Finite Group Databasebeta

← C3×#13108191 #72247243

Subgroups — C3×#13108191 #72247243

Order 4860.

484 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C3×D5×AΣL(1,𝔽27)C211yesmaximal, characteristic
2C32×C3⋊ΓL(1,𝔽16)—31nomaximal
3C3×3−1+2⋊AGL(1,𝔽5)—31nomaximal
4C3×3−1+2⋊AGL(1,𝔽5)—31nomaximal
5C3×(C3⋊ΓL(1,𝔽16))⋊C3—31nomaximal
6#13108191C311yesmaximal, characteristic, direct product factor
7C3×C5×AΣL(1,𝔽27)C411yescharacteristic, complemented, Hall
8C3×#66330—51nomaximal, Hall
9C34×D5D311yescharacteristic
10C3×D5×3−1+2D311yescharacteristic
11C3×D5×3−1+2D311yescharacteristic
12C3×D5×H(3,𝔽3)D311yescharacteristic
13D5×AΣL(1,𝔽27)—21no—
14D5×AΣL(1,𝔽27)—21no—
15D5×AΣL(1,𝔽27)—21no—
16D5×AΣL(1,𝔽27)—21no—
17D5×AΣL(1,𝔽27)C2×C311yescharacteristic
18C3×C3⋊ΓL(1,𝔽16)—31no—
19C3×C3⋊ΓL(1,𝔽16)—31no—
20C3×C3⋊ΓL(1,𝔽16)—31no—
21C3×C3⋊ΓL(1,𝔽16)—31no—
22C3×C3⋊ΓL(1,𝔽16)—31no—
23C32×ΓL(1,𝔽16)—91no—
24C32×ΓL(1,𝔽16)—91no—
25C32×ΓL(1,𝔽16)—91no—
26C32×ΓL(1,𝔽16)—91no—
273−1+2⋊AGL(1,𝔽5)—31no—
283−1+2⋊AGL(1,𝔽5)—31no—
293−1+2⋊AGL(1,𝔽5)—31no—
303−1+2⋊AGL(1,𝔽5)—31no—
313−1+2⋊AGL(1,𝔽5)—31no—
323−1+2⋊AGL(1,𝔽5)—31no—
33(C3⋊ΓL(1,𝔽16))⋊C3—31no—
34(C3⋊ΓL(1,𝔽16))⋊C3—31no—
35(C3⋊ΓL(1,𝔽16))⋊C3—31no—
36C3×C9⋊AGL(1,𝔽5)—31no—
37C3×C9⋊AGL(1,𝔽5)—31no—
38C2×C3×AΣL(1,𝔽27)—51no—
39C34×C5Dic311yescharacteristic, complemented
40C3×C5×3−1+2Dic311yescharacteristic, complemented
41C3×C5×3−1+2Dic311yescharacteristic, complemented
42C3×C5×H(3,𝔽3)Dic311yescharacteristic, complemented
43C5×AΣL(1,𝔽27)—21no—
44C5×AΣL(1,𝔽27)—21no—
45C5×AΣL(1,𝔽27)—21no—
46C5×AΣL(1,𝔽27)—21no—
47C5×AΣL(1,𝔽27)C3×C411yescharacteristic, complemented
48C32×C3⋊Dic3—151no—
49C3×3−1+2⋊C4—151no—
50C3×3−1+2⋊C4—151no—
51C3×(C3⋊Dic3)⋊C3—151no—
52#66330—51no—
53C33×D5—21no—
54C33×D5—31no—
55C33×D5—31no—
56C33×D5—61no—
57C33×D5—61no—
58C33×D5—61no—
59C33×D5—61no—
60C33×D5—61no—
61C33×D5—61no—
62C33×D5—61no—
63C33×D5—61no—
64C33×D5C3×D311yescharacteristic
65C33×D5C3⋊D311yescharacteristic
66D5×3−1+2—21no—
67D5×3−1+2—21no—
68D5×3−1+2—31no—
69D5×3−1+2—61no—
70D5×3−1+2—61no—
71D5×3−1+2—61no—
72D5×3−1+2C3×D311yescharacteristic
73D5×3−1+2C3×D311yescharacteristic
74D5×H(3,𝔽3)—21no—
75D5×H(3,𝔽3)—61no—
76D5×H(3,𝔽3)—61no—
77D5×H(3,𝔽3)C3×D311yescharacteristic
78C3×C9×D5—31no—
79C3×C9×D5—61no—
80C3×AΣL(1,𝔽27)AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
81C3⋊ΓL(1,𝔽16)—31no—
82C3⋊ΓL(1,𝔽16)—31no—
83C3×ΓL(1,𝔽16)—91no—
84C3×ΓL(1,𝔽16)—91no—
85C3×ΓL(1,𝔽16)—91no—
86C3×ΓL(1,𝔽16)—91no—
87C3×ΓL(1,𝔽16)—91no—
88C3×ΓL(1,𝔽16)—91no—
89C3×ΓL(1,𝔽16)—91no—
90C3×ΓL(1,𝔽16)—91no—
91C3×ΓL(1,𝔽16)—91no—
92C3×ΓL(1,𝔽16)—91no—
93C3×ΓL(1,𝔽16)—91no—
94C3×ΓL(1,𝔽16)—91no—
95C3×ΓL(1,𝔽16)—91no—
96C3×ΓL(1,𝔽16)—91no—
97C3×ΓL(1,𝔽16)—91no—
98C3×ΓL(1,𝔽16)—91no—
99C3×ΓL(1,𝔽16)—271no—
100C32×AGL(1,𝔽5)—271no—
101C9⋊AGL(1,𝔽5)—31no—
102C9⋊AGL(1,𝔽5)—31no—
103C2×C34—51no—
104C2×C3×3−1+2—51no—
105C2×C3×3−1+2—51no—
106C2×C3×H(3,𝔽3)—51no—
107C2×AΣL(1,𝔽27)—51no—
108C2×AΣL(1,𝔽27)—101no—
109C2×AΣL(1,𝔽27)—101no—
110C2×AΣL(1,𝔽27)—101no—
111C2×AΣL(1,𝔽27)—101no—
112C33×C5—21no—
113C33×C5—31no—
114C33×C5—31no—
115C33×C5—61no—
116C33×C5—61no—
117C33×C5—61no—
118C33×C5—61no—
119C33×C5—61no—
120C33×C5—61no—
121C33×C5—61no—
122C33×C5—61no—
123C33×C5C3×Dic311yescharacteristic, complemented
124C33×C5C3⋊Dic311yescharacteristic
125C5×3−1+2—21no—
126C5×3−1+2—21no—
127C5×3−1+2—31no—
128C5×3−1+2—61no—
129C5×3−1+2—61no—
130C5×3−1+2—61no—
131C5×3−1+2C3×Dic311yescharacteristic, complemented
132C5×3−1+2C3×Dic311yescharacteristic, complemented
133C5×H(3,𝔽3)—21no—
134C5×H(3,𝔽3)—61no—
135C5×H(3,𝔽3)—61no—
136C5×H(3,𝔽3)C3×Dic311yescharacteristic, complemented
137C3×C5×C9—31no—
138C3×C5×C9—61no—
139C3×C3⋊Dic3—151no—
140C3×C3⋊Dic3—151no—
141C3×C3⋊Dic3—151no—
142C3×C3⋊Dic3—151no—
143C3×C3⋊Dic3—151no—
144C32×Dic3—451no—
145C32×Dic3—451no—
146C32×Dic3—451no—
147C32×Dic3—451no—
1483−1+2⋊C4—151no—
1493−1+2⋊C4—151no—
1503−1+2⋊C4—151no—
1513−1+2⋊C4—151no—
1523−1+2⋊C4—151no—
1533−1+2⋊C4—151no—
154(C3⋊Dic3)⋊C3—151no—
155(C3⋊Dic3)⋊C3—151no—
156(C3⋊Dic3)⋊C3—151no—
157C3×Dic9—151no—
158C3×Dic9—151no—
159C32×D5—31no—
160C32×D5—31no—
161C32×D5—31no—
162C32×D5—31no—
163C32×D5—31no—
164C32×D5—31no—
165C32×D5—61no—
166C32×D5—61no—
167C32×D5—61no—
168C32×D5—61no—
169C32×D5—61no—
170C32×D5—61no—
171C32×D5—61no—
172C32×D5—61no—
173C32×D5—61no—
174C32×D5—61no—
175C32×D5—61no—
176C32×D5—61no—
177C32×D5—61no—
178C32×D5—61no—
179C32×D5—61no—
180C32×D5—61no—
181C32×D5—61no—
182C32×D5—61no—
183C32×D5—61no—
184C32×D5—61no—
185C32×D5—61no—
186C32×D5—61no—
187C32×D5—61no—
188C32×D5—61no—
189C32×D5—181no—
190C32×D5—181no—
191C32×D5#27211yescharacteristic
192C32×D5#27211yescharacteristic
193C32×D5#27211yescharacteristic
194C32×D5C3×C3⋊D311yescharacteristic
195C9×D5—61no—
196C9×D5—61no—
197C9×D5—61no—
198C9×D5—61no—
199C34ΓL(1,𝔽16)11yescharacteristic, complemented
200C3×3−1+2ΓL(1,𝔽16)11yescharacteristic, complemented
201C3×3−1+2ΓL(1,𝔽16)11yescharacteristic, complemented
202C3×H(3,𝔽3)ΓL(1,𝔽16)11yescharacteristic, complemented
203AΣL(1,𝔽27)—21no—
204AΣL(1,𝔽27)—21no—
205AΣL(1,𝔽27)—21no—
206AΣL(1,𝔽27)—21no—
207AΣL(1,𝔽27)C3×AGL(1,𝔽5)11yescharacteristic, complemented
208ΓL(1,𝔽16)—91no—
209ΓL(1,𝔽16)—91no—
210ΓL(1,𝔽16)—91no—
211ΓL(1,𝔽16)—91no—
212ΓL(1,𝔽16)—271no—
213C3×AGL(1,𝔽5)—271no—
214C3×AGL(1,𝔽5)—271no—
215C3×AGL(1,𝔽5)—271no—
216C3×AGL(1,𝔽5)—271no—
217C2×C33—51no—
218C2×C33—51no—
219C2×C33—101no—
220C2×C33—151no—
221C2×C33—301no—
222C2×C33—301no—
223C2×C33—301no—
224C2×C33—301no—
225C2×C33—301no—
226C2×C33—301no—
227C2×C33—301no—
228C2×C33—301no—
229C2×C33—301no—
230C2×3−1+2—51no—
231C2×3−1+2—51no—
232C2×3−1+2—101no—
233C2×3−1+2—101no—
234C2×3−1+2—151no—
235C2×3−1+2—151no—
236C2×3−1+2—151no—
237C2×3−1+2—301no—
238C2×H(3,𝔽3)—51no—
239C2×H(3,𝔽3)—101no—
240C2×H(3,𝔽3)—151no—
241C2×H(3,𝔽3)—301no—
242C2×C3×C9—301no—
243C2×C3×C9—301no—
244C32×C5—31no—
245C32×C5—31no—
246C32×C5—31no—
247C32×C5—31no—
248C32×C5—31no—
249C32×C5—31no—
250C32×C5—61no—
251C32×C5—61no—
252C32×C5—61no—
253C32×C5—61no—
254C32×C5—61no—
255C32×C5—61no—
256C32×C5—61no—
257C32×C5—61no—
258C32×C5—61no—
259C32×C5—61no—
260C32×C5—61no—
261C32×C5—61no—
262C32×C5—61no—
263C32×C5—61no—
264C32×C5—61no—
265C32×C5—61no—
266C32×C5—61no—
267C32×C5—61no—
268C32×C5—61no—
269C32×C5—61no—
270C32×C5—61no—
271C32×C5—61no—
272C32×C5—61no—
273C32×C5—61no—
274C32×C5—181no—
275C32×C5—181no—
276C32×C5#108411yescharacteristic
277C32×C5#108411yescharacteristic
278C32×C5#108411yescharacteristic
279C32×C5C3×C3⋊Dic311yescharacteristic
280C5×C9—61no—
281C5×C9—61no—
282C5×C9—61no—
283C5×C9—61no—
284C32×C4—1351no—
285C3⋊Dic3—151no—
286C3⋊Dic3—151no—
287C3×Dic3—451no—
288C3×Dic3—451no—
289C3×Dic3—451no—
290C3×Dic3—451no—
291C3×Dic3—451no—
292C3×Dic3—451no—
293C3×Dic3—451no—
294C3×Dic3—451no—
295C3×Dic3—451no—
296C3×Dic3—451no—
297C3×Dic3—451no—
298C3×Dic3—451no—
299C3×Dic3—451no—
300C3×Dic3—451no—
301C3×Dic3—451no—
302C3×Dic3—451no—
303C3×Dic3—1351no—
304Dic9—151no—
305Dic9—151no—
306C3×D5—21no—
307C3×D5—31no—
308C3×D5—31no—
309C3×D5—61no—
310C3×D5—61no—
311C3×D5—61no—
312C3×D5—61no—
313C3×D5—61no—
314C3×D5—61no—
315C3×D5—61no—
316C3×D5—91no—
317C3×D5—181no—
318C3×D5#418311yescharacteristic
319C3×D5C3×#27211yescharacteristic
320C33—21no—
321C33—31no—
322C33—61no—
323C33—61no—
324C33—61no—
325C33—61no—
326C33—61no—
327C33—61no—
328C33—61no—
329C33—61no—
330C33—61no—
331C33C3×ΓL(1,𝔽16)11yescharacteristic, complemented
332C33C3⋊ΓL(1,𝔽16)11yescharacteristic
3333−1+2—21no—
3343−1+2—21no—
3353−1+2—31no—
3363−1+2—31no—
3373−1+2—31no—
3383−1+2—61no—
3393−1+2C3×ΓL(1,𝔽16)11yescharacteristic, complemented
3403−1+2C3×ΓL(1,𝔽16)11yescharacteristic, complemented
341H(3,𝔽3)—21no—
342H(3,𝔽3)—31no—
343H(3,𝔽3)—61no—
344H(3,𝔽3)C3×ΓL(1,𝔽16)11yescharacteristic, complemented
345C3×C9—61no—
346C3×C9—61no—
347AGL(1,𝔽5)—271noHall
348C2×C32—51no—
349C2×C32—51no—
350C2×C32—51no—
351C2×C32—51no—
352C2×C32—151no—
353C2×C32—151no—
354C2×C32—151no—
355C2×C32—151no—
356C2×C32—151no—
357C2×C32—151no—
358C2×C32—301no—
359C2×C32—301no—
360C2×C32—301no—
361C2×C32—301no—
362C2×C32—301no—
363C2×C32—301no—
364C2×C32—301no—
365C2×C32—301no—
366C2×C32—301no—
367C2×C32—301no—
368C2×C32—301no—
369C2×C32—301no—
370C2×C32—301no—
371C2×C32—301no—
372C2×C32—301no—
373C2×C32—301no—
374C2×C32—301no—
375C2×C32—301no—
376C2×C32—301no—
377C2×C32—301no—
378C2×C32—301no—
379C2×C32—301no—
380C2×C32—301no—
381C2×C32—301no—
382C2×C32—451no—
383C2×C32—901no—
384C2×C9—151no—
385C2×C9—301no—
386C2×C9—301no—
387C2×C9—301no—
388C3×C5—21no—
389C3×C5—31no—
390C3×C5—31no—
391C3×C5—61no—
392C3×C5—61no—
393C3×C5—61no—
394C3×C5—61no—
395C3×C5—61no—
396C3×C5—61no—
397C3×C5—61no—
398C3×C5—91no—
399C3×C5—181no—
400C3×C5#6633011yescharacteristic, complemented
401C3×C5C3×#108411yescharacteristic
402C3×C4—1351no—
403C3×C4—1351no—
404C3×C4—1351no—
405C3×C4—1351no—
406Dic3—451no—
407Dic3—451no—
408Dic3—451no—
409Dic3—451no—
410Dic3—1351no—
411D5C3×#418311yescharacteristic
412C32—31no—
413C32—31no—
414C32—31no—
415C32—31no—
416C32—31no—
417C32—31no—
418C32—61no—
419C32—61no—
420C32—61no—
421C32—61no—
422C32—61no—
423C32—61no—
424C32—61no—
425C32—61no—
426C32—61no—
427C32—61no—
428C32—61no—
429C32—61no—
430C32—61no—
431C32—61no—
432C32—61no—
433C32—61no—
434C32—61no—
435C32—61no—
436C32—61no—
437C32—61no—
438C32—61no—
439C32—61no—
440C32—61no—
441C32—61no—
442C32—91no—
443C32—181no—
444C32#1058738011yescharacteristic
445C32#1058738011yescharacteristic
446C32#1058738011yescharacteristic
447C32C3×C3⋊ΓL(1,𝔽16)11yescharacteristic
448C9—31no—
449C9—61no—
450C9—61no—
451C9—61no—
452C2×C3—51no—
453C2×C3—51no—
454C2×C3—101no—
455C2×C3—151no—
456C2×C3—151no—
457C2×C3—151no—
458C2×C3—301no—
459C2×C3—301no—
460C2×C3—301no—
461C2×C3—301no—
462C2×C3—301no—
463C2×C3—301no—
464C2×C3—901no—
465C2×C3—901no—
466C5C3×#6633011yescharacteristic, complemented, Sylow
467C4—1351noSylow
468C3—21no—
469C3—31no—
470C3—31no—
471C3—31no—
472C3—61no—
473C3—61no—
474C3—61no—
475C3—61no—
476C3—61no—
477C3—61no—
478C3—181no—
479C3—181no—
480C3#1310819111yescharacteristic, direct product factor
481C3C3×#1058738011yescharacteristic
482C2—51no—
subgroupG/H
center, hypercenterC3#13108191
commutator, hypocenterC5×AΣL(1,𝔽27)C3×C4
FrattiniC32C3×C3⋊ΓL(1,𝔽16)
FittingC3×C5×AΣL(1,𝔽27)C4
socleC32×C5#1084
prime radicalC3×D5×AΣL(1,𝔽27)C2
cosocleD5×AΣL(1,𝔽27)C2×C3
LCSG ▷ C5×AΣL(1,𝔽27)
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
6. #13108191C3direct (N × H)
7. C3×C5×AΣL(1,𝔽27)C4semidirect
39. C34×C5Dic3semidirect
40. C3×C5×3−1+2Dic3semidirect
41. C3×C5×3−1+2Dic3semidirect
42. C3×C5×H(3,𝔽3)Dic3semidirect
47. C5×AΣL(1,𝔽27)C3×C4semidirect
80. C3×AΣL(1,𝔽27)AGL(1,𝔽5)semidirect
123. C33×C5C3×Dic3semidirect
131. C5×3−1+2C3×Dic3semidirect
132. C5×3−1+2C3×Dic3semidirect
136. C5×H(3,𝔽3)C3×Dic3semidirect
199. C34ΓL(1,𝔽16)semidirect
200. C3×3−1+2ΓL(1,𝔽16)semidirect
201. C3×3−1+2ΓL(1,𝔽16)semidirect
202. C3×H(3,𝔽3)ΓL(1,𝔽16)semidirect
207. AΣL(1,𝔽27)C3×AGL(1,𝔽5)semidirect
331. C33C3×ΓL(1,𝔽16)semidirect
339. 3−1+2C3×ΓL(1,𝔽16)semidirect
340. 3−1+2C3×ΓL(1,𝔽16)semidirect
344. H(3,𝔽3)C3×ΓL(1,𝔽16)semidirect
400. C3×C5#66330semidirect
466. C5C3×#66330semidirect
480. C3#13108191direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×AΣL(1,𝔽27)2
C32×C3⋊ΓL(1,𝔽16)3
C3×3−1+2⋊AGL(1,𝔽5)3
C3×(C3⋊ΓL(1,𝔽16))⋊C33
#131081913
C3×#663305