Finite Group Databasebeta

← #29580883

Subgroups — #29580883

Order 15616.

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

normal subgroups only

This group has 528 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#21455858C212yesmaximal
2#21446800C211yesmaximal, characteristic
3#21444248C212yesmaximal
4#21443648C211yesmaximal, characteristic
5C61×#2399C211yesmaximal, characteristic
6#16840319C2211yescharacteristic
7#16840097—22no—
8#16840057—22no—
9#16839972—22no—
10#16839918—22no—
11C4×#11758202—24no—
12#16839815C2212yes—
13#16839786C2211yescharacteristic
14#16839522—22no—
15Q8⋊#92418—24no—
16C2×C4×Q8×C61C2212yes—
17C61×#398—22no—
18C61×#395—22no—
19C4×C4⋊C4×C61—22no—
20C61×#376C2211yescharacteristic
21C2×#11758202—24no—
22C2×#11758194—21no—
23#14519777D412yes—
24#14519769—21no—
25#14519756—22no—
26C4×Dic122—22no—
27#14519736—21no—
28#14519710—24no—
29C4⋊#92418—22no—
30C4⋊#92418—22no—
31#14519701—24no—
32#14519700—24no—
33C22⋊#92418—22no—
34C4×#92418D414yes—
35C22×Q8×C61D412yes—
36C4×Q8×C61—22no—
37C4×Q8×C61—22no—
38C4×Q8×C61—22no—
39C4×Q8×C61—22no—
40C2×C4⋊C4×C61—24no—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61D412yes—
43C2×C42×C61—22no—
44C2×C42×C61D412yes—
45C2×C42×C61C2311yescharacteristic
46#95×C61—22no—
47#95×C61—22no—
48#95×C61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—22no—
51C2×Q8×C61—44no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61C2×D412yes—
56C22×C4×C61C2×D411yescharacteristic
57C2×Dic122—22no—
58C4⋊C4×C61—22no—
59C4⋊C4×C61—22no—
60C4⋊C4×C61—48no—
61C4⋊C4×C61—42no—
62C4⋊C4×C61—42no—
63C42×C61—22no—
64C42×C61—42no—
65C42×C61—42no—
66C42×C61Q1614yescomplemented
67C42×C61C2×D411yescharacteristic
68C42×C61C2×D412yes—
69C42×C61#4111yescharacteristic
70C22⋊Dic61—41no—
71#11758202—48no—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74#11758196—42no—
75C4×Dic61—41no—
76#11758194—42no—
77C2×#92418—24no—
78C2×#92418—41no—
79C23×C61#12011yescharacteristic
80Q8×C61—24no—
81Q8×C61—24no—
82Q8×C61—42no—
83Q8×C61—42no—
84C2×C4×C61—24no—
85C2×C4×C61—24no—
86C2×C4×C61—42no—
87C2×C4×C61—44no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61—44no—
91C2×C4×C61—42no—
92C2×C4×C61#12012yes—
93C2×C4×C61#12112yes—
94C2×C4×C61#12111yescharacteristic
95C2×C4×C61#12711yescharacteristic
96C2×C4×C61C2×Q1612yes—
97C2×C4×C61#13712yes—
98C2×Dic61—42no—
99Dic122—44no—
100#92418—44no—
101#92418—81no—
102#10017—611nomaximal, Sylow
103C22×C61—22no—
104C22×C61#39211yescharacteristic
105C22×C61#45112yes—
106C4×C61—22no—
107C4×C61—22no—
108C4×C61—44no—
109C4×C61—44no—
110C4×C61—82no—
111C4×C61#46212yes—
112C4×C61#49411yescharacteristic
113C4×C61#49711yescharacteristic
114Dic61—82no—
115#3104—612no—
116#2399D6111yescharacteristic
117#1995—611no—
118#1487—612no—
119#1274—611no—
120C2×C61—22no—
121C2×C61#201012yes—
122C2×C61#201611yescharacteristic
123#533—611no—
124C2×C4×Q8C2×D6112yes—
125#484—1222no—
126#477—1222no—
127#462—1222no—
128#451—1222no—
129C4×Q16—1224no—
130#431—612no—
131#423—611no—
132#398—22no—
133#395—22no—
134C4×C4⋊C4—22no—
135#376C2×D6111yescharacteristic
136#339—1222no—
137Q8⋊C8—1224no—
138C61#1001711yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D6112yes—
140C2×Q16—1224no—
141C2×M16—1221no—
142C4⋊Q8—612no—
143#125—1221no—
144C22⋊Q8—1222no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—1222no—
150#117—1221no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D6112yes—
154C2×C42—22no—
155C2×C42C22⋊D6112yes—
156C2×C42C22×D6111yescharacteristic
157#107—1224no—
158C4⋊C8—1222no—
159C4⋊C8—1222no—
160#103—1224no—
161#103—1224no—
162C22⋊C8—1222no—
163C4×C8—614no—
164#95—22no—
165#95—22no—
166#95—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—1222no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×C22⋊D6112yes—
175C22×C4C2×C22⋊D6111yescharacteristic
176Q16—2448no—
177M16—2442no—
178C2×C8—1224no—
179C2×C8—2441no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—1222no—
186C4⋊C4—1222no—
187C4⋊C4—2442no—
188C22⋊C4—2441no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—2441no—
193C42#1175820214yescomplemented
194C42D4×D6111yescharacteristic
195C42D4×D6112yes—
196C42#1175822411yescharacteristic
197C23#1451974611yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—2444no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—2442no—
212C2×C4#1451974512yes—
213C2×C4#1451974811yescharacteristic
214C2×C4#1451974912yes—
215C2×C4#1451977311yescharacteristic
216C2×C4C2×#1175820212yes—
217C2×C4#1451980812yes—
218C8—4881no—
219C8—2444no—
220C22—22no—
221C22#1683967011yescharacteristic
222C22#1683991812yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—4882no—
229C4#1683996912yes—
230C4#1684014111yescharacteristic
231C4#1684015411yescharacteristic
232C2—22no—
233C2#2144688412yes—
234C2#2144691911yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2399C2
socleC22×C61#392
prime radicalC23×C61#120
hypercenter#2399D61
hypocenterC61#10017
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
66. C42×C61Q16semidirect
138. C61#10017semidirect
193. C42#11758202semidirect

Maximal subgroups

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

subgroupindex
#214558582
#214468002
#214442482
#214436482
C61×#23992
#1001761