Finite Group Databasebeta

← #30604308

Subgroups — #30604308

Order 15616.

616 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461076C211yesmaximal, characteristic, complemented
2#21461056C211yesmaximal, characteristic, complemented
3#21460747C214yesmaximal, complemented
4#21450766C211yesmaximal, characteristic
5#16840733—21no—
6D61⋊(C22⋊C8)—21no—
7#16840607C2212yes—
8#16840607C2212yes—
9#16840606—21no—
10#16840606—21no—
11C2×#14519773C412yes—
12C2×#14519748—21no—
13C2×#14519748—21no—
14C2×C4×D244C2211yescharacteristic
15#16839670—21no—
16#16839670—21no—
17#16839665C2212yes—
18C4×C4⋊Dic61C411yescharacteristic
19C2×#127×C61C411yescharacteristic
20C2×#11758213—41no—
21C2×D61⋊C8—41no—
22#14519834—41no—
23#14519834—41no—
24C22⋊#92420—22no—
25C22⋊#92420—22no—
26C22×D244C2×C411yescharacteristic
27C22×C22⋊D61—22no—
28C22×C22⋊D61—22no—
29C22×C4×D61D411yescharacteristic
30C22×C4×D61D411yescharacteristic
31#14519773—44no—
32#14519748—44no—
33#14519748—44no—
34C4×D244—41no—
35C4×D244—41no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2311yescharacteristic
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×#11758198D411yescharacteristic
41C2×#11758198D411yescharacteristic
42C2×C4×Dic61C2×C412yes—
43#14519684—21no—
44C22×D4×C61—21no—
45C22×D4×C61—21no—
46C22×D4×C61C2×C412yes—
47#127×C61—41no—
48#127×C61—41no—
49C2×C42×C61C2×C411yescharacteristic
50C24×C61—22no—
51C23×D61C22⋊C411yescharacteristic
52C23×D61C22⋊C411yescharacteristic
53C2×D4×C61—24no—
54C2×D4×C61—42no—
55C2×D4×C61—42no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C2×D4×C61—42no—
59C2×D4×C61—41no—
60C2×D4×C61—41no—
61C2×D4×C61—82no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C2×C22⋊D61—44no—
66C2×C22⋊D61—44no—
67C2×C22⋊D61—44no—
68C2×C22⋊D61—44no—
69C2×C22⋊D61—82no—
70C2×C22⋊D61—82no—
71C22×Dic61—22no—
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×D244—22no—
75C2×D244—22no—
76C2×D244—41no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—42no—
82C2×C4×D61—42no—
83C2×#92420—42no—
84C2×#92420—42no—
85#11758213—82no—
86D61⋊C8—82no—
87C42×C61—21no—
88C42×C61—21no—
89C22⋊Dic61—44no—
90C22⋊Dic61—44no—
91#11758198—41no—
92#11758198—41no—
93C4⋊Dic61—24no—
94C4⋊Dic61—41no—
95C4×Dic61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61C2×C22⋊C411yescharacteristic
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61—41no—
112C22×D61—41no—
113C22×D61#10012yes—
114D4×C61—84no—
115D4×C61—82no—
116D4×C61—84no—
117D4×C61—82no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—22no—
121C2×C4×C61—21no—
122C2×C4×C61—41no—
123C2×C4×C61#9911yescharacteristic
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#13112yes—
126C22⋊D61—88no—
127C22⋊D61—88no—
128C2×Dic61—24no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135D244—44no—
136C4×D61—42no—
137C4×D61—42no—
138C4×D61—81no—
139C4×D61—81no—
140#92420—82no—
141#92420—82no—
142#11563—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—82no—
146C22×C61—82no—
147C22×C61—82no—
148C22×C61—82no—
149C22×C61—82no—
150C22×C61—82no—
151C22×C61—82no—
152C22×C61#40811yescharacteristic
153C22×C61C2×#9911yescharacteristic
154C22×C61C2×#10011yescharacteristic
155C2×D61—22no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—42no—
159C2×D61—42no—
160C2×D61—41no—
161C2×D61—41no—
162C2×D61—41no—
163C2×D61—81no—
164C2×D61—81no—
165C4×C61—21no—
166C4×C61—22no—
167C4×C61—41no—
168C4×C61#35312yes—
169Dic61—44no—
170Dic61—41no—
171Dic61—41no—
172#2541—611no—
173#1465—611no—
174#1459—611no—
175#1318—614no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61—162no—
179C2×C61#147111yescharacteristic
180C2×C61C2×#35311yescharacteristic
181C2×C61#213411yescharacteristic
182D61—42no—
183D61—81no—
184C2×#127—612no—
185C2×#127#692311yescharacteristic
186C2×#121—1221no—
187C2×#121—1221no—
188C2×C4×D4—611no—
189#407—1221no—
190C2×C22⋊C8—1221no—
191#392—1221no—
192#392—1221no—
193#390—612no—
194C4×C4⋊C4—611no—
195#324—612no—
196#324—612no—
197#323—1221no—
198#323—1221no—
199C61#1156311yescharacteristic, complemented, Sylow
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—611no—
203C22×D4—1222no—
204C22×D4—1222no—
205C22×D4C2×#692312yes—
206C23×C4—611no—
207C23×C4—611no—
208C2×M16—2441no—
209C22×C8—2441no—
210#127—41no—
211#127—41no—
212#127—2444no—
213#121—2444no—
214#121—2444no—
215C4×D4—2441no—
216C4×D4—2441no—
217C2×C4⋊C4—611no—
218C2×C4⋊C4—1221no—
219C2×C22⋊C4—611no—
220C2×C22⋊C4—611no—
221C2×C22⋊C4—1222no—
222C2×C22⋊C4—1222no—
223C2×C42—612no—
224C2×C42C2×#692311yescharacteristic
225C22⋊C8—1222no—
226C22⋊C8—1222no—
227C22⋊C8—2441no—
228C22⋊C8—2441no—
229#95—1221no—
230C24—22no—
231C24—611no—
232C24—611no—
233C2×D4—24no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—82no—
242C2×D4—1222no—
243C2×D4—1222no—
244C2×D4—4882no—
245C2×D4—4882no—
246C2×D4—2441no—
247C2×D4—2444no—
248C2×D4—2444no—
249C2×D4—2444no—
250C2×D4—2444no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—1222no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—2442no—
259C22×C4—2442no—
260C22×C4C22⋊#692311yescharacteristic
261C22×C4C22⋊#692311yescharacteristic
262C22×C4C22×#692311yescharacteristic
263M16—4882no—
264C2×C8—4882no—
265C2×C8—2442no—
266C2×C8—2442no—
267C4⋊C4—1224no—
268C4⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C22⋊C4—2444no—
272C22⋊C4—2444no—
273C42—21no—
274C42—21no—
275C42—2442no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—612no—
284C23—1221no—
285C23—1221no—
286C23—1221no—
287C23—1221no—
288C23—1221no—
289C23—2441no—
290C23—2441no—
291C23—2441no—
292C23—2441no—
293C23C2×C22⋊#692311yescharacteristic
294D4—84no—
295D4—82no—
296D4—84no—
297D4—82no—
298D4—4888no—
299D4—4888no—
300D4—2444no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—1224no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—4881no—
310C2×C4—4881no—
311C2×C4—2442no—
312C2×C4—2442no—
313C2×C4—2442no—
314C2×C4—2442no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4#1451983511yescharacteristic
318C2×C4#1451983511yescharacteristic
319C2×C4#1451986412yes—
320C8—4882no—
321C8—4882no—
322C22—21no—
323C22—21no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—1222no—
332C22—4881no—
333C22—4881no—
334C22—2442no—
335C22—2442no—
336C22—2441no—
337C22—2441no—
338C22—2441no—
339C22—2441no—
340C22—2441no—
341C22#1684073611yescharacteristic
342C22C2×#1451983511yescharacteristic
343C22D61⋊#10011yescharacteristic
344C4—21no—
345C4—22no—
346C4—41no—
347C4—2444no—
348C4—2441no—
349C4—2441no—
350C4#1684066612yes—
351C2—21no—
352C2—21no—
353C2—162no—
354C2—4881no—
355C2—2442no—
356C2#2146109411yescharacteristic
357C2C2×#1684066611yescharacteristic
358C2#2146211811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC22×C61C2×#99
prime radical#21450766C2
cosocleC2×C4⋊Dic61C23
hypercenterC2×#127#6923
hypocenterC61#11563
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461076C2semidirect
2. #21461056C2semidirect
3. #21460747C2semidirect
199. C61#11563semidirect

Maximal subgroups

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

subgroupindex
#214610762
#214610562
#214607472
#214507662
#1156361