Finite Group Databasebeta

← #29682306

Subgroups — #29682306

Order 15616.

1178 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×#16839789C211yesmaximal, characteristic, complemented
2C2×#16839730C211yesmaximal, characteristic
3C2×#16839728C211yesmaximal, characteristic
4C22×C8×Dic61C211yesmaximal, characteristic, complemented
5#21448307C211yesmaximal, characteristic
6#21446426C218yesmaximal, complemented
7#21445512C211yesmaximal, characteristic, complemented
8C61×#1729C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12C22×C4×Dic61C2211yescharacteristic
13#16839789—24no—
14C2×C4⋊#92418C2214yes—
15#16839730—24no—
16#16839728—24no—
17C2×#14519689C2214yes—
18C2×#14519689C2214yes—
19C2×C8×Dic61—24no—
20C2×C8×Dic61C2214yes—
21#16839667C412yes—
22#16839663C412yes—
23#16839639C414yes—
24C4×#11758198C2214yes—
25#16839516C2214yes—
26C22×M16×C61C2211yescharacteristic
27C23×C8×C61C2211yescharacteristic
28C2×#117×C61C2211yescharacteristic
29C61×#336C2214yes—
30C2×#11758222—22no—
31C2×#11758222—22no—
32C2×#11758222—22no—
33C2×#11758222—22no—
34C2×#11758222—22no—
35C2×#11758222—22no—
36C22×Dic122C2×C411yescharacteristic
37C22×D244C2×C411yescharacteristic
38C22×C22⋊D61C2×C412yes—
39C23×Dic61C2×C412yes—
40C22×C4×D61C2312yes—
41C2×#11758194—22no—
42C2×#11758194—22no—
43C2×#11758194—22no—
44C22×#92418—22no—
45C22×#92418—22no—
46C22×#92418D414yes—
47C22×#92418C2312yes—
48C22×#92418C2312yes—
49C2×#11758198C2×C418yes—
50C2×C4×Dic61—22no—
51C2×C4×Dic61—22no—
52C2×C4×Dic61C2×C412yes—
53C2×C4×Dic61C2312yes—
54C4⋊#92418—28no—
55#14519689—28no—
56#14519689—28no—
57C8×Dic61—28no—
58#14519684C2×C418yes—
59C23×C4×C61C2311yescharacteristic
60C2×M16×C61—22no—
61C2×M16×C61—22no—
62C2×M16×C61—22no—
63C22×C8×C61—22no—
64C22×C8×C61—22no—
65C22×C8×C61D414yes—
66C22×C8×C61C2312yes—
67C22×C8×C61C2312yes—
68#117×C61—22no—
69#117×C61—22no—
70C2×C4⋊C4×C61C2×C412yes—
71C2×C22⋊C4×C61C2×C412yes—
72C2×C42×C61C2312yes—
73C24×C61C22×C411yescharacteristic
74C23×D61C22×C412yes—
75C22×C4×C61C22×C414yes—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D412yes—
79C22×C4×C61#4112yes—
80C22×C4×C61#4112yes—
81C22×C4×C61#4112yes—
82C22×C4×C61#4112yes—
83C22×C4×C61C2411yescharacteristic
84C2×C22⋊D61—24no—
85C2×C22⋊D61—24no—
86C2×C22⋊D61—24no—
87C22×Dic61—24no—
88C22×Dic61—28no—
89C22×Dic61C22×C412yes—
90C22×Dic61C22×C414yes—
91#11758222—44no—
92#11758222—44no—
93#11758222—44no—
94#11758222—44no—
95C2×D244—22no—
96C2×D244—22no—
97C2×D244—22no—
98C2×C4×D61—22no—
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×Dic122—22no—
105C2×Dic122—22no—
106C2×Dic122—22no—
107M16×C61—44no—
108C2×C8×C61—28no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61C2×D414yes—
115C4⋊C4×C61—24no—
116C22⋊C4×C61—24no—
117C42×C61—22no—
118C42×C61—22no—
119#11758198—216no—
120C4×Dic61—24no—
121C4×Dic61—22no—
122C4×Dic61—22no—
123#11758194—44no—
124C2×#92418—28no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—22no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418C2×D414yes—
131C23×C61—24no—
132C23×C61#11712yes—
133C23×C61C4×D412yes—
134C23×C61C4×D412yes—
135C23×C61C23×C411yescharacteristic
136C22×D61—22no—
137C22×D61—22no—
138C22×D61—22no—
139C22×D61—28no—
140C2×C4×C61—24no—
141C2×C4×C61—28no—
142C2×C4×C61#11712yes—
143C2×C4×C61C4×D412yes—
144C2×C4×C61C4×D412yes—
145C2×C4×C61#12114yes—
146C2×C4×C61#12314yes—
147C2×C4×C61#12414yes—
148C2×C4×C61#12614yes—
149C2×C4×C61C22×D411yescharacteristic
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C2×#4111yescharacteristic
152C2×C4×C61C2×#4111yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C2×C4×C61C2×#4111yescharacteristic
155C22⋊D61—48no—
156C2×Dic61—216no—
157C2×Dic61—24no—
158C2×Dic61—24no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61#13118yes—
163D244—44no—
164C4×D61—42no—
165C4×D61—42no—
166C4×D61—42no—
167C4×D61—42no—
168Dic122—44no—
169C8×C61—24no—
170C8×C61—42no—
171#92418—24no—
172#92418—42no—
173#21046—611nomaximal, Sylow
174C22×C61—24no—
175C22×C61—24no—
176C22×C61—24no—
177C22×C61#38814yes—
178C22×C61C2×#11711yescharacteristic
179C22×C61C2×C4×D411yescharacteristic
180C22×C61C2×C4×D411yescharacteristic
181C22×C61C2×#13114yes—
182C2×D61—44no—
183C2×D61—44no—
184C2×D61—44no—
185C2×D61—42no—
186C4×C61—44no—
187C4×C61#38814yes—
188C4×C61C2×#12111yescharacteristic
189C4×C61C2×#12311yescharacteristic
190C4×C61C2×#12411yescharacteristic
191C4×C61C2×#12611yescharacteristic
192Dic61—28no—
193Dic61—42no—
194C2×#424—611no—
195C2×#408—611no—
196C2×#408—611no—
197C22×C4×C8—611no—
198#2297—611no—
199#1923—618no—
200#1729D6111yescharacteristic, complemented
201#1728—611no—
202C2×C61—24no—
203C2×C61C2×#38811yescharacteristic
204C2×C61#292212yes—
205C2×C61#292912yes—
206C2×C61#293112yes—
207D61—82no—
208C22×#41—611no—
209C22×M16—611no—
210C22×M16C2×D6111yescharacteristic
211C23×C8—611no—
212C23×C8C2×D6111yescharacteristic
213C2×#117C2×D6111yescharacteristic
214C22×C42—611no—
215#424—1224no—
216C2×C4⋊C8—614no—
217#408—1224no—
218#408—1224no—
219C2×C22⋊C8—614no—
220C2×C22⋊C8—614no—
221C2×C4×C8—614no—
222C2×C4×C8—1224no—
223#391—612no—
224#390—612no—
225#386—614no—
226C4×C22⋊C4—614no—
227#336—614no—
228#336C2×D6114yes—
229C61#2104611yescharacteristic, complemented, Sylow
230C2×#41—1222no—
231C2×#41—1222no—
232C2×#41—1222no—
233C2×#41—1222no—
234C2×#41—1222no—
235C2×#41—1222no—
236C22×Q8—611no—
237C22×D4—612no—
238C22×D4—611no—
239C23×C4—612no—
240C23×C4—612no—
241C23×C4C22×D6111yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—1222no—
246C2×M16—1222no—
247C2×M16—1222no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—614no—
251C22×C8—612no—
252C22×C8—612no—
253C22×C8—1222no—
254C22×C8—1222no—
255C22×C8D24414yes—
256C22×C8C22×D6112yes—
257C22×C8C22×D6112yes—
258#117—22no—
259#117—22no—
260C2×C4⋊C4C4×D6112yes—
261C2×C22⋊C4—618no—
262C2×C22⋊C4C4×D6112yes—
263C2×C42—612no—
264C2×C42—612no—
265C2×C42—1222no—
266C2×C42—1222no—
267C2×C42C22×D6112yes—
268C4⋊C8—1228no—
269C22⋊C8—1228no—
270C22⋊C8—1228no—
271C4×C8—1228no—
272#95—618no—
273C24—612no—
274C24C2×C4×D6111yescharacteristic
275#41—2444no—
276#41—2444no—
277#41—2444no—
278#41—2444no—
279C2×Q8—1222no—
280C2×Q8—1222no—
281C2×Q8—1222no—
282C2×D4—1224no—
283C2×D4—1224no—
284C2×D4—1224no—
285C2×D4—1222no—
286C2×D4—1222no—
287C2×D4—1222no—
288C22×C4—614no—
289C22×C4—612no—
290C22×C4—1228no—
291C22×C4—1224no—
292C22×C4—1222no—
293C22×C4—1222no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4C2×C4×D6114yes—
299C22×C4C2×C4×D6111yescharacteristic
300C22×C4C2×D24412yes—
301C22×C4#1175822212yes—
302C22×C4D4×D6112yes—
303C22×C4#1175822412yes—
304C22×C4#1175822412yes—
305C22×C4#1175822612yes—
306C22×C4C23×D6111yescharacteristic
307M16—44no—
308M16—2444no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—28no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—614no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—1228no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8C2×D24414yes—
323C4⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—12216no—
326C42—22no—
327C42—22no—
328C42—1222no—
329C42—1222no—
330C42—1224no—
331C23—24no—
332C23—1222no—
333C23—1222no—
334C23—1222no—
335C23—1228no—
336C23#1451973512yes—
337C23#1451973712yes—
338C23C4×D24412yes—
339C23C22×C4×D6111yescharacteristic
340Q8—2444no—
341D4—2448no—
342D4—2444no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—618no—
346C2×C4—12216no—
347C2×C4—1224no—
348C2×C4—1224no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4#1451973512yes—
357C2×C4#1451973712yes—
358C2×C4C4×D24412yes—
359C2×C4#1451975214yes—
360C2×C4#1451976114yes—
361C2×C4#1451976314yes—
362C2×C4#1451977214yes—
363C2×C4C2×D4×D6111yescharacteristic
364C2×C4C22×D24411yescharacteristic
365C2×C4C2×#1175822411yescharacteristic
366C2×C4C2×#1175822411yescharacteristic
367C2×C4C2×#1175822211yescharacteristic
368C2×C4C2×#1175822611yescharacteristic
369C8—24no—
370C8—42no—
371C8—1224no—
372C8—2442no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—2442no—
377C22—2444no—
378C22—2444no—
379C22—2444no—
380C22#1683965414yes—
381C22C2×#1451973511yescharacteristic
382C22C2×#1451973711yescharacteristic
383C22C2×C4×D24411yescharacteristic
384C22#131×D6114yes—
385C4—44no—
386C4—1228no—
387C4—2442no—
388C4#1683965414yes—
389C4C2×#1451975211yescharacteristic
390C4C2×#1451976111yescharacteristic
391C4C2×#1451976311yescharacteristic
392C4C2×#1451977211yescharacteristic
393C2—24no—
394C2—4882no—
395C2C2×#1683965411yescharacteristic
396C2#2145446212yes—
397C2#2145451512yes—
398C2#2145452712yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1729C2
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1729D61
hypocenterC61#21046
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
1. C2×#16839789C2semidirect
4. C22×C8×Dic61C2semidirect
6. #21446426C2semidirect
7. #21445512C2semidirect
8. C61×#1729C2semidirect
200. #1729D61semidirect
229. C61#21046semidirect

Maximal subgroups

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

subgroupindex
C2×#168397892
C2×#168397302
C2×#168397282
C22×C8×Dic612
#214483072
#214464262
#214455122
C61×#17292
#2104661