Finite Group Databasebeta

← #29581009

Subgroups — #29581009

Order 15616.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#21456343C211yesmaximal, characteristic
2#21455859C212yesmaximal
3#21446759C211yesmaximal, characteristic
4#21444223C212yesmaximal
5C61×#1275C211yesmaximal, characteristic
6#16840319C2212yes—
7C2×C4×Dic122C2211yescharacteristic
8#16840171—22no—
9#16840171—22no—
10#16840098—22no—
11#16840058—22no—
12#16839973—22no—
13#16839920—22no—
14C4×Dic244—24no—
15#16839800—22no—
16C2×C4⋊#92418C2211yescharacteristic
17#16839516—22no—
18#16839494—24no—
19C61×#431C2212yes—
20C2×#97×C61C2211yescharacteristic
21C22×Dic122D411yescharacteristic
22C2×Dic244—24no—
23C22×#92418—21no—
24#14519777D412yes—
25#14519777D412yes—
26#14519769—22no—
27#14519756—22no—
28#14519756—22no—
29C4×Dic122—22no—
30C4×Dic122—22no—
31C4×Dic122—22no—
32C4×Dic122—22no—
33#14519736—22no—
34C2×C4⋊Dic61D411yescharacteristic
35C2×#11758196—21no—
36C2×C4×Dic61—21no—
37#14519711—22no—
38#14519711—22no—
39#14519711—24no—
40#14519709—24no—
41C4⋊#92418—22no—
42#14519702—24no—
43#14519702—24no—
44C22×C8×C61D412yes—
45C2×C42×C61C2311yescharacteristic
46C4⋊C8×C61—22no—
47C22⋊C8×C61—22no—
48#97×C61—22no—
49C4×C8×C61D414yes—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D411yescharacteristic
52C22×C4×C61#4111yescharacteristic
53C22×Dic61—21no—
54C22×Dic61—21no—
55C2×Dic122—22no—
56C2×Dic122—22no—
57C2×Dic122—22no—
58C2×Dic122—42no—
59C2×C8×C61—22no—
60C2×C8×C61—22no—
61C2×C8×C61—24no—
62C2×C8×C61Q1614yescomplemented
63C42×C61C2×D412yes—
64C42×C61C2×D412yes—
65C22⋊Dic61—42no—
66C4⋊Dic61—22no—
67C4⋊Dic61—22no—
68C4⋊Dic61—22no—
69C4⋊Dic61—22no—
70C4⋊Dic61—22no—
71#11758196—42no—
72#11758196—42no—
73#11758196—42no—
74C4×Dic61—41no—
75C4×Dic61—41no—
76C4×Dic61—42no—
77C2×#92418—41no—
78C2×#92418—42no—
79Dic244—48no—
80C23×C61#12111yescharacteristic
81C2×C4×C61—41no—
82C2×C4×C61—41no—
83C2×C4×C61#12012yes—
84C2×C4×C61#12011yescharacteristic
85C2×C4×C61#12112yes—
86C2×C4×C61#12711yescharacteristic
87C2×C4×C61C2×Q1612yes—
88C2×C4×C61#13712yes—
89C2×Dic61—22no—
90C2×Dic61—41no—
91C2×Dic61—41no—
92C2×Dic61—42no—
93C2×Dic61—41no—
94C2×Dic61—42no—
95C2×Dic61—42no—
96Dic122—24no—
97Dic122—42no—
98Dic122—44no—
99Dic122—44no—
100C8×C61—24no—
101C8×C61—44no—
102#92418—81no—
103#10037—611nomaximal, Sylow
104C22×C61—21no—
105C22×C61—21no—
106C22×C61#39211yescharacteristic
107C22×C61#46212yes—
108C4×C61—22no—
109C4×C61—22no—
110C4×C61#45112yes—
111C4×C61#49411yescharacteristic
112C4×C61#49711yescharacteristic
113Dic61—42no—
114Dic61—82no—
115Dic61—82no—
116Dic61—81no—
117#3162—611no—
118#3104—612no—
119#1985—611no—
120#1481—612no—
121#1275D6111yescharacteristic
122C2×C61—22no—
123C2×C61#201011yescharacteristic
124C2×C61#201611yescharacteristic
125C2×C61#201711yescharacteristic
126#533—612no—
127C2×C4×Q8—611no—
128#501—1222no—
129#501—1222no—
130#484—1222no—
131#477—1222no—
132#462—1222no—
133#451—1222no—
134C4×Q16—1224no—
135#431C2×D6112yes—
136#427—1222no—
137C2×C4⋊C8—611no—
138C2×#97C2×D6111yescharacteristic
139#336—1222no—
140Q8⋊C8—1224no—
141C61#1003711yescharacteristic, complemented, Sylow
142C22×Q8—611no—
143C2×Q16—1224no—
144C22×C8—1221no—
145C22×C8D24412yes—
146C4⋊Q8—612no—
147C4⋊Q8—612no—
148#125—1222no—
149C22⋊Q8—1222no—
150C22⋊Q8—1222no—
151C4×Q8—1222no—
152C4×Q8—1222no—
153C4×Q8—1222no—
154C4×Q8—1222no—
155#117—1222no—
156C2×C4⋊C4—611no—
157C2×C4⋊C4—1221no—
158C2×C42—1221no—
159C2×C42C22×D6111yescharacteristic
160#107—1222no—
161#107—1222no—
162#107—1224no—
163#106—1224no—
164C4⋊C8—22no—
165C4⋊C8—1222no—
166#103—1224no—
167#103—1224no—
168C22⋊C8—22no—
169#97—22no—
170C4×C8D24414yes—
171C2×Q8—1222no—
172C2×Q8—1222no—
173C2×Q8—1222no—
174C2×Q8—2442no—
175C22×C4—1221no—
176C22×C4—1221no—
177C22×C4C2×D24411yescharacteristic
178C22×C4D4×D6111yescharacteristic
179C22×C4#1175822611yescharacteristic
180Q16—2448no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—24no—
184C2×C8—2442no—
185C2×C8—2441no—
186C2×C8Dic24414yescomplemented
187C4⋊C4—1222no—
188C4⋊C4—1222no—
189C4⋊C4—1222no—
190C4⋊C4—1222no—
191C4⋊C4—1222no—
192C4⋊C4—2442no—
193C4⋊C4—2442no—
194C4⋊C4—2442no—
195C22⋊C4—2442no—
196C42—2441no—
197C42—2441no—
198C42—2442no—
199C42C2×D24412yes—
200C42D4×D6112yes—
201C23#1451975211yescharacteristic
202Q8—1224no—
203Q8—2444no—
204Q8—2444no—
205Q8—2442no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—1222no—
209C2×C4—2442no—
210C2×C4—2441no—
211C2×C4—2442no—
212C2×C4—2442no—
213C2×C4—2441no—
214C2×C4—2441no—
215C2×C4#1451974712yes—
216C2×C4#1451974711yescharacteristic
217C2×C4#1451975212yes—
218C2×C4#1451977411yescharacteristic
219C2×C4C2×Dic24412yes—
220C2×C4#1451981012yes—
221C8—24no—
222C8—44no—
223C8—4881no—
224C22—21no—
225C22—21no—
226C22#1683967211yescharacteristic
227C22#1683997312yes—
228C4—22no—
229C4—22no—
230C4—4881no—
231C4—4882no—
232C4—4882no—
233C4—2442no—
234C4#1683992012yes—
235C4#1684014211yescharacteristic
236C4#1684015611yescharacteristic
237C2—22no—
238C2#2144688711yescharacteristic
239C2#2144692111yescharacteristic
240C2#2144692411yescharacteristic
subgroupG/H
centerC2×C4#14519774
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1275C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#1275D61
hypocenterC61#10037
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
62. C2×C8×C61Q16semidirect
141. C61#10037semidirect
186. C2×C8Dic244semidirect

Maximal subgroups

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

subgroupindex
#214563432
#214558592
#214467592
#214442232
C61×#12752
#1003761