Finite Group Databasebeta

← #29581006

Subgroups — #29581006

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#21455854C212yesmaximal
2#21446759C211yesmaximal, characteristic
3#21444222C212yesmaximal
4#21443650C211yesmaximal, characteristic
5C61×#3162C211yesmaximal, characteristic
6C2×C4×Dic122C2211yescharacteristic
7#16840095—22no—
8#16840054—22no—
9#16839969—22no—
10#16839917—22no—
11C4×#11758202—24no—
12#16839815C2212yes—
13C2×C4⋊#92418C2211yescharacteristic
14#16839516—22no—
15Q8⋊#92418—24no—
16C61×#533C2212yes—
17C61×#501—22no—
18C61×#501—22no—
19C61×#427—22no—
20C2×#97×C61C2211yescharacteristic
21C22×Dic122D411yescharacteristic
22C2×#11758202—24no—
23C22×#92418—21no—
24C4×Dic122—22no—
25C4×Dic122—22no—
26C2×C4⋊Dic61D411yescharacteristic
27C2×#11758196—21no—
28C2×C4×Dic61—21no—
29#14519710—24no—
30C4⋊#92418—22no—
31C4⋊#92418—22no—
32#14519701—24no—
33#14519700—24no—
34C22⋊#92418—22no—
35C4×#92418D414yes—
36C22×C8×C61D412yes—
37C4⋊Q8×C61D412yes—
38C4⋊Q8×C61D412yes—
39#125×C61—22no—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—22no—
42C4×Q8×C61—22no—
43C4×Q8×C61—22no—
44#117×C61—22no—
45C2×C42×C61C2311yescharacteristic
46#107×C61—22no—
47#107×C61—22no—
48#106×C61—24no—
49#97×C61—22no—
50C2×Q8×C61—22no—
51C2×Q8×C61—22no—
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61C2×D411yescharacteristic
54C22×C4×C61#4111yescharacteristic
55C22×Dic61—21no—
56C22×Dic61—21no—
57C2×Dic122—22no—
58C2×Dic122—42no—
59C2×C8×C61—22no—
60C2×C8×C61—22no—
61C2×C8×C61Q1614yescomplemented
62C4⋊C4×C61—22no—
63C4⋊C4×C61—22no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—22no—
66C4⋊C4×C61—42no—
67C4⋊C4×C61—42no—
68C22⋊C4×C61—42no—
69C42×C61—42no—
70C42×C61C2×D412yes—
71C42×C61C2×D412yes—
72#11758202—48no—
73C4⋊Dic61—22no—
74#11758196—42no—
75C4×Dic61—41no—
76C4×Dic61—41no—
77C2×#92418—24no—
78C2×#92418—41no—
79C2×#92418—42no—
80C23×C61#12111yescharacteristic
81Q8×C61—44no—
82Q8×C61—44no—
83C2×C4×C61—41no—
84C2×C4×C61—41no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61#12012yes—
88C2×C4×C61#12011yescharacteristic
89C2×C4×C61#12112yes—
90C2×C4×C61#12711yescharacteristic
91C2×C4×C61C2×Q1612yes—
92C2×C4×C61#13712yes—
93C2×Dic61—22no—
94C2×Dic61—41no—
95C2×Dic61—41no—
96C2×Dic61—42no—
97C2×Dic61—41no—
98Dic122—24no—
99Dic122—42no—
100C8×C61—24no—
101#92418—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—82no—
111C4×C61—82no—
112C4×C61#45112yes—
113C4×C61#49411yescharacteristic
114C4×C61#49711yescharacteristic
115Dic61—42no—
116Dic61—81no—
117#3162D6111yescharacteristic
118#3104—612no—
119#1985—611no—
120#1481—612no—
121#1275—611no—
122C2×C61—22no—
123C2×C61#201011yescharacteristic
124C2×C61#201611yescharacteristic
125C2×C61#201711yescharacteristic
126#533C2×D6112yes—
127C2×C4×Q8—611no—
128#501—22no—
129#501—22no—
130#484—1222no—
131#477—1222no—
132#462—1222no—
133#451—1222no—
134C4×Q16—1224no—
135#431—612no—
136#427—22no—
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×C8C22⋊D6112yes—
146C4⋊Q8C22⋊D6112yes—
147C4⋊Q8C22⋊D6112yes—
148#125—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1222no—
154C4×Q8—1222no—
155#117—22no—
156C2×C4⋊C4—611no—
157C2×C4⋊C4—1221no—
158C2×C42—1221no—
159C2×C42C22×D6111yescharacteristic
160#107—22no—
161#107—22no—
162#107—1224no—
163#106—24no—
164C4⋊C8—1222no—
165C4⋊C8—1222no—
166#103—1224no—
167#103—1224no—
168C22⋊C8—1222no—
169#97—22no—
170C4×C8—614no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—1222no—
174C2×Q8—2442no—
175C22×C4—1221no—
176C22×C4—1221no—
177C22×C4D4×D6111yescharacteristic
178C22×C4#1175822411yescharacteristic
179C22×C4C2×C22⋊D6111yescharacteristic
180Q16—2448no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—1224no—
184C2×C8—2442no—
185C2×C8—2441no—
186C2×C8#1175820214yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—1222no—
194C4⋊C4—2442no—
195C22⋊C4—42no—
196C42—42no—
197C42—2441no—
198C42—2441no—
199C42D4×D6112yes—
200C42C2×C22⋊D6112yes—
201C23#1451974811yescharacteristic
202Q8—44no—
203Q8—44no—
204Q8—1224no—
205Q8—2442no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—1222no—
211C2×C4—2442no—
212C2×C4—2441no—
213C2×C4—2441no—
214C2×C4—2441no—
215C2×C4#1451974512yes—
216C2×C4#1451974611yescharacteristic
217C2×C4#1451974912yes—
218C2×C4#1451977311yescharacteristic
219C2×C4C2×#1175820212yes—
220C2×C4#1451980812yes—
221C8—24no—
222C8—4881no—
223C8—2444no—
224C22—21no—
225C22—21no—
226C22#1683967011yescharacteristic
227C22#1683996912yes—
228C4—22no—
229C4—22no—
230C4—82no—
231C4—82no—
232C4—4881no—
233C4—2442no—
234C4#1683991812yes—
235C4#1684014111yescharacteristic
236C4#1684015411yescharacteristic
237C2—22no—
238C2#2144688311yescharacteristic
239C2#2144691911yescharacteristic
240C2#2144692211yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3162C2
socleC22×C61#392
prime radicalC23×C61#121
hypercenter#3162D61
hypocenterC61#10037
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
61. C2×C8×C61Q16semidirect
141. C61#10037semidirect
186. C2×C8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214558542
#214467592
#214442222
#214436502
C61×#31622
#1003761