Finite Group Databasebeta

← #29580640

Subgroups — #29580640

Order 15616.

492 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 85 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#21455763C212yesmaximal
2#21445888C211yesmaximal, characteristic
3#21444183C212yesmaximal
4#21443696C211yesmaximal, characteristic
5C61×#3189C211yesmaximal, characteristic
6C2×#14519736C2211yescharacteristic
7#16840100—24no—
8#16840095—22no—
9#16840090—22no—
10#16839802—22no—
11#16839771—22no—
12#16839726C2212yes—
13C2×C22⋊#92418C2211yescharacteristic
14#16839516—22no—
15#16839488—24no—
16C2×C22⋊Q8×C61C2212yes—
17C61×#484—22no—
18C61×#467—22no—
19C61×#451—22no—
20C2×C22⋊C8×C61C2211yescharacteristic
21C2×#11758194—22no—
22C22×#92418—21no—
23#14519736—22no—
24#14519736—22no—
25C2×C4⋊Dic61D411yescharacteristic
26C2×C4⋊Dic61D411yescharacteristic
27C2×C22⋊Dic61—21no—
28C2×C4×Dic61—21no—
29#14519710—24no—
30#14519708—24no—
31#14519701—24no—
32#14519701—24no—
33C22⋊#92418—22no—
34C22⋊#92418—22no—
35C22⋊#92418D414yes—
36C22×Q8×C61D412yes—
37C23×C4×C61C2311yescharacteristic
38C2×Q16×C61—22no—
39C22×C8×C61D412yes—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—22no—
43C22⋊Q8×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61D412yes—
46C2×C22⋊C4×C61—22no—
47#107×C61—22no—
48#103×C61—24no—
49C22⋊C8×C61—22no—
50C24×C61#4111yescharacteristic
51C2×Q8×C61—22no—
52C2×Q8×C61—22no—
53C2×Q8×C61—44no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—42no—
57C22×C4×C61C2×D412yes—
58C22×C4×C61C2×D411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D412yes—
61C22×Dic61—21no—
62C22×Dic61—21no—
63Q16×C61—82no—
64C2×C8×C61—22no—
65C2×C8×C61—42no—
66C4⋊C4×C61—22no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—42no—
69C4⋊C4×C61—42no—
70C22⋊C4×C61—42no—
71C22⋊C4×C61—42no—
72C22⋊Dic61—42no—
73C4⋊Dic61—22no—
74C4⋊Dic61—22no—
75C4×Dic61—41no—
76C4×Dic61—41no—
77#11758194—44no—
78C2×#92418—24no—
79C2×#92418—42no—
80C2×#92418—41no—
81C23×C61—22no—
82C23×C61—41no—
83C23×C61—41no—
84C23×C61#12112yes—
85C23×C61#12111yescharacteristic
86Q8×C61—44no—
87Q8×C61—44no—
88C2×C4×C61—22no—
89C2×C4×C61—24no—
90C2×C4×C61—41no—
91C2×C4×C61—42no—
92C2×C4×C61—44no—
93C2×C4×C61—42no—
94C2×C4×C61—42no—
95C2×C4×C61—82no—
96C2×C4×C61#12012yes—
97C2×C4×C61#12011yescharacteristic
98C2×C4×C61#12711yescharacteristic
99C2×Dic61—42no—
100C2×Dic61—41no—
101C2×Dic61—41no—
102C2×Dic61—42no—
103C8×C61—42no—
104#92418—44no—
105#92418—81no—
106#9979—611nomaximal, Sylow
107C22×C61—21no—
108C22×C61—21no—
109C22×C61—22no—
110C22×C61—22no—
111C22×C61—22no—
112C22×C61—82no—
113C22×C61#39211yescharacteristic
114C22×C61#47312yes—
115C4×C61—22no—
116C4×C61—21no—
117C4×C61—82no—
118C4×C61—82no—
119C4×C61#45812yes—
120Dic61—81no—
121Dic61—81no—
122#3189D6111yescharacteristic
123#3095—612no—
124#1818—611no—
125#1472—612no—
126#1293—611no—
127C2×C61—22no—
128C2×C61—42no—
129C2×C61#201711yescharacteristic
130C2×C61#202011yescharacteristic
131C2×C61#202211yescharacteristic
132C2×C22⋊Q8C2×D6112yes—
133C2×#117—611no—
134#485—1224no—
135#484—22no—
136#484—1222no—
137#483—1222no—
138#467—22no—
139#451—22no—
140#428—1222no—
141#419—1222no—
142#407—612no—
143C2×C22⋊C8—611no—
144C2×C22⋊C8C2×D6111yescharacteristic
145#336—1222no—
146#324—1224no—
147C61#997911yescharacteristic, complemented, Sylow
148C22×Q8C22⋊D6112yes—
149C23×C4C22×D6111yescharacteristic
150C2×Q16—22no—
151C2×M16—1222no—
152C22×C8—1221no—
153C22×C8C22⋊D6112yes—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158#117—1222no—
159#117—1222no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—611no—
162C2×C4⋊C4—611no—
163C2×C4⋊C4C22⋊D6112yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—1221no—
166C2×C42—1221no—
167#107—22no—
168#107—1224no—
169#106—1224no—
170#103—24no—
171#103—1224no—
172#103—1224no—
173C22⋊C8—22no—
174C22⋊C8—614no—
175C22⋊C8—1222no—
176C22⋊C8—1222no—
177C24#1175822411yescharacteristic
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—44no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—1221no—
185C22×C4—1221no—
186C22×C4D4×D6111yescharacteristic
187C22×C4D4×D6112yes—
188C22×C4C2×C22⋊D6112yes—
189C22×C4C2×C22⋊D6111yescharacteristic
190Q16—82no—
191M16—2444no—
192C2×C8—22no—
193C2×C8—42no—
194C2×C8—1224no—
195C2×C8—2442no—
196C2×C8—2441no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—1222no—
202C4⋊C4—1222no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—2442no—
206C42—2441no—
207C42—2441no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23#1451974811yescharacteristic
212C23#1451974912yes—
213Q8—44no—
214Q8—44no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—82no—
223C2×C4—2442no—
224C2×C4—2442no—
225C2×C4—2441no—
226C2×C4—2441no—
227C2×C4#1451974512yes—
228C2×C4#1451974611yescharacteristic
229C2×C4#1451977311yescharacteristic
230C8—42no—
231C8—4881no—
232C8—2444no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—82no—
239C22#1683967011yescharacteristic
240C22#1684003412yes—
241C4—21no—
242C4—22no—
243C4—82no—
244C4—82no—
245C4—4881no—
246C4—4881no—
247C4#1683995212yes—
248C2—22no—
249C2—42no—
250C2#2144692211yescharacteristic
251C2#2144693811yescharacteristic
252C2#2144694811yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3189C2
socleC22×C61#392
prime radicalC24×C61#41
hypercenter#3189D61
hypocenterC61#9979
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
147. C61#9979semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214458882
#214441832
#214436962
C61×#31892
#997961