Finite Group Databasebeta

← #29592788

Subgroups — #29592788

Order 15616.

460 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 63 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#21446980C211yesmaximal, characteristic, complemented
2#21444758C212yesmaximal, complemented
3#21443822C212yesmaximal, complemented
4#21443753C211yesmaximal, characteristic, complemented
5C61×#2733C211yesmaximal, characteristic
6#16840393C2211yescharacteristic
7#16839913—22no—
8#16839771—21no—
9#16839531C2211yescharacteristic
10#16839522—21no—
11#16839498—22no—
12#16839488C2212yes—
13C2×#124×C61—21no—
14C61×#394C2212yes—
15C61×#392—22no—
16C61×#391C2211yescharacteristic
17C61×#390—21no—
18C61×#388—22no—
19C2×#11758200—42no—
20C2×#11758194—21no—
21#14519776—41no—
22#14519768—41no—
23#14519756—22no—
24#14519736—21no—
25#14519701—41no—
26#14519701—41no—
27#14519698—42no—
28C22⋊#92418—22no—
29C22×Q8×C61D411yescharacteristic
30C22×D4×C61—21no—
31C22×D4×C61—21no—
32C22×D4×C61D412yes—
33#124×C61—42no—
34#124×C61—42no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61C2311yescharacteristic
37C2×C22⋊C4×C61—22no—
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—22no—
41C2×C42×C61—21no—
42C2×C42×C61D411yescharacteristic
43#95×C61—22no—
44#95×C61D412yes—
45C24×C61—22no—
46C2×Q8×C61—21no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C2×Q8×C61SD1612yescomplemented
50C2×D4×C61—42no—
51C2×D4×C61—42no—
52C2×D4×C61—42no—
53C2×D4×C61—82no—
54C2×D4×C61—81no—
55C2×D4×C61—81no—
56C2×D4×C61—81no—
57C2×D4×C61—81no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—22no—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63C2×Dic122—22no—
64C4⋊C4×C61—41no—
65C4⋊C4×C61—81no—
66C22⋊C4×C61—44no—
67C22⋊C4×C61—44no—
68C22⋊C4×C61—42no—
69C22⋊C4×C61—42no—
70C22⋊C4×C61—82no—
71C42×C61—42no—
72C42×C61—41no—
73C22⋊Dic61—41no—
74#11758200—84no—
75C4⋊Dic61—22no—
76C4⋊Dic61—21no—
77C4⋊Dic61—21no—
78#11758196—42no—
79C4×Dic61—41no—
80#11758194—41no—
81#11758194—41no—
82C2×#92418—42no—
83C2×#92418—41no—
84C23×C61—42no—
85C23×C61—42no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61—42no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61#12011yescharacteristic
92Q8×C61—42no—
93Q8×C61—81no—
94D4×C61—44no—
95D4×C61—82no—
96D4×C61—84no—
97D4×C61—84no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61C2×SD1611yescharacteristic
112C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
113C2×Dic61—41no—
114C2×Dic61—41no—
115Dic122—82no—
116#92418—81no—
117#92418—82no—
118#12212—611nomaximal, Sylow
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—82no—
122C22×C61—82no—
123C22×C61—82no—
124C22×C61—82no—
125C22×C61—82no—
126C22×C61—82no—
127C22×C61—82no—
128C22×C61#45011yescharacteristic
129C22×C61#45511yescharacteristic
130C22×C61B(4,𝔽2)11yescharacteristic
131C4×C61—21no—
132C4×C61—21no—
133C4×C61—42no—
134C4×C61—42no—
135C4×C61—42no—
136C4×C61—82no—
137Dic61—81no—
138Dic61—81no—
139#2733D6111yescharacteristic
140#2028—611no—
141#1599—612no—
142#1345—612no—
143#1319—611no—
144C2×C61—21no—
145C2×C61—21no—
146C2×C61—162no—
147C2×C61#160111yescharacteristic
148C2×C61#219911yescharacteristic
149C2×C61#220411yescharacteristic
150#544—611no—
151C2×#124—21no—
152#450—1222no—
153#419—1221no—
154#394C2×D6112yes—
155#392—22no—
156#391C2×D6111yescharacteristic
157#390—21no—
158#388—22no—
159#343—611no—
160#339—1221no—
161#328—1222no—
162#324—612no—
163C61#1221211yescharacteristic, complemented, Sylow
164C22×Q8C22⋊D6111yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22⋊D6112yes—
168C2×SD16—2442no—
169C2×M16—1221no—
170C4⋊Q8—2441no—
171#125—2441no—
172#124—42no—
173#124—42no—
174C22⋊Q8—1222no—
175#117—1221no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22×D6111yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—21no—
183C2×C42C22⋊D6111yescharacteristic
184#103—2441no—
185#103—2441no—
186#102—2442no—
187C22⋊C8—1222no—
188#95—22no—
189#95C22⋊D6112yes—
190C24—22no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—1222no—
195C2×Q8#1175820012yescomplemented
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C2×C22⋊D6111yescharacteristic
208C22×C4C2×C22⋊D6112yes—
209SD16—4884no—
210M16—2441no—
211M16—2441no—
212C2×C8—2442no—
213C2×C8—2441no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—1222no—
217C4⋊C4—1221no—
218C4⋊C4—1221no—
219C4⋊C4—2442no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C22⋊C4—2441no—
226C42—41no—
227C42—42no—
228C42—2441no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23#1451974611yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—4882no—
240D4—44no—
241D4—82no—
242D4—84no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—2441no—
258C2×C4—2441no—
259C2×C4C2×#1175820011yescharacteristic
260C2×C4#1451980311yescharacteristic
261C8—4881no—
262C8—4882no—
263C22—21no—
264C22—21no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22#1683991311yescharacteristic
273C22#1683994011yescharacteristic
274C22#1683994911yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4—4881no—
282C4—4881no—
283C2—21no—
284C2—21no—
285C2—162no—
286C2#2144477111yescharacteristic
287C2#2144782011yescharacteristic
288C2#2144784511yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
Fitting, prime radicalC61×#2733C2
socleC22×C61B(4,𝔽2)
hypercenter#2733D61
hypocenterC61#12212
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #21446980C2semidirect
2. #21444758C2semidirect
3. #21443822C2semidirect
4. #21443753C2semidirect
49. C2×Q8×C61SD16semidirect
163. C61#12212semidirect
195. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214469802
#214447582
#214438222
#214437532
C61×#27332
#1221261