Finite Group Databasebeta

← #29592858

Subgroups — #29592858

Order 15616.

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

normal subgroups only

This group has 380 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
2#21446980C211yesmaximal, characteristic
3#21444795C212yesmaximal
4#21443850C212yesmaximal
5C61×#1319C211yesmaximal, characteristic
6#16840393C2211yescharacteristic
7#16839920—22no—
8#16839771—21no—
9#16839683—22no—
10#16839682C2212yes—
11#16839669—21no—
12#16839668—21no—
13#16839666—22no—
14#16839531C2211yescharacteristic
15#16839522—21no—
16#16839500—22no—
17C61×#391C2211yescharacteristic
18C61×#324C2212yes—
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×Dic244—42no—
22C2×#11758194—21no—
23#14519776—41no—
24#14519768—41no—
25#14519756—22no—
26#14519736—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—21no—
29C2×#11758196—22no—
30C2×C4×Dic61—21no—
31#14519702—42no—
32#14519701—41no—
33#14519701—41no—
34#14519684—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684D412yes—
38C22×Q8×C61D411yescharacteristic
39C2×C4⋊C4×C61C2311yescharacteristic
40C2×C42×C61D411yescharacteristic
41C22⋊C8×C61—22no—
42#95×C61—22no—
43C2×Q8×C61—21no—
44C2×Q8×C61—41no—
45C2×Q8×C61—41no—
46C2×Q8×C61Q1612yescomplemented
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61C2×D411yescharacteristic
50C22×C4×C61C2×D412yes—
51C22×Dic61—22no—
52C22×Dic61—22no—
53C2×Dic122—24no—
54C2×Dic122—22no—
55C2×Dic122—42no—
56C2×Dic122—82no—
57C2×Dic122—81no—
58C2×Dic122—81no—
59C2×C8×C61—42no—
60C4⋊C4×C61—41no—
61C42×C61—41no—
62C22⋊Dic61—41no—
63C4⋊Dic61—21no—
64C4⋊Dic61—21no—
65C4⋊Dic61—22no—
66C4⋊Dic61—81no—
67C4⋊Dic61—81no—
68#11758196—42no—
69#11758196—82no—
70C4×Dic61—41no—
71C4×Dic61—81no—
72#11758194—41no—
73#11758194—41no—
74C2×#92418—41no—
75Dic244—84no—
76C23×C61#12011yescharacteristic
77Q8×C61—42no—
78Q8×C61—81no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—22no—
84C2×C4×C61—41no—
85C2×C4×C61—41no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61—42no—
89C2×C4×C61C2×Q1611yescharacteristic
90C2×C4×C61#13711yescharacteristic
91C2×Dic61—24no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—42no—
96C2×Dic61—41no—
97C2×Dic61—41no—
98C2×Dic61—42no—
99Dic122—44no—
100Dic122—84no—
101Dic122—82no—
102Dic122—82no—
103C8×C61—82no—
104#92418—81no—
105#12224—611nomaximal, Sylow
106C22×C61—21no—
107C22×C61—21no—
108C22×C61#45111yescharacteristic
109C22×C61#45511yescharacteristic
110C22×C61B(4,𝔽2)11yescharacteristic
111C4×C61—21no—
112C4×C61—21no—
113C4×C61—42no—
114C4×C61—42no—
115C4×C61—42no—
116Dic61—44no—
117Dic61—82no—
118Dic61—81no—
119Dic61—81no—
120#2811—611no—
121#2028—611no—
122#1605—612no—
123#1354—612no—
124#1319D6111yescharacteristic
125C2×C61—21no—
126C2×C61—21no—
127C2×C61#160211yescharacteristic
128C2×C61#219911yescharacteristic
129C2×C61#220511yescharacteristic
130#544—611no—
131#451—1222no—
132#419—1221no—
133#395—612no—
134#395—1222no—
135#391—1222no—
136#391—1221no—
137#391—1221no—
138#391C2×D6111yescharacteristic
139#343—611no—
140#339—1221no—
141#328—1222no—
142#324C2×D6112yes—
143C61#1222411yescharacteristic, complemented, Sylow
144C22×Q8—612no—
145C22×Q8—1221no—
146C22×Q8D24411yescharacteristic
147C2×Q16—2442no—
148C2×M16—1221no—
149C4⋊Q8—2441no—
150#125—2441no—
151C22⋊Q8—1222no—
152#117—1221no—
153C2×C4⋊C4—1221no—
154C2×C4⋊C4—1221no—
155C2×C4⋊C4—1222no—
156C2×C4⋊C4C22×D6111yescharacteristic
157C2×C42—1221no—
158C2×C42D24411yescharacteristic
159#103—2442no—
160#103—2441no—
161#103—2441no—
162C22⋊C8—22no—
163#95—22no—
164#95—612no—
165#95—1222no—
166#95—1222no—
167#95—1222no—
168C2×Q8—21no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—1224no—
172C2×Q8—1222no—
173C2×Q8—4881no—
174C2×Q8—4881no—
175C2×Q8—4882no—
176C2×Q8—2442no—
177C2×Q8Dic24412yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—1222no—
181C22×C4—1222no—
182C22×C4C2×D24411yescharacteristic
183C22×C4D4×D6112yes—
184Q16—4884no—
185M16—2441no—
186M16—2441no—
187C2×C8—42no—
188C2×C8—2441no—
189C4⋊C4—41no—
190C4⋊C4—1221no—
191C4⋊C4—1221no—
192C4⋊C4—1222no—
193C4⋊C4—4882no—
194C4⋊C4—4881no—
195C4⋊C4—4881no—
196C4⋊C4—2442no—
197C22⋊C4—2441no—
198C42—41no—
199C42—4881no—
200C42—2441no—
201C23#1451974711yescharacteristic
202Q8—42no—
203Q8—81no—
204Q8—4882no—
205Q8—4882no—
206Q8—4884no—
207Q8—2444no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—1224no—
219C2×C4—2442no—
220C2×C4—2442no—
221C2×C4—2441no—
222C2×C4—2441no—
223C2×C4—2442no—
224C2×C4—2442no—
225C2×C4—2442no—
226C2×C4C2×Dic24411yescharacteristic
227C2×C4#1451981011yescharacteristic
228C8—82no—
229C8—4881no—
230C22—21no—
231C22—21no—
232C22#1683992011yescharacteristic
233C22#1683994211yescharacteristic
234C22#1683995011yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—4882no—
241C4—4881no—
242C4—4881no—
243C4—2444no—
244C2—21no—
245C2—21no—
246C2#2144477811yescharacteristic
247C2#2144782211yescharacteristic
248C2#2144785011yescharacteristic
subgroupG/H
centerC22#16839950
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1319C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#1319D61
hypocenterC61#12224
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
46. C2×Q8×C61Q16semidirect
143. C61#12224semidirect
177. C2×Q8Dic244semidirect

Maximal subgroups

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

subgroupindex
#214538122
#214469802
#214447952
#214438502
C61×#13192
#1222461