Finite Group Databasebeta

← #29592831

Subgroups — #29592831

Order 15616.

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

normal subgroups only

This group has 460 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#21452191C211yesmaximal, characteristic
2#21444793C212yesmaximal, complemented
3#21443848C212yesmaximal, complemented
4#21443751C211yesmaximal, characteristic
5C61×#2027C211yesmaximal, characteristic
6C2×#14519754—22no—
7#16839917—22no—
8#16839685—22no—
9#16839682C2212yes—
10#16839660—21no—
11#16839642—21no—
12#16839499—22no—
13#16839488C2212yes—
14C61×#544C2211yescharacteristic
15C61×#418—21no—
16C61×#390C2211yescharacteristic
17C61×#343C2211yescharacteristic
18C61×#339—21no—
19C22×Dic122D412yes—
20C23×Dic61—21no—
21C2×#11758202—42no—
22#14519754—48no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×#11758196—22no—
26C2×#11758196—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—21no—
29#14519700—42no—
30C22⋊#92418—22no—
31#14519684—22no—
32#14519684D412yes—
33C22×D4×C61D411yescharacteristic
34C2×M16×C61—21no—
35C4⋊Q8×C61—41no—
36#125×C61—41no—
37C22⋊Q8×C61—22no—
38#117×C61—21no—
39C2×C4⋊C4×C61C2311yescharacteristic
40C2×C22⋊C4×C61—22no—
41C2×C42×C61D411yescharacteristic
42#102×C61—41no—
43#102×C61—41no—
44C24×C61—21no—
45C2×Q8×C61—22no—
46C2×D4×C61—21no—
47C2×D4×C61—41no—
48C2×D4×C61—41no—
49C2×D4×C61Q1612yescomplemented
50C22×C4×C61—21no—
51C22×C4×C61C2×D411yescharacteristic
52C22×C4×C61C2×D412yes—
53C22×Dic61—22no—
54C22×Dic61—22no—
55C22×Dic61—44no—
56C22×Dic61—81no—
57C2×Dic122—42no—
58C2×Dic122—44no—
59M16×C61—41no—
60M16×C61—41no—
61C2×C8×C61—41no—
62C4⋊C4×C61—21no—
63C4⋊C4×C61—21no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—41no—
66C4⋊C4×C61—42no—
67C22⋊C4×C61—44no—
68C22⋊C4×C61—41no—
69C42×C61—41no—
70C42×C61—41no—
71C22⋊Dic61—44no—
72C22⋊Dic61—81no—
73#11758202—84no—
74C4⋊Dic61—24no—
75C4⋊Dic61—81no—
76#11758196—44no—
77#11758196—44no—
78C2×#92418—42no—
79C23×C61—22no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—41no—
83C23×C61—41no—
84C23×C61#12011yescharacteristic
85Q8×C61—82no—
86D4×C61—42no—
87D4×C61—81no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61—42no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61C2×Q1611yescharacteristic
98C2×C4×C61#13711yescharacteristic
99C2×Dic61—24no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—84no—
106Dic122—44no—
107Dic122—82no—
108C8×C61—81no—
109#92418—82no—
110#12220—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—21no—
113C22×C61—44no—
114C22×C61—42no—
115C22×C61—82no—
116C22×C61—81no—
117C22×C61—81no—
118C22×C61#45111yescharacteristic
119C22×C61#45511yescharacteristic
120C22×C61B(4,𝔽2)11yescharacteristic
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—42no—
124C4×C61—42no—
125C4×C61—81no—
126C4×C61—81no—
127Dic61—44no—
128Dic61—82no—
129#2683—611no—
130#2027D6111yescharacteristic
131#1605—612no—
132#1354—612no—
133#1318—611no—
134C2×C61—21no—
135C2×C61—21no—
136C2×C61—82no—
137C2×C61#160211yescharacteristic
138C2×C61#219611yescharacteristic
139C2×C61#220311yescharacteristic
140#544C2×D6111yescharacteristic
141C2×C22⋊Q8—1222no—
142#451—1222no—
143#418—21no—
144#396—1222no—
145#395—612no—
146#390C2×D6111yescharacteristic
147#389—1221no—
148#386—1221no—
149#343C2×D6111yescharacteristic
150#339—21no—
151#328—1222no—
152#324—612no—
153C61#1222011yescharacteristic, complemented, Sylow
154C22×Q8—612no—
155C22×D4C22⋊D6111yescharacteristic
156C23×C4—1221no—
157C2×Q16—2442no—
158C2×M16—21no—
159C4⋊Q8—41no—
160#125—41no—
161C22⋊Q8—22no—
162C22⋊Q8—2448no—
163#117—21no—
164C2×C4⋊C4—1221no—
165C2×C4⋊C4—1221no—
166C2×C4⋊C4—1222no—
167C2×C4⋊C4—1222no—
168C2×C4⋊C4C22×D6111yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—1221no—
171C2×C22⋊C4—1222no—
172C2×C42C22⋊D6111yescharacteristic
173#103—2442no—
174#102—41no—
175#102—41no—
176C22⋊C8—1222no—
177#95—612no—
178#95—1222no—
179C24—21no—
180C2×Q8—22no—
181C2×Q8—2442no—
182C2×Q8—2444no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1175820212yescomplemented
187C22×C4—21no—
188C22×C4—1222no—
189C22×C4—1222no—
190C22×C4—4881no—
191C22×C4—2444no—
192C22×C4D4×D6112yes—
193C22×C4C2×C22⋊D6111yescharacteristic
194Q16—4884no—
195M16—41no—
196M16—41no—
197C2×C8—41no—
198C2×C8—2442no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—1224no—
205C4⋊C4—4881no—
206C4⋊C4—2444no—
207C4⋊C4—2444no—
208C22⋊C4—44no—
209C22⋊C4—41no—
210C22⋊C4—4881no—
211C22⋊C4—2444no—
212C42—41no—
213C42—41no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1451974511yescharacteristic
220Q8—82no—
221Q8—4882no—
222Q8—2444no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—1224no—
235C2×C4—4884no—
236C2×C4—2442no—
237C2×C4—2442no—
238C2×C4—2442no—
239C2×C4—2442no—
240C2×C4—2442no—
241C2×C4C2×#1175820211yescharacteristic
242C2×C4#1451980711yescharacteristic
243C8—81no—
244C8—4882no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1683991711yescharacteristic
253C22#1683993911yescharacteristic
254C22#1683994811yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—81no—
260C4—81no—
261C4—4882no—
262C4—2444no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2144477511yescharacteristic
267C2#2144780211yescharacteristic
268C2#2144783911yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2027C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D4×C61D4
hypercenter#2027D61
hypocenterC61#12220
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2027

Complements

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

NH ≅ G/Ntype
2. #21444793C2semidirect
3. #21443848C2semidirect
49. C2×D4×C61Q16semidirect
153. C61#12220semidirect
186. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214521912
#214447932
#214438482
#214437512
C61×#20272
#1222061