Finite Group Databasebeta

← #29600333

Subgroups — #29600333

Order 15616.

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

normal subgroups only

This group has 406 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#21456273C211yesmaximal, characteristic
2#21447481C212yesmaximal
3#21443984C211yesmaximal, characteristic, complemented
4#21443983C212yesmaximal, complemented
5C61×#1409C211yesmaximal, characteristic, complemented
6C2×#14519806—22no—
7#16840310C2211yescharacteristic
8C4⋊Q8×D61C411yescharacteristic, complemented
9#16840153—22no—
10#16840142C411yescharacteristic, complemented
11#16840141C411yescharacteristic, complemented
12#16840134C2211yescharacteristic
13#16839734—22no—
14C4×#11758189C411yescharacteristic, complemented
15#16839556—22no—
16#16839555C2212yes—
17C61×#492C2211yescharacteristic
18C61×#353C2212yes—
19C2×#11758224D412yes—
20C2×Q8×D61—21no—
21#14519806—44no—
22#14519806—44no—
23C2×Dic244—22no—
24C2×#11758202—22no—
25C2×#11758200—22no—
26C2×#11758189—21no—
27#14519777C2×C411yescharacteristic
28#14519777C2×C411yescharacteristic
29#14519776—41no—
30#14519775—21no—
31#14519764D412yes—
32#14519748—41no—
33D4×Dic61—41no—
34#14519733—42no—
35C4⋊C4×D61—41no—
36C42×D61C2×C411yescharacteristic
37#14519692—22no—
38#14519691—42no—
39#14519687—21no—
40C4×#92418C2×C411yescharacteristic
41C2×SD16×C61—22no—
42C4⋊Q8×C61C2×C411yescharacteristic
43#127×C61C2311yescharacteristic
44#99×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C2×Q8×C61—21no—
47C2×D4×C61—41no—
48C2×D4×C61C2×D412yes—
49C2×C22⋊D61—22no—
50C22×Dic61—22no—
51Q8×D61—42no—
52Q8×D61—42no—
53#11758224—42no—
54#11758224—42no—
55C2×C4×D61—21no—
56C2×C4×D61—41no—
57C2×C4×D61C22⋊C411yescharacteristic
58C2×Dic122—21no—
59C2×Dic122—41no—
60C2×Dic122C22⋊C412yes—
61SD16×C61—44no—
62C2×C8×C61—21no—
63C4⋊C4×C61—41no—
64C22⋊C4×C61—42no—
65C42×C61C22×C411yescharacteristic
66C22⋊Dic61—42no—
67C22⋊Dic61—42no—
68#11758202—44no—
69#11758200—44no—
70C4⋊Dic61—41no—
71C4⋊Dic61—41no—
72#11758196—42no—
73#11758196—82no—
74C4×Dic61—21no—
75C4×Dic61—82no—
76C4×Dic61C22⋊C411yescharacteristic
77C2×#92418—21no—
78Dic244—44no—
79#11758189—24no—
80C23×C61—22no—
81C22×D61#9911yescharacteristic
82Q8×C61—42no—
83D4×C61—42no—
84D4×C61—82no—
85C2×C4×C61—21no—
86C2×C4×C61—41no—
87C2×C4×C61—82no—
88C2×C4×C61C2×C22⋊C411yescharacteristic
89C22⋊D61—82no—
90C2×Dic61—21no—
91C2×Dic61—22no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—41no—
95C2×Dic61—82no—
96C2×Dic61#9911yescharacteristic
97C4×D61—22no—
98C4×D61—42no—
99C4×D61—81no—
100C4×D61—81no—
101Dic122—42no—
102Dic122—42no—
103Dic122—82no—
104C8×C61—22no—
105#92418—22no—
106#13586—611nomaximal, Sylow
107C22×C61—42no—
108C22×C61—82no—
109C22×C61C2×#9911yescharacteristic
110C2×D61—21no—
111C2×D61—41no—
112C4×C61—22no—
113C4×C61—21no—
114C4×C61—81no—
115C4×C61—162no—
116Dic61—41no—
117Dic61—42no—
118Dic61—81no—
119Dic61—162no—
120Dic61#35312yes—
121#3152—611no—
122#2126—612no—
123#1409—611no—
124#1409—612no—
125#1409D6111yescharacteristic, complemented
126C2×C61—21no—
127C2×C61—82no—
128C2×C61C2×#35311yescharacteristic
129D61—41no—
130C2×#137—1222no—
131#532—611no—
132C2×C4⋊Q8—611no—
133#497—1222no—
134#494—611no—
135#494—611no—
136#492—611no—
137#492C2×D6111yescharacteristic
138#410—1222no—
139C4×M16—611no—
140#354—1222no—
141#353—612no—
142#353C2×D6112yes—
143C61#1358611yescharacteristic, complemented, Sylow
144C2×#41—612no—
145C22×Q8—1221no—
146#137—2444no—
147#137—2444no—
148C2×Q16—1222no—
149C2×Q16—1222no—
150C2×SD16—22no—
151C2×SD16—1222no—
152C2×M16—1221no—
153C4⋊Q8—611no—
154C4⋊Q8—611no—
155C4⋊Q8—1221no—
156C4⋊Q8—2441no—
157C4⋊Q8C4×D6111yescharacteristic
158#127C22×D6111yescharacteristic
159#124—612no—
160#121—2441no—
161C4×D4—2441no—
162#117—2442no—
163C2×C4⋊C4—2441no—
164C2×C42—611no—
165#99—22no—
166#99—1222no—
167#99—2442no—
168#97—1221no—
169C4×C8—611no—
170C4×C8C4×D6111yescharacteristic
171#41—2442no—
172#41—2442no—
173C2×Q8—21no—
174C2×Q8—612no—
175C2×Q8—1221no—
176C2×Q8—2441no—
177C2×Q8—2442no—
178C2×Q8—2442no—
179C2×D4—41no—
180C2×D4—1222no—
181C2×D4D4×D6112yes—
182C22×C4—611no—
183C22×C4—1221no—
184C22×C4—1222no—
185C22×C4—2441no—
186Q16—2444no—
187Q16—2444no—
188SD16—44no—
189SD16—2444no—
190M16—1224no—
191C2×C8—21no—
192C2×C8—1221no—
193C4⋊C4—41no—
194C4⋊C4—4882no—
195C4⋊C4—2441no—
196C4⋊C4—2441no—
197C4⋊C4—2442no—
198C22⋊C4—42no—
199C22⋊C4—2442no—
200C22⋊C4—2442no—
201C42—611no—
202C42—1221no—
203C42—4882no—
204C42C2×C4×D6111yescharacteristic
205C23—22no—
206C23—611no—
207Q8—42no—
208Q8—4882no—
209Q8—2442no—
210Q8—2442no—
211D4—42no—
212D4—82no—
213D4—4882no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—82no—
217C2×C4—611no—
218C2×C4—1222no—
219C2×C4—1221no—
220C2×C4—1222no—
221C2×C4—4881no—
222C2×C4—4881no—
223C2×C4—4882no—
224C2×C4—2442no—
225C2×C4—2442no—
226C2×C4—2442no—
227C2×C4—2441no—
228C2×C4C22⋊C4×D6111yescharacteristic
229C8—22no—
230C8—1222no—
231C22—42no—
232C22—82no—
233C22—1221no—
234C22—2441no—
235C22#99×D6111yescharacteristic
236C4—22no—
237C4—21no—
238C4—81no—
239C4—162no—
240C4—612no—
241C4—9762no—
242C4—4881no—
243C4—2441no—
244C4—2442no—
245C2—21no—
246C2—82no—
247C2—2441no—
248C2#353×D6111yescharacteristic
subgroupG/H
centerC2#353×D61
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1409C2
socleC2×C61C2×#353
prime radical#16840310C22
cosocle#127×C61C23
hypercenter#1409D61
hypocenterC61#13586
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1409

Complements

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

NH ≅ G/Ntype
3. #21443984C2semidirect
4. #21443983C2semidirect
5. C61×#1409C2semidirect
8. C4⋊Q8×D61C4semidirect
10. #16840142C4semidirect
11. #16840141C4semidirect
14. C4×#11758189C4semidirect
125. #1409D61semidirect
143. C61#13586semidirect

Maximal subgroups

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

subgroupindex
#214562732
#214474812
#214439842
#214439832
C61×#14092
#1358661