Finite Group Databasebeta

← #29600418

Subgroups — #29600418

Order 15616.

348 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 57 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#21456870C211yesmaximal, characteristic
2#21447489C211yesmaximal, characteristic, complemented
3#21447485C211yesmaximal, characteristic, complemented
4#21443990C211yesmaximal, characteristic, complemented
5#21443981C211yesmaximal, characteristic, complemented
6#21443981C211yesmaximal, characteristic, complemented
7C61×#1412C211yesmaximal, characteristic, complemented
8#16840368C411yescharacteristic, complemented
9#16840325C2211yescharacteristic
10#16840151C411yescharacteristic, complemented
11#16840138C2211yescharacteristic
12#16840119C411yescharacteristic, complemented
13#16839976—21no—
14#16839956—21no—
15#16839908—21no—
16#16839897—21no—
17#16839824C411yescharacteristic, complemented
18#16839734—21no—
19#99×D61—21no—
20#16839556—21no—
21#16839555C2211yescharacteristic
22#16839555C2211yescharacteristic
23#16839554—21no—
24C61×#490C2211yescharacteristic
25C61×#353C2211yescharacteristic
26C61×#353C2211yescharacteristic
27C2×#11758224D411yescharacteristic
28C2×D4×D61D411yescharacteristic
29C2×#11758190—21no—
30C2×D488—21no—
31C2×#11758199—21no—
32C2×#11758199—21no—
33#14519774C2×C411yescharacteristic, complemented
34#14519773D411yescharacteristic
35#14519765C2×C411yescharacteristic
36#14519764D411yescharacteristic
37#14519762—41no—
38#14519752—21no—
39#14519752—41no—
40#14519749—41no—
41#14519745—41no—
42#14519739—21no—
43#14519734C2×C411yescharacteristic
44#14519733—41no—
45C22⋊C4×D61—41no—
46#14519706—21no—
47#14519700—21no—
48#14519697—21no—
49#14519692—21no—
50#14519692—21no—
51#14519691—41no—
52#14519691—41no—
53#14519689—21no—
54C4×#92418C2×C411yescharacteristic, complemented
55#127×C61C2311yescharacteristic
56#125×C61C2×C411yescharacteristic, complemented
57#102×C61—21no—
58#102×C61—21no—
59#99×C61—21no—
60#99×C61—21no—
61#97×C61C2×C411yescharacteristic, complemented
62C23×D61—21no—
63C2×D4×C61—41no—
64C2×D4×C61C2×D411yescharacteristic
65C2×D4×C61C2×D411yescharacteristic
66C2×C22⋊D61—21no—
67C2×C22⋊D61—21no—
68C2×C22⋊D61—41no—
69C22×Dic61—21no—
70#11758224—81no—
71D4×D61—81no—
72C2×D244—21no—
73C2×D244—41no—
74C2×D244—41no—
75C2×D244C22⋊C411yescharacteristic
76C2×C4×D61—41no—
77C2×C4×D61—81no—
78C2×C4×D61C22⋊C411yescharacteristic
79C2×Dic122C22⋊C411yescharacteristic
80C2×C8×C61—21no—
81C4⋊C4×C61—21no—
82C4⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C42×C61C22×C411yescharacteristic
86C22⋊Dic61—41no—
87C22⋊Dic61—41no—
88C22⋊Dic61—41no—
89#11758199—81no—
90#11758199—81no—
91#11758198—21no—
92#11758198—41no—
93#11758198—41no—
94#11758198—41no—
95#11758198—81no—
96#11758196—21no—
97#11758196—81no—
98C4×Dic61—81no—
99C4×Dic61C22⋊C411yescharacteristic
100C2×#92418—21no—
101D488—81no—
102#11758190—81no—
103C23×C61—21no—
104C23×C61—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—161no—
110C22×D61#9911yescharacteristic
111D4×C61—41no—
112D4×C61—41no—
113D4×C61—161no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—81no—
117C2×C4×C61—81no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C22⋊D61—81no—
120C22⋊D61—81no—
121C22⋊D61—161no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—81no—
127C2×Dic61—81no—
128C2×Dic61#9911yescharacteristic
129D244—41no—
130D244—41no—
131D244—41no—
132D244—161no—
133D244—161no—
134C4×D61—41no—
135C4×D61—161no—
136C4×D61—162no—
137Dic122—41no—
138C8×C61—41no—
139#92418—41no—
140#13609—611nomaximal, Sylow
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61C2×#9911yescharacteristic
146C2×D61—21no—
147C2×D61—41no—
148C2×D61—41no—
149C2×D61—82no—
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—161no—
155C4×C61—21no—
156C4×C61—41no—
157C4×C61—81no—
158C4×C61—161no—
159C4×C61—161no—
160Dic61—21no—
161Dic61—41no—
162Dic61—41no—
163Dic61—161no—
164Dic61—161no—
165#3220—611no—
166#2128—611no—
167#2127—611no—
168#1412—611no—
169#1412D6111yescharacteristic, complemented
170#1408—611no—
171#1408—611no—
172C2×C61—21no—
173C2×C61—81no—
174C2×C61—81no—
175C2×C61#212711yescharacteristic
176D61—41no—
177D61—81no—
178D61—161no—
179#540—611no—
180#535—611no—
181#496—611no—
182#493—611no—
183#490C2×D6111yescharacteristic
184#488—611no—
185#463—1221no—
186#459—1221no—
187#449—1221no—
188#447—1221no—
189#433—611no—
190#410—1221no—
191C2×#99—1221no—
192#354—1221no—
193#353—611no—
194#353—611no—
195#353—1221no—
196#353C2×D6111yescharacteristic
197#353C2×D6111yescharacteristic
198C61#1360911yescharacteristic, complemented, Sylow
199C2×#41—611no—
200C22×D4—611no—
201C2×SD16—1221no—
202C2×D8—1221no—
203C2×D8—1221no—
204C2×D8—1221no—
205#127—611no—
206#127—611no—
207#127C22×D6111yescharacteristic
208#125C4×D6111yescharacteristic, complemented
209#124—611no—
210#124—611no—
211#123—2441no—
212#121—1221no—
213#121—2441no—
214#121—2441no—
215#120—2441no—
216C4×D4—1221no—
217#117—611no—
218#117—2441no—
219C2×C22⋊C4—2441no—
220C4⋊C8—1221no—
221#103—1221no—
222#102—21no—
223#102—21no—
224#102—1221no—
225#99—21no—
226#99—21no—
227#99—1221no—
228#99—1221no—
229#99—2441no—
230#99—2441no—
231C22⋊C8—1221no—
232#97C4×D6111yescharacteristic, complemented
233C4×C8—611no—
234C24—1221no—
235#41—4881no—
236C2×Q8—611no—
237C2×D4—41no—
238C2×D4—611no—
239C2×D4—1221no—
240C2×D4—1221no—
241C2×D4—1221no—
242C2×D4—4881no—
243C2×D4—2441no—
244C2×D4—2441no—
245C2×D4—2441no—
246C2×D4D4×D6111yescharacteristic
247C2×D4D4×D6111yescharacteristic
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—4881no—
251C22×C4—2441no—
252SD16—4881no—
253D8—4881no—
254D8—4881no—
255D8—4881no—
256C2×C8—21no—
257C2×C8—1221no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—1221no—
261C4⋊C4—4881no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—1221no—
265C22⋊C4—4881no—
266C22⋊C4—2441no—
267C22⋊C4—2441no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C42—611no—
273C42—4881no—
274C42C2×C4×D6111yescharacteristic
275C23—21no—
276C23—21no—
277C23—611no—
278C23—1221no—
279C23—9761no—
280C23—2441no—
281C23—2441no—
282C23—2441no—
283Q8—2441no—
284D4—41no—
285D4—41no—
286D4—161no—
287D4—9761no—
288D4—9761no—
289D4—9761no—
290D4—4881no—
291D4—4881no—
292D4—2441no—
293D4—2441no—
294D4—2441no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—611no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—9761no—
303C2×C4—9762no—
304C2×C4—4881no—
305C2×C4—4881no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4C22⋊C4×D6111yescharacteristic
310C8—41no—
311C8—2441no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—81no—
316C22—1221no—
317C22—9761no—
318C22—4881no—
319C22—4881no—
320C22—4881no—
321C22—4882no—
322C22—4881no—
323C22—2441no—
324C22—2441no—
325C22#99×D6111yescharacteristic
326C4—21no—
327C4—41no—
328C4—81no—
329C4—161no—
330C4—161no—
331C4—1221no—
332C4—9761no—
333C4—9761no—
334C4—2441no—
335C4—2441no—
336C2—21no—
337C2—81no—
338C2—81no—
339C2—9761no—
340C2—4881no—
341C2—2441no—
342C2#2144748711yescharacteristic
subgroupG/H
centerC2#21447487
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1412C2
socleC2×C61#2127
prime radical#21456870C2
cosocle#127×C61C23
hypercenter#1412D61
hypocenterC61#13609
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
2. #21447489C2semidirect
3. #21447485C2semidirect
4. #21443990C2semidirect
5. #21443981C2semidirect
6. #21443981C2semidirect
7. C61×#1412C2semidirect
8. #16840368C4semidirect
10. #16840151C4semidirect
12. #16840119C4semidirect
17. #16839824C4semidirect
33. #14519774C2×C4semidirect
54. C4×#92418C2×C4semidirect
56. #125×C61C2×C4semidirect
61. #97×C61C2×C4semidirect
169. #1412D61semidirect
198. C61#13609semidirect
208. #125C4×D61semidirect
232. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214568702
#214474892
#214474852
#214439902
#214439812
C61×#14122
#1360961