Finite Group Databasebeta

← #30595149

Subgroups — #30595149

Order 15616.

298 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 44 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#21461824C211yesmaximal, characteristic
2D61⋊#329C211yesmaximal, characteristic
3#355×D61C211yesmaximal, characteristic
4#16840829C411yescharacteristic
5#16840755—21no—
6D61⋊#100C411yescharacteristic
7#16840721—21no—
8#16840633—21no—
9#16840618—21no—
10#16840617C412yescomplemented
11#124×D61C2211yescharacteristic
12#101×D61—21no—
13#16839559C411yescharacteristic, complemented
14#16839558—21no—
15C61×#355C411yescharacteristic, complemented
16D61⋊M16—21no—
17C2×D61⋊C8—41no—
18D4×#6923—42no—
19#14519856—21no—
20C2×C22⋊#6923—41no—
21#14519840—42no—
22#14519837—41no—
23#14519832—22no—
24#14519832—41no—
25C2×Q8×D61D411yescharacteristic
26C2×D4×D61C2×C411yescharacteristic
27M16×D61—41no—
28#14519766D411yescharacteristic
29#14519765Q811yescharacteristic
30#14519764C2×C411yescharacteristic
31#14519763—41no—
32C22⋊C4×D61—41no—
33C42×D61D411yescharacteristic
34#14519696—21no—
35#14519695—41no—
36#124×C61C2×C411yescharacteristic
37#101×C61—21no—
38C23×D61—21no—
39C22×#6923—42no—
40C2×Q8×C61C22⋊C411yescharacteristic
41C2×D4×C61C4211yescharacteristic
42C2×C22⋊D61—21no—
43Q8×D61—41no—
44Q8×D61—41no—
45D4×D61—41no—
46D4×D61—41no—
47C2×D244C4⋊C411yescharacteristic
48C2×C4×D61—21no—
49C2×C4×D61—21no—
50C2×C4×D61C22⋊C411yescharacteristic
51C2×Dic122—21no—
52C2×Dic122C4⋊C411yescharacteristic
53C22⋊#6923—42no—
54C22⋊#6923—41no—
55#11758217—41no—
56C2×#92420—42no—
57C4⋊#6923—22no—
58C4×#6923—41no—
59#11758213—41no—
60D61⋊C8—44no—
61D61⋊C8—41no—
62M16×C61—41no—
63C22⋊C4×C61—41no—
64C42×C61C4⋊C411yescharacteristic
65C22⋊Dic61—41no—
66#11758198—41no—
67#11758198—41no—
68C4×Dic61—21no—
69C4×Dic61C22⋊C411yescharacteristic
70#11758194—41no—
71#11758189—81no—
72C8×D61—81no—
73C23×C61—21no—
74C22×D61—21no—
75C22×D61—41no—
76C22×D61—41no—
77C22×D61—82no—
78C22×D61#9911yescharacteristic
79C2×#6923—42no—
80C2×#6923—162no—
81Q8×C61—41no—
82D4×C61—41no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61#9511yescharacteristic
86C22⋊D61—42no—
87C2×Dic61—21no—
88C2×Dic61—21no—
89C2×Dic61#9911yescharacteristic
90D244—41no—
91C4×D61—21no—
92C4×D61—21no—
93C4×D61—42no—
94C4×D61—41no—
95C4×D61—41no—
96Dic122—41no—
97Dic122—81no—
98#92420—44no—
99#92420—81no—
100C8×C61—81no—
101#92418—81no—
102#7487—611nomaximal, Sylow
103C22×C61—41no—
104C22×C61—81no—
105C22×C61#34211yescharacteristic
106C2×D61—22no—
107C2×D61—41no—
108C2×D61—82no—
109C2×D61—81no—
110C2×D61—161no—
111C2×D61AΣL(1,𝔽16)11yescharacteristic
112C2×D61#35211yescharacteristic
113#6923—82no—
114C4×C61—21no—
115C4×C61—41no—
116C4×C61—41no—
117Dic61—21no—
118Dic61—41no—
119Dic61—41no—
120C2×#355—611no—
121#1885—611no—
122C2×#329—611no—
123C2×C61—21no—
124C2×C61—81no—
125C2×C61#139011yescharacteristic
126D61—21no—
127D61—81no—
128D61#140312yes—
129C2×#124—611no—
130#518—611no—
131#419—1221no—
132C2×#101—1221no—
133C2×#100—611no—
134C2×#97—1221no—
135#355—611no—
136#355—1221no—
137#355#692311yescharacteristic, complemented
138#339—1221no—
139#329—612no—
140#329—1221no—
141C61#748711yescharacteristic, complemented, Sylow
142C22×Q8—611no—
143C22×D4—611no—
144C2×M16—1221no—
145C2×M16—2441no—
146C22×C8—2441no—
147#124—611no—
148#124—611no—
149#124—611no—
150#124—2441no—
151#124C2×#692311yescharacteristic
152C4×D4—2442no—
153#117—1221no—
154C2×C22⋊C4—2441no—
155C2×C22⋊C4—2441no—
156C2×C42—611no—
157#103—2442no—
158#101—21no—
159#101—1221no—
160#101—2441no—
161#100—2441no—
162#97—1222no—
163#97—2441no—
164C24—1221no—
165C2×Q8—611no—
166C2×Q8—1221no—
167C2×Q8—2441no—
168C2×Q8—2441no—
169C2×Q8C22⋊#692311yescharacteristic
170C2×D4—611no—
171C2×D4—1221no—
172C2×D4—2441no—
173C2×D4—2441no—
174C2×D4C4×#692311yescharacteristic
175C22×C4—611no—
176C22×C4—1221no—
177C22×C4—1221no—
178C22×C4—2442no—
179M16—41no—
180M16—4881no—
181M16—2441no—
182M16—2441no—
183M16—2441no—
184C2×C8—4881no—
185C2×C8—2441no—
186C2×C8—2442no—
187C2×C8—2444no—
188C4⋊C4—1222no—
189C22⋊C4—41no—
190C22⋊C4—2441no—
191C22⋊C4—2441no—
192C22⋊C4—2441no—
193C22⋊C4—2441no—
194C22⋊C4—2442no—
195C42—611no—
196C42—1221no—
197C42—2441no—
198C42C4⋊#692311yescharacteristic
199C23—21no—
200C23—611no—
201C23—1221no—
202C23—4882no—
203C23—2441no—
204C23—2441no—
205Q8—41no—
206Q8—4881no—
207Q8—2441no—
208D4—41no—
209D4—2441no—
210D4—2442no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—611no—
214C2×C4—1221no—
215C2×C4—1221no—
216C2×C4—1221no—
217C2×C4—1221no—
218C2×C4—9762no—
219C2×C4—2441no—
220C2×C4—2441no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4(C2×#6923)⋊C411yescharacteristic
224C8—81no—
225C8—4881no—
226C8—4881no—
227C8—2444no—
228C22—41no—
229C22—81no—
230C22—611no—
231C22—611no—
232C22—1222no—
233C22—9761no—
234C22—4881no—
235C22—4882no—
236C22—2441no—
237C22#1684063911yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—1221no—
242C4—4882no—
243C4—2441no—
244C4—2441no—
245C2—21no—
246C2—81no—
247C2—612no—
248C2—1221no—
249C2—4881no—
250C2#2146090211yescharacteristic
subgroupG/H
centerC2#21460902
commutatorC2×D4×C61C42
Frattini#124C2×#6923
FittingC61×#355C4
socleC2×C61#1390
prime radicalC2×D4×D61C2×C4
cosocle#124×D61C22
hypercenter#355#6923
hypocenterC61#7487
LCSG ▷ C2×D4×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #355

Complements

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

NH ≅ G/Ntype
10. #16840617C4semidirect
13. #16839559C4semidirect
15. C61×#355C4semidirect
137. #355#6923semidirect
141. C61#7487semidirect

Maximal subgroups

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

subgroupindex
#214618242
D61⋊#3292
#355×D612
#748761