Finite Group Databasebeta

← C73×#11759 #32921163

Subgroups — C73×#11759 #32921163

Order 18688.

586 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2561C211yesmaximal, characteristic
2C2×#342×C73C211yesmaximal, characteristic
3C73×#1462C211yesmaximal, characteristic
4C73×#1351C212yesmaximal, complemented
5C73×#1344C212yesmaximal, complemented
6C2×C22⋊Q8×C73C411yescharacteristic, complemented
7C22×C22⋊C4×C73—21no—
8C22×C22⋊C4×C73C2211yescharacteristic
9#407×C73—21no—
10#397×C73C411yescharacteristic, complemented
11#397×C73C2211yescharacteristic
12#393×C73C411yescharacteristic, complemented
13#393×C73C2211yescharacteristic
14#388×C73—21no—
15#388×C73—21no—
16#381×C73—21no—
17#379×C73—21no—
18C4×C22⋊C4×C73C411yescharacteristic, complemented
19#342×C73—21no—
20#342×C73C2212yes—
21#323×C73—21no—
22#323×C73C2212yes—
23C25×C73D411yescharacteristic
24C22×Q8×C73C2×C411yescharacteristic, complemented
25C23×C4×C73—41no—
26C23×C4×C73C2×C411yescharacteristic
27C23×C4×C73D411yescharacteristic
28C2×M16×C73—41no—
29C22⋊Q8×C73—41no—
30C22⋊Q8×C73—41no—
31C2×C4⋊C4×C73—21no—
32C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
33C2×C22⋊C4×C73—24no—
34C2×C22⋊C4×C73—22no—
35C2×C22⋊C4×C73—21no—
36C2×C22⋊C4×C73—21no—
37C2×C22⋊C4×C73—21no—
38C2×C22⋊C4×C73—21no—
39C2×C22⋊C4×C73—21no—
40C2×C22⋊C4×C73—21no—
41C2×C22⋊C4×C73—41no—
42C2×C22⋊C4×C73—41no—
43C2×C22⋊C4×C73—41no—
44C2×C22⋊C4×C73—42no—
45C2×C22⋊C4×C73C2×C411yescharacteristic
46C2×C22⋊C4×C73D411yescharacteristic
47C2×C22⋊C4×C73D411yescharacteristic
48C2×C22⋊C4×C73C2311yescharacteristic
49C2×C42×C73—21no—
50C22⋊C8×C73—22no—
51C22⋊C8×C73—41no—
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73C2×C411yescharacteristic, complemented
57#95×C73C2×C411yescharacteristic, complemented
58C24×C73—21no—
59C24×C73—21no—
60C24×C73—22no—
61C24×C73—21no—
62C24×C73—21no—
63C24×C73—42no—
64C24×C73—81no—
65C24×C73C22⋊C411yescharacteristic
66C24×C73C22⋊C411yescharacteristic
67C24×C73C2×D411yescharacteristic
68C2×Q8×C73—41no—
69C2×Q8×C73—81no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—42no—
79C22×C4×C73—44no—
80C22×C4×C73—42no—
81C22×C4×C73—81no—
82C22×C4×C73—82no—
83C22×C4×C73—82no—
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22×C411yescharacteristic
87C22×C4×C73C2×D411yescharacteristic
88M16×C73—82no—
89C2×C8×C73—42no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—42no—
92C22⋊C4×C73—24no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C22⋊C4×C73—41no—
97C22⋊C4×C73—42no—
98C22⋊C4×C73—44no—
99C22⋊C4×C73—41no—
100C22⋊C4×C73—41no—
101C22⋊C4×C73—42no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—42no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—81no—
106C22⋊C4×C73—81no—
107C42×C73—81no—
108C23×C73—22no—
109C23×C73—22no—
110C23×C73—22no—
111C23×C73—22no—
112C23×C73—22no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—41no—
120C23×C73—42no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—42no—
126C23×C73—81no—
127C23×C73—81no—
128C23×C73—84no—
129C23×C73—82no—
130C23×C73—82no—
131C23×C73#9911yescharacteristic
132C23×C73#9911yescharacteristic
133C23×C73#10412yes—
134C23×C73C2×C22⋊C411yescharacteristic
135C23×C73AGL(1,ℤ/8)11yescharacteristic
136C23×C73#13711yescharacteristic
137Q8×C73—82no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—42no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—44no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—81no—
161C2×C4×C73—81no—
162C2×C4×C73—81no—
163C2×C4×C73—88no—
164C2×C4×C73—82no—
165C2×C4×C73—82no—
166C8×C73—82no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—22no—
176C22×C73—21no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73—42no—
180C22×C73—41no—
181C22×C73—41no—
182C22×C73—44no—
183C22×C73—44no—
184C22×C73—41no—
185C22×C73—41no—
186C22×C73—41no—
187C22×C73—41no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73—82no—
191C22×C73—82no—
192C22×C73#35412yes—
193C22×C73C2×#9911yescharacteristic
194C22×C73#41711yescharacteristic
195C22×C73C2×#10411yescharacteristic
196C4×C73—41no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73—41no—
200C4×C73—81no—
201C4×C73—81no—
202C4×C73—84no—
203C4×C73—81no—
204C4×C73—81no—
205#11759C7311yesmaximal, characteristic, direct product factor, Sylow
206C2×C73—21no—
207C2×C73—21no—
208C2×C73—22no—
209C2×C73—41no—
210C2×C73—41no—
211C2×C73—42no—
212C2×C73—41no—
213C2×C73#150411yescharacteristic
214C2×C73#212111yescharacteristic
215C2×C73C2×#35411yescharacteristic
216#2561C2×C7311yescharacteristic
217C2×#342C2×C7311yescharacteristic
218#1462C2×C7311yescharacteristic
219#1351C2×C7312yescomplemented
220#1344C2×C7312yescomplemented
221C73#1175911yescharacteristic, direct product factor, Sylow
222C2×C22⋊Q8C4×C7311yescharacteristic, complemented
223C22×C22⋊C4—21no—
224C22×C22⋊C4C22×C7311yescharacteristic
225#407—21no—
226#397C4×C7311yescharacteristic, complemented
227#397C22×C7311yescharacteristic
228#393C4×C7311yescharacteristic, complemented
229#393C22×C7311yescharacteristic
230#388—21no—
231#388—21no—
232#381—21no—
233#379—21no—
234C4×C22⋊C4C4×C7311yescharacteristic, complemented
235#342—21no—
236#342C22×C7312yes—
237#323—21no—
238#323C22×C7312yes—
239C25D4×C7311yescharacteristic
240C22×Q8C2×C4×C7311yescharacteristic, complemented
241C23×C4—41no—
242C23×C4C2×C4×C7311yescharacteristic
243C23×C4D4×C7311yescharacteristic
244C2×M16—41no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C2×C4×C7311yescharacteristic, complemented
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4C2×C4×C7311yescharacteristic
262C2×C22⋊C4D4×C7311yescharacteristic
263C2×C22⋊C4D4×C7311yescharacteristic
264C2×C22⋊C4C23×C7311yescharacteristic
265C2×C42—21no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95C2×C4×C7311yescharacteristic, complemented
273#95C2×C4×C7311yescharacteristic, complemented
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—42no—
280C24—81no—
281C24C22⋊C4×C7311yescharacteristic
282C24C22⋊C4×C7311yescharacteristic
283C24C2×D4×C7311yescharacteristic
284C2×Q8—41no—
285C2×Q8—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C22⋊C4×C7311yescharacteristic
301C22×C4C22⋊C4×C7311yescharacteristic
302C22×C4C22×C4×C7311yescharacteristic
303C22×C4C2×D4×C7311yescharacteristic
304M16—82no—
305C2×C8—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C22⋊C4—24no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—81no—
322C22⋊C4—81no—
323C42—81no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23#99×C7311yescharacteristic
348C23#99×C7311yescharacteristic
349C23#104×C7312yes—
350C23C2×C22⋊C4×C7311yescharacteristic
351C23AGL(1,ℤ/8)×C7311yescharacteristic
352C23#137×C7311yescharacteristic
353Q8—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—88no—
380C2×C4—82no—
381C2×C4—82no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—44no—
399C22—44no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22—82no—
408C22#354×C7312yes—
409C22C2×#99×C7311yescharacteristic
410C22#417×C7311yescharacteristic
411C22C2×#104×C7311yescharacteristic
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—81no—
417C4—81no—
418C4—84no—
419C4—81no—
420C4—81no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2—41no—
428C2C73×#150411yescharacteristic
429C2C73×#212111yescharacteristic
430C2C2×#354×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#99
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C22⋊C4C23×C73
prime radicalC25×C73D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C24×C73 ◁ C22×C22⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
4. C73×#1351C2semidirect
5. C73×#1344C2semidirect
6. C2×C22⋊Q8×C73C4semidirect
10. #397×C73C4semidirect
12. #393×C73C4semidirect
18. C4×C22⋊C4×C73C4semidirect
24. C22×Q8×C73C2×C4semidirect
32. C2×C4⋊C4×C73C2×C4semidirect
56. #95×C73C2×C4semidirect
57. #95×C73C2×C4semidirect
205. #11759C73direct (N × H)
219. #1351C2×C73semidirect
220. #1344C2×C73semidirect
221. C73#11759direct (N × H)
222. C2×C22⋊Q8C4×C73semidirect
226. #397C4×C73semidirect
228. #393C4×C73semidirect
234. C4×C22⋊C4C4×C73semidirect
240. C22×Q8C2×C4×C73semidirect
248. C2×C4⋊C4C2×C4×C73semidirect
272. #95C2×C4×C73semidirect
273. #95C2×C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#25612
C2×#342×C732
C73×#14622
C73×#13512
C73×#13442
#1175973