Finite Group Databasebeta

← #29592269

Subgroups — #29592269

Order 15616.

442 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 59 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#21453812C211yesmaximal, characteristic, complemented
2#21444788C212yesmaximal
3#21443848C212yesmaximal, complemented
4#21443753C211yesmaximal, characteristic, complemented
5C61×#2007C211yesmaximal, characteristic
6#16839900—22no—
7#16839683—22no—
8#16839682C2212yes—
9#16839669—21no—
10#16839668—21no—
11#16839666—22no—
12#16839496—22no—
13#16839488C2212yes—
14C2×#137×C61—21no—
15C61×#537C2211yescharacteristic
16C2×#104×C61—21no—
17C61×#391C2211yescharacteristic
18C61×#343C2211yescharacteristic
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×#11758200—42no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—21no—
24C2×#11758196—22no—
25C2×C4×Dic61—21no—
26#14519700—42no—
27C22⋊#92418—22no—
28#14519684—22no—
29#14519684—22no—
30#14519684—22no—
31#14519684D412yes—
32C2×#41×C61—21no—
33C22×Q8×C61D411yescharacteristic
34#137×C61—42no—
35#137×C61—42no—
36C2×Q16×C61—41no—
37C2×SD16×C61—41no—
38C2×M16×C61—21no—
39C22⋊Q8×C61—41no—
40#121×C61—22no—
41#121×C61—41no—
42C2×C4⋊C4×C61C2311yescharacteristic
43C2×C42×C61D411yescharacteristic
44#104×C61—42no—
45#95×C61—22no—
46#41×C61—41no—
47#41×C61—41no—
48#41×C61—41no—
49#41×C61—41no—
50C2×Q8×C61—21no—
51C2×Q8×C61—21no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61SD1612yescomplemented
55C2×D4×C61—21no—
56C2×D4×C61—22no—
57C2×D4×C61—41no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—41no—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63C22×Dic61—22no—
64C22×Dic61—22no—
65C2×Dic122—24no—
66C2×Dic122—42no—
67C2×Dic122—82no—
68C2×Dic122—81no—
69C2×Dic122—81no—
70Q16×C61—44no—
71SD16×C61—44no—
72M16×C61—22no—
73M16×C61—41no—
74C2×C8×C61—41no—
75C4⋊C4×C61—22no—
76C4⋊C4×C61—41no—
77C22⋊C4×C61—42no—
78C42×C61—41no—
79#11758200—84no—
80C4⋊Dic61—81no—
81C4⋊Dic61—81no—
82#11758196—82no—
83C4×Dic61—81no—
84C2×#92418—42no—
85C23×C61—41no—
86C23×C61#12011yescharacteristic
87Q8×C61—42no—
88Q8×C61—41no—
89Q8×C61—41no—
90Q8×C61—81no—
91D4×C61—41no—
92D4×C61—41no—
93D4×C61—82no—
94D4×C61—81no—
95D4×C61—81no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61—41no—
107C2×C4×C61—81no—
108C2×C4×C61—81no—
109C2×C4×C61C2×SD1611yescharacteristic
110C2×C4×C61#13711yescharacteristic
111C2×Dic61—24no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117Dic122—44no—
118Dic122—84no—
119Dic122—82no—
120C8×C61—42no—
121#92418—82no—
122#12130—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—81no—
126C22×C61—81no—
127C22×C61—81no—
128C22×C61#44811yescharacteristic
129C22×C61#45311yescharacteristic
130C22×C61B(4,𝔽2)11yescharacteristic
131C4×C61—21no—
132C4×C61—21no—
133C4×C61—42no—
134C4×C61—42no—
135C4×C61—42no—
136C4×C61—81no—
137Dic61—44no—
138Dic61—82no—
139#2811—611no—
140#2007D6111yescharacteristic
141#1604—612no—
142#1354—612no—
143#1319—611no—
144C2×C61—21no—
145C2×C61—21no—
146C2×C61—161no—
147C2×C61#159611yescharacteristic
148C2×C61#219911yescharacteristic
149C2×C61#220511yescharacteristic
150C2×#137—21no—
151#537C2×D6111yescharacteristic
152#448—1222no—
153C2×#104—21no—
154#395—612no—
155#395—1222no—
156#391—1222no—
157#391—1221no—
158#391—1221no—
159#391C2×D6111yescharacteristic
160#343C2×D6111yescharacteristic
161#327—1222no—
162#324—612no—
163C61#1213011yescharacteristic, complemented, Sylow
164C2×#41—21no—
165C22×Q8—612no—
166C22×Q8—1221no—
167C22×Q8C22⋊D6111yescharacteristic
168#137—42no—
169#137—42no—
170C2×Q16—41no—
171C2×SD16—41no—
172C2×SD16—2442no—
173C2×M16—21no—
174C22⋊Q8—41no—
175#121—22no—
176#121—41no—
177C2×C4⋊C4—1221no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4—1222no—
180C2×C4⋊C4C22×D6111yescharacteristic
181C2×C42—1221no—
182C2×C42C22⋊D6111yescharacteristic
183#104—42no—
184#103—2442no—
185C22⋊C8—1222no—
186#95—22no—
187#95—612no—
188#95—1222no—
189#95—1222no—
190#95—1222no—
191#41—41no—
192#41—41no—
193#41—41no—
194#41—41no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—1224no—
200C2×Q8—4881no—
201C2×Q8—4881no—
202C2×Q8—4882no—
203C2×Q8—2442no—
204C2×Q8#1175820012yescomplemented
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—41no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4D4×D6112yes—
214C22×C4C2×C22⋊D6111yescharacteristic
215Q16—44no—
216SD16—44no—
217SD16—4884no—
218M16—22no—
219M16—41no—
220C2×C8—41no—
221C2×C8—2442no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—4882no—
225C4⋊C4—4881no—
226C4⋊C4—4881no—
227C22⋊C4—42no—
228C42—41no—
229C42—4881no—
230C23—41no—
231C23#1451974511yescharacteristic
232Q8—42no—
233Q8—41no—
234Q8—41no—
235Q8—81no—
236Q8—4882no—
237Q8—4884no—
238Q8—2444no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
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—41no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—1224no—
258C2×C4—2442no—
259C2×C4—2442no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4C2×#1175820011yescharacteristic
264C2×C4#1451980711yescharacteristic
265C8—42no—
266C8—4882no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22#1683990011yescharacteristic
273C22#1683992911yescharacteristic
274C22#1683994811yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—81no—
281C4—4882no—
282C4—2444no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2#2144474111yescharacteristic
287C2#2144781911yescharacteristic
288C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2007C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#2007D61
hypocenterC61#12130
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #21453812C2semidirect
3. #21443848C2semidirect
4. #21443753C2semidirect
54. C2×Q8×C61SD16semidirect
163. C61#12130semidirect
204. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214538122
#214447882
#214438482
#214437532
C61×#20072
#1213061