Finite Group Databasebeta

← #29600406

Subgroups — #29600406

Order 15616.

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

normal subgroups only

This group has 332 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#21456886C211yesmaximal, characteristic
2#21447509C211yesmaximal, characteristic
3#21447487C211yesmaximal, characteristic
4#21443995C211yesmaximal, characteristic, complemented
5#21443993C211yesmaximal, characteristic, complemented
6#21443979C211yesmaximal, characteristic, complemented
7C61×#1407C211yesmaximal, characteristic, complemented
8#16840373C411yescharacteristic, complemented
9#16840330C2211yescharacteristic
10#16840136C411yescharacteristic, complemented
11#16840127C411yescharacteristic, complemented
12#16840116C2211yescharacteristic
13#16839986—21no—
14#16839960—21no—
15#16839923—21no—
16#16839911—21no—
17#16839824C411yescharacteristic, complemented
18#16839745—21no—
19#99×D61—21no—
20#16839559C2211yescharacteristic
21#16839558—21no—
22#16839557C2211yescharacteristic
23#16839554—21no—
24C61×#497C2211yescharacteristic
25C61×#355C2211yescharacteristic
26C61×#354C2211yescharacteristic
27C2×#11758226D411yescharacteristic
28C2×D4×D61D411yescharacteristic
29C2×#11758201—21no—
30C2×#11758201—21no—
31M16×D61—41no—
32#14519774C2×C411yescharacteristic
33#14519773D411yescharacteristic
34#14519772—41no—
35#14519769C2×C411yescharacteristic, complemented
36#14519767—41no—
37#14519766D411yescharacteristic
38#14519753—21no—
39#14519750—41no—
40#14519747—41no—
41Q8×Dic61—21no—
42#14519734C2×C411yescharacteristic
43C22⋊C4×D61—41no—
44#14519706—21no—
45#14519701—21no—
46#14519699—21no—
47#14519699—21no—
48#14519698—21no—
49#14519696—21no—
50#14519693—41no—
51#14519692—21no—
52#14519691—41no—
53#14519689—21no—
54C4×#92418C2×C411yescharacteristic, complemented
55C2×Q16×C61—21no—
56C2×SD16×C61—21no—
57C4⋊Q8×C61C2×C411yescharacteristic, complemented
58#124×C61C2311yescharacteristic
59#101×C61—21no—
60#99×C61—21no—
61#97×C61C2×C411yescharacteristic, complemented
62C23×D61—21no—
63C2×Q8×C61—21no—
64C2×Q8×C61C2×D411yescharacteristic
65C2×D4×C61C2×D411yescharacteristic
66C2×C22⋊D61—21no—
67C2×C22⋊D61—41no—
68#11758226—81no—
69D4×D61—81no—
70C2×D244—21no—
71C2×D244—41no—
72C2×D244C22⋊C411yescharacteristic
73C2×D244C22⋊C411yescharacteristic
74C2×C4×D61—21no—
75C2×C4×D61—81no—
76C2×C4×D61C22⋊C411yescharacteristic
77Q16×C61—81no—
78SD16×C61—81no—
79M16×C61—41no—
80C2×C8×C61—21no—
81C4⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C42×C61C22×C411yescharacteristic
85C22⋊Dic61—41no—
86#11758201—81no—
87#11758201—81no—
88#11758198—21no—
89#11758198—41no—
90#11758198—41no—
91#11758198—81no—
92C4⋊Dic61—21no—
93C4⋊Dic61—41no—
94#11758196—21no—
95#11758196—41no—
96#11758196—41no—
97C4×Dic61—41no—
98C4×Dic61C22⋊C411yescharacteristic
99#11758194—41no—
100C2×#92418—21no—
101#11758189—81no—
102C8×D61—81no—
103C23×C61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—161no—
109C22×D61#9911yescharacteristic
110Q8×C61—41no—
111Q8×C61—41no—
112Q8×C61—41no—
113D4×C61—41no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—81no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C22⋊D61—81no—
120C22⋊D61—161no—
121C2×Dic61—21no—
122C2×Dic61—41no—
123C2×Dic61—81no—
124C2×Dic61#9911yescharacteristic
125D244—41no—
126D244—41no—
127D244—81no—
128D244—161no—
129C4×D61—41no—
130C4×D61—41no—
131C4×D61—41no—
132C4×D61—162no—
133C8×C61—41no—
134C8×C61—81no—
135#92418—41no—
136#92418—81no—
137#13607—611nomaximal, Sylow
138C22×C61—41no—
139C22×C61—81no—
140C22×C61C2×#9911yescharacteristic
141C2×D61—21no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—41no—
145C2×D61—82no—
146C2×D61—81no—
147C2×D61—81no—
148C2×D61—161no—
149C4×C61—21no—
150C4×C61—41no—
151C4×C61—41no—
152C4×C61—81no—
153C4×C61—161no—
154Dic61—21no—
155Dic61—41no—
156Dic61—81no—
157Dic61—161no—
158#3222—611no—
159#2134—611no—
160#2127—611no—
161#1414—611no—
162#1414—611no—
163#1407—611no—
164#1407D6111yescharacteristic, complemented
165C2×C61—21no—
166C2×C61—81no—
167C2×C61#212711yescharacteristic
168D61—41no—
169D61—81no—
170D61—81no—
171#541—611no—
172#535—611no—
173#497C2×D6111yescharacteristic
174#492—611no—
175#490—611no—
176#488—611no—
177#464—1221no—
178#460—1221no—
179#452—1221no—
180#450—1221no—
181#433—611no—
182#413—1221no—
183C2×#99—1221no—
184#355—611no—
185#355—1221no—
186#355C2×D6111yescharacteristic
187#354—611no—
188#354C2×D6111yescharacteristic
189#353—1221no—
190C61#1360711yescharacteristic, complemented, Sylow
191C2×#41—611no—
192C22×D4—611no—
193C2×Q16—21no—
194C2×SD16—21no—
195C2×SD16—1221no—
196C2×SD16—1221no—
197C2×M16—2441no—
198C4⋊Q8C4×D6111yescharacteristic, complemented
199#127—611no—
200#127—611no—
201#126—2441no—
202#125—611no—
203#125—2441no—
204#124—611no—
205#124C22×D6111yescharacteristic
206C22⋊Q8—1221no—
207#121—2441no—
208#120—2441no—
209C4×Q8—1221no—
210#117—611no—
211C2×C22⋊C4—2441no—
212C4⋊C8—1221no—
213#103—1221no—
214#102—1221no—
215#102—1221no—
216#102—1221no—
217#101—21no—
218#101—1221no—
219#100—2441no—
220#99—21no—
221#99—1221no—
222#99—2441no—
223C22⋊C8—1221no—
224#97C4×D6111yescharacteristic, complemented
225C4×C8—611no—
226C24—1221no—
227#41—4881no—
228C2×Q8—21no—
229C2×Q8D4×D6111yescharacteristic
230C2×D4—611no—
231C2×D4—611no—
232C2×D4—1221no—
233C2×D4—1221no—
234C2×D4—4881no—
235C2×D4—2441no—
236C2×D4—2441no—
237C2×D4D4×D6111yescharacteristic
238C22×C4—611no—
239C22×C4—1221no—
240C22×C4—4881no—
241Q16—81no—
242SD16—81no—
243SD16—4881no—
244SD16—4881no—
245M16—41no—
246M16—4881no—
247M16—2441no—
248C2×C8—21no—
249C2×C8—1221no—
250C2×C8—4881no—
251C4⋊C4—41no—
252C4⋊C4—1221no—
253C4⋊C4—1221no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—1221no—
260C22⋊C4—4881no—
261C22⋊C4—2441no—
262C22⋊C4—2441no—
263C22⋊C4—2441no—
264C42—611no—
265C42—2441no—
266C42C2×C4×D6111yescharacteristic
267C23—21no—
268C23—611no—
269C23—1221no—
270C23—1221no—
271C23—9761no—
272C23—2441no—
273C23—2441no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277D4—41no—
278D4—9761no—
279D4—9761no—
280D4—4881no—
281D4—4881no—
282D4—2441no—
283D4—2441no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—611no—
289C2×C4—1221no—
290C2×C4—9762no—
291C2×C4—4881no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4C22⋊C4×D6111yescharacteristic
297C8—41no—
298C8—81no—
299C8—4881no—
300C8—2441no—
301C22—41no—
302C22—81no—
303C22—1221no—
304C22—9761no—
305C22—4881no—
306C22—4881no—
307C22—4882no—
308C22—2441no—
309C22—2441no—
310C22—2441no—
311C22#99×D6111yescharacteristic
312C4—21no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—161no—
317C4—1221no—
318C4—9761no—
319C4—4881no—
320C4—2441no—
321C2—21no—
322C2—81no—
323C2—4881no—
324C2—4881no—
325C2—2441no—
326C2#2144748511yescharacteristic
subgroupG/H
centerC2#21447485
commutatorC42×C61C22×C4
Frattini#124C22×D61
FittingC61×#1407C2
socleC2×C61#2127
prime radical#16840330C22
cosocle#124×C61C23
hypercenter#1407D61
hypocenterC61#13607
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #1407

Complements

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

NH ≅ G/Ntype
4. #21443995C2semidirect
5. #21443993C2semidirect
6. #21443979C2semidirect
7. C61×#1407C2semidirect
8. #16840373C4semidirect
10. #16840136C4semidirect
11. #16840127C4semidirect
17. #16839824C4semidirect
35. #14519769C2×C4semidirect
54. C4×#92418C2×C4semidirect
57. C4⋊Q8×C61C2×C4semidirect
61. #97×C61C2×C4semidirect
164. #1407D61semidirect
190. C61#13607semidirect
198. C4⋊Q8C4×D61semidirect
224. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214568862
#214475092
#214474872
#214439952
#214439932
#214439792
C61×#14072
#1360761