Finite Group Databasebeta

← #29591027

Subgroups — #29591027

Order 15616.

776 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 99 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#21449175C211yesmaximal, characteristic
2#21444140C211yesmaximal, characteristic, complemented
3#21443815C214yesmaximal, complemented
4C61×#1799C211yesmaximal, characteristic, complemented
5C23×D244C411yescharacteristic, complemented
6C2×#14519751—21no—
7C2×#14519745—21no—
8#16839815—21no—
9#16839679C412yescomplemented
10#16839671C2212yes—
11#16839664—21no—
12#16839622C411yescharacteristic, complemented
13#16839617—21no—
14#16839486—21no—
15#16839486C2212yescomplemented
16C2×C4×D4×C61C2211yescharacteristic
17C2×#99×C61—21no—
18C61×#380—21no—
19C61×#342C2212yescomplemented
20C24×D61—21no—
21C22×D244—21no—
22C22×D244—44no—
23C22×D244—41no—
24C22×D244—41no—
25C22×D244C2×C412yes—
26C22×C22⋊D61—22no—
27#14519751—44no—
28#14519745—42no—
29#14519745—42no—
30C2×C4⋊Dic61—21no—
31C2×C22⋊Dic61—21no—
32C2×#11758198—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C4⋊#92418—41no—
36C22⋊#92418—22no—
37C4×#92418—41no—
38#14519684C2×C412yes—
39C22×D4×C61D411yescharacteristic
40C23×C4×C61C2×C411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C4×D4×C61—41no—
43C4×D4×C61—41no—
44C2×C4⋊C4×C61—41no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61D411yescharacteristic
48C2×C22⋊C4×C61C2311yescharacteristic
49C2×C42×C61D411yescharacteristic
50#99×C61—42no—
51#95×C61—41no—
52C24×C61C22⋊C411yescharacteristic
53C24×C61C2×D411yescharacteristic
54C23×D61—22no—
55C23×D61—44no—
56C23×D61—42no—
57C23×D61—42no—
58C23×D61—42no—
59C23×D61—82no—
60C2×D4×C61—41no—
61C2×D4×C61—41no—
62C2×D4×C61D812yes—
63C2×D4×C61SD1612yes—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C2×C22⋊D61—44no—
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—82no—
80C22×Dic61—22no—
81C2×D244—24no—
82C2×D244—42no—
83C2×D244—42no—
84C2×D244—48no—
85C2×D244—44no—
86C2×D244—41no—
87C2×D244—41no—
88C2×D244—44no—
89C2×D244—44no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—82no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—42no—
95C22⋊C4×C61—42no—
96C42×C61—21no—
97C42×C61—21no—
98C22⋊Dic61—42no—
99#11758198—44no—
100#11758198—44no—
101#11758198—82no—
102C4⋊Dic61—42no—
103C2×#92418—42no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10212yes—
113C23×C61C2×C22⋊C411yescharacteristic
114C23×C61C2×D811yescharacteristic
115C23×C61C2×SD1611yescharacteristic
116C22×D61—24no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—42no—
120C22×D61—48no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—88no—
125C22×D61—88no—
126C22×D61—84no—
127C22×D61—84no—
128D4×C61—42no—
129D4×C61—42no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—22no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—81no—
143C2×C4×C61—82no—
144C2×C4×C61—82no—
145C2×C4×C61—82no—
146C2×C4×C61#9911yescharacteristic
147C2×C4×C61#9911yescharacteristic
148C2×C4×C61#10212yes—
149C22⋊D61—84no—
150C22⋊D61—84no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154D244—44no—
155D244—44no—
156D244—44no—
157D244—44no—
158#92418—82no—
159#11873—611nomaximal, Sylow
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—21no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—81no—
174C22×C61C2×#9911yescharacteristic
175C22×C61C2×#10211yescharacteristic
176C22×C61#42111yescharacteristic
177C2×D61—44no—
178C2×D61—44no—
179C2×D61—44no—
180C2×D61—82no—
181C2×D61—82no—
182C2×D61—816no—
183C2×D61—82no—
184C2×D61—82no—
185C4×C61—21no—
186C4×C61—22no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—41no—
190C4×C61—162no—
191C4×C61AΣL(1,𝔽16)12yes—
192Dic61—82no—
193#2408—611no—
194#1799D6111yescharacteristic, complemented
195#1460—611no—
196#1343—614no—
197C2×C61—21no—
198C2×C61—21no—
199C2×C61—41no—
200C2×C61—42no—
201C2×C61—41no—
202C2×C61#151911yescharacteristic
203C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
204C2×C61#212811yescharacteristic
205D61—84no—
206C23×D4—611no—
207C2×#121—1221no—
208C2×#120—1221no—
209C2×C4×D4C2×D6111yescharacteristic
210#431—1221no—
211C2×#99—21no—
212#394—612no—
213#392—612no—
214#390—1221no—
215#380—21no—
216#380—611no—
217#379—1221no—
218#342C2×D6112yescomplemented
219#323—612no—
220#323—1221no—
221C61#1187311yescharacteristic, complemented, Sylow
222C25—1221no—
223C22×D4—612no—
224C22×D4—1222no—
225C22×D4—1221no—
226C22×D4—2441no—
227C22×D4—2441no—
228C22×D4—2444no—
229C22×D4C22⋊D6111yescharacteristic
230C23×C4C4×D6111yescharacteristic
231C23×C4C22⋊D6111yescharacteristic
232#121—2444no—
233#120—2442no—
234#120—2442no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4C4×D6111yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1221no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4D24411yescharacteristic
246C2×C22⋊C4C22×D6111yescharacteristic
247C2×C42D24411yescharacteristic
248C4⋊C8—2441no—
249#99—42no—
250C22⋊C8—1222no—
251C4×C8—2441no—
252#95—41no—
253#95—612no—
254C24—1222no—
255C24—4882no—
256C24—2442no—
257C24—2442no—
258C24—2444no—
259C24—2442no—
260C24#1175819811yescharacteristic
261C24C2×C22⋊D6111yescharacteristic
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—1224no—
265C2×D4—4882no—
266C2×D4—2442no—
267C2×D4—2442no—
268C2×D4—2448no—
269C2×D4—2444no—
270C2×D4—2444no—
271C2×D4—2444no—
272C2×D4—2444no—
273C2×D4—2442no—
274C2×D4—2442no—
275C2×D4—2441no—
276C2×D4—2441no—
277C2×D4#1175819912yes—
278C2×D4#1175820112yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—1222no—
287C22×C4#1175819811yescharacteristic
288C22×C4#1175819811yescharacteristic
289C22×C4#1175819811yescharacteristic
290C22×C4C2×C4×D6111yescharacteristic
291C22×C4C2×D24411yescharacteristic
292C2×C8—2442no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—2442no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—4882no—
301C22⋊C4—2444no—
302C22⋊C4—2442no—
303C22⋊C4—2444no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—1224no—
315C23—4884no—
316C23—4884no—
317C23—4888no—
318C23—4888no—
319C23—2442no—
320C23—2442no—
321C23—2442no—
322C23—2448no—
323C23—2442no—
324C23—2442no—
325C23—2442no—
326C23#1451969712yes—
327C23C2×#1175819811yescharacteristic
328C23C2×#1175819911yescharacteristic
329C23C2×#1175820111yescharacteristic
330D4—42no—
331D4—42no—
332D4—4884no—
333D4—4884no—
334D4—2444no—
335D4—2444no—
336D4—2444no—
337D4—2444no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4#1451969111yescharacteristic
358C2×C4#1451969111yescharacteristic
359C2×C4#1451969712yes—
360C8—4882no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—4882no—
376C22—4882no—
377C22—48816no—
378C22—4882no—
379C22—4882no—
380C22—2444no—
381C22—2444no—
382C22—2444no—
383C22C2×#1451969111yescharacteristic
384C22C2×#1451969711yescharacteristic
385C22#1683978011yescharacteristic
386C4—21no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—162no—
392C4—4882no—
393C4#1683954812yes—
394C2—21no—
395C2—21no—
396C2—41no—
397C2—42no—
398C2—41no—
399C2—4884no—
400C2#2144438611yescharacteristic
401C2C2×#1683954811yescharacteristic
402C2#2144749011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radical#21449175C2
cosocleC2×C22⋊C4×C61C23
hypercenter#1799D61
hypocenterC61#11873
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21444140C2semidirect
3. #21443815C2semidirect
4. C61×#1799C2semidirect
5. C23×D244C4semidirect
9. #16839679C4semidirect
12. #16839622C4semidirect
15. #16839486C22semidirect
19. C61×#342C22semidirect
194. #1799D61semidirect
218. #342C2×D61semidirect
221. C61#11873semidirect

Maximal subgroups

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

subgroupindex
#214491752
#214441402
#214438152
C61×#17992
#1187361