Finite Group Databasebeta

← C2×C3×#11925356 #72230751

Subgroups — C2×C3×#11925356 #72230751

Order 7200.

1168 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#10597126C211yesmaximal, characteristic
2C22×C3×C3⋊#1044C211yesmaximal, characteristic
3C2×#11925356C311yesmaximal, characteristic, direct product factor
4C2×C3×#87832—31nomaximal
5C3×#11925356C214yesmaximal, direct product factor
6C22×C32×#1043C211yesmaximal, characteristic
7C22×C32×C5⋊D5C2211yescharacteristic
8C2×C3×C3⋊#1044—22no—
9C2×C3×C3⋊#1044C2212yes—
10C2×C32×#1043—22no—
11C2×C32×#1043C2212yes—
12C2×C3×C5×Dic15C412yescomplemented
13C3×#10597126C412yescomplemented
14C3×#10597126C2212yes—
15C2×C3×Dic3⋊AGL(1,𝔽5)—52nomaximal
16C22×C3⋊#1044C2×C311yescharacteristic
17C22×C3×#1044—31no—
18C22×C3×#1043—21no—
19C22×C3×#1043D311yescharacteristic
20C22×C3×#1043C2×C311yescharacteristic
21C2×#10597126C2×C311yescharacteristic
22C2×C3×#6036—31no—
23#11925356C2×C314yesdirect product factor
24C3×#87832—34no—
25C22×C32×C52C2×C411yescharacteristic
26C2×C32×C5⋊D5C2×C411yescharacteristic
27C2×C32×C5⋊D5D412yes—
28C2×C32×C5⋊D5Q812yes—
29C2×C32×C5⋊D5C2311yescharacteristic
30C3×C3⋊#1044—22no—
31C32×#1043—22no—
32C3×C5×Dic15C2×C414yescomplemented
33C2×#87832—31noHall
34C22×C3×ΓL(1,𝔽16)—52no—
35C22×C32×AGL(1,𝔽5)—52no—
36C22×C32×AGL(1,𝔽5)—102no—
37C2×C3×D5×Dic3—52no—
38C3×Dic3⋊AGL(1,𝔽5)—58no—
39C22×C3×C5⋊D5—21no—
40C22×C3×C5⋊D5C2×D311yescharacteristic
41C22×C3×C5⋊D5C22×C311yescharacteristic
42C2×C3⋊#1044—22no—
43C2×C3⋊#1044C22×C312yes—
44C2×C3×#1044—32no—
45C2×C3×#1044—62no—
46C2×C3×#1043—22no—
47C2×C3×#1043—22no—
48C2×C3×#1043—22no—
49C2×C3×#1043—22no—
50C2×C3×#1043—22no—
51C2×C3×#1043C2×D312yes—
52C2×C3×#1043C22×C312yes—
53C2×C5×Dic15C3×C412yescomplemented
54C2×C3×C5×Dic5—32no—
55#10597126C3×C412yescomplemented
56#10597126C22×C312yes—
57C3×#6036—32no—
58C3×#6036—32no—
59C2×C3×C4⋊AGL(1,𝔽5)—152no—
60C2×Dic3⋊AGL(1,𝔽5)—52no—
61C2×C32×C52C4⋊C412yescomplemented
62C2×C32×C52C22×C411yescharacteristic
63C32×C5⋊D5C4⋊C412yescomplemented
64C32×C5⋊D5C2×D411yescharacteristic
65C32×C5⋊D5C2×Q811yescharacteristic
66C22×#1044—31no—
67C22×#1043C3×D311yescharacteristic
68C2×#6036—31no—
69#87832—34no—
70C22×C32×D5—52no—
71C22×C32×D5—102no—
72C2×C3×ΓL(1,𝔽16)—54no—
73C2×C3×ΓL(1,𝔽16)—104no—
74C2×C32×AGL(1,𝔽5)—54no—
75C2×C32×AGL(1,𝔽5)—104no—
76C2×C32×AGL(1,𝔽5)—104no—
77C2×C32×AGL(1,𝔽5)—104no—
78C2×C32×AGL(1,𝔽5)—104no—
79C2×C3×Dic15AGL(1,𝔽5)12yescomplemented
80C2×C3×C5×Dic3—52no—
81C3×D5×Dic3—54no—
82C3×D5×Dic3—54no—
83C22×C3×C52—21no—
84C22×C3×C52C4×D311yescharacteristic
85C22×C3×C52C2×C3×C411yescharacteristic
86C2×C3×C5⋊D5—22no—
87C2×C3×C5⋊D5—21no—
88C2×C3×C5⋊D5—22no—
89C2×C3×C5⋊D5—21no—
90C2×C3×C5⋊D5Dic612yes—
91C2×C3×C5⋊D5C4×D311yescharacteristic
92C2×C3×C5⋊D5#6712yes—
93C2×C3×C5⋊D5C2×C3×C411yescharacteristic
94C2×C3×C5⋊D5C3×D412yes—
95C2×C3×C5⋊D5C3×Q812yes—
96C2×C3×C5⋊D5C22×D311yescharacteristic
97C2×C3×C5⋊D5C23×C311yescharacteristic
98C3⋊#1044—22no—
99C3×#1044—62no—
100C3×#1043—22no—
101C3×#1043—22no—
102C3×#1043—22no—
103C3×#1043—22no—
104C5×Dic15C2×C3×C414yescomplemented
105C3×C5×Dic5—34no—
106C2×C3×Dic3⋊C4—251noHall
107C22×ΓL(1,𝔽16)—52no—
108C22×C3×AGL(1,𝔽5)—52no—
109C22×C3×AGL(1,𝔽5)—52no—
110C22×C3×AGL(1,𝔽5)—102no—
111C22×C3×AGL(1,𝔽5)—102no—
112C22×C3×AGL(1,𝔽5)—102no—
113C22×C3×AGL(1,𝔽5)—102no—
114C22×C3×AGL(1,𝔽5)—102no—
115C22×C3×AGL(1,𝔽5)—152no—
116C2×C3×C4×D5—152no—
117C2×D5×Dic3—52no—
118C3×C4⋊AGL(1,𝔽5)—158no—
119Dic3⋊AGL(1,𝔽5)—58no—
120C32×C52C2×C4⋊C411yescharacteristic, complemented, Hall
121C22×C5⋊D5C2×C3×D311yescharacteristic
122C2×#1044—32no—
123C2×#1044—62no—
124C2×#1043—22no—
125C2×#1043C2×C3×D312yes—
126C2×C5×Dic5—32no—
127#6036—32no—
128#6036—32no—
129C22×C32×C5—22no—
130C22×C32×C5C2×AGL(1,𝔽5)12yes—
131C2×C32×D5—54no—
132C2×C32×D5—54no—
133C2×C32×D5—52no—
134C2×C32×D5—52no—
135C2×C32×D5—104no—
136C2×C32×D5—102no—
137C2×C32×D5—102no—
138C2×C32×D5—104no—
139C3×ΓL(1,𝔽16)—104no—
140C32×AGL(1,𝔽5)—104no—
141C32×AGL(1,𝔽5)—104no—
142C32×AGL(1,𝔽5)—104no—
143C3×Dic15C2×AGL(1,𝔽5)14yescomplemented
144C3×C5×Dic3—54no—
145C2×C4⋊AGL(1,𝔽5)—152no—
146C2×C3×C52—22no—
147C2×C3×C52—21no—
148C2×C3×C52Dic3⋊C412yescomplemented
149C2×C3×C52C3×C4⋊C412yescomplemented
150C2×C3×C52C2×C4×D311yescharacteristic
151C2×C3×C52C22×C3×C411yescharacteristic
152C3×C5⋊D5—22no—
153C3×C5⋊D5—21no—
154C3×C5⋊D5—21no—
155C3×C5⋊D5Dic3⋊C412yescomplemented
156C3×C5⋊D5C3×C4⋊C412yescomplemented
157C3×C5⋊D5C2×Dic611yescharacteristic
158C3×C5⋊D5C2×#6711yescharacteristic
159C3×C5⋊D5C2×C3×D411yescharacteristic
160C3×C5⋊D5C2×C3×Q811yescharacteristic
161C22×C32×C4—251no—
162C22×C3×Dic3—251no—
163C22×C3×Dic3—251no—
164C3×Dic3⋊C4—254no—
165C22×C3×D5—52no—
166C22×C3×D5—52no—
167C22×C3×D5—102no—
168C22×C3×D5—102no—
169C22×C3×D5—102no—
170C22×C3×D5—102no—
171C22×C3×D5—102no—
172C2×ΓL(1,𝔽16)—54no—
173C2×ΓL(1,𝔽16)—104no—
174C2×C3×AGL(1,𝔽5)—54no—
175C2×C3×AGL(1,𝔽5)—54no—
176C2×C3×AGL(1,𝔽5)—104no—
177C2×C3×AGL(1,𝔽5)—104no—
178C2×C3×AGL(1,𝔽5)—104no—
179C2×C3×AGL(1,𝔽5)—104no—
180C2×C3×AGL(1,𝔽5)—104no—
181C2×C3×AGL(1,𝔽5)—104no—
182C2×C3×AGL(1,𝔽5)—104no—
183C2×C3×AGL(1,𝔽5)—104no—
184C2×C3×AGL(1,𝔽5)—104no—
185C2×C3×AGL(1,𝔽5)—104no—
186C2×C3×AGL(1,𝔽5)—104no—
187C2×C3×AGL(1,𝔽5)—104no—
188C2×C3×AGL(1,𝔽5)—104no—
189C2×C3×AGL(1,𝔽5)—104no—
190C2×C3×AGL(1,𝔽5)—104no—
191C2×C3×AGL(1,𝔽5)—104no—
192C2×C3×AGL(1,𝔽5)—104no—
193C2×C3×AGL(1,𝔽5)—154no—
194C2×C3×AGL(1,𝔽5)—304no—
195C2×C3×C4×C5—152no—
196C2×Dic15C3×AGL(1,𝔽5)12yescomplemented
197C2×C5×Dic3—52no—
198C2×C3×Dic5—32no—
199C3×C4×D5—154no—
200C3×C4×D5—154no—
201D5×Dic3—54no—
202D5×Dic3—54no—
203C22×C52C3×C4×D311yescharacteristic
204C2×C5⋊D5C3×Dic612yes—
205C2×C5⋊D5C3×C4×D311yescharacteristic
206C2×C5⋊D5C3×#6712yes—
207C2×C5⋊D5C22×C3×D311yescharacteristic
208#1044—62no—
209#1043—22no—
210C5×Dic5—34no—
211C2×C3×C4⋊C4—751no—
212C2×Dic3⋊C4—251no—
213C2×C32×C5—24no—
214C2×C32×C5—22no—
215C2×C32×C5C4⋊AGL(1,𝔽5)14yescomplemented
216C2×C32×C5C22×AGL(1,𝔽5)12yes—
217C32×D5—52no—
218C32×D5—54no—
219C32×D5—52no—
220C32×D5—102no—
221C32×D5—104no—
222C32×D5—102no—
223C22×AGL(1,𝔽5)—52no—
224C22×AGL(1,𝔽5)—102no—
225C22×AGL(1,𝔽5)—152no—
226C2×C4×D5—152no—
227C4⋊AGL(1,𝔽5)—158no—
228C3×C52—21no—
229C3×C52C2×Dic3⋊C411yescharacteristic, complemented
230C3×C52C2×C3×C4⋊C411yescharacteristic, complemented
231C23×C32—251no—
232C2×C32×C4—252no—
233C2×C32×C4—502no—
234C2×C3×Dic3—52no—
235C2×C3×Dic3—252no—
236C2×C3×Dic3—252no—
237C2×C3×Dic3—252no—
238C2×C3×Dic3—502no—
239C22×C3×C5—22no—
240C22×C3×C5—22no—
241C22×C3×C5—22no—
242C22×C3×C5—22no—
243C22×C3×C5—22no—
244C22×C3×C5D3×AGL(1,𝔽5)12yes—
245C22×C3×C5C2×C3×AGL(1,𝔽5)12yes—
246C2×C3×D5—54no—
247C2×C3×D5—54no—
248C2×C3×D5—54no—
249C2×C3×D5—54no—
250C2×C3×D5—52no—
251C2×C3×D5—52no—
252C2×C3×D5—52no—
253C2×C3×D5—52no—
254C2×C3×D5—104no—
255C2×C3×D5—104no—
256C2×C3×D5—102no—
257C2×C3×D5—102no—
258C2×C3×D5—104no—
259C2×C3×D5—102no—
260C2×C3×D5—102no—
261C2×C3×D5—104no—
262C2×C3×D5—102no—
263C2×C3×D5—104no—
264C2×C3×D5—102no—
265C2×C3×D5—104no—
266C2×C3×D5—104no—
267C2×C3×D5—104no—
268C2×C3×D5—102no—
269C2×C3×D5—102no—
270C2×C3×D5—104no—
271C2×C3×D5—104no—
272C2×C3×D5—102no—
273C2×C3×D5—102no—
274ΓL(1,𝔽16)—104no—
275C3×AGL(1,𝔽5)—104no—
276C3×AGL(1,𝔽5)—104no—
277C3×AGL(1,𝔽5)—104no—
278C3×AGL(1,𝔽5)—104no—
279C3×AGL(1,𝔽5)—104no—
280C3×AGL(1,𝔽5)—104no—
281C3×AGL(1,𝔽5)—104no—
282C3×AGL(1,𝔽5)—104no—
283C3×AGL(1,𝔽5)—104no—
284C3×AGL(1,𝔽5)—104no—
285C3×AGL(1,𝔽5)—104no—
286C3×AGL(1,𝔽5)—104no—
287C3×AGL(1,𝔽5)—304no—
288C3×C4×C5—154no—
289Dic15C2×C3×AGL(1,𝔽5)14yescomplemented
290C3×Dic5—34no—
291C5×Dic3—54no—
292C2×C52C3×Dic3⋊C412yescomplemented
293C2×C52C2×C3×C4×D311yescharacteristic
294C5⋊D5C3×Dic3⋊C412yescomplemented
295C5⋊D5C2×C3×Dic611yescharacteristic
296C5⋊D5C2×C3×#6711yescharacteristic
297C22×C3×C4—251no—
298C22×C3×C4—251no—
299C22×C3×C4—501no—
300C22×C3×C4—751no—
301C22×C3×C4—751no—
302C22×Dic3—251no—
303C22×Dic3—251no—
304C3×C4⋊C4—754no—
305Dic3⋊C4—254no—
306C32×C5—22no—
307C32×C5C2×C4⋊AGL(1,𝔽5)12yescomplemented
308C22×D5—52no—
309C22×D5—102no—
310C2×AGL(1,𝔽5)—54no—
311C2×AGL(1,𝔽5)—104no—
312C2×AGL(1,𝔽5)—104no—
313C2×AGL(1,𝔽5)—104no—
314C2×AGL(1,𝔽5)—104no—
315C2×AGL(1,𝔽5)—154no—
316C2×AGL(1,𝔽5)—304no—
317C2×C4×C5—152no—
318C2×Dic5—32no—
319C4×D5—154no—
320C4×D5—154no—
321C22×C32—252no—
322C22×C32—252no—
323C22×C32—251no—
324C22×C32—251no—
325C22×C32D5⋊AGL(1,𝔽5)11yescharacteristic
326C32×C4—502no—
327C3×Dic3—54no—
328C3×Dic3—502no—
329C2×C4⋊C4—751noSylow
330C2×C3×C5—22no—
331C2×C3×C5—22no—
332C2×C3×C5—24no—
333C2×C3×C5—24no—
334C2×C3×C5—22no—
335C2×C3×C5—24no—
336C2×C3×C5—22no—
337C2×C3×C5—22no—
338C2×C3×C5—24no—
339C2×C3×C5—24no—
340C2×C3×C5Dic3⋊AGL(1,𝔽5)14yescomplemented
341C2×C3×C5C3×C4⋊AGL(1,𝔽5)14yescomplemented
342C2×C3×C5C2×D3×AGL(1,𝔽5)12yes—
343C2×C3×C5C22×C3×AGL(1,𝔽5)12yes—
344C3×D5—54no—
345C3×D5—52no—
346C3×D5—52no—
347C3×D5—54no—
348C3×D5—52no—
349C3×D5—52no—
350C3×D5—104no—
351C3×D5—104no—
352C3×D5—102no—
353C3×D5—102no—
354C3×D5—102no—
355C3×D5—102no—
356C3×D5—102no—
357C3×D5—104no—
358C3×D5—102no—
359C3×D5—102no—
360C3×D5—102no—
361C3×D5—104no—
362C3×D5—104no—
363C3×D5—102no—
364C3×D5—102no—
365C52C2×C3×Dic3⋊C411yescharacteristic, complemented, Sylow
366C23×C3—251no—
367C23×C3—251no—
368C23×C3—501no—
369C2×C3×C4—152no—
370C2×C3×C4—252no—
371C2×C3×C4—252no—
372C2×C3×C4—502no—
373C2×C3×C4—502no—
374C2×C3×C4—502no—
375C2×C3×C4—502no—
376C2×C3×C4—502no—
377C2×C3×C4—752no—
378C2×C3×C4—752no—
379C2×C3×C4—752no—
380C2×C3×C4—1502no—
381C2×Dic3—52no—
382C2×Dic3—252no—
383C2×Dic3—252no—
384C2×Dic3—252no—
385C2×Dic3—502no—
386C22×C5—22no—
387C22×C5C3×D3×AGL(1,𝔽5)12yes—
388C2×D5—52no—
389C2×D5—52no—
390C2×D5—54no—
391C2×D5—54no—
392C2×D5—104no—
393C2×D5—102no—
394C2×D5—104no—
395C2×D5—102no—
396AGL(1,𝔽5)—104no—
397AGL(1,𝔽5)—104no—
398AGL(1,𝔽5)—104no—
399AGL(1,𝔽5)—304no—
400C4×C5—154no—
401Dic5—34no—
402C2×C32—251no—
403C2×C32—252no—
404C2×C32—251no—
405C2×C32#8783212yescomplemented
406C2×C32C2×D5⋊AGL(1,𝔽5)11yescharacteristic
407C22×C4—251no—
408C22×C4—751no—
409C22×C4—751no—
410C4⋊C4—754no—
411C3×C5—22no—
412C3×C5—22no—
413C3×C5—22no—
414C3×C5—22no—
415C3×C5—22no—
416C3×C5C2×Dic3⋊AGL(1,𝔽5)12yescomplemented
417C3×C5C2×C3×C4⋊AGL(1,𝔽5)12yescomplemented
418C22×C3—21no—
419C22×C3—251no—
420C22×C3—251no—
421C22×C3—251no—
422C22×C3—251no—
423C22×C3—252no—
424C22×C3—252no—
425C22×C3—252no—
426C22×C3—252no—
427C22×C3—501no—
428C22×C3—502no—
429C22×C3—501no—
430C22×C3—502no—
431C22×C3C3×D5⋊AGL(1,𝔽5)11yescharacteristic
432C22×C3#1059718211yescharacteristic
433C3×C4—154no—
434C3×C4—502no—
435C3×C4—502no—
436C3×C4—502no—
437C3×C4—502no—
438C3×C4—1502no—
439Dic3—54no—
440Dic3—502no—
441C2×C5—22no—
442C2×C5—24no—
443C2×C5C3×Dic3⋊AGL(1,𝔽5)14yescomplemented
444C2×C5C2×C3×D3×AGL(1,𝔽5)12yes—
445D5—54no—
446D5—52no—
447D5—52no—
448D5—104no—
449D5—102no—
450D5—102no—
451C32C2×#8783211yescharacteristic, complemented, Sylow
452C23—251no—
453C2×C4—152no—
454C2×C4—252no—
455C2×C4—502no—
456C2×C4—752no—
457C2×C4—752no—
458C2×C4—752no—
459C2×C4—1502no—
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—251no—
463C2×C3—251no—
464C2×C3—252no—
465C2×C3—251no—
466C2×C3—251no—
467C2×C3—252no—
468C2×C3—501no—
469C2×C3—501no—
470C2×C3—502no—
471C2×C3C3×#8783212yescomplemented
472C2×C3#1192535612yesdirect product factor
473C2×C3C2×C3×D5⋊AGL(1,𝔽5)11yescharacteristic
474C2×C3C2×#1059718211yescharacteristic
475C5—22no—
476C5C2×C3×Dic3⋊AGL(1,𝔽5)12yescomplemented
477C22—251no—
478C22—251no—
479C22—252no—
480C22—252no—
481C22C3×#1059718211yescharacteristic
482C4—154no—
483C4—502no—
484C4—1502no—
485C3—21no—
486C3C2×C3×#8783211yescharacteristic, complemented
487C3C2×#1192535611yescharacteristic, direct product factor
488C2—251no—
489C2—251no—
490C2—252no—
491C2C2×C3×#1059718211yescharacteristic
492C2C3×#1192535612yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#10597182
commutatorC2×C3×C52C22×C3×C4
FrattiniC2C2×C3×#10597182
Fitting, socleC22×C32×C52C2×C4
prime radicalC22×C32×C5⋊D5C22
cosocleC2×C3×C5⋊D5C23×C3
hypocenterC3×C52C2×C3×C4⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
3. C2×#11925356C3direct (N × H)
5. C3×#11925356C2direct (N × H)
12. C2×C3×C5×Dic15C4semidirect
13. C3×#10597126C4semidirect
23. #11925356C2×C3direct (N × H)
32. C3×C5×Dic15C2×C4semidirect
53. C2×C5×Dic15C3×C4semidirect
55. #10597126C3×C4semidirect
61. C2×C32×C52C4⋊C4semidirect
63. C32×C5⋊D5C4⋊C4semidirect
79. C2×C3×Dic15AGL(1,𝔽5)semidirect
104. C5×Dic15C2×C3×C4semidirect
120. C32×C52C2×C4⋊C4semidirect
143. C3×Dic15C2×AGL(1,𝔽5)semidirect
148. C2×C3×C52Dic3⋊C4semidirect
149. C2×C3×C52C3×C4⋊C4semidirect
155. C3×C5⋊D5Dic3⋊C4semidirect
156. C3×C5⋊D5C3×C4⋊C4semidirect
196. C2×Dic15C3×AGL(1,𝔽5)semidirect
215. C2×C32×C5C4⋊AGL(1,𝔽5)semidirect
229. C3×C52C2×Dic3⋊C4semidirect
230. C3×C52C2×C3×C4⋊C4semidirect
289. Dic15C2×C3×AGL(1,𝔽5)semidirect
292. C2×C52C3×Dic3⋊C4semidirect
294. C5⋊D5C3×Dic3⋊C4semidirect
307. C32×C5C2×C4⋊AGL(1,𝔽5)semidirect
340. C2×C3×C5Dic3⋊AGL(1,𝔽5)semidirect
341. C2×C3×C5C3×C4⋊AGL(1,𝔽5)semidirect
365. C52C2×C3×Dic3⋊C4semidirect
405. C2×C32#87832semidirect
416. C3×C5C2×Dic3⋊AGL(1,𝔽5)semidirect
417. C3×C5C2×C3×C4⋊AGL(1,𝔽5)semidirect
443. C2×C5C3×Dic3⋊AGL(1,𝔽5)semidirect
451. C32C2×#87832semidirect
471. C2×C3C3×#87832semidirect
472. C2×C3#11925356direct (N × H)
476. C5C2×C3×Dic3⋊AGL(1,𝔽5)semidirect
486. C3C2×C3×#87832semidirect
487. C3C2×#11925356direct (N × H)
492. C2C3×#11925356direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#105971262
C22×C3×C3⋊#10442
C2×#119253563
C2×C3×#878323
C3×#119253562
C22×C32×#10432
C2×C3×Dic3⋊AGL(1,𝔽5)5