Finite Group Databasebeta

← #29592855

Subgroups — #29592855

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#21444793C212yesmaximal
3#21443848C212yesmaximal
4#21443753C211yesmaximal, characteristic
5C61×#2028C211yesmaximal, characteristic
6#16839917—22no—
7#16839683—22no—
8#16839682C2212yes—
9#16839669—21no—
10#16839668—21no—
11#16839666—22no—
12#16839499—22no—
13#16839488C2212yes—
14C61×#544C2211yescharacteristic
15C61×#419—21no—
16C61×#391C2211yescharacteristic
17C61×#343C2211yescharacteristic
18C61×#339—21no—
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×#11758202—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—
32C22×Q8×C61D411yescharacteristic
33C2×M16×C61—21no—
34C4⋊Q8×C61—41no—
35#125×C61—41no—
36C22⋊Q8×C61—22no—
37#117×C61—21no—
38C2×C4⋊C4×C61C2311yescharacteristic
39C2×C42×C61D411yescharacteristic
40#103×C61—41no—
41#103×C61—41no—
42#95×C61—22no—
43C2×Q8×C61—21no—
44C2×Q8×C61—22no—
45C2×Q8×C61—41no—
46C2×Q8×C61—41no—
47C2×Q8×C61Q1612yescomplemented
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D412yes—
52C22×Dic61—22no—
53C22×Dic61—22no—
54C2×Dic122—24no—
55C2×Dic122—42no—
56C2×Dic122—82no—
57C2×Dic122—81no—
58C2×Dic122—81no—
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—41no—
68C42×C61—41no—
69C42×C61—41no—
70#11758202—84no—
71C4⋊Dic61—81no—
72C4⋊Dic61—81no—
73#11758196—82no—
74C4×Dic61—81no—
75C2×#92418—42no—
76C23×C61#12011yescharacteristic
77Q8×C61—42no—
78Q8×C61—81no—
79Q8×C61—82no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—22no—
84C2×C4×C61—22no—
85C2×C4×C61—41no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—42no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61C2×Q1611yescharacteristic
93C2×C4×C61#13711yescharacteristic
94C2×Dic61—24no—
95C2×Dic61—42no—
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100Dic122—44no—
101Dic122—84no—
102Dic122—82no—
103C8×C61—81no—
104#92418—82no—
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—
116C4×C61—81no—
117C4×C61—81no—
118Dic61—44no—
119Dic61—82no—
120#2811—611no—
121#2028D6111yescharacteristic
122#1605—612no—
123#1354—612no—
124#1319—611no—
125C2×C61—21no—
126C2×C61—21no—
127C2×C61#160211yescharacteristic
128C2×C61#219911yescharacteristic
129C2×C61#220511yescharacteristic
130#544C2×D6111yescharacteristic
131#451—1222no—
132#419—21no—
133#395—612no—
134#395—1222no—
135#391—1222no—
136#391—1221no—
137#391—1221no—
138#391C2×D6111yescharacteristic
139#343C2×D6111yescharacteristic
140#339—21no—
141#328—1222no—
142#324—612no—
143C61#1222411yescharacteristic, complemented, Sylow
144C22×Q8—612no—
145C22×Q8—1221no—
146C22×Q8C22⋊D6111yescharacteristic
147C2×Q16—2442no—
148C2×M16—21no—
149C4⋊Q8—41no—
150#125—41no—
151C22⋊Q8—22no—
152#117—21no—
153C2×C4⋊C4—1221no—
154C2×C4⋊C4—1221no—
155C2×C4⋊C4—1222no—
156C2×C4⋊C4C22×D6111yescharacteristic
157C2×C42—1221no—
158C2×C42C22⋊D6111yescharacteristic
159#103—41no—
160#103—41no—
161#103—2442no—
162C22⋊C8—1222no—
163#95—22no—
164#95—612no—
165#95—1222no—
166#95—1222no—
167#95—1222no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—1224no—
173C2×Q8—4881no—
174C2×Q8—4881no—
175C2×Q8—4882no—
176C2×Q8—2442no—
177C2×Q8#1175820212yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—1222no—
181C22×C4—1222no—
182C22×C4D4×D6112yes—
183C22×C4C2×C22⋊D6111yescharacteristic
184Q16—4884no—
185M16—41no—
186M16—41no—
187C2×C8—41no—
188C2×C8—2442no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—4882no—
195C4⋊C4—4881no—
196C4⋊C4—4881no—
197C22⋊C4—41no—
198C42—41no—
199C42—41no—
200C42—4881no—
201C23#1451974511yescharacteristic
202Q8—42no—
203Q8—81no—
204Q8—82no—
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—41no—
219C2×C4—41no—
220C2×C4—1224no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4—2442no—
224C2×C4—2442no—
225C2×C4—2442no—
226C2×C4C2×#1175820211yescharacteristic
227C2×C4#1451980711yescharacteristic
228C8—81no—
229C8—4882no—
230C22—21no—
231C22—21no—
232C22#1683991711yescharacteristic
233C22#1683993911yescharacteristic
234C22#1683994811yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—4882no—
243C4—2444no—
244C2—21no—
245C2—21no—
246C2#2144477511yescharacteristic
247C2#2144781911yescharacteristic
248C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2028C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#2028D61
hypocenterC61#12224
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C61Q16semidirect
143. C61#12224semidirect
177. C2×Q8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214538122
#214447932
#214438482
#214437532
C61×#20282
#1222461