Finite Group Databasebeta

← C73×#12806 #32922210

Subgroups — C73×#12806 #32922210

Order 18688.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2930C211yesmaximal, characteristic
2C73×#2120C212yesmaximal
3C73×#1813C211yesmaximal, characteristic, complemented
4C73×#1812C211yesmaximal, characteristic, complemented
5C73×#1395C212yesmaximal, complemented
6C2×#131×C73C2211yescharacteristic
7C22×M16×C73C411yescharacteristic, complemented
8#525×C73C2211yescharacteristic
9#436×C73—21no—
10#436×C73—21no—
11#435×C73—21no—
12#435×C73—21no—
13C2×#108×C73—21no—
14#427×C73—21no—
15#424×C73C411yescharacteristic, complemented
16#413×C73C2212yes—
17#410×C73C2212yes—
18#408×C73C2211yescharacteristic
19#407×C73C411yescharacteristic, complemented
20#405×C73C411yescharacteristic, complemented
21#352×C73—22no—
22AΣL(1,𝔽16)×C73—22no—
23#338×C73—21no—
24H(3,ℤ/4)×C73—21no—
25C2×#41×C73C2311yescharacteristic
26C23×C4×C73C2×C411yescharacteristic
27#131×C73—41no—
28#131×C73—41no—
29C2×M16×C73—21no—
30C2×M16×C73—21no—
31C2×M16×C73—22no—
32C2×M16×C73—41no—
33C2×M16×C73—41no—
34C2×M16×C73C2×C411yescharacteristic
35C2×M16×C73C2×C411yescharacteristic
36C2×M16×C73D411yescharacteristic
37C2×M16×C73D411yescharacteristic
38C22×C8×C73C2×C411yescharacteristic
39C22×C8×C73C2×C411yescharacteristic
40C22×C8×C73D411yescharacteristic
41C22×C8×C73D411yescharacteristic
42#123×C73D411yescharacteristic
43#123×C73Q811yescharacteristic
44C22⋊Q8×C73—21no—
45#121×C73—21no—
46#120×C73D411yescharacteristic
47#120×C73Q811yescharacteristic
48C4×D4×C73—21no—
49C4×D4×C73—21no—
50#117×C73—22no—
51#117×C73C2×C411yescharacteristic
52#108×C73—41no—
53#108×C73—41no—
54#107×C73—41no—
55#106×C73—41no—
56C4⋊C8×C73—21no—
57C4⋊C8×C73—21no—
58#101×C73—22no—
59#100×C73—22no—
60#99×C73—22no—
61#99×C73—22no—
62C22⋊C8×C73—21no—
63C22⋊C8×C73—21no—
64C22⋊C8×C73—21no—
65C22⋊C8×C73—21no—
66#97×C73—21no—
67C4×C8×C73—21no—
68C24×C73C4⋊C411yescharacteristic
69#41×C73—41no—
70#41×C73—41no—
71C2×Q8×C73#4111yescharacteristic
72C2×D4×C73—21no—
73C2×D4×C73—21no—
74C2×D4×C73C2×D411yescharacteristic
75C2×D4×C73C2×Q811yescharacteristic
76C2×D4×C73#4111yescharacteristic
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—41no—
80C22×C4×C73—41no—
81C22×C4×C73C4⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85M16×C73—41no—
86M16×C73—41no—
87M16×C73—41no—
88M16×C73—42no—
89M16×C73—41no—
90M16×C73—41no—
91M16×C73—41no—
92M16×C73—42no—
93M16×C73—42no—
94C2×C8×C73—21no—
95C2×C8×C73—21no—
96C2×C8×C73—21no—
97C2×C8×C73—21no—
98C2×C8×C73—21no—
99C2×C8×C73—21no—
100C2×C8×C73—21no—
101C2×C8×C73—21no—
102C2×C8×C73—41no—
103C2×C8×C73—42no—
104C2×C8×C73C22⋊C411yescharacteristic
105C2×C8×C73C22⋊C411yescharacteristic
106C2×C8×C73C22⋊C411yescharacteristic
107C2×C8×C73C22⋊C411yescharacteristic
108C4⋊C4×C73—21no—
109C4⋊C4×C73—21no—
110C4⋊C4×C73—21no—
111C4⋊C4×C73—42no—
112C22⋊C4×C73—21no—
113C22⋊C4×C73—21no—
114C22⋊C4×C73—21no—
115C22⋊C4×C73—21no—
116C22⋊C4×C73—42no—
117C22⋊C4×C73C4⋊C411yescharacteristic
118C22⋊C4×C73C4⋊C411yescharacteristic
119C42×C73—21no—
120C42×C73—42no—
121C23×C73—21no—
122C23×C73—41no—
123C23×C73—81no—
124C23×C73C2×C4⋊C411yescharacteristic
125C23×C73#12111yescharacteristic
126C23×C73C22⋊Q811yescharacteristic
127Q8×C73—41no—
128D4×C73—41no—
129D4×C73—41no—
130D4×C73—41no—
131D4×C73—81no—
132D4×C73—81no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
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—82no—
147C2×C4×C73C2×C22⋊C411yescharacteristic
148C2×C4×C73#11711yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73C22⋊Q811yescharacteristic
151C8×C73—21no—
152C8×C73—21no—
153C8×C73—21no—
154C8×C73—41no—
155C8×C73—41no—
156C8×C73—82no—
157C8×C73#9911yescharacteristic
158C8×C73#9911yescharacteristic
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73—41no—
167C22×C73—41no—
168C22×C73—82no—
169C22×C73#38011yescharacteristic
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—41no—
174C4×C73—41no—
175C4×C73—41no—
176C4×C73—162no—
177C4×C73C2×#9911yescharacteristic
178C4×C73#41011yescharacteristic
179#12806C7311yesmaximal, characteristic, direct product factor, Sylow
180C2×C73—21no—
181C2×C73—41no—
182C2×C73—41no—
183C2×C73—41no—
184C2×C73—81no—
185C2×C73#179911yescharacteristic
186#2930C2×C7311yescharacteristic
187#2120C2×C7312yes—
188#1813C2×C7311yescharacteristic, complemented
189#1812C2×C7311yescharacteristic, complemented
190#1395C2×C7312yescomplemented
191C73#1280611yescharacteristic, direct product factor, Sylow
192C2×#131C22×C7311yescharacteristic
193C22×M16C4×C7311yescharacteristic, complemented
194#525C22×C7311yescharacteristic
195#436—21no—
196#436—21no—
197#435—21no—
198#435—21no—
199C2×#108—21no—
200#427—21no—
201#424C4×C7311yescharacteristic, complemented
202#413C22×C7312yes—
203#410C22×C7312yes—
204#408C22×C7311yescharacteristic
205#407C4×C7311yescharacteristic, complemented
206#405C4×C7311yescharacteristic, complemented
207#352—22no—
208AΣL(1,𝔽16)—22no—
209#338—21no—
210H(3,ℤ/4)—21no—
211C2×#41C23×C7311yescharacteristic
212C23×C4C2×C4×C7311yescharacteristic
213#131—41no—
214#131—41no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—22no—
218C2×M16—41no—
219C2×M16—41no—
220C2×M16C2×C4×C7311yescharacteristic
221C2×M16C2×C4×C7311yescharacteristic
222C2×M16D4×C7311yescharacteristic
223C2×M16D4×C7311yescharacteristic
224C22×C8C2×C4×C7311yescharacteristic
225C22×C8C2×C4×C7311yescharacteristic
226C22×C8D4×C7311yescharacteristic
227C22×C8D4×C7311yescharacteristic
228#123D4×C7311yescharacteristic
229#123Q8×C7311yescharacteristic
230C22⋊Q8—21no—
231#121—21no—
232#120D4×C7311yescharacteristic
233#120Q8×C7311yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236#117—22no—
237#117C2×C4×C7311yescharacteristic
238#108—41no—
239#108—41no—
240#107—41no—
241#106—41no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244#101—22no—
245#100—22no—
246#99—22no—
247#99—22no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#97—21no—
253C4×C8—21no—
254C24C4⋊C4×C7311yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8#41×C7311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C2×D4×C7311yescharacteristic
261C2×D4C2×Q8×C7311yescharacteristic
262C2×D4#41×C7311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C4⋊C4×C7311yescharacteristic
268C22×C4C22×C4×C7311yescharacteristic
269C22×C4C2×D4×C7311yescharacteristic
270C22×C4C2×D4×C7311yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—42no—
279M16—42no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8C22⋊C4×C7311yescharacteristic
291C2×C8C22⋊C4×C7311yescharacteristic
292C2×C8C22⋊C4×C7311yescharacteristic
293C2×C8C22⋊C4×C7311yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—42no—
303C22⋊C4C4⋊C4×C7311yescharacteristic
304C22⋊C4C4⋊C4×C7311yescharacteristic
305C42—21no—
306C42—42no—
307C23—21no—
308C23—41no—
309C23—81no—
310C23C2×C4⋊C4×C7311yescharacteristic
311C23#121×C7311yescharacteristic
312C23C22⋊Q8×C7311yescharacteristic
313Q8—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4C2×C22⋊C4×C7311yescharacteristic
334C2×C4#117×C7311yescharacteristic
335C2×C4#121×C7311yescharacteristic
336C2×C4C22⋊Q8×C7311yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—41no—
341C8—41no—
342C8—82no—
343C8#99×C7311yescharacteristic
344C8#99×C7311yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22#380×C7311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—162no—
363C4C2×#99×C7311yescharacteristic
364C4#410×C7311yescharacteristic
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2C73×#179911yescharacteristic
subgroupG/H
centerC4×C73#410
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×#41C23×C73
socleC2×C73#1799
prime radical#525×C73C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C73 ◁ C2×C8×C73 ◁ C2×#131×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#1813C2semidirect
4. C73×#1812C2semidirect
5. C73×#1395C2semidirect
7. C22×M16×C73C4semidirect
15. #424×C73C4semidirect
19. #407×C73C4semidirect
20. #405×C73C4semidirect
179. #12806C73direct (N × H)
188. #1813C2×C73semidirect
189. #1812C2×C73semidirect
190. #1395C2×C73semidirect
191. C73#12806direct (N × H)
193. C22×M16C4×C73semidirect
201. #424C4×C73semidirect
205. #407C4×C73semidirect
206. #405C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#29302
C73×#21202
C73×#18132
C73×#18122
C73×#13952
#1280673