Finite Group Databasebeta

← C73×#61435 #32970839

Subgroups — C73×#61435 #32970839

Order 18688.

1852 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×#3482C212yesmaximal, complemented
2C73×#3395C211yesmaximal, characteristic, complemented
3Q8×D8×C73C211yesmaximal, characteristic, complemented
4C73×#3359C212yesmaximal, complemented
5C73×#3342C213yesmaximal, complemented
6C73×#3302C216yesmaximal, complemented
7C73×#3283C213yesmaximal, complemented
8C73×#3170C213yesmaximal, complemented
9C73×#3107C216yesmaximal, complemented
10C8×#41×C73C211yesmaximal, characteristic, complemented
11C4×#135×C73C213yesmaximal, complemented
12C2×#135×C73C2213yes—
13#557×C73C2212yes—
14#556×C73C2216yes—
15#552×C73—26no—
16D4×Q8×C73C2212yes—
17#548×C73—22no—
18#547×C73C2216yes—
19#533×C73C2216yes—
20#529×C73—26no—
21C4×#41×C73C2216yes—
22C4×#41×C73C2211yescharacteristic
23C2×C4×Q8×C73—22no—
24#500×C73C2213yes—
25#494×C73C2213yes—
26#487×C73C2216yes—
27#485×C73C22112yes—
28#475×C73C2216yes—
29#474×C73C2216yes—
30#467×C73C2216yes—
31#467×C73C2216yes—
32#462×C73C2216yes—
33#461×C73C2216yes—
34#452×C73C22112yes—
35C8×Q8×C73C2211yescharacteristic
36C8×Q8×C73C2211yescharacteristic
37C4×Q16×C73C2213yes—
38C4×Q16×C73C2216yes—
39C4×Q16×C73C2213yes—
40C4×SD16×C73C2216yes—
41C4×SD16×C73C2216yes—
42C4×D8×C73C2213yes—
43C4×D8×C73C2211yescharacteristic
44C8×D4×C73C2213yes—
45C8×D4×C73C2213yes—
46#431×C73C2213yes—
47#427×C73C2213yes—
48#416×C73C2216yes—
49C2×C4×C8×C73C2213yes—
50C2×#41×C73D411yescharacteristic
51C2×#41×C73C2316yes—
52C22×Q8×C73—22no—
53#135×C73—212no—
54C2×Q16×C73C2313yes—
55C2×Q16×C73C2316yes—
56C2×SD16×C73C2316yes—
57C2×D8×C73C2311yescharacteristic
58C22×C8×C73C2313yes—
59C22×C8×C73C2313yes—
60C4⋊Q8×C73C2316yes—
61C4⋊Q8×C73C2316yes—
62#126×C73—212no—
63#125×C73—26no—
64#125×C73—26no—
65#124×C73—26no—
66#123×C73—212no—
67C22⋊Q8×C73—26no—
68C22⋊Q8×C73—26no—
69C22⋊Q8×C73C23112yes—
70#121×C73—26no—
71C4×Q8×C73—22no—
72C4×Q8×C73—26no—
73C4×Q8×C73—26no—
74C4×Q8×C73D411yescharacteristic
75C4×Q8×C73C2316yes—
76C4×Q8×C73C2316yes—
77C4×Q8×C73C2311yescharacteristic
78C4×D4×C73—26no—
79C4×D4×C73—26no—
80C4×D4×C73D413yes—
81C4×D4×C73C2312yes—
82C4×D4×C73C2316yes—
83C4×D4×C73C2313yes—
84#117×C73—26no—
85#117×C73—26no—
86#117×C73D413yes—
87#117×C73C2316yes—
88C2×C4⋊C4×C73—26no—
89C2×C42×C73—26no—
90C2×C42×C73C2313yes—
91#107×C73C2311yescharacteristic
92#107×C73C2313yes—
93#107×C73C2316yes—
94#106×C73C2316yes—
95C4⋊C8×C73C2313yes—
96C4⋊C8×C73C2313yes—
97#103×C73C2316yes—
98#103×C73C2316yes—
99#103×C73C23112yes—
100#102×C73C2312yes—
101#102×C73C2316yes—
102C22⋊C8×C73C2316yes—
103C4×C8×C73C2311yescharacteristic
104C4×C8×C73C2313yes—
105C4×C8×C73C2313yes—
106C4×C8×C73C2313yes—
107#41×C73—24no—
108#41×C73—212no—
109#41×C73—46no—
110C2×Q8×C73—26no—
111C2×Q8×C73—48no—
112C2×Q8×C73C2×D411yescharacteristic
113C2×Q8×C73C2416yes—
114C2×D4×C73—26no—
115C2×D4×C73C2×D413yes—
116C2×D4×C73C2412yes—
117C22×C4×C73—26no—
118C22×C4×C73—26no—
119C22×C4×C73—22no—
120C22×C4×C73C2×D413yes—
121C22×C4×C73C2×D413yes—
122C22×C4×C73C2413yes—
123Q16×C73—26no—
124Q16×C73—46no—
125SD16×C73—212no—
126D8×C73#4114yescomplemented
127C2×C8×C73—26no—
128C2×C8×C73—43no—
129C2×C8×C73C2411yescharacteristic
130C2×C8×C73C2413yes—
131C2×C8×C73C2413yes—
132C2×C8×C73C2411yescharacteristic
133C4⋊C4×C73—26no—
134C4⋊C4×C73—212no—
135C4⋊C4×C73—26no—
136C4⋊C4×C73—26no—
137C4⋊C4×C73—26no—
138C4⋊C4×C73—26no—
139C4⋊C4×C73C2×D413yes—
140C4⋊C4×C73C2×D413yes—
141C4⋊C4×C73C2412yes—
142C4⋊C4×C73C2416yes—
143C22⋊C4×C73—212no—
144C22⋊C4×C73—26no—
145C22⋊C4×C73—22no—
146C22⋊C4×C73—26no—
147C22⋊C4×C73C2×D416yes—
148C42×C73—26no—
149C42×C73—26no—
150C42×C73—26no—
151C42×C73C2×D413yes—
152C42×C73C2×D413yes—
153C42×C73C2413yes—
154C42×C73C2411yescharacteristic
155C23×C73—22no—
156C23×C73C22×D413yes—
157Q8×C73—26no—
158Q8×C73—26no—
159Q8×C73—412no—
160Q8×C73#13514yescomplemented
161D4×C73—26no—
162D4×C73—22no—
163D4×C73—412no—
164D4×C73C2×#4114yes—
165C2×C4×C73—26no—
166C2×C4×C73—26no—
167C2×C4×C73—23no—
168C2×C4×C73—23no—
169C2×C4×C73—22no—
170C2×C4×C73—26no—
171C2×C4×C73—26no—
172C2×C4×C73—412no—
173C2×C4×C73—412no—
174C2×C4×C73—44no—
175C2×C4×C73C22×D413yes—
176C2×C4×C73C22×D411yescharacteristic
177C2×C4×C73C22×D413yes—
178C2×C4×C73C22×D413yes—
179C2×C4×C73C22×D411yescharacteristic
180C2×C4×C73C2511yescharacteristic
181C8×C73—23no—
182C8×C73—21no—
183C8×C73—23no—
184C8×C73C2×#4112yes—
185C22×C73—23no—
186C22×C73—24no—
187C22×C73—23no—
188C22×C73—23no—
189C22×C73—42no—
190C22×C73—42no—
191C22×C73C23×D411yescharacteristic
192C4×C73—23no—
193C4×C73—21no—
194C4×C73—23no—
195C4×C73—46no—
196C4×C73—46no—
197C4×C73—42no—
198C4×C73C2×#13516yes—
199C4×C73#57812yes—
200C4×C73C22×#4111yescharacteristic
201C4×C73#58511yescharacteristic
202#61435C7311yesmaximal, characteristic, direct product factor, Sylow
203C2×C73—43no—
204C2×C73—44no—
205C2×C73D4×#4111yescharacteristic
206C2×C73C22×#13511yescharacteristic
207C2×C73C2×#57811yescharacteristic
208#3482C2×C7312yescomplemented
209#3395C2×C7311yescharacteristic, complemented
210Q8×D8C2×C7311yescharacteristic, complemented
211#3359C2×C7312yescomplemented
212#3342C2×C7313yescomplemented
213#3302C2×C7316yescomplemented
214#3283C2×C7313yescomplemented
215#3170C2×C7313yescomplemented
216#3107C2×C7316yescomplemented
217C8×#41C2×C7311yescharacteristic, complemented
218C4×#135C2×C7313yescomplemented
219C73#6143511yescharacteristic, direct product factor, Sylow
220C2×#135C22×C7313yes—
221#557C22×C7312yes—
222#556C22×C7316yes—
223#552—26no—
224D4×Q8C22×C7312yes—
225#548—22no—
226#547C22×C7316yes—
227#533C22×C7316yes—
228#529—26no—
229C4×#41C22×C7316yes—
230C4×#41C22×C7311yescharacteristic
231C2×C4×Q8—22no—
232#500C22×C7313yes—
233#494C22×C7313yes—
234#487C22×C7316yes—
235#485C22×C73112yes—
236#475C22×C7316yes—
237#474C22×C7316yes—
238#467C22×C7316yes—
239#467C22×C7316yes—
240#462C22×C7316yes—
241#461C22×C7316yes—
242#452C22×C73112yes—
243C8×Q8C22×C7311yescharacteristic
244C8×Q8C22×C7311yescharacteristic
245C4×Q16C22×C7313yes—
246C4×Q16C22×C7316yes—
247C4×Q16C22×C7313yes—
248C4×SD16C22×C7316yes—
249C4×SD16C22×C7316yes—
250C4×D8C22×C7313yes—
251C4×D8C22×C7311yescharacteristic
252C8×D4C22×C7313yes—
253C8×D4C22×C7313yes—
254#431C22×C7313yes—
255#427C22×C7313yes—
256#416C22×C7316yes—
257C2×C4×C8C22×C7313yes—
258C2×#41D4×C7311yescharacteristic
259C2×#41C23×C7316yes—
260C22×Q8—22no—
261#135—212no—
262C2×Q16C23×C7313yes—
263C2×Q16C23×C7316yes—
264C2×SD16C23×C7316yes—
265C2×D8C23×C7311yescharacteristic
266C22×C8C23×C7313yes—
267C22×C8C23×C7313yes—
268C4⋊Q8C23×C7316yes—
269C4⋊Q8C23×C7316yes—
270#126—212no—
271#125—26no—
272#125—26no—
273#124—26no—
274#123—212no—
275C22⋊Q8—26no—
276C22⋊Q8—26no—
277C22⋊Q8C23×C73112yes—
278#121—26no—
279C4×Q8—22no—
280C4×Q8—26no—
281C4×Q8—26no—
282C4×Q8D4×C7311yescharacteristic
283C4×Q8C23×C7316yes—
284C4×Q8C23×C7316yes—
285C4×Q8C23×C7311yescharacteristic
286C4×D4—26no—
287C4×D4—26no—
288C4×D4D4×C7313yes—
289C4×D4C23×C7312yes—
290C4×D4C23×C7316yes—
291C4×D4C23×C7313yes—
292#117—26no—
293#117—26no—
294#117D4×C7313yes—
295#117C23×C7316yes—
296C2×C4⋊C4—26no—
297C2×C42—26no—
298C2×C42C23×C7313yes—
299#107C23×C7311yescharacteristic
300#107C23×C7313yes—
301#107C23×C7316yes—
302#106C23×C7316yes—
303C4⋊C8C23×C7313yes—
304C4⋊C8C23×C7313yes—
305#103C23×C7316yes—
306#103C23×C7316yes—
307#103C23×C73112yes—
308#102C23×C7312yes—
309#102C23×C7316yes—
310C22⋊C8C23×C7316yes—
311C4×C8C23×C7311yescharacteristic
312C4×C8C23×C7313yes—
313C4×C8C23×C7313yes—
314C4×C8C23×C7313yes—
315#41—24no—
316#41—212no—
317#41—46no—
318C2×Q8—26no—
319C2×Q8—48no—
320C2×Q8C2×D4×C7311yescharacteristic
321C2×Q8C24×C7316yes—
322C2×D4—26no—
323C2×D4C2×D4×C7313yes—
324C2×D4C24×C7312yes—
325C22×C4—22no—
326C22×C4—26no—
327C22×C4—26no—
328C22×C4C2×D4×C7313yes—
329C22×C4C2×D4×C7313yes—
330C22×C4C24×C7313yes—
331Q16—26no—
332Q16—46no—
333SD16—212no—
334D8#41×C7314yescomplemented
335C2×C8—26no—
336C2×C8—43no—
337C2×C8C24×C7311yescharacteristic
338C2×C8C24×C7313yes—
339C2×C8C24×C7313yes—
340C2×C8C24×C7311yescharacteristic
341C4⋊C4—26no—
342C4⋊C4—212no—
343C4⋊C4—26no—
344C4⋊C4—26no—
345C4⋊C4—26no—
346C4⋊C4—26no—
347C4⋊C4C2×D4×C7313yes—
348C4⋊C4C2×D4×C7313yes—
349C4⋊C4C24×C7312yes—
350C4⋊C4C24×C7316yes—
351C22⋊C4—212no—
352C22⋊C4—26no—
353C22⋊C4—22no—
354C22⋊C4—26no—
355C22⋊C4C2×D4×C7316yes—
356C42—26no—
357C42—26no—
358C42—26no—
359C42C2×D4×C7313yes—
360C42C2×D4×C7313yes—
361C42C24×C7311yescharacteristic
362C42C24×C7313yes—
363C23—22no—
364C23C22×D4×C7313yes—
365Q8—26no—
366Q8—26no—
367Q8—412no—
368Q8#135×C7314yescomplemented
369D4—26no—
370D4—22no—
371D4—412no—
372D4C2×#41×C7314yes—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—23no—
376C2×C4—23no—
377C2×C4—22no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—412no—
381C2×C4—412no—
382C2×C4—44no—
383C2×C4C22×D4×C7313yes—
384C2×C4C22×D4×C7311yescharacteristic
385C2×C4C22×D4×C7313yes—
386C2×C4C22×D4×C7313yes—
387C2×C4C22×D4×C7311yescharacteristic
388C2×C4C25×C7311yescharacteristic
389C8—23no—
390C8—21no—
391C8—23no—
392C8C2×#41×C7312yes—
393C22—23no—
394C22—24no—
395C22—23no—
396C22—23no—
397C22—42no—
398C22—42no—
399C22C23×D4×C7311yescharacteristic
400C4—23no—
401C4—21no—
402C4—23no—
403C4—46no—
404C4—46no—
405C4—42no—
406C4C2×#135×C7316yes—
407C4#578×C7312yes—
408C4C22×#41×C7311yescharacteristic
409C4#585×C7311yescharacteristic
410C2—43no—
411C2—44no—
412C2D4×#41×C7311yescharacteristic
413C2C22×#135×C7311yescharacteristic
414C2C2×#578×C7311yescharacteristic
subgroupG/H
centerC2×C4×C73C22×D4
commutator, Frattini, cosocleC2×C4C25×C73
socleC22×C73C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C73 ◁ C4×#41×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3482C2semidirect
2. C73×#3395C2semidirect
3. Q8×D8×C73C2semidirect
4. C73×#3359C2semidirect
5. C73×#3342C2semidirect
6. C73×#3302C2semidirect
7. C73×#3283C2semidirect
8. C73×#3170C2semidirect
9. C73×#3107C2semidirect
10. C8×#41×C73C2semidirect
11. C4×#135×C73C2semidirect
126. D8×C73#41semidirect
160. Q8×C73#135semidirect
202. #61435C73direct (N × H)
208. #3482C2×C73semidirect
209. #3395C2×C73semidirect
210. Q8×D8C2×C73semidirect
211. #3359C2×C73semidirect
212. #3342C2×C73semidirect
213. #3302C2×C73semidirect
214. #3283C2×C73semidirect
215. #3170C2×C73semidirect
216. #3107C2×C73semidirect
217. C8×#41C2×C73semidirect
218. C4×#135C2×C73semidirect
219. C73#61435direct (N × H)
334. D8#41×C73semidirect
368. Q8#135×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34822
C73×#33952
Q8×D8×C732
C73×#33592
C73×#33422
C73×#33022
C73×#32832
C73×#31702
C73×#31072
C8×#41×C732
C4×#135×C732
#6143573