Finite Group Databasebeta

← C73×#10256 #32919660

Subgroups — C73×#10256 #32919660

Order 18688.

580 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3246C211yesmaximal, characteristic, complemented
2C73×#3077C211yesmaximal, characteristic, complemented
3C73×#3036C211yesmaximal, characteristic
4C73×#1954C211yesmaximal, characteristic, complemented
5C73×#1493C211yesmaximal, characteristic, complemented
6C73×#1492C211yesmaximal, characteristic, complemented
7C73×#1292C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C73—21no—
9#533×C73C2211yescharacteristic
10#532×C73C2211yescharacteristic
11C2×C4×D4×C73C2211yescharacteristic
12#478×C73—22no—
13#476×C73—21no—
14#476×C73—21no—
15#475×C73—21no—
16#474×C73—21no—
17#465×C73—22no—
18#463×C73—22no—
19#461×C73—21no—
20#461×C73—21no—
21#460×C73—21no—
22#459×C73—21no—
23#428×C73—21no—
24#423×C73C2211yescharacteristic
25#423×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27#418×C73—21no—
28#417×C73—21no—
29C2×C22⋊C8×C73—22no—
30C2×C4×C8×C73C2211yescharacteristic
31Q8⋊C8×C73—22no—
32D4⋊C8×C73—22no—
33C2×#41×C73—21no—
34C22×D4×C73D411yescharacteristic
35C23×C4×C73—21no—
36AGL(1,ℤ/8)×C73—42no—
37AGL(1,ℤ/8)×C73—42no—
38C2×SD16×C73—22no—
39C2×SD16×C73—22no—
40C2×D8×C73—22no—
41C2×M16×C73—21no—
42C2×M16×C73—21no—
43C22×C8×C73—21no—
44C22×C8×C73D412yes—
45C4⋊Q8×C73D411yescharacteristic
46C4⋊Q8×C73D411yescharacteristic
47C4⋊Q8×C73D411yescharacteristic
48#127×C73D411yescharacteristic
49#125×C73—21no—
50#124×C73—21no—
51C22⋊Q8×C73—21no—
52C22⋊Q8×C73—21no—
53#121×C73—21no—
54#121×C73—21no—
55C4×Q8×C73—21no—
56C4×Q8×C73—21no—
57C4×D4×C73—21no—
58C4×D4×C73—21no—
59C4×D4×C73—21no—
60C4×D4×C73—21no—
61C4×D4×C73—21no—
62C4×D4×C73—21no—
63#117×C73—21no—
64C2×C4⋊C4×C73D411yescharacteristic
65C2×C22⋊C4×C73—21no—
66C2×C42×C73C2311yescharacteristic
67#107×C73—22no—
68#106×C73—22no—
69#106×C73—22no—
70C4⋊C8×C73—21no—
71C4⋊C8×C73—21no—
72C4⋊C8×C73—21no—
73C4⋊C8×C73—21no—
74C4⋊C8×C73D412yes—
75C4⋊C8×C73D412yes—
76#103×C73—22no—
77#103×C73—22no—
78#102×C73—22no—
79#102×C73—22no—
80#102×C73—22no—
81#102×C73—22no—
82C22⋊C8×C73—44no—
83C4×C8×C73—21no—
84C4×C8×C73—21no—
85C24×C73—21no—
86#41×C73—41no—
87#41×C73—41no—
88#41×C73—41no—
89#41×C73—41no—
90C2×Q8×C73—21no—
91C2×Q8×C73—21no—
92C2×Q8×C73—21no—
93C2×D4×C73—21no—
94C2×D4×C73—21no—
95C2×D4×C73—21no—
96C2×D4×C73—21no—
97C2×D4×C73—21no—
98C2×D4×C73—21no—
99C2×D4×C73—21no—
100C2×D4×C73—21no—
101C2×D4×C73—41no—
102C2×D4×C73SD1612yescomplemented
103C22×C4×C73—21no—
104C22×C4×C73—42no—
105C22×C4×C73—42no—
106C22×C4×C73—41no—
107C22×C4×C73—41no—
108C22×C4×C73—41no—
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73#4111yescharacteristic
112SD16×C73—44no—
113SD16×C73—82no—
114D8×C73—44no—
115M16×C73—41no—
116M16×C73—41no—
117M16×C73—41no—
118M16×C73—41no—
119C2×C8×C73—22no—
120C2×C8×C73—22no—
121C2×C8×C73—22no—
122C2×C8×C73—22no—
123C2×C8×C73—41no—
124C2×C8×C73—41no—
125C2×C8×C73—42no—
126C4⋊C4×C73—21no—
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—21no—
130C4⋊C4×C73—21no—
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—41no—
135C4⋊C4×C73—41no—
136C22⋊C4×C73—41no—
137C22⋊C4×C73—41no—
138C22⋊C4×C73—41no—
139C22⋊C4×C73—41no—
140C22⋊C4×C73—41no—
141C42×C73—41no—
142C42×C73C2×D411yescharacteristic
143C42×C73C2×D411yescharacteristic
144C42×C73C2×D411yescharacteristic
145C42×C73C2×D411yescharacteristic
146C23×C73—22no—
147C23×C73—42no—
148C23×C73—42no—
149C23×C73—41no—
150C23×C73—41no—
151C23×C73—41no—
152C23×C73#12111yescharacteristic
153Q8×C73—42no—
154Q8×C73—41no—
155Q8×C73—41no—
156Q8×C73—41no—
157Q8×C73—41no—
158D4×C73—22no—
159D4×C73—22no—
160D4×C73—42no—
161D4×C73—41no—
162D4×C73—41no—
163D4×C73—41no—
164D4×C73—41no—
165D4×C73—41no—
166D4×C73—41no—
167D4×C73—81no—
168D4×C73—81no—
169C2×C4×C73—21no—
170C2×C4×C73—21no—
171C2×C4×C73—41no—
172C2×C4×C73—42no—
173C2×C4×C73—41no—
174C2×C4×C73—41no—
175C2×C4×C73—41no—
176C2×C4×C73—41no—
177C2×C4×C73—82no—
178C2×C4×C73—81no—
179C2×C4×C73—81no—
180C2×C4×C73—81no—
181C2×C4×C73—81no—
182C2×C4×C73#12011yescharacteristic
183C2×C4×C73#12011yescharacteristic
184C2×C4×C73#12011yescharacteristic
185C2×C4×C73#12111yescharacteristic
186C2×C4×C73#12111yescharacteristic
187C2×C4×C73#12711yescharacteristic
188C2×C4×C73C2×SD1611yescharacteristic
189C2×C4×C73#13511yescharacteristic
190C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C73#13711yescharacteristic
192C8×C73—42no—
193C8×C73—42no—
194C8×C73—42no—
195C8×C73—42no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—44no—
199C22×C73—42no—
200C22×C73—82no—
201C22×C73—81no—
202C22×C73—81no—
203C22×C73—81no—
204C22×C73—81no—
205C22×C73—81no—
206C22×C73#39211yescharacteristic
207C22×C73#46511yescharacteristic
208C22×C73#46811yescharacteristic
209C4×C73—21no—
210C4×C73—21no—
211C4×C73—21no—
212C4×C73—81no—
213C4×C73—81no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73#44811yescharacteristic
217C4×C73#44911yescharacteristic
218C4×C73#45011yescharacteristic
219C4×C73#45211yescharacteristic
220C4×C73#45311yescharacteristic
221C4×C73#45411yescharacteristic
222#10256C7311yesmaximal, characteristic, direct product factor, Sylow
223C2×C73—21no—
224C2×C73—21no—
225C2×C73—82no—
226C2×C73—161no—
227C2×C73#200011yescharacteristic
228C2×C73#200611yescharacteristic
229C2×C73#201711yescharacteristic
230#3246C2×C7311yescharacteristic, complemented
231#3077C2×C7311yescharacteristic, complemented
232#3036C2×C7311yescharacteristic
233#1954C2×C7311yescharacteristic, complemented
234#1493C2×C7311yescharacteristic, complemented
235#1492C2×C7311yescharacteristic, complemented
236#1292C2×C7311yescharacteristic, complemented
237C73#1025611yescharacteristic, direct product factor, Sylow
238C2×AGL(1,ℤ/8)—21no—
239#533C22×C7311yescharacteristic
240#532C22×C7311yescharacteristic
241C2×C4×D4C22×C7311yescharacteristic
242#478—22no—
243#476—21no—
244#476—21no—
245#475—21no—
246#474—21no—
247#465—22no—
248#463—22no—
249#461—21no—
250#461—21no—
251#460—21no—
252#459—21no—
253#428—21no—
254#423C22×C7311yescharacteristic
255#423C22×C7311yescharacteristic
256C2×C4⋊C8C22×C7311yescharacteristic
257#418—21no—
258#417—21no—
259C2×C22⋊C8—22no—
260C2×C4×C8C22×C7311yescharacteristic
261Q8⋊C8—22no—
262D4⋊C8—22no—
263C2×#41—21no—
264C22×D4D4×C7311yescharacteristic
265C23×C4—21no—
266AGL(1,ℤ/8)—42no—
267AGL(1,ℤ/8)—42no—
268C2×SD16—22no—
269C2×SD16—22no—
270C2×D8—22no—
271C2×M16—21no—
272C2×M16—21no—
273C22×C8—21no—
274C22×C8D4×C7312yes—
275C4⋊Q8D4×C7311yescharacteristic
276C4⋊Q8D4×C7311yescharacteristic
277C4⋊Q8D4×C7311yescharacteristic
278#127D4×C7311yescharacteristic
279#125—21no—
280#124—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283#121—21no—
284#121—21no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293#117—21no—
294C2×C4⋊C4D4×C7311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C23×C7311yescharacteristic
297#107—22no—
298#106—22no—
299#106—22no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8D4×C7312yes—
305C4⋊C8D4×C7312yes—
306#103—22no—
307#103—22no—
308#102—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312C22⋊C8—44no—
313C4×C8—21no—
314C4×C8—21no—
315C24—21no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4SD16×C7312yescomplemented
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4C2×D4×C7311yescharacteristic
340C22×C4C2×D4×C7311yescharacteristic
341C22×C4#41×C7311yescharacteristic
342SD16—44no—
343SD16—82no—
344D8—44no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C42—41no—
372C42C2×D4×C7311yescharacteristic
373C42C2×D4×C7311yescharacteristic
374C42C2×D4×C7311yescharacteristic
375C42C2×D4×C7311yescharacteristic
376C23—22no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23#121×C7311yescharacteristic
383Q8—42no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—41no—
388D4—22no—
389D4—22no—
390D4—42no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—81no—
398D4—81no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4#120×C7311yescharacteristic
413C2×C4#120×C7311yescharacteristic
414C2×C4#120×C7311yescharacteristic
415C2×C4#121×C7311yescharacteristic
416C2×C4#121×C7311yescharacteristic
417C2×C4#127×C7311yescharacteristic
418C2×C4C2×SD16×C7311yescharacteristic
419C2×C4#135×C7311yescharacteristic
420C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
421C2×C4#137×C7311yescharacteristic
422C8—42no—
423C8—42no—
424C8—42no—
425C8—42no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22—42no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#392×C7311yescharacteristic
437C22#465×C7311yescharacteristic
438C22#468×C7311yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4#448×C7311yescharacteristic
447C4#449×C7311yescharacteristic
448C4#450×C7311yescharacteristic
449C4#452×C7311yescharacteristic
450C4#453×C7311yescharacteristic
451C4#454×C7311yescharacteristic
452C2—21no—
453C2—21no—
454C2—82no—
455C2—161no—
456C2C73×#200011yescharacteristic
457C2C73×#200611yescharacteristic
458C2C73×#201711yescharacteristic
subgroupG/H
center, socleC22×C73#392
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC73×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3246C2semidirect
2. C73×#3077C2semidirect
4. C73×#1954C2semidirect
5. C73×#1493C2semidirect
6. C73×#1492C2semidirect
7. C73×#1292C2semidirect
102. C2×D4×C73SD16semidirect
222. #10256C73direct (N × H)
230. #3246C2×C73semidirect
231. #3077C2×C73semidirect
233. #1954C2×C73semidirect
234. #1493C2×C73semidirect
235. #1492C2×C73semidirect
236. #1292C2×C73semidirect
237. C73#10256direct (N × H)
332. C2×D4SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32462
C73×#30772
C73×#30362
C73×#19542
C73×#14932
C73×#14922
C73×#12922
#1025673