Finite Group Databasebeta

← C61×#17223 #29520656

Subgroups — C61×#17223 #29520656

Order 15616.

1270 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#395C211yesmaximal, characteristic, complemented
2C2×C61×#393C211yesmaximal, characteristic
3C61×#1446C211yesmaximal, characteristic, complemented
4C61×#1443C213yesmaximal, complemented
5C61×#1442C213yesmaximal, complemented
6C61×#1440C213yesmaximal
7C61×#1439C213yesmaximal
8C23×Q8×C61C2211yescharacteristic
9C22×C4⋊C4×C61C2213yes—
10C22×C22⋊C4×C61C2213yes—
11C22×C22⋊C4×C61C2213yes—
12C22×C22⋊C4×C61C2211yescharacteristic
13C61×#395—24no—
14C61×#393—24no—
15C2×#95×C61C2213yes—
16C2×#95×C61C2213yes—
17C2×#95×C61C2213yes—
18C2×#95×C61C2213yes—
19C2×#95×C61C2213yes—
20C2×#95×C61C2213yes—
21C2×#95×C61C2213yes—
22C2×#95×C61C2213yes—
23C25×C61C2311yescharacteristic
24C22×Q8×C61—43no—
25C22×Q8×C61—43no—
26C22×Q8×C61D414yes—
27C23×C4×C61C2313yes—
28C23×C4×C61C2311yescharacteristic
29C23×C4×C61C2313yes—
30C23×C4×C61C2311yescharacteristic
31C23×C4×C61C2313yes—
32C23×C4×C61C2313yes—
33C2×C4⋊C4×C61—26no—
34C2×C4⋊C4×C61—43no—
35C2×C4⋊C4×C61—43no—
36C2×C22⋊C4×C61—26no—
37C2×C22⋊C4×C61—26no—
38C2×C22⋊C4×C61—26no—
39C2×C22⋊C4×C61—43no—
40C2×C22⋊C4×C61—43no—
41C2×C22⋊C4×C61—43no—
42C2×C22⋊C4×C61—43no—
43#95×C61—26no—
44#95×C61—26no—
45#95×C61—26no—
46#95×C61—26no—
47#95×C61—43no—
48#95×C61—43no—
49#95×C61—43no—
50#95×C61—43no—
51C24×C61—21no—
52C24×C61—23no—
53C24×C61—23no—
54C24×C61—23no—
55C24×C61—23no—
56C24×C61—21no—
57C24×C61#4112yes—
58C24×C61C2411yescharacteristic
59C2×Q8×C61—412no—
60C2×Q8×C61—412no—
61C2×Q8×C61—44no—
62C22×C4×C61—23no—
63C22×C4×C61—23no—
64C22×C4×C61—23no—
65C22×C4×C61—23no—
66C22×C4×C61—23no—
67C22×C4×C61—23no—
68C22×C4×C61—23no—
69C22×C4×C61—23no—
70C22×C4×C61—23no—
71C22×C4×C61—23no—
72C22×C4×C61—23no—
73C22×C4×C61—23no—
74C22×C4×C61—23no—
75C22×C4×C61—21no—
76C22×C4×C61—23no—
77C22×C4×C61—23no—
78C22×C4×C61—23no—
79C22×C4×C61—23no—
80C22×C4×C61—23no—
81C22×C4×C61—23no—
82C22×C4×C61—23no—
83C22×C4×C61—23no—
84C22×C4×C61—23no—
85C22×C4×C61—23no—
86C22×C4×C61—23no—
87C22×C4×C61—23no—
88C22×C4×C61—23no—
89C22×C4×C61—23no—
90C22×C4×C61—21no—
91C22×C4×C61—23no—
92C22×C4×C61—23no—
93C22×C4×C61—23no—
94C22×C4×C61C2×D416yes—
95C22×C4×C61#4116yes—
96C4⋊C4×C61—412no—
97C22⋊C4×C61—412no—
98C22⋊C4×C61—412no—
99C22⋊C4×C61—44no—
100C23×C61—26no—
101C23×C61—22no—
102C23×C61—26no—
103C23×C61—43no—
104C23×C61—43no—
105C23×C61—43no—
106C23×C61—43no—
107C23×C61—43no—
108C23×C61—43no—
109C23×C61—43no—
110C23×C61—41no—
111C23×C61—43no—
112C23×C61—43no—
113C23×C61C22×D411yescharacteristic
114C23×C61C2×#4111yescharacteristic
115C23×C61C2×#4113yes—
116C23×C61H(5,𝔽2)13yes—
117C23×C61H(5,𝔽2)13yes—
118C23×C612−1+411yescharacteristic
119C23×C612−1+413yes—
120Q8×C61—416no—
121C2×C4×C61—26no—
122C2×C4×C61—26no—
123C2×C4×C61—26no—
124C2×C4×C61—26no—
125C2×C4×C61—26no—
126C2×C4×C61—26no—
127C2×C4×C61—43no—
128C2×C4×C61—43no—
129C2×C4×C61—43no—
130C2×C4×C61—43no—
131C2×C4×C61—43no—
132C2×C4×C61—43no—
133C2×C4×C61—43no—
134C2×C4×C61—43no—
135C2×C4×C61—43no—
136C2×C4×C61—43no—
137C2×C4×C61—43no—
138C2×C4×C61—43no—
139C2×C4×C61—43no—
140C2×C4×C61—43no—
141C2×C4×C61—43no—
142C2×C4×C61—43no—
143C2×C4×C61—43no—
144C2×C4×C61—43no—
145C2×C4×C61—43no—
146C2×C4×C61—43no—
147C2×C4×C61—43no—
148C2×C4×C61—43no—
149C2×C4×C61—43no—
150C2×C4×C61—43no—
151C2×C4×C61—43no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—43no—
155#17223C6111yesmaximal, characteristic, direct product factor, Sylow
156C22×C61—42no—
157C22×C61—46no—
158C22×C61—46no—
159C22×C61—83no—
160C22×C61—83no—
161C22×C61—81no—
162C22×C61—81no—
163C22×C61#53413yes—
164C22×C61#53913yes—
165C22×C61#53913yes—
166C22×C61#54213yes—
167C22×C61#54213yes—
168C22×C61#54811yescharacteristic
169C22×C61#54813yes—
170C22×C61#55113yes—
171C22×C61#55513yes—
172C22×C61#56013yes—
173C22×C61H(3,𝔽4)11yescharacteristic
174C22×C61#56313yes—
175C22×C61#56313yes—
176C4×C61—46no—
177C4×C61—46no—
178C4×C61—83no—
179C4×C61—83no—
180C4×C61—81no—
181C4×C61—81no—
182C2×#395C2×C6111yescharacteristic, complemented
183C2×#393C2×C6111yescharacteristic
184#1446C2×C6111yescharacteristic, complemented
185#1443C2×C6113yescomplemented
186#1442C2×C6113yescomplemented
187#1440C2×C6113yes—
188#1439C2×C6113yes—
189C2×C61—82no—
190C2×C61#272813yes—
191C2×C61#273113yes—
192C2×C61#273713yes—
193C2×C61#278511yescharacteristic
194C2×C61#281211yescharacteristic
195C2×C61#283511yescharacteristic
196C2×C61#283813yes—
197C23×Q8C22×C6111yescharacteristic
198C22×C4⋊C4C22×C6113yes—
199C22×C22⋊C4C22×C6113yes—
200C22×C22⋊C4C22×C6113yes—
201C22×C22⋊C4C22×C6111yescharacteristic
202#395—24no—
203#393—24no—
204C2×#95C22×C6113yes—
205C2×#95C22×C6113yes—
206C2×#95C22×C6113yes—
207C2×#95C22×C6113yes—
208C2×#95C22×C6113yes—
209C2×#95C22×C6113yes—
210C2×#95C22×C6113yes—
211C2×#95C22×C6113yes—
212C61#1722311yescharacteristic, direct product factor, Sylow
213C25C23×C6111yescharacteristic
214C22×Q8—43no—
215C22×Q8—43no—
216C22×Q8D4×C6114yes—
217C23×C4C23×C6111yescharacteristic
218C23×C4C23×C6113yes—
219C23×C4C23×C6111yescharacteristic
220C23×C4C23×C6113yes—
221C23×C4C23×C6113yes—
222C23×C4C23×C6113yes—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—43no—
225C2×C4⋊C4—43no—
226C2×C22⋊C4—26no—
227C2×C22⋊C4—26no—
228C2×C22⋊C4—26no—
229C2×C22⋊C4—43no—
230C2×C22⋊C4—43no—
231C2×C22⋊C4—43no—
232C2×C22⋊C4—43no—
233#95—26no—
234#95—26no—
235#95—26no—
236#95—26no—
237#95—43no—
238#95—43no—
239#95—43no—
240#95—43no—
241C24—21no—
242C24—23no—
243C24—23no—
244C24—23no—
245C24—23no—
246C24—21no—
247C24#41×C6112yes—
248C24C24×C6111yescharacteristic
249C2×Q8—412no—
250C2×Q8—412no—
251C2×Q8—44no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—21no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—23no—
271C22×C4—23no—
272C22×C4—23no—
273C22×C4—23no—
274C22×C4—23no—
275C22×C4—23no—
276C22×C4—23no—
277C22×C4—23no—
278C22×C4—23no—
279C22×C4—23no—
280C22×C4—21no—
281C22×C4—23no—
282C22×C4—23no—
283C22×C4—23no—
284C22×C4C2×D4×C6116yes—
285C22×C4#41×C6116yes—
286C4⋊C4—412no—
287C22⋊C4—412no—
288C22⋊C4—412no—
289C22⋊C4—44no—
290C23—26no—
291C23—22no—
292C23—26no—
293C23—43no—
294C23—43no—
295C23—43no—
296C23—43no—
297C23—43no—
298C23—43no—
299C23—43no—
300C23—41no—
301C23—43no—
302C23—43no—
303C23C22×D4×C6111yescharacteristic
304C23C2×#41×C6111yescharacteristic
305C23C2×#41×C6113yes—
306C23H(5,𝔽2)×C6113yes—
307C23H(5,𝔽2)×C6113yes—
308C232−1+4×C6111yescharacteristic
309C232−1+4×C6113yes—
310Q8—416no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—43no—
318C2×C4—43no—
319C2×C4—43no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—43no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—43no—
328C2×C4—43no—
329C2×C4—43no—
330C2×C4—43no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—43no—
337C2×C4—43no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—43no—
345C22—42no—
346C22—46no—
347C22—46no—
348C22—81no—
349C22—81no—
350C22—83no—
351C22—83no—
352C22C61×#53413yes—
353C22C61×#53913yes—
354C22C61×#53913yes—
355C22C61×#54213yes—
356C22C61×#54213yes—
357C22C61×#54811yescharacteristic
358C22C61×#54813yes—
359C22C61×#55113yes—
360C22C61×#55513yes—
361C22C61×#56013yes—
362C22C61×H(3,𝔽4)11yescharacteristic
363C22C61×#56313yes—
364C22C61×#56313yes—
365C4—46no—
366C4—46no—
367C4—81no—
368C4—81no—
369C4—83no—
370C4—83no—
371C2—82no—
372C2C61×#272813yes—
373C2C61×#273113yes—
374C2C61×#273713yes—
375C2C61×#278511yescharacteristic
376C2C61×#281211yescharacteristic
377C2C61×#283511yescharacteristic
378C2C61×#283813yes—
subgroupG/H
center, socleC24×C61C24
commutator, Frattini, cosocleC24C24×C61
prime radicalC25×C61C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C61×#395C2semidirect
3. C61×#1446C2semidirect
4. C61×#1443C2semidirect
5. C61×#1442C2semidirect
155. #17223C61direct (N × H)
182. C2×#395C2×C61semidirect
184. #1446C2×C61semidirect
185. #1443C2×C61semidirect
186. #1442C2×C61semidirect
212. C61#17223direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#3952
C2×C61×#3932
C61×#14462
C61×#14432
C61×#14422
C61×#14402
C61×#14392
#1722361