Finite Group Databasebeta

← C2×C61×#1344 #29515123

Subgroups — C2×C61×#1344 #29515123

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#397C211yesmaximal, characteristic
2C2×C61×#342C211yesmaximal, characteristic
3C2×C61×#323C211yesmaximal, characteristic
4C61×#1344C214yesmaximal, direct product factor
5C22×C4⋊C4×C61C411yescharacteristic, complemented
6C22×C22⋊C4×C61—21no—
7C22×C22⋊C4×C61—21no—
8C22×C22⋊C4×C61C2211yescharacteristic
9C2×C22⋊C8×C61—21no—
10C61×#397—21no—
11C61×#397—21no—
12C61×#397C412yescomplemented
13C61×#397C2212yes—
14C2×#95×C61—21no—
15C2×#95×C61C411yescharacteristic, complemented
16C61×#342—21no—
17C61×#342C2212yes—
18C61×#323—21no—
19C61×#323C2212yes—
20C25×C61D411yescharacteristic
21C23×C4×C61—21no—
22C23×C4×C61—21no—
23C23×C4×C61—41no—
24C23×C4×C61C2×C411yescharacteristic
25C23×C4×C61D411yescharacteristic
26C22×C8×C61—41no—
27C2×C4⋊C4×C61—21no—
28C2×C4⋊C4×C61—41no—
29C2×C4⋊C4×C61—41no—
30C2×C4⋊C4×C61C2×C412yescomplemented
31C2×C22⋊C4×C61—24no—
32C2×C22⋊C4×C61—22no—
33C2×C22⋊C4×C61—22no—
34C2×C22⋊C4×C61—22no—
35C2×C22⋊C4×C61—41no—
36C2×C22⋊C4×C61—41no—
37C2×C22⋊C4×C61—41no—
38C2×C22⋊C4×C61—42no—
39C2×C22⋊C4×C61—42no—
40C2×C22⋊C4×C61—42no—
41C2×C22⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61C2311yescharacteristic
45C22⋊C8×C61—22no—
46C22⋊C8×C61—41no—
47#95×C61—22no—
48#95×C61—21no—
49#95×C61—22no—
50#95×C61C2×C412yescomplemented
51C24×C61—21no—
52C24×C61—21no—
53C24×C61—22no—
54C24×C61—42no—
55C24×C61—42no—
56C24×C61C22⋊C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C24×C61D812yes—
59C24×C61SD1612yes—
60C24×C61C2×D411yescharacteristic
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61—44no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61—82no—
79C22×C4×C61—82no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C2×C8×C61—42no—
85C2×C8×C61—82no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—42no—
88C22⋊C4×C61—24no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—44no—
94C22⋊C4×C61—44no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—42no—
97C22⋊C4×C61—42no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—44no—
111C23×C61—44no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—84no—
117C23×C61—81no—
118C23×C61—81no—
119C23×C61#9911yescharacteristic
120C23×C61#9911yescharacteristic
121C23×C61#10212yes—
122C23×C61#10212yes—
123C23×C61#10412yes—
124C23×C61C2×C22⋊C411yescharacteristic
125C23×C61C2×D811yescharacteristic
126C23×C61C2×SD1611yescharacteristic
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—44no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—81no—
146C2×C4×C61—84no—
147C2×C4×C61—88no—
148C2×C4×C61—82no—
149C2×C4×C61—82no—
150C2×C4×C61—82no—
151C2×C4×C61—82no—
152C8×C61—82no—
153C2×#1344C6111yesmaximal, characteristic, direct product factor, Sylow
154C22×C61—24no—
155C22×C61—24no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—22no—
162C22×C61—21no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—44no—
167C22×C61—44no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—82no—
173C22×C61—82no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61#32712yes—
177C22×C61#32712yes—
178C22×C61#35212yes—
179C22×C61#35412yes—
180C22×C61C2×#9911yescharacteristic
181C22×C61C2×#10211yescharacteristic
182C22×C61C2×#10411yescharacteristic
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—41no—
186C4×C61—41no—
187C4×C61—82no—
188C4×C61—84no—
189C4×C61—82no—
190C2×#397C2×C6111yescharacteristic
191C2×#342C2×C6111yescharacteristic
192C2×#323C2×C6111yescharacteristic
193#1344C2×C6114yesdirect product factor
194C2×C61—21no—
195C2×C61—22no—
196C2×C61—21no—
197C2×C61—41no—
198C2×C61—42no—
199C2×C61—41no—
200C2×C61#134414yesdirect product factor
201C2×C61C2×#32711yescharacteristic
202C2×C61C2×#35211yescharacteristic
203C2×C61C2×#35411yescharacteristic
204C22×C4⋊C4C4×C6111yescharacteristic, complemented
205C22×C22⋊C4—21no—
206C22×C22⋊C4—21no—
207C22×C22⋊C4C22×C6111yescharacteristic
208C2×C22⋊C8—21no—
209#397—21no—
210#397—21no—
211#397C4×C6112yescomplemented
212#397C22×C6112yes—
213C2×#95—21no—
214C2×#95C4×C6111yescharacteristic, complemented
215#342—21no—
216#342C22×C6112yes—
217#323—21no—
218#323C22×C6112yes—
219C61C2×#134411yescharacteristic, direct product factor, Sylow
220C25D4×C6111yescharacteristic
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—41no—
224C23×C4C2×C4×C6111yescharacteristic
225C23×C4D4×C6111yescharacteristic
226C22×C8—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4C2×C4×C6112yescomplemented
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4C2×C4×C6111yescharacteristic
242C2×C22⋊C4D4×C6111yescharacteristic
243C2×C22⋊C4D4×C6111yescharacteristic
244C2×C22⋊C4C23×C6111yescharacteristic
245C22⋊C8—22no—
246C22⋊C8—41no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95C2×C4×C6112yescomplemented
251C24—21no—
252C24—21no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24C22⋊C4×C6111yescharacteristic
257C24C22⋊C4×C6111yescharacteristic
258C24D8×C6112yes—
259C24SD16×C6112yes—
260C24C2×D4×C6111yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C22⋊C4×C6111yescharacteristic
281C22×C4C22⋊C4×C6111yescharacteristic
282C22×C4C22×C4×C6111yescharacteristic
283C22×C4C2×D4×C6111yescharacteristic
284C2×C8—42no—
285C2×C8—82no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—24no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—44no—
311C23—44no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—84no—
317C23—81no—
318C23—81no—
319C23#99×C6111yescharacteristic
320C23#99×C6111yescharacteristic
321C23#102×C6112yes—
322C23#102×C6112yes—
323C23#104×C6112yes—
324C23C2×C22⋊C4×C6111yescharacteristic
325C23C2×D8×C6111yescharacteristic
326C23C2×SD16×C6111yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—84no—
347C2×C4—88no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C8—82no—
353C22—24no—
354C22—24no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—44no—
366C22—44no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—82no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22C61×#32712yes—
376C22C61×#32712yes—
377C22C61×#35212yes—
378C22C61×#35412yes—
379C22C2×#99×C6111yescharacteristic
380C22C2×#102×C6111yescharacteristic
381C22C2×#104×C6111yescharacteristic
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—82no—
387C4—84no—
388C4—82no—
389C2—21no—
390C2—22no—
391C2—21no—
392C2—41no—
393C2—42no—
394C2—41no—
395C2C61×#134414yesdirect product factor
396C2C2×C61×#32711yescharacteristic
397C2C2×C61×#35211yescharacteristic
398C2C2×C61×#35411yescharacteristic
subgroupG/H
center, socleC23×C61#99
commutatorC22×C4C22×C4×C61
Frattini, cosocleC2×C22⋊C4C23×C61
prime radicalC25×C61D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C61 ◁ C24×C61 ◁ C22×C22⋊C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
4. C61×#1344C2direct (N × H)
5. C22×C4⋊C4×C61C4semidirect
12. C61×#397C4semidirect
15. C2×#95×C61C4semidirect
30. C2×C4⋊C4×C61C2×C4semidirect
50. #95×C61C2×C4semidirect
153. C2×#1344C61direct (N × H)
193. #1344C2×C61direct (N × H)
200. C2×C61#1344direct (N × H)
204. C22×C4⋊C4C4×C61semidirect
211. #397C4×C61semidirect
214. C2×#95C4×C61semidirect
219. C61C2×#1344direct (N × H)
230. C2×C4⋊C4C2×C4×C61semidirect
250. #95C2×C4×C61semidirect
395. C2C61×#1344direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#3972
C2×C61×#3422
C2×C61×#3232
C61×#13442
C2×#134461