Finite Group Databasebeta

← C73×#33785 #32943189

Subgroups — C73×#33785 #32943189

Order 18688.

594 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3168C211yesmaximal, characteristic, complemented
2C73×#3147C211yesmaximal, characteristic, complemented
3C73×#2251C211yesmaximal, characteristic, complemented
4C73×#2250C211yesmaximal, characteristic, complemented
5C73×#2245C211yesmaximal, characteristic, complemented
6C73×#2244C211yesmaximal, characteristic, complemented
7C73×#2211C212yesmaximal, complemented
8C73×#2210C212yesmaximal, complemented
9C73×#2189C212yesmaximal, complemented
10C73×#2186C212yesmaximal, complemented
11C73×#2163C211yesmaximal, characteristic, complemented
12C2×#135×C73—21no—
13C2×#135×C73C2212yes—
14#532×C73—21no—
15#529×C73—21no—
16C2×Q32×C73C2212yes—
17C2×SD32×C73C2212yes—
18C2×SD32×C73C2212yes—
19C2×D16×C73C2212yes—
20#499×C73C2211yescharacteristic
21#495×C73—21no—
22#495×C73C2212yes—
23#495×C73C2211yescharacteristic
24#494×C73C2211yescharacteristic
25#493×C73C2211yescharacteristic
26#492×C73—21no—
27#488×C73—21no—
28#467×C73C2211yescharacteristic
29#466×C73C2211yescharacteristic
30#465×C73—21no—
31C4×Q16×C73C2211yescharacteristic
32C4×SD16×C73—21no—
33C4×D8×C73C2211yescharacteristic
34C2×C4×C8×C73C2211yescharacteristic
35C4⋊C16×C73C2212yes—
36#358×C73C2212yes—
37#358×C73C2212yes—
38#357×C73C2212yes—
39#357×C73C2212yes—
40C22⋊C16×C73C2212yes—
41#346×C73C2211yescharacteristic
42#346×C73C2211yescharacteristic
43C2×#41×C73—22no—
44#135×C73—42no—
45#135×C73—42no—
46#135×C73—44no—
47C2×Q16×C73—21no—
48C2×Q16×C73C2311yescharacteristic
49C2×Q16×C73C2312yes—
50C2×SD16×C73—21no—
51C2×SD16×C73—22no—
52C2×SD16×C73—22no—
53C2×D8×C73—21no—
54C2×D8×C73C2311yescharacteristic
55C2×D8×C73C2312yes—
56C22×C8×C73D411yescharacteristic
57C22×C8×C73C2311yescharacteristic
58C4⋊Q8×C73—21no—
59#127×C73—21no—
60#126×C73—41no—
61#125×C73—21no—
62#124×C73—21no—
63#124×C73—22no—
64#123×C73—41no—
65C22⋊Q8×C73—21no—
66#121×C73—21no—
67#121×C73—41no—
68#121×C73—41no—
69C4×Q8×C73—21no—
70C4×D4×C73—21no—
71C4×D4×C73—41no—
72C4×D4×C73—41no—
73C4×D4×C73—41no—
74#117×C73—41no—
75C2×C42×C73D411yescharacteristic
76Q32×C73—42no—
77SD32×C73—42no—
78SD32×C73—42no—
79D16×C73—42no—
80C2×C16×C73C2312yes—
81C2×C16×C73C2312yes—
82#107×C73C2311yescharacteristic
83#107×C73C2311yescharacteristic
84#106×C73—21no—
85#103×C73—21no—
86#103×C73—21no—
87#102×C73—21no—
88#102×C73—21no—
89C4×C8×C73D411yescharacteristic
90C4×C8×C73D411yescharacteristic
91C4×C8×C73C2311yescharacteristic
92C4×C8×C73C2311yescharacteristic
93#41×C73—44no—
94#41×C73—44no—
95C2×Q8×C73—21no—
96C2×Q8×C73—22no—
97C2×D4×C73—21no—
98C2×D4×C73—22no—
99C2×D4×C73—41no—
100C2×D4×C73—42no—
101C2×D4×C73—41no—
102C2×D4×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73—42no—
105C22×C4×C73D811yescharacteristic
106C22×C4×C73D811yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108Q16×C73—22no—
109Q16×C73—22no—
110Q16×C73—22no—
111Q16×C73—42no—
112SD16×C73—44no—
113SD16×C73—44no—
114SD16×C73—42no—
115D8×C73—22no—
116D8×C73—22no—
117D8×C73—22no—
118D8×C73—42no—
119C2×C8×C73—22no—
120C2×C8×C73—22no—
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73C2411yescharacteristic
125C4⋊C4×C73—21no—
126C4⋊C4×C73—21no—
127C4⋊C4×C73—41no—
128C4⋊C4×C73—41no—
129C4⋊C4×C73—41no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73—41no—
132C4⋊C4×C73—41no—
133C22⋊C4×C73—41no—
134C22⋊C4×C73—41no—
135C22⋊C4×C73—41no—
136C22⋊C4×C73—41no—
137C22⋊C4×C73—42no—
138C22⋊C4×C73—42no—
139C22⋊C4×C73—41no—
140C42×C73—41no—
141C42×C73D811yescharacteristic
142C42×C73D811yescharacteristic
143C42×C73C2×D411yescharacteristic
144C42×C73C2×D411yescharacteristic
145C16×C73—22no—
146C16×C73—22no—
147C23×C73—41no—
148C23×C73—42no—
149C23×C73C2×D811yescharacteristic
150Q8×C73—42no—
151Q8×C73—41no—
152Q8×C73—42no—
153Q8×C73—41no—
154D4×C73—42no—
155D4×C73—41no—
156D4×C73—42no—
157D4×C73—41no—
158D4×C73—82no—
159D4×C73—82no—
160D4×C73—82no—
161D4×C73—84no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—42no—
170C2×C4×C73—82no—
171C2×C4×C73—84no—
172C2×C4×C73C2×D811yescharacteristic
173C2×C4×C73C2×D811yescharacteristic
174C2×C4×C73C2×D811yescharacteristic
175C2×C4×C73C2×D811yescharacteristic
176C2×C4×C73C2×D811yescharacteristic
177C2×C4×C73C22×D411yescharacteristic
178C8×C73—21no—
179C8×C73—21no—
180C8×C73AGL(1,ℤ/8)12yes—
181C8×C73H(5,𝔽2)12yes—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—81no—
186C22×C73—82no—
187C22×C73—82no—
188C22×C73—81no—
189C22×C73—82no—
190C22×C73—81no—
191C22×C73C22×D811yescharacteristic
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—21no—
197C4×C73—81no—
198C4×C73—81no—
199C4×C73—82no—
200C4×C73—81no—
201C4×C73#53511yescharacteristic
202C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
203#33785C7311yesmaximal, characteristic, direct product factor, Sylow
204C2×C73—41no—
205C2×C73—161no—
206C2×C73—162no—
207C2×C73#320511yescharacteristic
208C2×C73#341411yescharacteristic
209C2×C73#341411yescharacteristic
210#3168C2×C7311yescharacteristic, complemented
211#3147C2×C7311yescharacteristic, complemented
212#2251C2×C7311yescharacteristic, complemented
213#2250C2×C7311yescharacteristic, complemented
214#2245C2×C7311yescharacteristic, complemented
215#2244C2×C7311yescharacteristic, complemented
216#2211C2×C7312yescomplemented
217#2210C2×C7312yescomplemented
218#2189C2×C7312yescomplemented
219#2186C2×C7312yescomplemented
220#2163C2×C7311yescharacteristic, complemented
221C73#3378511yescharacteristic, direct product factor, Sylow
222C2×#135—21no—
223C2×#135C22×C7312yes—
224#532—21no—
225#529—21no—
226C2×Q32C22×C7312yes—
227C2×SD32C22×C7312yes—
228C2×SD32C22×C7312yes—
229C2×D16C22×C7312yes—
230#499C22×C7311yescharacteristic
231#495—21no—
232#495C22×C7312yes—
233#495C22×C7311yescharacteristic
234#494C22×C7311yescharacteristic
235#493C22×C7311yescharacteristic
236#492—21no—
237#488—21no—
238#467C22×C7311yescharacteristic
239#466C22×C7311yescharacteristic
240#465—21no—
241C4×Q16C22×C7311yescharacteristic
242C4×SD16—21no—
243C4×D8C22×C7311yescharacteristic
244C2×C4×C8C22×C7311yescharacteristic
245C4⋊C16C22×C7312yes—
246#358C22×C7312yes—
247#358C22×C7312yes—
248#357C22×C7312yes—
249#357C22×C7312yes—
250C22⋊C16C22×C7312yes—
251#346C22×C7311yescharacteristic
252#346C22×C7311yescharacteristic
253C2×#41—22no—
254#135—42no—
255#135—42no—
256#135—44no—
257C2×Q16—21no—
258C2×Q16C23×C7312yes—
259C2×Q16C23×C7311yescharacteristic
260C2×SD16—22no—
261C2×SD16—22no—
262C2×SD16—21no—
263C2×D8—21no—
264C2×D8C23×C7312yes—
265C2×D8C23×C7311yescharacteristic
266C22×C8D4×C7311yescharacteristic
267C22×C8C23×C7311yescharacteristic
268C4⋊Q8—21no—
269#127—21no—
270#126—41no—
271#125—21no—
272#124—21no—
273#124—22no—
274#123—41no—
275C22⋊Q8—21no—
276#121—21no—
277#121—41no—
278#121—41no—
279C4×Q8—21no—
280C4×D4—21no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284#117—41no—
285C2×C42D4×C7311yescharacteristic
286Q32—42no—
287SD32—42no—
288SD32—42no—
289D16—42no—
290C2×C16C23×C7312yes—
291C2×C16C23×C7312yes—
292#107C23×C7311yescharacteristic
293#107C23×C7311yescharacteristic
294#106—21no—
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299C4×C8D4×C7311yescharacteristic
300C4×C8D4×C7311yescharacteristic
301C4×C8C23×C7311yescharacteristic
302C4×C8C23×C7311yescharacteristic
303#41—44no—
304#41—44no—
305C2×Q8—21no—
306C2×Q8—22no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4D8×C7311yescharacteristic
316C22×C4D8×C7311yescharacteristic
317C22×C4C2×D4×C7311yescharacteristic
318Q16—22no—
319Q16—22no—
320Q16—22no—
321Q16—42no—
322SD16—44no—
323SD16—44no—
324SD16—42no—
325D8—22no—
326D8—22no—
327D8—22no—
328D8—42no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8C2×D4×C7311yescharacteristic
332C2×C8C2×D4×C7311yescharacteristic
333C2×C8C2×D4×C7311yescharacteristic
334C2×C8C24×C7311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—41no—
350C42—41no—
351C42D8×C7311yescharacteristic
352C42D8×C7311yescharacteristic
353C42C2×D4×C7311yescharacteristic
354C42C2×D4×C7311yescharacteristic
355C16—22no—
356C16—22no—
357C23—41no—
358C23—42no—
359C23C2×D8×C7311yescharacteristic
360Q8—42no—
361Q8—41no—
362Q8—42no—
363Q8—41no—
364D4—42no—
365D4—41no—
366D4—42no—
367D4—41no—
368D4—82no—
369D4—82no—
370D4—84no—
371D4—82no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4C2×D8×C7311yescharacteristic
383C2×C4C2×D8×C7311yescharacteristic
384C2×C4C2×D8×C7311yescharacteristic
385C2×C4C2×D8×C7311yescharacteristic
386C2×C4C2×D8×C7311yescharacteristic
387C2×C4C22×D4×C7311yescharacteristic
388C8—21no—
389C8—21no—
390C8AGL(1,ℤ/8)×C7312yes—
391C8H(5,𝔽2)×C7312yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—81no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—82no—
400C22—81no—
401C22C22×D8×C7311yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—81no—
408C4—81no—
409C4—82no—
410C4—81no—
411C4#535×C7311yescharacteristic
412C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
413C2—41no—
414C2—161no—
415C2—162no—
416C2C73×#320511yescharacteristic
417C2C73×#341411yescharacteristic
418C2C73×#341411yescharacteristic
subgroupG/H
center, socleC22×C73C22×D8
commutator, Frattini, cosocleC2×C8C24×C73
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3168C2semidirect
2. C73×#3147C2semidirect
3. C73×#2251C2semidirect
4. C73×#2250C2semidirect
5. C73×#2245C2semidirect
6. C73×#2244C2semidirect
7. C73×#2211C2semidirect
8. C73×#2210C2semidirect
9. C73×#2189C2semidirect
10. C73×#2186C2semidirect
11. C73×#2163C2semidirect
203. #33785C73direct (N × H)
210. #3168C2×C73semidirect
211. #3147C2×C73semidirect
212. #2251C2×C73semidirect
213. #2250C2×C73semidirect
214. #2245C2×C73semidirect
215. #2244C2×C73semidirect
216. #2211C2×C73semidirect
217. #2210C2×C73semidirect
218. #2189C2×C73semidirect
219. #2186C2×C73semidirect
220. #2163C2×C73semidirect
221. C73#33785direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31682
C73×#31472
C73×#22512
C73×#22502
C73×#22452
C73×#22442
C73×#22112
C73×#22102
C73×#21892
C73×#21862
C73×#21632
#3378573