Finite Group Databasebeta

← #30625086

Subgroups — #30625086

Order 15616.

1800 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462365C213yesmaximal
2#21462354C216yesmaximal
3#21462353C216yesmaximal
4#21462319C216yesmaximal
5#21462313C216yesmaximal
6#125×#6923C213yesmaximal
7#564×D61C211yesmaximal, characteristic
8#16840713C2213yes—
9#16840713C2213yes—
10#16840713C2216yes—
11#16840713C2216yes—
12#16840713C2216yes—
13#16840713C2216yes—
14#16840711C2216yes—
15#16840711C2216yes—
16#16840711C2216yes—
17#16840711C2216yes—
18#16840711C2216yes—
19#16840711C2216yes—
20#16840711C2216yes—
21#16840711C2216yes—
22#16840711C2216yes—
23#16840711C2216yes—
24#16840708C2212yes—
25#16840707C2212yes—
26#16840706C2213yes—
27C4⋊C4×#6923C2216yes—
28C4⋊C4×#6923C2213yes—
29C4⋊C4×#6923C2216yes—
30C4⋊C4×#6923C2213yes—
31C4⋊C4×#6923C2216yes—
32C4⋊C4×#6923C2216yes—
33C4×C4⋊#6923C2216yes—
34#16840698C2216yes—
35C42×#6923C2211yescharacteristic
36C61⋊#564C413yescomplemented
37C61⋊#564C416yescomplemented
38C61⋊#564C416yescomplemented
39#125×D61C2213yes—
40#125×D61C2216yes—
41#125×D61C2216yes—
42C61×#564C411yescharacteristic, complemented
43C2×C4⋊#6923C2316yes—
44C2×C4⋊#6923C2313yes—
45C2×C4⋊#6923C2313yes—
46C2×C4⋊#6923C2316yes—
47C2×C4⋊#6923C2316yes—
48C2×C4⋊#6923C2316yes—
49C2×C4×#6923C2313yes—
50C2×C4×#6923C2316yes—
51C2×C4×#6923C2316yes—
52C2×C4×#6923C2313yes—
53C2×C4×#6923C2316yes—
54C2×C4×#6923C2316yes—
55(C2×#6923)⋊C4C2316yes—
56(C2×#6923)⋊C4C2316yes—
57(C2×#6923)⋊C4C2316yes—
58(C2×#6923)⋊C4C2316yes—
59(C2×#6923)⋊C4C2316yes—
60(C2×#6923)⋊C4C2316yes—
61(C2×#6923)⋊C4C2316yes—
62(C2×#6923)⋊C4C2316yes—
63(C2×#6923)⋊C4C2316yes—
64(C2×#6923)⋊C4C2316yes—
65#14519769C2×C413yes—
66#14519769C2×C416yes—
67#14519769C2×C416yes—
68#14519768C2×C413yes—
69#14519768C2×C413yes—
70#14519768C2×C416yes—
71#14519768C2×C416yes—
72#14519768C2×C416yes—
73#14519768C2×C416yes—
74#14519767C2×C416yes—
75#14519767C2×C416yes—
76#14519767C2×C416yes—
77#14519767C2×C416yes—
78#14519767C2×C416yes—
79#14519767C2×C416yes—
80#14519767C2×C416yes—
81#14519767C2×C416yes—
82#14519767C2×C416yes—
83#14519767C2×C416yes—
84C4⋊C4×D61C2316yes—
85C4⋊C4×D61C2316yes—
86C4⋊C4×D61C2316yes—
87C4⋊C4×D61C2316yes—
88C4⋊C4×D61C2313yes—
89C4⋊C4×D61C2313yes—
90C42×D61C2311yescharacteristic
91C42×D61C2312yes—
92C42×D61C2312yes—
93#125×C61C2×C413yes—
94#125×C61C2×C416yes—
95#125×C61C2×C416yes—
96C22×#6923C2416yes—
97C22×#6923C2416yes—
98C22×#6923C2411yescharacteristic
99C22×#6923C2413yes—
100C2×C4×D61C2413yes—
101C2×C4×D61C2416yes—
102C2×C4×D61C2416yes—
103C4⋊#6923—212no—
104C4⋊#6923—46no—
105C4⋊#6923—43no—
106C4⋊#6923—46no—
107C4⋊#6923—46no—
108C4⋊#6923—43no—
109C4×#6923—26no—
110C4×#6923—212no—
111C4×#6923—46no—
112C4×#6923—46no—
113C4×#6923—43no—
114C4×#6923—46no—
115C4⋊C4×C61C22×C413yes—
116C4⋊C4×C61C22×C416yes—
117C4⋊C4×C61C22×C416yes—
118C4⋊C4×C61C22×C413yes—
119C4⋊C4×C61C22×C416yes—
120C4⋊C4×C61C22×C416yes—
121C42×C61C22×C411yescharacteristic
122C42×C61C22×C412yes—
123C42×C61C22×C412yes—
124C4⋊Dic61C22×C416yes—
125C4⋊Dic61C22×C416yes—
126C4⋊Dic61C22×C416yes—
127C4⋊Dic61C22×C416yes—
128C4⋊Dic61C22×C413yes—
129C4⋊Dic61C22×C413yes—
130#11758196C22×C416yes—
131#11758196C22×C416yes—
132#11758196C22×C416yes—
133#11758196C22×C416yes—
134#11758196C22×C416yes—
135#11758196C22×C416yes—
136#11758196C22×C416yes—
137#11758196C22×C416yes—
138#11758196C22×C416yes—
139#11758196C22×C416yes—
140C4×Dic61C22×C413yes—
141C4×Dic61C22×C416yes—
142C4×Dic61C22×C416yes—
143C22×D61C2511yescharacteristic
144C2×#6923—23no—
145C2×#6923—23no—
146C2×#6923—26no—
147C2×#6923—26no—
148C2×#6923—26no—
149C2×#6923—26no—
150C2×#6923—26no—
151C2×#6923—23no—
152C2×#6923—26no—
153C2×#69232−1+416yes—
154C2×C4×C61C23×C413yes—
155C2×C4×C61C23×C416yes—
156C2×C4×C61C23×C416yes—
157C2×Dic61C23×C413yes—
158C2×Dic61C23×C416yes—
159C2×Dic61C23×C416yes—
160C4×D61—23no—
161C4×D61—26no—
162C4×D61—26no—
163C4×D61—26no—
164C4×D61—23no—
165C4×D61—26no—
166#34530—611nomaximal, Sylow
167C22×C61C24×C411yescharacteristic
168C2×D61C2×2−1+413yes—
169C2×D61#58513yes—
170#6923—26no—
171#6923—46no—
172#6923—46no—
173#6923—41no—
174C4×C61—23no—
175C4×C61—26no—
176C4×C61—26no—
177Dic61—23no—
178Dic61—26no—
179Dic61—26no—
180C2×#564—611no—
181#2382—613no—
182#2371—616no—
183#2370—616no—
184#2336—616no—
185#2330—616no—
186C4×#125—613no—
187C2×C61C4×2−1+413yes—
188D61#356313yes—
189D61#356711yescharacteristic
190#564—616no—
191#564—616no—
192#564—613no—
193#564#692311yescharacteristic, complemented
194C2×#125—613no—
195C2×#125—616no—
196C2×#125—616no—
197#389—613no—
198#389—613no—
199#389—616no—
200#389—616no—
201#389—616no—
202#389—616no—
203#387—616no—
204#387—616no—
205#387—616no—
206#387—616no—
207#387—616no—
208#387—616no—
209#387—616no—
210#387—616no—
211#387—616no—
212#387—616no—
213#384—612no—
214#383—612no—
215#382—613no—
216C4×C4⋊C4—616no—
217C4×C4⋊C4—613no—
218C4×C4⋊C4—616no—
219C4×C4⋊C4—613no—
220C4×C4⋊C4—616no—
221C4×C4⋊C4—616no—
222C4×C4⋊C4—616no—
223#376—616no—
224C43—611no—
225C61#3453011yescharacteristic, complemented, Sylow
226#125—616no—
227#125—616no—
228#125—616no—
229#125—616no—
230#125—616no—
231#125—616no—
232#125—616no—
233#125—616no—
234#125—616no—
235#125—616no—
236#125—613no—
237#125—613no—
238#125—616no—
239#125—616no—
240#125—616no—
241#125—616no—
242#125—613no—
243#125—616no—
244#125—616no—
245#125C2×#692313yes—
246#125C2×#692316yes—
247#125C2×#692316yes—
248C2×C4⋊C4—613no—
249C2×C4⋊C4—616no—
250C2×C4⋊C4—616no—
251C2×C4⋊C4—613no—
252C2×C4⋊C4—616no—
253C2×C4⋊C4—616no—
254C2×C4⋊C4—616no—
255C2×C4⋊C4—613no—
256C2×C4⋊C4—613no—
257C2×C4⋊C4—616no—
258C2×C4⋊C4—616no—
259C2×C4⋊C4—616no—
260C2×C42—611no—
261C2×C42—612no—
262C2×C42—612no—
263C2×C42—613no—
264C2×C42—616no—
265C2×C42—613no—
266C2×C42—616no—
267C2×C42—616no—
268C2×C42—616no—
269#95—616no—
270#95—616no—
271#95—616no—
272#95—616no—
273#95—616no—
274#95—616no—
275#95—616no—
276#95—616no—
277#95—616no—
278#95—616no—
279C22×C4—613no—
280C22×C4—616no—
281C22×C4—613no—
282C22×C4—616no—
283C22×C4—616no—
284C22×C4—616no—
285C22×C4—611no—
286C4⋊C4—616no—
287C4⋊C4—616no—
288C4⋊C4—616no—
289C4⋊C4—616no—
290C4⋊C4—616no—
291C4⋊C4—616no—
292C4⋊C4—616no—
293C4⋊C4—616no—
294C4⋊C4—616no—
295C4⋊C4—616no—
296C4⋊C4—613no—
297C4⋊C4—616no—
298C4⋊C4—616no—
299C4⋊C4—613no—
300C4⋊C4—616no—
301C4⋊C4—616no—
302C4⋊C4—12212no—
303C4⋊C4—2446no—
304C4⋊C4—2443no—
305C4⋊C4—2446no—
306C4⋊C4—2446no—
307C4⋊C4—2443no—
308C4⋊C4C22×#692313yes—
309C4⋊C4C22×#692316yes—
310C4⋊C4C22×#692316yes—
311C4⋊C4C22×#692313yes—
312C4⋊C4C22×#692316yes—
313C4⋊C4C22×#692316yes—
314C42—613no—
315C42—616no—
316C42—616no—
317C42—1226no—
318C42—12212no—
319C42—2446no—
320C42—2446no—
321C42—2443no—
322C42—2446no—
323C42C22×#692311yescharacteristic
324C42C22×#692312yes—
325C42C22×#692312yes—
326C23—611no—
327C2×C4—616no—
328C2×C4—613no—
329C2×C4—616no—
330C2×C4—616no—
331C2×C4—1223no—
332C2×C4—1226no—
333C2×C4—1226no—
334C2×C4—1223no—
335C2×C4—1226no—
336C2×C4—1226no—
337C2×C4—1223no—
338C2×C4—1223no—
339C2×C4—1226no—
340C2×C4—1226no—
341C2×C4—1226no—
342C2×C4—1226no—
343C2×C4—1223no—
344C2×C4—1226no—
345C2×C4—1226no—
346C2×C4C23×#692313yes—
347C2×C4C23×#692316yes—
348C2×C4C23×#692316yes—
349C22—613no—
350C22—613no—
351C22C24×#692311yescharacteristic
352C4—23no—
353C4—26no—
354C4—26no—
355C4—1226no—
356C4—1223no—
357C4—1226no—
358C4—1226no—
359C4—2446no—
360C4—2446no—
361C4—2441no—
362C2—613no—
363C2—611no—
364C22−1+4×#692313yes—
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#564C4
prime radical, cosocleC22×D61C25
hypercenter#564#6923
hypocenterC61#34530
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
36. C61⋊#564C4semidirect
37. C61⋊#564C4semidirect
38. C61⋊#564C4semidirect
42. C61×#564C4semidirect
193. #564#6923semidirect
225. C61#34530semidirect

Maximal subgroups

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

subgroupindex
#214623652
#214623542
#214623532
#214623192
#214623132
#125×#69232
#564×D612
#3453061