Finite Group Databasebeta

← C61×#15171 #29518604

Subgroups — C61×#15171 #29518604

Order 15616.

1302 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#401C211yesmaximal, characteristic, complemented
2C2×C61×#396C211yesmaximal, characteristic
3C61×#1442C213yesmaximal, complemented
4C61×#1440C213yesmaximal
5C61×#1437C213yesmaximal
6C61×#1436C213yesmaximal, complemented
7C4×#95×C61C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C61C2213yes—
9C22×C22⋊C4×C61C2211yescharacteristic
10C22×C22⋊C4×C61C2213yes—
11C22×C22⋊C4×C61C2213yes—
12C22×C42×C61C2213yes—
13C61×#401—24no—
14C61×#396—24no—
15C2×#95×C61C2213yes—
16C2×#95×C61C2213yes—
17C2×#95×C61C2213yes—
18C2×#95×C61C2213yes—
19C2×#95×C61C2213yes—
20C2×#95×C61C2213yes—
21C2×#95×C61C2213yes—
22C2×#95×C61C2211yescharacteristic
23C25×C61C2311yescharacteristic
24C23×C4×C61C2311yescharacteristic
25C23×C4×C61C2313yes—
26C23×C4×C61C2313yes—
27C23×C4×C61C2313yes—
28C23×C4×C61C2313yes—
29C23×C4×C61C2311yescharacteristic
30C2×C4⋊C4×C61—26no—
31C2×C4⋊C4×C61—26no—
32C2×C4⋊C4×C61—43no—
33C2×C22⋊C4×C61—26no—
34C2×C22⋊C4×C61—26no—
35C2×C22⋊C4×C61—26no—
36C2×C22⋊C4×C61—26no—
37C2×C22⋊C4×C61—26no—
38C2×C22⋊C4×C61—43no—
39C2×C22⋊C4×C61—43no—
40C2×C42×C61—26no—
41C2×C42×C61—43no—
42C2×C42×C61—43no—
43#95×C61—26no—
44#95×C61—26no—
45#95×C61—26no—
46#95×C61—26no—
47#95×C61—26no—
48#95×C61—26no—
49#95×C61—43no—
50#95×C61D414yes—
51C24×C61—21no—
52C24×C61—23no—
53C24×C61—23no—
54C24×C61—23no—
55C24×C61—23no—
56C24×C61—21no—
57C24×C61#4112yes—
58C24×C61C2411yescharacteristic
59C22×C4×C61—23no—
60C22×C4×C61—21no—
61C22×C4×C61—23no—
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—23no—
76C22×C4×C61—23no—
77C22×C4×C61—23no—
78C22×C4×C61—23no—
79C22×C4×C61—21no—
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×C61C2×D416yes—
90C22×C4×C61#4116yes—
91C22×C4×C61#4116yes—
92C22×C4×C61#4116yes—
93C4⋊C4×C61—412no—
94C22⋊C4×C61—412no—
95C22⋊C4×C61—412no—
96C22⋊C4×C61—44no—
97C42×C61—412no—
98C23×C61—26no—
99C23×C61—22no—
100C23×C61—26no—
101C23×C61—41no—
102C23×C61—43no—
103C23×C61—43no—
104C23×C61—43no—
105C23×C61—43no—
106C23×C61—43no—
107C23×C61—43no—
108C23×C61—43no—
109C23×C61—43no—
110C23×C61—43no—
111C23×C61C22×D411yescharacteristic
112C23×C61C2×#4113yes—
113C23×C61C2×#4113yes—
114C23×C61C2×#4111yescharacteristic
115C23×C61C2×#4113yes—
116C23×C61H(5,𝔽2)13yes—
117C23×C612−1+411yescharacteristic
118C2×C4×C61—26no—
119C2×C4×C61—26no—
120C2×C4×C61—26no—
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—26no—
128C2×C4×C61—26no—
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—41no—
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#12614yes—
152#15171C6111yesmaximal, characteristic, direct product factor, Sylow
153C22×C61—46no—
154C22×C61—46no—
155C22×C61—42no—
156C22×C61—83no—
157C22×C61—81no—
158C22×C61—81no—
159C22×C61—83no—
160C22×C61C2×#12611yescharacteristic
161C22×C61#52913yes—
162C22×C61#52913yes—
163C22×C61#52913yes—
164C22×C61#53213yes—
165C22×C61#53813yes—
166C22×C61#53913yes—
167C22×C61#54613yes—
168C22×C61#54811yescharacteristic
169C22×C61#55113yes—
170C22×C61#55313yes—
171C22×C61#55313yes—
172C22×C61#55613yes—
173C4×C61—24no—
174C4×C61—46no—
175C4×C61—46no—
176C4×C61—46no—
177C4×C61—83no—
178C4×C61—81no—
179C2×#401C2×C6111yescharacteristic, complemented
180C2×#396C2×C6111yescharacteristic
181#1442C2×C6113yescomplemented
182#1440C2×C6113yes—
183#1437C2×C6113yes—
184#1436C2×C6113yescomplemented
185C4×#95C2×C6111yescharacteristic, complemented
186C2×C61—82no—
187C2×C61#240111yescharacteristic
188C2×C61#255913yes—
189C2×C61#256013yes—
190C2×C61#257613yes—
191C2×C61#259413yes—
192C2×C61#264311yescharacteristic
193C2×C61#286011yescharacteristic
194C22×C4⋊C4C22×C6113yes—
195C22×C22⋊C4C22×C6111yescharacteristic
196C22×C22⋊C4C22×C6113yes—
197C22×C22⋊C4C22×C6113yes—
198C22×C42C22×C6113yes—
199#401—24no—
200#396—24no—
201C2×#95C22×C6113yes—
202C2×#95C22×C6113yes—
203C2×#95C22×C6113yes—
204C2×#95C22×C6113yes—
205C2×#95C22×C6113yes—
206C2×#95C22×C6113yes—
207C2×#95C22×C6113yes—
208C2×#95C22×C6111yescharacteristic
209C61#1517111yescharacteristic, direct product factor, Sylow
210C25C23×C6111yescharacteristic
211C23×C4C23×C6111yescharacteristic
212C23×C4C23×C6113yes—
213C23×C4C23×C6113yes—
214C23×C4C23×C6113yes—
215C23×C4C23×C6113yes—
216C23×C4C23×C6111yescharacteristic
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—43no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—26no—
222C2×C22⋊C4—26no—
223C2×C22⋊C4—26no—
224C2×C22⋊C4—26no—
225C2×C22⋊C4—43no—
226C2×C22⋊C4—43no—
227C2×C42—26no—
228C2×C42—43no—
229C2×C42—43no—
230#95—26no—
231#95—26no—
232#95—26no—
233#95—26no—
234#95—26no—
235#95—26no—
236#95—43no—
237#95D4×C6114yes—
238C24—21no—
239C24—23no—
240C24—23no—
241C24—23no—
242C24—23no—
243C24—21no—
244C24#41×C6112yes—
245C24C24×C6111yescharacteristic
246C22×C4—23no—
247C22×C4—21no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—23no—
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—23no—
266C22×C4—21no—
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×C4C2×D4×C6116yes—
277C22×C4#41×C6116yes—
278C22×C4#41×C6116yes—
279C22×C4#41×C6116yes—
280C4⋊C4—412no—
281C22⋊C4—412no—
282C22⋊C4—412no—
283C22⋊C4—44no—
284C42—412no—
285C23—22no—
286C23—26no—
287C23—26no—
288C23—43no—
289C23—43no—
290C23—43no—
291C23—43no—
292C23—43no—
293C23—43no—
294C23—43no—
295C23—41no—
296C23—43no—
297C23—43no—
298C23C22×D4×C6111yescharacteristic
299C23C2×#41×C6113yes—
300C23C2×#41×C6113yes—
301C23C2×#41×C6113yes—
302C23C2×#41×C6111yescharacteristic
303C23H(5,𝔽2)×C6113yes—
304C232−1+4×C6111yescharacteristic
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
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—41no—
336C2×C4—43no—
337C2×C4—43no—
338C2×C4#126×C6114yes—
339C22—46no—
340C22—46no—
341C22—42no—
342C22—83no—
343C22—83no—
344C22—81no—
345C22—81no—
346C22C2×#126×C6111yescharacteristic
347C22C61×#52913yes—
348C22C61×#52913yes—
349C22C61×#52913yes—
350C22C61×#53213yes—
351C22C61×#53813yes—
352C22C61×#53913yes—
353C22C61×#54613yes—
354C22C61×#54811yescharacteristic
355C22C61×#55113yes—
356C22C61×#55313yes—
357C22C61×#55313yes—
358C22C61×#55613yes—
359C4—24no—
360C4—46no—
361C4—46no—
362C4—46no—
363C4—83no—
364C4—81no—
365C2—82no—
366C2C61×#240111yescharacteristic
367C2C61×#255913yes—
368C2C61×#256013yes—
369C2C61×#257613yes—
370C2C61×#259413yes—
371C2C61×#264311yescharacteristic
372C2C61×#286011yescharacteristic
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×#401C2semidirect
3. C61×#1442C2semidirect
6. C61×#1436C2semidirect
7. C4×#95×C61C2semidirect
152. #15171C61direct (N × H)
179. C2×#401C2×C61semidirect
181. #1442C2×C61semidirect
184. #1436C2×C61semidirect
185. C4×#95C2×C61semidirect
209. C61#15171direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#4012
C2×C61×#3962
C61×#14422
C61×#14402
C61×#14372
C61×#14362
C4×#95×C612
#1517161