylow
202C2×C73—41no—
203C2×C73—81no—
204C2×C73—161no—
205C2×C73#188111yescharacteristic
206C2×C73#190011yescharacteristic
207C2×C73#190611yescharacteristic
208#3040C2×C7311yescharacteristic
209#2971C2×C7311yescharacteristic, complemented
210#1527C2×C7311yescharacteristic, complemented
211#1488C2×C7311yescharacteristic, complemented
212#1469C2×C7311yescharacteristic, complemented
213#1307C2×C7311yescharacteristic, complemented
214#1299C2×C7311yescharacteristic, complemented
215C73#935011yescharacteristic, direct product factor, Sylow
216C2×#137C4×C7311yescharacteristic
217C2×AGL(1,ℤ/8)C4×C7311yescharacteristic
218C2×#124C22×C7311yescharacteristic
219C4×#41C22×C7311yescharacteristic
220#464—21no—
221#464—21no—
222#463—21no—
223#463—21no—
224C8×Q8—21no—
225#436—21no—
226#436—22no—
227#432—21no—
228#432C22×C7311yescharacteristic
229#432C22×C7311yescharacteristic
230#431C22×C7311yescharacteristic
231#423C22×C7311yescharacteristic
232#419C4×C7311yescharacteristic
233#418C4×C7311yescharacteristic
234C4×M16—21no—
235C4×M16C22×C7311yescharacteristic
236#329—21no—
237#329—21no—
238Q8⋊C8—21no—
239D4⋊C8—21no—
240#323—22no—
241C2×#41C2×C4×C7311yescharacteristic
242C22×Q8C2×C4×C7311yescharacteristic
243C22×D4C2×C4×C7311yescharacteristic
244#137—42no—
245AGL(1,ℤ/8)—42no—
246C2×Q16—21no—
247C2×SD16—21no—
248C2×SD16—21no—
249C2×D8—21no—
250C2×M16—22no—
251C2×M16—21no—
252C2×M16C2×C4×C7311yescharacteristic
253C2×M16C2×C4×C7311yescharacteristic
254#124—21no—
255#124—21no—
256#124—21no—
257#124—21no—
258C4×Q8D4×C7311yescharacteristic
259C4×Q8D4×C7311yescharacteristic
260C4×D4—21no—
261C4×D4—21no—
262C4×D4D4×C7311yescharacteristic
263C4×D4D4×C7311yescharacteristic
264#117—21no—
265#117C2×C4×C7311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42C23×C7311yescharacteristic
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—22no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8D4×C7312yes—
284#97—21no—
285#97—21no—
286#97D4×C7311yescharacteristic
287#97D4×C7311yescharacteristic
288C4×C8—22no—
289C4×C8—21no—
290C4×C8—21no—
291C24—21no—
292#41—22no—
293#41—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8C22⋊C4×C7311yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C22⋊C4×C7311yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4C22×C4×C7311yescharacteristic
310C22×C4C2×D4×C7311yescharacteristic
311C22×C4#41×C7311yescharacteristic
312Q16—42no—
313SD16—42no—
314SD16—42no—
315D8—42no—
316M16—42no—
317M16—44no—
318M16—41no—
319M16—41no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—41no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C22⋊C4×C7311yescharacteristic
330C4⋊C4C22⋊C4×C7311yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42C2×D4×C7311yescharacteristic
342C42C2×D4×C7311yescharacteristic
343C42C2×D4×C7311yescharacteristic
344C42#41×C7311yescharacteristic
345C23—21no—
346C23—42no—
347C23—41no—
348C23—81no—
349C23—81no—
350C23C4×D4×C7311yescharacteristic
351Q8—21no—
352Q8—41no—
353Q8—41no—
354Q8—81no—
355Q8#100×C7312yes—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C2×C22⋊C4×C7311yescharacteristic
375C2×C4C4×D4×C7311yescharacteristic
376C2×C4#120×C7311yescharacteristic
377C2×C4#121×C7311yescharacteristic
378C2×C4#121×C7311yescharacteristic
379C2×C4#123×C7311yescharacteristic
380C8—22no—
381C8—42no—
382C8—41no—
383C8—41no—
384C8—41no—
385C8—81no—
386C8—81no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—162no—
397C22#386×C7311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—21no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4C2×#100×C7311yescharacteristic
408C4#413×C7311yescharacteristic
409C2—41no—
410C2—81no—
411C2—161no—
412C2C73×#188111yescharacteristic
413C2C73×#190011yescharacteristic
414C2C73×#190611yescharacteristic
subgroupG/H
center, socleC22×C73#386
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C42C23×C73
prime radicalC2×AGL(1,ℤ/8)×C73C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2971C2semidirect
3. C73×#1527C2semidirect
4. C73×#1488C2semidirect
5. C73×#1469C2semidirect
6. C73×#1307C2semidirect
7. C73×#1299C2semidirect
201. #9350C73direct (N × H)
209. #2971C2×C73semidirect
210. #1527C2×C73semidirect
211. #1488C2×C73semidirect
212. #1469C2×C73semidirect
213. #1307C2×C73semidirect
214. #1299C2×C73semidirect
215. C73#9350direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#30402
C73×#29712
C73×#15272
C73×#14882
C73×#14692
C73×#13072
C73×#12992
#935073