Finite Group Databasebeta

← C61×#12987 #29516420

Subgroups — C61×#12987 #29516420

Order 15616.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C61×#504C211yesmaximal, characteristic, complemented
2C61×#2955C211yesmaximal, characteristic
3C61×#2117C211yesmaximal, characteristic, complemented
4C2×C61×#350C211yesmaximal, characteristic, complemented
5C61×#2115C211yesmaximal, characteristic, complemented
6C61×#1379C211yesmaximal, characteristic, complemented
7C61×#1378C211yesmaximal, characteristic, complemented
8C61×#575C411yescharacteristic
9C2×#131×C61C2211yescharacteristic
10C2×#131×C61C2211yescharacteristic
11C61×#504—22no—
12C61×#504—22no—
13C2×M32×C61—21no—
14C2×M32×C61C2211yescharacteristic
15C2×M32×C61C2211yescharacteristic
16C2×M32×C61C2211yescharacteristic
17C22×C16×C61—21no—
18C22×C16×C61C2211yescharacteristic
19C61×#444—22no—
20C61×#443—22no—
21C61×#430C411yescharacteristic
22C61×#421C411yescharacteristic
23C61×#421C411yescharacteristic
24C61×#405C2211yescharacteristic
25C61×#350—22no—
26C61×#350—21no—
27C61×#350—21no—
28C61×#349—22no—
29C2×#41×C61C2×C411yescharacteristic
30C2×#41×C61C2×C411yescharacteristic
31#137×C61—21no—
32#137×C61C812yes—
33AGL(1,ℤ/8)×C61—21no—
34AGL(1,ℤ/8)×C61C812yes—
35#135×C61—22no—
36#131×C61—21no—
37#131×C61—21no—
38#131×C61—21no—
39#131×C61—21no—
40#131×C61—21no—
41#131×C61—21no—
42#131×C61D412yes—
43#131×C61D412yes—
44C2×M16×C61—21no—
45C2×M16×C61—21no—
46C2×M16×C61C2×C411yescharacteristic
47C2×M16×C61C2×C411yescharacteristic
48C2×M16×C61C2×C411yescharacteristic
49C22×C8×C61—21no—
50C22×C8×C61—21no—
51C22×C8×C61C2311yescharacteristic
52#117×C61C2×C411yescharacteristic
53M32×C61—24no—
54M32×C61—21no—
55M32×C61—21no—
56M32×C61—21no—
57M32×C61—21no—
58M32×C61—21no—
59M32×C61D412yes—
60C2×C16×C61—24no—
61C2×C16×C61—21no—
62C2×C16×C61—21no—
63C2×C16×C61—21no—
64C2×C16×C61—21no—
65C2×C16×C61—21no—
66C2×C16×C61D411yescharacteristic
67C2×C16×C61D411yescharacteristic
68#108×C61—22no—
69#104×C61—22no—
70#104×C61—22no—
71#97×C61—21no—
72C4×C8×C61—21no—
73#41×C61—21no—
74#41×C61—21no—
75#41×C61—21no—
76#41×C61—21no—
77#41×C61—21no—
78#41×C61—21no—
79#41×C61C22⋊C412yes—
80#41×C61C2×C812yes—
81C2×Q8×C61C2×C811yescharacteristic
82C2×Q8×C61M1611yescharacteristic
83C2×D4×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61C2×C811yescharacteristic
86C2×D4×C61M1611yescharacteristic
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61C22×C411yescharacteristic
90Q16×C61—22no—
91SD16×C61—22no—
92SD16×C61—22no—
93D8×C61—22no—
94M16×C61—21no—
95M16×C61—21no—
96M16×C61—22no—
97M16×C61—22no—
98M16×C61—21no—
99M16×C61—21no—
100M16×C61—41no—
101M16×C61—41no—
102M16×C61C22⋊C412yes—
103M16×C61C2×C812yes—
104C2×C8×C61—21no—
105C2×C8×C61—21no—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C2×C8×C61—21no—
109C2×C8×C61—41no—
110C2×C8×C61—41no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61#4111yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C4⋊C4×C61—21no—
118C22⋊C4×C61—21no—
119C42×C61—21no—
120C16×C61—21no—
121C16×C61—22no—
122C16×C61—21no—
123C16×C61—41no—
124C16×C61—41no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61#13111yescharacteristic
128Q8×C61—21no—
129Q8×C61—21no—
130Q8×C61—21no—
131Q8×C61C22⋊C812yes—
132D4×C61—22no—
133D4×C61—22no—
134D4×C61—21no—
135D4×C61—21no—
136D4×C61—21no—
137D4×C61—41no—
138D4×C61—41no—
139D4×C61C22⋊C812yes—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C4×D411yescharacteristic
150C2×C4×C61C22×C811yescharacteristic
151C2×C4×C61C2×M1611yescharacteristic
152C2×C4×C61#13111yescharacteristic
153C8×C61—22no—
154C8×C61—22no—
155C8×C61—41no—
156C8×C61#12011yescharacteristic
157C8×C61#12111yescharacteristic
158C8×C61#12111yescharacteristic
159C8×C61#12311yescharacteristic
160#12987C6111yesmaximal, characteristic, direct product factor, Sylow
161C22×C61—22no—
162C22×C61—41no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—81no—
166C22×C61#40811yescharacteristic
167C22×C61C8×D411yescharacteristic
168C22×C61#43511yescharacteristic
169C4×C61—22no—
170C4×C61—41no—
171C4×C61—81no—
172C4×C61#38611yescharacteristic
173C4×C61C2×C22⋊C811yescharacteristic
174C4×C61C8×D411yescharacteristic
175C4×C61#43511yescharacteristic
176C2×#504C2×C6111yescharacteristic, complemented
177#2955C2×C6111yescharacteristic
178#2117C2×C6111yescharacteristic, complemented
179C2×#350C2×C6111yescharacteristic, complemented
180#2115C2×C6111yescharacteristic, complemented
181#1379C2×C6111yescharacteristic, complemented
182#1378C2×C6111yescharacteristic, complemented
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61—42no—
187C2×C61—81no—
188C2×C61#186911yescharacteristic
189#575C4×C6111yescharacteristic
190C2×#131C22×C6111yescharacteristic
191C2×#131C22×C6111yescharacteristic
192#504—22no—
193#504—22no—
194C2×M32—21no—
195C2×M32C22×C6111yescharacteristic
196C2×M32C22×C6111yescharacteristic
197C2×M32C22×C6111yescharacteristic
198C22×C16—21no—
199C22×C16C22×C6111yescharacteristic
200#444—22no—
201#443—22no—
202#430C4×C6111yescharacteristic
203#421C4×C6111yescharacteristic
204#421C4×C6111yescharacteristic
205#405C22×C6111yescharacteristic
206#350—22no—
207#350—21no—
208#350—21no—
209#349—22no—
210C61#1298711yescharacteristic, direct product factor, Sylow
211C2×#41C2×C4×C6111yescharacteristic
212C2×#41C2×C4×C6111yescharacteristic
213#137—21no—
214#137C8×C6112yes—
215AGL(1,ℤ/8)—21no—
216AGL(1,ℤ/8)C8×C6112yes—
217#135—22no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—21no—
223#131—21no—
224#131D4×C6112yes—
225#131D4×C6112yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16C2×C4×C6111yescharacteristic
229C2×M16C2×C4×C6111yescharacteristic
230C2×M16C2×C4×C6111yescharacteristic
231C22×C8—21no—
232C22×C8—21no—
233C22×C8C23×C6111yescharacteristic
234#117C2×C4×C6111yescharacteristic
235M32—24no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—21no—
241M32D4×C6112yes—
242C2×C16—24no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16D4×C6111yescharacteristic
249C2×C16D4×C6111yescharacteristic
250#108—22no—
251#104—22no—
252#104—22no—
253#97—21no—
254C4×C8—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260#41—21no—
261#41C22⋊C4×C6112yes—
262#41C2×C8×C6112yes—
263C2×Q8C2×C8×C6111yescharacteristic
264C2×Q8M16×C6111yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C2×C8×C6111yescharacteristic
268C2×D4M16×C6111yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×C4×C6111yescharacteristic
272Q16—22no—
273SD16—22no—
274SD16—22no—
275D8—22no—
276M16—21no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—21no—
281M16—21no—
282M16—41no—
283M16—41no—
284M16C22⋊C4×C6112yes—
285M16C2×C8×C6112yes—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8C2×D4×C6111yescharacteristic
294C2×C8C2×D4×C6111yescharacteristic
295C2×C8C2×D4×C6111yescharacteristic
296C2×C8C2×D4×C6111yescharacteristic
297C2×C8#41×C6111yescharacteristic
298C2×C8#41×C6111yescharacteristic
299C4⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C16—21no—
303C16—22no—
304C16—21no—
305C16—41no—
306C16—41no—
307C23—21no—
308C23—21no—
309C23#131×C6111yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8C22⋊C8×C6112yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4C22⋊C8×C6112yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊C4×C6111yescharacteristic
330C2×C4C4×D4×C6111yescharacteristic
331C2×C4C4×D4×C6111yescharacteristic
332C2×C4C22×C8×C6111yescharacteristic
333C2×C4C2×M16×C6111yescharacteristic
334C2×C4#131×C6111yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8#120×C6111yescharacteristic
339C8#121×C6111yescharacteristic
340C8#121×C6111yescharacteristic
341C8#123×C6111yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—81no—
347C22C61×#40811yescharacteristic
348C22C8×D4×C6111yescharacteristic
349C22C61×#43511yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4C61×#38611yescharacteristic
354C4C2×C22⋊C8×C6111yescharacteristic
355C4C8×D4×C6111yescharacteristic
356C4C61×#43511yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2C61×#186911yescharacteristic
subgroupG/H
centerC8×C61#121
commutatorC2×C4C22×C8×C61
Frattini, cosocleC22×C8C23×C61
socleC2×C61#1869
prime radicalC61×#575C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8×C61 ◁ C22×C8×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C61×#504C2semidirect
3. C61×#2117C2semidirect
4. C2×C61×#350C2semidirect
5. C61×#2115C2semidirect
6. C61×#1379C2semidirect
7. C61×#1378C2semidirect
160. #12987C61direct (N × H)
176. C2×#504C2×C61semidirect
178. #2117C2×C61semidirect
179. C2×#350C2×C61semidirect
180. #2115C2×C61semidirect
181. #1379C2×C61semidirect
182. #1378C2×C61semidirect
210. C61#12987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5042
C61×#29552
C61×#21172
C2×C61×#3502
C61×#21152
C61×#13792
C61×#13782
#1298761