Finite Group Databasebeta

← #29592832

Subgroups — #29592832

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#21446975C211yesmaximal, characteristic
2#21444792C212yesmaximal, complemented
3#21443849C212yesmaximal, complemented
4#21443751C211yesmaximal, characteristic
5C61×#2683C211yesmaximal, characteristic
6#16840393C2211yescharacteristic
7#16839918—22no—
8#16839767—21no—
9#16839531C2211yescharacteristic
10#16839522—21no—
11#16839498—22no—
12#16839488C2212yes—
13C2×C22⋊Q8×C61—22no—
14C61×#396—22no—
15C61×#395C2212yes—
16C61×#390C2211yescharacteristic
17C61×#389—21no—
18C61×#386—21no—
19C2×#11758202—42no—
20C2×#11758194—21no—
21#14519776—41no—
22#14519768—41no—
23#14519756—22no—
24#14519736—21no—
25#14519701—42no—
26#14519698—41no—
27#14519698—41no—
28C22⋊#92418—22no—
29C22×Q8×C61D412yes—
30C22×D4×C61D411yescharacteristic
31C23×C4×C61—21no—
32C22⋊Q8×C61—48no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—22no—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61C2311yescharacteristic
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—21no—
41C2×C42×C61D411yescharacteristic
42#95×C61—22no—
43#95×C61D412yes—
44C24×C61—21no—
45C2×Q8×C61—44no—
46C2×Q8×C61—42no—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×D4×C61—41no—
50C2×D4×C61Q1612yescomplemented
51C22×C4×C61—22no—
52C22×C4×C61—21no—
53C22×C4×C61—22no—
54C22×C4×C61—44no—
55C22×C4×C61—81no—
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D412yes—
58C2×Dic122—22no—
59C4⋊C4×C61—24no—
60C4⋊C4×C61—41no—
61C4⋊C4×C61—44no—
62C4⋊C4×C61—44no—
63C4⋊C4×C61—81no—
64C22⋊C4×C61—44no—
65C22⋊C4×C61—44no—
66C22⋊C4×C61—81no—
67C42×C61—41no—
68C22⋊Dic61—41no—
69#11758202—84no—
70C4⋊Dic61—21no—
71C4⋊Dic61—21no—
72C4⋊Dic61—22no—
73#11758196—42no—
74C4×Dic61—41no—
75#11758194—41no—
76#11758194—41no—
77C2×#92418—42no—
78C2×#92418—41no—
79C23×C61—22no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—41no—
83C23×C61—41no—
84C23×C61#12011yescharacteristic
85Q8×C61—44no—
86Q8×C61—82no—
87D4×C61—42no—
88D4×C61—81no—
89C2×C4×C61—21no—
90C2×C4×C61—21no—
91C2×C4×C61—22no—
92C2×C4×C61—24no—
93C2×C4×C61—22no—
94C2×C4×C61—42no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—84no—
103C2×C4×C61C2×Q1611yescharacteristic
104C2×C4×C61#13711yescharacteristic
105C2×Dic61—41no—
106C2×Dic61—41no—
107Dic122—82no—
108#92418—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—44no—
126C4×C61—82no—
127Dic61—81no—
128Dic61—81no—
129#2683D6111yescharacteristic
130#2027—611no—
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#544—611no—
141C2×C22⋊Q8—22no—
142#451—1222no—
143#418—1221no—
144#396—22no—
145#395C2×D6112yes—
146#390C2×D6111yescharacteristic
147#389—21no—
148#386—21no—
149#343—611no—
150#339—1221no—
151#328—1222no—
152#324—612no—
153C61#1222011yescharacteristic, complemented, Sylow
154C22×Q8C22⋊D6112yes—
155C22×D4C22⋊D6111yescharacteristic
156C23×C4—21no—
157C2×Q16—2442no—
158C2×M16—1221no—
159C4⋊Q8—2441no—
160#125—2441no—
161C22⋊Q8—48no—
162C22⋊Q8—1222no—
163#117—1221no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×D6111yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C42C22⋊D6111yescharacteristic
173#103—2442no—
174#102—2441no—
175#102—2441no—
176C22⋊C8—1222no—
177#95—22no—
178#95C22⋊D6112yes—
179C24—21no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×Q8—1222no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1175820212yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×C22⋊D6111yescharacteristic
193C22×C4C2×C22⋊D6112yes—
194Q16—4884no—
195M16—2441no—
196M16—2441no—
197C2×C8—2442no—
198C2×C8—2441no—
199C4⋊C4—24no—
200C4⋊C4—41no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—1221no—
205C4⋊C4—1221no—
206C4⋊C4—1222no—
207C4⋊C4—2442no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—81no—
211C22⋊C4—2441no—
212C42—41no—
213C42—2441no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1451974611yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—4882no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—84no—
239C2×C4—2441no—
240C2×C4—2441no—
241C2×C4C2×#1175820211yescharacteristic
242C2×C4#1451980811yescharacteristic
243C8—4881no—
244C8—4882no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1683991811yescharacteristic
253C22#1683994011yescharacteristic
254C22#1683994911yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—4881no—
262C4—4881no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2144477611yescharacteristic
267C2#2144780311yescharacteristic
268C2#2144784011yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2683C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D4×C61D4
hypercenter#2683D61
hypocenterC61#12220
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #21444792C2semidirect
3. #21443849C2semidirect
50. 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
#214469752
#214447922
#214438492
#214437512
C61×#26832
#1222061