Finite Group Databasebeta

← #29600396

Subgroups — #29600396

Order 15616.

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

normal subgroups only

This group has 344 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#21456864C211yesmaximal, characteristic
2#21447509C211yesmaximal, characteristic, complemented
3#21447487C211yesmaximal, characteristic, complemented
4#21443995C211yesmaximal, characteristic, complemented
5#21443993C211yesmaximal, characteristic, complemented
6#21443976C211yesmaximal, characteristic, complemented
7C61×#1406C211yesmaximal, characteristic, complemented
8#16840366C411yescharacteristic, complemented
9#16840330C2211yescharacteristic
10#16840138C411yescharacteristic, complemented
11#16840127C411yescharacteristic, complemented
12#16840116C2211yescharacteristic
13#16839979—21no—
14#16839955—21no—
15#16839906—21no—
16#16839894—21no—
17#16839824C411yescharacteristic, complemented
18#16839745—21no—
19#99×D61—21no—
20#16839559C2211yescharacteristic
21#16839558—21no—
22#16839557C2211yescharacteristic
23#16839554—21no—
24C61×#496C2211yescharacteristic
25C61×#355C2211yescharacteristic
26C61×#354C2211yescharacteristic
27C2×#11758226D411yescharacteristic
28C2×D4×D61D411yescharacteristic
29C2×#11758199—21no—
30C2×#11758199—21no—
31M16×D61—41no—
32#14519774C2×C411yescharacteristic
33#14519773D411yescharacteristic
34#14519769C2×C411yescharacteristic, complemented
35#14519766D411yescharacteristic
36#14519758—41no—
37#14519750—41no—
38#14519749—41no—
39#14519748—21no—
40#14519747—41no—
41D4×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×SD16×C61—21no—
56C2×D8×C61—21no—
57#127×C61C2×C411yescharacteristic, complemented
58#124×C61C2311yescharacteristic
59#101×C61—21no—
60#99×C61—21no—
61#97×C61C2×C411yescharacteristic, complemented
62C23×D61—21no—
63C2×Q8×C61C2×D411yescharacteristic
64C2×D4×C61—21no—
65C2×D4×C61—41no—
66C2×D4×C61C2×D411yescharacteristic
67C2×C22⋊D61—21no—
68C2×C22⋊D61—41no—
69C2×C22⋊D61—41no—
70C22×Dic61—41no—
71#11758226—81no—
72D4×D61—81no—
73C2×D244—21no—
74C2×D244—41no—
75C2×D244C22⋊C411yescharacteristic
76C2×D244C22⋊C411yescharacteristic
77C2×C4×D61—21no—
78C2×C4×D61—81no—
79C2×C4×D61C22⋊C411yescharacteristic
80SD16×C61—81no—
81D8×C61—81no—
82M16×C61—41no—
83C2×C8×C61—21no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C42×C61C22×C411yescharacteristic
87C22⋊Dic61—41no—
88C22⋊Dic61—41no—
89C22⋊Dic61—41no—
90#11758199—81no—
91#11758199—81no—
92#11758198—21no—
93#11758198—41no—
94#11758198—81no—
95C4⋊Dic61—21no—
96#11758196—21no—
97#11758196—41no—
98C4×Dic61C22⋊C411yescharacteristic
99#11758194—41no—
100C2×#92418—21no—
101#11758189—81no—
102C8×D61—81no—
103C23×C61—21no—
104C23×C61—41no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—161no—
110C22×D61#9911yescharacteristic
111Q8×C61—41no—
112D4×C61—41no—
113D4×C61—41no—
114D4×C61—41no—
115D4×C61—161no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—81no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C22⋊D61—81no—
121C22⋊D61—161no—
122C22⋊D61—161no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—81no—
126C2×Dic61—161no—
127C2×Dic61#9911yescharacteristic
128D244—41no—
129D244—41no—
130D244—81no—
131D244—161no—
132C4×D61—41no—
133C4×D61—41no—
134C4×D61—41no—
135C4×D61—162no—
136C8×C61—41no—
137C8×C61—81no—
138#92418—41no—
139#92418—81no—
140#13605—611nomaximal, Sylow
141C22×C61—41no—
142C22×C61—81no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61—81no—
146C22×C61C2×#9911yescharacteristic
147C2×D61—21no—
148C2×D61—41no—
149C2×D61—41no—
150C2×D61—41no—
151C2×D61—82no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—161no—
155C4×C61—21no—
156C4×C61—41no—
157C4×C61—41no—
158C4×C61—161no—
159Dic61—21no—
160Dic61—41no—
161Dic61—81no—
162Dic61—161no—
163#3220—611no—
164#2134—611no—
165#2127—611no—
166#1414—611no—
167#1414—611no—
168#1406—611no—
169#1406D6111yescharacteristic, complemented
170C2×C61—21no—
171C2×C61—81no—
172C2×C61—161no—
173C2×C61#212711yescharacteristic
174D61—41no—
175D61—81no—
176D61—81no—
177#540—611no—
178#535—611no—
179#496C2×D6111yescharacteristic
180#493—611no—
181#490—611no—
182#488—611no—
183#463—1221no—
184#459—1221no—
185#449—1221no—
186#447—1221no—
187#433—611no—
188#413—1221no—
189C2×#99—1221no—
190#355—611no—
191#355—1221no—
192#355C2×D6111yescharacteristic
193#354—611no—
194#354C2×D6111yescharacteristic
195#353—1221no—
196C61#1360511yescharacteristic, complemented, Sylow
197C2×#41—611no—
198C22×D4—611no—
199C2×SD16—21no—
200C2×D8—21no—
201C2×D8—1221no—
202C2×D8—1221no—
203C2×M16—2441no—
204#127—611no—
205#127—611no—
206#127C4×D6111yescharacteristic, complemented
207#125—611no—
208#124—611no—
209#124C22×D6111yescharacteristic
210#123—2441no—
211#121—1221no—
212#121—2441no—
213#121—2441no—
214#120—2441no—
215C4×D4—1221no—
216#117—611no—
217C2×C22⋊C4—2441no—
218C4⋊C8—1221no—
219#103—1221no—
220#102—1221no—
221#102—1221no—
222#102—1221no—
223#101—21no—
224#101—1221no—
225#100—2441no—
226#99—21no—
227#99—1221no—
228#99—2441no—
229C22⋊C8—1221no—
230#97C4×D6111yescharacteristic, complemented
231C4×C8—611no—
232C24—1221no—
233#41—4881no—
234C2×Q8D4×D6111yescharacteristic
235C2×D4—21no—
236C2×D4—41no—
237C2×D4—611no—
238C2×D4—611no—
239C2×D4—1221no—
240C2×D4—1221no—
241C2×D4—4881no—
242C2×D4—2441no—
243C2×D4—2441no—
244C2×D4—2441no—
245C2×D4D4×D6111yescharacteristic
246C22×C4—611no—
247C22×C4—1221no—
248C22×C4—4881no—
249C22×C4—2441no—
250SD16—81no—
251D8—81no—
252D8—4881no—
253D8—4881no—
254M16—41no—
255M16—4881no—
256M16—2441no—
257C2×C8—21no—
258C2×C8—1221no—
259C2×C8—4881no—
260C4⋊C4—1221no—
261C4⋊C4—1221no—
262C4⋊C4—2441no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—1221no—
266C22⋊C4—4881no—
267C22⋊C4—2441no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C42—611no—
272C42C2×C4×D6111yescharacteristic
273C23—21no—
274C23—41no—
275C23—611no—
276C23—1221no—
277C23—1221no—
278C23—9761no—
279C23—2441no—
280C23—2441no—
281Q8—41no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—161no—
286D4—9761no—
287D4—9761no—
288D4—9761no—
289D4—4881no—
290D4—4881no—
291D4—2441no—
292D4—2441no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—611no—
297C2×C4—1221no—
298C2×C4—9762no—
299C2×C4—9761no—
300C2×C4—4881no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4C22⋊C4×D6111yescharacteristic
306C8—41no—
307C8—81no—
308C8—4881no—
309C8—2441no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—1221no—
316C22—9761no—
317C22—4881no—
318C22—4881no—
319C22—4882no—
320C22—2441no—
321C22—2441no—
322C22—2441no—
323C22#99×D6111yescharacteristic
324C4—21no—
325C4—41no—
326C4—41no—
327C4—161no—
328C4—1221no—
329C4—9761no—
330C4—4881no—
331C4—2441no—
332C2—21no—
333C2—81no—
334C2—161no—
335C2—4881no—
336C2—4881no—
337C2—2441no—
338C2#2144748511yescharacteristic
subgroupG/H
centerC2#21447485
commutatorC42×C61C22×C4
Frattini#124C22×D61
FittingC61×#1406C2
socleC2×C61#2127
prime radical#21456864C2
cosocle#124×C61C23
hypercenter#1406D61
hypocenterC61#13605
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #1406

Complements

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

NH ≅ G/Ntype
2. #21447509C2semidirect
3. #21447487C2semidirect
4. #21443995C2semidirect
5. #21443993C2semidirect
6. #21443976C2semidirect
7. C61×#1406C2semidirect
8. #16840366C4semidirect
10. #16840138C4semidirect
11. #16840127C4semidirect
17. #16839824C4semidirect
34. #14519769C2×C4semidirect
54. C4×#92418C2×C4semidirect
57. #127×C61C2×C4semidirect
61. #97×C61C2×C4semidirect
169. #1406D61semidirect
196. C61#13605semidirect
206. #127C4×D61semidirect
230. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214568642
#214475092
#214474872
#214439952
#214439932
#214439762
C61×#14062
#1360561