Finite Group Databasebeta

← #29600401

Subgroups — #29600401

Order 15616.

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

normal subgroups only

This group has 320 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#21456875C211yesmaximal, characteristic
2#21447505C211yesmaximal, characteristic, complemented
3#21447492C211yesmaximal, characteristic, complemented
4#21443995C211yesmaximal, characteristic, complemented
5#21443993C211yesmaximal, characteristic, complemented
6#21443976C211yesmaximal, characteristic, complemented
7C61×#1406C211yesmaximal, characteristic, complemented
8#16840366C411yescharacteristic, complemented
9#16840332C2211yescharacteristic
10#16840134C411yescharacteristic, complemented
11#16840131C411yescharacteristic, complemented
12#16840116C2211yescharacteristic
13#16839975—21no—
14#16839965—21no—
15#16839904—21no—
16#16839899—21no—
17#16839824C411yescharacteristic, complemented
18#101×D61—21no—
19#16839734—21no—
20#16839560—21no—
21#16839559C2211yescharacteristic
22#16839557C2211yescharacteristic
23#16839556—21no—
24C61×#496C2211yescharacteristic
25C61×#355C2211yescharacteristic
26C61×#354C2211yescharacteristic
27C2×#11758224D411yescharacteristic
28C2×Q8×D61D411yescharacteristic
29C2×#11758200—21no—
30C2×#11758200—21no—
31M16×D61—41no—
32#14519777C2×C411yescharacteristic
33#14519775D411yescharacteristic
34#14519769C2×C411yescharacteristic
35#14519764D411yescharacteristic
36#14519759—41no—
37#14519758—41no—
38#14519754—41no—
39#14519749—41no—
40#14519748—21no—
41D4×Dic61—21no—
42#14519734C2×C411yescharacteristic
43#14519733—41no—
44#14519706—21no—
45#14519702—21no—
46#14519702—21no—
47#14519701—21no—
48#14519698—21no—
49#14519696—21no—
50#14519695—41no—
51#14519692—21no—
52#14519691—41no—
53#14519689—21no—
54C4×#92418C2×C411yescharacteristic
55C2×SD16×C61—21no—
56C2×D8×C61—21no—
57#127×C61C2×C411yescharacteristic
58#124×C61C2311yescharacteristic
59#101×C61—21no—
60#99×C61—21no—
61#97×C61C2×C411yescharacteristic
62C2×Q8×C61C2×D411yescharacteristic
63C2×D4×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61C2×D411yescharacteristic
66C2×C22⋊D61—21no—
67C2×C22⋊D61—41no—
68C22×Dic61—21no—
69C22×Dic61—41no—
70Q8×D61—81no—
71#11758224—81no—
72C2×C4×D61—21no—
73C2×C4×D61C22⋊C411yescharacteristic
74C2×Dic122—21no—
75C2×Dic122C22⋊C411yescharacteristic
76C2×Dic122C22⋊C411yescharacteristic
77SD16×C61—81no—
78D8×C61—81no—
79M16×C61—41no—
80C2×C8×C61—21no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C42×C61C22×C411yescharacteristic
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87C22⋊Dic61—41no—
88#11758200—81no—
89#11758200—81no—
90#11758198—21no—
91C4⋊Dic61—21no—
92C4⋊Dic61—41no—
93#11758196—21no—
94#11758196—41no—
95#11758196—41no—
96#11758196—81no—
97C4×Dic61—81no—
98C4×Dic61C22⋊C411yescharacteristic
99#11758194—41no—
100C2×#92418—21no—
101#11758189—81no—
102C8×D61—81no—
103C23×C61—21no—
104C23×C61—41no—
105C22×D61#9911yescharacteristic
106Q8×C61—41no—
107D4×C61—41no—
108D4×C61—41no—
109D4×C61—41no—
110D4×C61—161no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—81no—
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C22⋊D61—81no—
116C22⋊D61—161no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—81no—
124C2×Dic61—161no—
125C2×Dic61#9911yescharacteristic
126C4×D61—41no—
127C4×D61—41no—
128C4×D61—41no—
129Dic122—41no—
130Dic122—41no—
131Dic122—81no—
132C8×C61—41no—
133C8×C61—81no—
134#92418—41no—
135#92418—81no—
136#13606—611nomaximal, Sylow
137C22×C61—41no—
138C22×C61—81no—
139C22×C61—81no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61C2×#9911yescharacteristic
143C2×D61—21no—
144C2×D61—41no—
145C4×C61—21no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—161no—
149Dic61—21no—
150Dic61—41no—
151Dic61—41no—
152Dic61—41no—
153Dic61—81no—
154Dic61—161no—
155#3221—611no—
156#2133—611no—
157#2129—611no—
158#1414—611no—
159#1414—611no—
160#1406—611no—
161#1406D6111yescharacteristic, complemented
162C2×C61—21no—
163C2×C61—81no—
164C2×C61—161no—
165C2×C61#212711yescharacteristic
166D61—41no—
167#540—611no—
168#536—611no—
169#496C2×D6111yescharacteristic
170#492—611no—
171#491—611no—
172#488—611no—
173#463—1221no—
174#461—1221no—
175#449—1221no—
176#448—1221no—
177#433—611no—
178C2×#101—1221no—
179#410—1221no—
180#356—1221no—
181#355—611no—
182#355C2×D6111yescharacteristic
183#354—611no—
184#354—1221no—
185#354C2×D6111yescharacteristic
186C61#1360611yescharacteristic, complemented, Sylow
187C2×#41—611no—
188C22×Q8—611no—
189C2×SD16—21no—
190C2×SD16—1221no—
191C2×SD16—1221no—
192C2×D8—21no—
193C2×M16—2441no—
194C4⋊Q8—611no—
195C4⋊Q8—611no—
196#127C4×D6111yescharacteristic
197#125—611no—
198#124—611no—
199#124C22×D6111yescharacteristic
200#123—2441no—
201#123—2441no—
202C22⋊Q8—2441no—
203#121—1221no—
204#121—2441no—
205C4×D4—1221no—
206#117—611no—
207#117—2441no—
208C4⋊C8—1221no—
209#103—1221no—
210#103—1221no—
211#103—1221no—
212#102—1221no—
213#101—21no—
214#101—1221no—
215#101—2441no—
216#99—21no—
217#99—1221no—
218#99—2441no—
219C22⋊C8—1221no—
220#97C4×D6111yescharacteristic
221C4×C8—611no—
222#41—4881no—
223C2×Q8—611no—
224C2×Q8—611no—
225C2×Q8—1221no—
226C2×Q8—4881no—
227C2×Q8D4×D6111yescharacteristic
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—1221no—
231C2×D4—2441no—
232C2×D4D4×D6111yescharacteristic
233C22×C4—611no—
234C22×C4—1221no—
235C22×C4—1221no—
236C22×C4—2441no—
237SD16—81no—
238SD16—4881no—
239SD16—4881no—
240D8—81no—
241M16—41no—
242M16—4881no—
243M16—2441no—
244C2×C8—21no—
245C2×C8—1221no—
246C2×C8—4881no—
247C4⋊C4—1221no—
248C4⋊C4—1221no—
249C4⋊C4—4881no—
250C4⋊C4—2441no—
251C4⋊C4—2441no—
252C4⋊C4—2441no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—1221no—
256C22⋊C4—2441no—
257C22⋊C4—2441no—
258C22⋊C4—2441no—
259C22⋊C4—2441no—
260C42—611no—
261C42—4881no—
262C42C2×C4×D6111yescharacteristic
263C23—21no—
264C23—41no—
265C23—611no—
266Q8—41no—
267Q8—4881no—
268Q8—2441no—
269Q8—2441no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—161no—
274D4—9761no—
275D4—4881no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—81no—
279C2×C4—611no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—9761no—
284C2×C4—4881no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4C22⋊C4×D6111yescharacteristic
292C8—41no—
293C8—81no—
294C8—4881no—
295C8—2441no—
296C22—41no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—1221no—
302C22—2441no—
303C22#99×D6111yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—161no—
308C4—1221no—
309C4—9761no—
310C4—4881no—
311C4—2441no—
312C4—2441no—
313C4—2441no—
314C2—21no—
315C2—81no—
316C2—161no—
317C2—2441no—
318C2#2144748511yescharacteristic
subgroupG/H
centerC2#21447485
commutatorC42×C61C22×C4
Frattini#124C22×D61
FittingC61×#1406C2
socleC2×C61#2127
prime radical#21456875C2
cosocle#124×C61C23
hypercenter#1406D61
hypocenterC61#13606
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. #21447505C2semidirect
3. #21447492C2semidirect
4. #21443995C2semidirect
5. #21443993C2semidirect
6. #21443976C2semidirect
7. C61×#1406C2semidirect
8. #16840366C4semidirect
10. #16840134C4semidirect
11. #16840131C4semidirect
17. #16839824C4semidirect
161. #1406D61semidirect
186. C61#13606semidirect

Maximal subgroups

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

subgroupindex
#214568752
#214475052
#214474922
#214439952
#214439932
#214439762
C61×#14062
#1360661