Finite Group Databasebeta

← #30601763

Subgroups — #30601763

Order 15616.

770 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461546C211yesmaximal, characteristic, complemented
2C2×#16840613C214yesmaximal, complemented
3#21460995C211yesmaximal, characteristic, complemented
4#21454161C211yesmaximal, characteristic
5C2×#14519834C2212yes—
6C2×#14519832—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840613—28no—
12C2×#14519748—21no—
13C2×#14519748—21no—
14C2×#14519748C2212yes—
15C2×C42×D61C2211yescharacteristic
16#16839665—21no—
17#16839665—21no—
18#16839665C412yescomplemented
19#16839632C411yescharacteristic, complemented
20C2×#127×C61C411yescharacteristic, complemented
21C23×#6923—22no—
22C22×#92420—22no—
23#14519834—22no—
24#14519834D414yescomplemented
25#14519832—42no—
26(C2×#6923)⋊C4—24no—
27(C2×#6923)⋊C4—24no—
28(C2×#6923)⋊C4—42no—
29C22×C22⋊D61—22no—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×C4×D61C2311yescharacteristic
33#14519748—24no—
34#14519748—24no—
35#14519748—44no—
36#14519748—44no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C412yes—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C4×Dic61C2×C411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46C42×D61—42no—
47C22×D4×C61—21no—
48C22×D4×C61—21no—
49C22×D4×C61C2×C412yes—
50#127×C61—42no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—22no—
53C23×D61C2×D411yescharacteristic
54C22×#6923—24no—
55C22×#6923—42no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×#6923—42no—
59C22×#6923—42no—
60C22×#6923—42no—
61C2×D4×C61—24no—
62C2×D4×C61—42no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C2×D4×C61—42no—
67C2×D4×C61—42no—
68C2×D4×C61—42no—
69C2×D4×C61—42no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—44no—
76C2×C22⋊D61—44no—
77C22×Dic61—22no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—41no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61C2×D412yes—
90C2×C4×D61#4112yes—
91C2×#92420—24no—
92C2×#92420—42no—
93C2×#92420—42no—
94C42×C61—22no—
95C22⋊Dic61—44no—
96C22⋊Dic61—44no—
97C22⋊Dic61—44no—
98C4⋊Dic61—24no—
99C4⋊Dic61—42no—
100C4⋊Dic61—42no—
101C4×Dic61—22no—
102C4×Dic61—41no—
103C4×Dic61—41no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—42no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61C2×C22⋊C411yescharacteristic
112C22×D61—21no—
113C22×D61—21no—
114C22×D61—21no—
115C22×D61—21no—
116C22×D61#12411yescharacteristic
117C22×D61#12711yescharacteristic
118C22×D61AGL(1,ℤ/8)12yes—
119C22×D61AGL(1,ℤ/8)12yes—
120C2×#6923—44no—
121C2×#6923—44no—
122C2×#6923—44no—
123C2×#6923—82no—
124C2×#6923—82no—
125C2×#6923—82no—
126C2×#6923—82no—
127D4×C61—84no—
128D4×C61—84no—
129D4×C61—84no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61#9912yes—
137C2×C4×C61#10412yes—
138C2×C4×C61C4×D412yes—
139C22⋊D61—84no—
140C22⋊D61—84no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61#12112yes—
153C4×D61—42no—
154C4×D61—42no—
155C4×D61—44no—
156C4×D61—44no—
157#92420—44no—
158#9684—611nomaximal, Sylow
159C22×C61—82no—
160C22×C61—82no—
161C22×C61—82no—
162C22×C61—82no—
163C22×C61—82no—
164C22×C61—82no—
165C22×C61—82no—
166C22×C61#39011yescharacteristic
167C22×C61C2×#9911yescharacteristic
168C22×C61#41011yescharacteristic
169C22×C61#41811yescharacteristic
170C22×C61#41811yescharacteristic
171C22×C61C2×#10411yescharacteristic
172C22×C61#42111yescharacteristic
173C2×D61—21no—
174C2×D61—21no—
175C2×D61—21no—
176C2×D61—21no—
177C2×D61—22no—
178C2×D61—22no—
179C2×D61—22no—
180C2×D61—22no—
181C2×D61#49012yes—
182C2×D61#49612yes—
183#6923—84no—
184C4×C61—22no—
185C4×C61—22no—
186C4×C61—22no—
187Dic61—22no—
188Dic61—41no—
189Dic61—41no—
190Dic61—82no—
191#2856—611no—
192#1749—611no—
193C2×#327—614no—
194#1434—611no—
195C2×C61—162no—
196C2×C61#150511yescharacteristic
197C2×C61#150711yescharacteristic
198C2×C61#151011yescharacteristic
199C2×C61#151411yescharacteristic
200C2×C61#196111yescharacteristic
201C2×C61#196411yescharacteristic
202C2×C61#198011yescharacteristic
203D61—22no—
204D61—22no—
205C2×#127#692311yescharacteristic, complemented
206C2×#121—612no—
207C2×#121—1221no—
208C2×#121—1221no—
209C22×C42—611no—
210C2×C22⋊C8—612no—
211C2×#97—1221no—
212#390—612no—
213#390—1221no—
214#390—1221no—
215#384—611no—
216C2×#95—612no—
217C2×#95—1221no—
218C2×#95—1221no—
219#336—1221no—
220#327—1228no—
221C61#968411yescharacteristic, complemented, Sylow
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—1222no—
225C22×D4C2×#692312yes—
226C23×C4—611no—
227C23×C4—611no—
228C23×C4—611no—
229C23×C4—1222no—
230C22×C8—1222no—
231#127—42no—
232#121—1224no—
233#121—1224no—
234#121—2444no—
235#121—2444no—
236C2×C4⋊C4—612no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C42—611no—
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—2442no—
246C2×C42C2×#692311yescharacteristic
247C22⋊C8—614no—
248C22⋊C8—1222no—
249#97—2442no—
250#95—1224no—
251#95—1224no—
252#95—2442no—
253C24—22no—
254C24—611no—
255C2×D4—24no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—2442no—
265C2×D4—2442no—
266C2×D4—2444no—
267C2×D4—2444no—
268C22×C4—611no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4—1222no—
277C22×C4—1224no—
278C22×C4—1222no—
279C22×C4—2441no—
280C22×C4—2441no—
281C22×C4—2441no—
282C22×C4—2441no—
283C22×C4—2442no—
284C22×C4—2442no—
285C22×C4—2442no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4C22⋊#692311yescharacteristic
290C22×C4C22⋊#692311yescharacteristic
291C22×C4C22×#692311yescharacteristic
292C2×C8—1224no—
293C2×C8—2442no—
294C2×C8—2442no—
295C4⋊C4—1224no—
296C4⋊C4—2442no—
297C4⋊C4—2442no—
298C22⋊C4—2444no—
299C22⋊C4—2444no—
300C22⋊C4—2444no—
301C42—22no—
302C42—1222no—
303C42—2441no—
304C42—2441no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—611no—
313C23—611no—
314C23—612no—
315C23—612no—
316C23—1221no—
317C23—1221no—
318C23—1221no—
319C23—1221no—
320C23C2×C22⋊#692311yescharacteristic
321D4—84no—
322D4—84no—
323D4—84no—
324D4—4884no—
325D4—4884no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—612no—
333C2×C4—1221no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—4882no—
342C2×C4—4882no—
343C2×C4—4882no—
344C2×C4—4882no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2444no—
348C2×C4—2444no—
349C2×C4—2444no—
350C2×C4—2444no—
351C2×C4—2444no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4#1451983512yes—
356C2×C4#1451984112yes—
357C2×C4D4×#692312yes—
358C8—2444no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—612no—
367C22—612no—
368C22—1221no—
369C22—1221no—
370C22—1222no—
371C22—1222no—
372C22—1221no—
373C22—1221no—
374C22—1222no—
375C22—1222no—
376C22#1684071411yescharacteristic
377C22C2×#1451983511yescharacteristic
378C22#1684074211yescharacteristic
379C22#1684075411yescharacteristic
380C22#1684075411yescharacteristic
381C22C2×#1451984111yescharacteristic
382C22#1684076011yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—1222no—
387C4—4884no—
388C4—4882no—
389C4—2441no—
390C4—2441no—
391C2—162no—
392C2—1222no—
393C2—1222no—
394C2#2146120811yescharacteristic
395C2#2146121511yescharacteristic
396C2#2146122411yescharacteristic
397C2#2146123911yescharacteristic
398C2#2146195111yescharacteristic
399C2#2146195811yescharacteristic
400C2#2146198311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC23×C61C2×C22⋊C4
prime radical#21454161C2
cosocleC22×C4×D61C23
hypercenterC2×#127#6923
hypocenterC61#9684
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461546C2semidirect
2. C2×#16840613C2semidirect
3. #21460995C2semidirect
18. #16839665C4semidirect
19. #16839632C4semidirect
20. C2×#127×C61C4semidirect
24. #14519834D4semidirect
205. C2×#127#6923semidirect
221. C61#9684semidirect

Maximal subgroups

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

subgroupindex
#214615462
C2×#168406132
#214609952
#214541612
#968461