Finite Group Databasebeta

← #29589512

Subgroups — #29589512

Order 15616.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#21454516C211yesmaximal, characteristic
2#21447632C212yesmaximal, complemented
3#21443757C211yesmaximal, characteristic, complemented
4#21443754C212yesmaximal, complemented
5C61×#1321C211yesmaximal, characteristic, complemented
6C2×#14519784C411yescharacteristic
7C22×C8×D61C411yescharacteristic
8#16840269C411yescharacteristic
9#16839836—22no—
10#16839825—22no—
11#16839788C411yescharacteristic
12#16839725C2211yescharacteristic
13#16839720C2211yescharacteristic
14#16839582—22no—
15#16839544C2212yes—
16#16839539—22no—
17C61×#408C2211yescharacteristic
18C61×#349C2212yes—
19C22×C4×D61C2×C411yescharacteristic
20#14519784—42no—
21C2×#11758194—21no—
22C2×#11758194C2×C411yescharacteristic
23C22×#92418—21no—
24C22×#92418C2×C411yescharacteristic
25C2×C8×D61—21no—
26C2×C8×D61—21no—
27C2×C8×D61—42no—
28#14519758—21no—
29#14519753—21no—
30#14519748—21no—
31#14519745—21no—
32C4×C22⋊D61—22no—
33#14519736C2×C411yescharacteristic
34#14519717—22no—
35#14519706—22no—
36#14519689—22no—
37#14519687D412yes—
38C8×Dic61D412yes—
39C2×#41×C61C2×C411yescharacteristic
40C2×M16×C61C2311yescharacteristic
41C22×C8×C61C2×C411yescharacteristic
42M32×C61—22no—
43C22⋊C8×C61—22no—
44C23×D61M1611yescharacteristic
45#41×C61—41no—
46#41×C61—41no—
47C2×Q8×C61M1611yescharacteristic
48C2×D4×C61—21no—
49C2×D4×C61M1611yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61C22×C411yescharacteristic
52C2×C22⋊D61—22no—
53C22×Dic61M1611yescharacteristic
54C2×C4×D61—21no—
55C2×C4×D61—21no—
56C2×C4×D61—41no—
57C2×C4×D61—41no—
58M16×C61—41no—
59C2×C8×C61—21no—
60C2×C8×C61—41no—
61C2×C8×C61C2×D412yes—
62C22⋊Dic61—22no—
63C22⋊Dic61—21no—
64#11758198—22no—
65C4⋊Dic61—21no—
66#11758196—22no—
67C4×Dic61C22⋊C412yes—
68#11758194—42no—
69#11758194—41no—
70C2×#92418—22no—
71C2×#92418—21no—
72C2×#92418—42no—
73C2×#92418C22⋊C411yescharacteristic
74C2×#92418C22⋊C411yescharacteristic
75C8×D61—42no—
76C8×D61—42no—
77C16×C61—42no—
78#11758185—42no—
79C23×C61—21no—
80C23×C61C2×M1611yescharacteristic
81C22×D61—21no—
82C22×D61—42no—
83C22×D61—41no—
84Q8×C61—41no—
85D4×C61—41no—
86D4×C61—42no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—21no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61C2×C22⋊C411yescharacteristic
93C2×C4×C61C2×M1611yescharacteristic
94C22⋊D61—44no—
95C2×Dic61—22no—
96C2×Dic61—21no—
97C2×Dic61—41no—
98C4×D61—42no—
99C4×D61—41no—
100C4×D61—41no—
101C8×C61—22no—
102C8×C61—41no—
103#92418—21no—
104#92418—41no—
105#92418#10012yes—
106#11537—611nomaximal, Sylow
107C22×C61—21no—
108C22×C61—41no—
109C22×C61—42no—
110C22×C61—41no—
111C22×C61#40711yescharacteristic
112C2×D61—22no—
113C2×D61—41no—
114C2×D61—42no—
115C2×D61—41no—
116C2×D61—42no—
117C4×C61—21no—
118C4×C61—41no—
119C4×C61#41011yescharacteristic
120C4×C61C2×#10011yescharacteristic
121Dic61—42no—
122Dic61—41no—
123#2929—611no—
124#2166—612no—
125#1321—611no—
126#1321D6111yescharacteristic, complemented
127#1320—612no—
128C2×C61—21no—
129C2×C61—41no—
130C2×C61—81no—
131C2×C61#146611yescharacteristic
132D61—42no—
133C2×#131—611no—
134C23×C8—611no—
135#525—611no—
136#435—1222no—
137C8×D4—1222no—
138#424—611no—
139#408C2×D6111yescharacteristic
140#407—611no—
141#405—611no—
142#364—1222no—
143#349—612no—
144#349C2×D6112yes—
145#347—1222no—
146C61#1153711yescharacteristic, complemented, Sylow
147C2×#41C4×D6111yescharacteristic
148C23×C4—611no—
149#131—2442no—
150C2×M16—611no—
151C2×M16—1221no—
152C2×M16C22×D6111yescharacteristic
153C22×C8—611no—
154C22×C8—1221no—
155C22×C8—1221no—
156C22×C8—1221no—
157C22×C8—2442no—
158C22×C8C4×D6111yescharacteristic
159#123—1221no—
160C22⋊Q8—1221no—
161#121—1221no—
162#120—1221no—
163C4×D4—1222no—
164#117—611no—
165M32—22no—
166M32—1222no—
167C4⋊C8—1222no—
168C22⋊C8—22no—
169C22⋊C8—1222no—
170#97—612no—
171C4×C8—612no—
172C24—611no—
173#41—41no—
174#41—41no—
175C2×Q8#1175818911yescharacteristic
176C2×D4—21no—
177C2×D4—1222no—
178C2×D4#1175818911yescharacteristic
179C22×C4—21no—
180C22×C4—611no—
181C22×C4—1221no—
182C22×C4—1221no—
183C22×C4—2441no—
184C22×C4—2441no—
185C22×C4C2×C4×D6111yescharacteristic
186M16—41no—
187M16—2442no—
188M16—2441no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—611no—
192C2×C8—611no—
193C2×C8—1222no—
194C2×C8—1221no—
195C2×C8—2442no—
196C2×C8—2442no—
197C2×C8—2442no—
198C2×C8D4×D6112yes—
199C4⋊C4—1222no—
200C4⋊C4—1221no—
201C22⋊C4—1222no—
202C22⋊C4—1222no—
203C22⋊C4—1221no—
204C42—612no—
205C16—42no—
206C16—2442no—
207C23—21no—
208C23—1221no—
209C23—2441no—
210C23—2442no—
211C23M16×D6111yescharacteristic
212Q8—41no—
213D4—41no—
214D4—42no—
215D4—2444no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—1222no—
222C2×C4—1221no—
223C2×C4—2442no—
224C2×C4—2441no—
225C2×C4—2441no—
226C2×C4—2441no—
227C2×C4C22⋊C4×D6111yescharacteristic
228C2×C4C2×#1175818911yescharacteristic
229C8—22no—
230C8—41no—
231C8—612no—
232C8—1221no—
233C8—2441no—
234C22—21no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—1222no—
239C22—2441no—
240C22—2442no—
241C22—2442no—
242C22—2441no—
243C22#1683972411yescharacteristic
244C4—21no—
245C4—41no—
246C4—2442no—
247C4—2441no—
248C4#1683973411yescharacteristic
249C4#100×D6111yescharacteristic
250C2—21no—
251C2—41no—
252C2—81no—
253C2—2442no—
254C2#2144415811yescharacteristic
subgroupG/H
centerC4#16839734
commutatorC22×C4×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1321C2
socleC2×C61#1466
prime radical#16840269C4
cosocleC2×M16×C61C23
hypercenter#1321D61
hypocenterC61#11537
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #21447632C2semidirect
3. #21443757C2semidirect
4. #21443754C2semidirect
5. C61×#1321C2semidirect
126. #1321D61semidirect
146. C61#11537semidirect

Maximal subgroups

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

subgroupindex
#214545162
#214476322
#214437572
#214437542
C61×#13212
#1153761