Finite Group Databasebeta

← #29596897

Subgroups — #29596897

Order 15616.

488 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 135 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#21454511C211yesmaximal, characteristic
2#21447440C211yesmaximal, characteristic, complemented
3#21447429C212yesmaximal, complemented
4#21443904C212yesmaximal, complemented
5C61×#3407C211yesmaximal, characteristic, complemented
6C22×C8×D61C2211yescharacteristic
7#16840269C411yescharacteristic
8C2×#14519717C2212yes—
9#16839831—22no—
10C8×C22⋊D61—22no—
11#16839790C411yescharacteristic
12#16839729C411yescharacteristic
13#16839725C411yescharacteristic
14#16839717C2211yescharacteristic
15#16839544—22no—
16#16839541—24no—
17C2×#131×C61C2211yescharacteristic
18C61×#504—24no—
19C2×M32×C61—22no—
20C2×M32×C61C2212yes—
21C22×C4×D61C2×C411yescharacteristic
22C2×#11758194C2×C411yescharacteristic
23C22×#92418C2×C411yescharacteristic
24C2×C8×D61—22no—
25C2×C8×D61—21no—
26C2×C8×D61—21no—
27C2×C8×D61—22no—
28#14519758C812yes—
29#14519753—21no—
30#14519748—21no—
31#14519745C812yes—
32C4×C22⋊D61—22no—
33#14519736C2×C411yescharacteristic
34#14519717—22no—
35#14519717—22no—
36C2×#11758185—22no—
37#14519706—22no—
38C22⋊#92418—22no—
39#14519689—22no—
40#14519687—21no—
41C8×Dic61—21no—
42C2×#41×C61C2×C411yescharacteristic
43#131×C61—22no—
44#131×C61—22no—
45C2×M16×C61C2×C411yescharacteristic
46C2×M16×C61D412yes—
47C22×C8×C61D412yes—
48C22×C8×C61C2311yescharacteristic
49M32×C61—24no—
50M32×C61—22no—
51M32×C61—22no—
52C2×C16×C61—24no—
53C2×C16×C61D414yescomplemented
54C23×D61C2×C811yescharacteristic
55#41×C61—22no—
56#41×C61—22no—
57C2×Q8×C61M1611yescharacteristic
58C2×D4×C61C2×C812yes—
59C2×D4×C61M1611yescharacteristic
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C2×C22⋊D61—22no—
63C22×Dic61C2×C811yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61—22no—
66C2×C4×D61—21no—
67C2×C4×D61—22no—
68M16×C61—22no—
69M16×C61—24no—
70C2×C8×C61—22no—
71C2×C8×C61—22no—
72C2×C8×C61C22⋊C412yes—
73C2×C8×C61C2×D412yes—
74C2×C8×C61C2×D411yescharacteristic
75C2×C8×C61C2×D411yescharacteristic
76C2×C8×C61#4112yes—
77C22⋊Dic61—22no—
78C22⋊Dic61C2×C812yes—
79#11758198—22no—
80C4⋊Dic61—21no—
81#11758196—22no—
82C4×Dic61—21no—
83#11758194—22no—
84C2×#92418—21no—
85C2×#92418—21no—
86C2×#92418—22no—
87C8×D61—41no—
88C8×D61—42no—
89C8×D61—41no—
90C16×C61—24no—
91#11758185—42no—
92C23×C61C22⋊C812yes—
93C23×C61C22×C811yescharacteristic
94C22×D61—21no—
95C22×D61—22no—
96C22×D61—42no—
97Q8×C61—22no—
98D4×C61—24no—
99D4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61C22⋊C812yes—
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61C4×D412yes—
105C2×C4×C61C2×M1611yescharacteristic
106C2×C4×C61#13112yes—
107C22⋊D61—44no—
108C2×Dic61—22no—
109C2×Dic61—21no—
110C2×Dic61—22no—
111C4×D61—41no—
112C4×D61—41no—
113C4×D61—42no—
114C8×C61—22no—
115C8×C61#12011yescharacteristic
116C8×C61#12112yes—
117C8×C61#12311yescharacteristic
118#92418—41no—
119#92418—41no—
120#12981—611nomaximal, Sylow
121C22×C61—22no—
122C22×C61—41no—
123C22×C61—42no—
124C22×C61C2×C22⋊C811yescharacteristic
125C22×C61C8×D412yes—
126C2×D61—22no—
127C2×D61—24no—
128C2×D61—41no—
129C2×D61—42no—
130C2×D61—41no—
131C4×C61—22no—
132C4×C61#38611yescharacteristic
133C4×C61#40811yescharacteristic
134C4×C61#43512yes—
135Dic61—42no—
136Dic61—41no—
137#3407D6111yescharacteristic, complemented
138#2929—611no—
139#2115—611no—
140#2112—612no—
141#1378—612no—
142C2×C61—21no—
143C2×C61—22no—
144C2×C61—42no—
145C2×C61#186911yescharacteristic
146D61—42no—
147C2×#131C2×D6111yescharacteristic
148C23×C8—611no—
149#525—611no—
150#504—24no—
151C2×M32—22no—
152C2×M32—612no—
153C2×M32C2×D6112yes—
154#435—1222no—
155C8×D4—1222no—
156#424—611no—
157#408—611no—
158#407—611no—
159#405—611no—
160#349—1222no—
161C22⋊C16—1224no—
162C61#1298111yescharacteristic, complemented, Sylow
163C2×#41C4×D6111yescharacteristic
164C23×C4—611no—
165#131—22no—
166#131—22no—
167C2×M16—611no—
168C2×M16C4×D6111yescharacteristic
169C2×M16D24412yes—
170C22×C8—611no—
171C22×C8—1221no—
172C22×C8—1222no—
173C22×C8—1222no—
174C22×C8—1221no—
175C22×C8C22⋊D6112yes—
176C22×C8C22×D6111yescharacteristic
177#123—612no—
178C22⋊Q8—1221no—
179#121—1221no—
180#120—612no—
181C4×D4—1222no—
182#117—611no—
183M32—24no—
184M32—22no—
185M32—22no—
186M32—1222no—
187M32—1222no—
188C2×C16—24no—
189C2×C16—1222no—
190C2×C16C22⋊D6114yescomplemented
191C4⋊C8—1222no—
192C22⋊C8—1222no—
193C22⋊C8—1222no—
194#97—1221no—
195C4×C8—1221no—
196C24—611no—
197#41—22no—
198#41—22no—
199C2×Q8#1175818911yescharacteristic
200C2×D4—1222no—
201C2×D4C8×D6112yes—
202C2×D4#1175818911yescharacteristic
203C22×C4—611no—
204C22×C4—1221no—
205C22×C4—1222no—
206C22×C4—1221no—
207C22×C4—1222no—
208C22×C4#1175819812yes—
209C22×C4C2×C4×D6111yescharacteristic
210M16—22no—
211M16—24no—
212M16—1222no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—1221no—
216C2×C8—1221no—
217C2×C8—1222no—
218C2×C8—2441no—
219C2×C8—2442no—
220C2×C8—2441no—
221C2×C8#1175819812yes—
222C2×C8C2×D24411yescharacteristic
223C2×C8#1175822212yes—
224C2×C8C2×C22⋊D6112yes—
225C2×C8C2×C22⋊D6111yescharacteristic
226C4⋊C4—1222no—
227C4⋊C4—1221no—
228C22⋊C4—612no—
229C22⋊C4—1222no—
230C22⋊C4—1222no—
231C42—1221no—
232C16—24no—
233C16—2442no—
234C23—1221no—
235C23—1222no—
236C23—2442no—
237C23#1451968912yes—
238C23C2×C8×D6111yescharacteristic
239Q8—22no—
240D4—24no—
241D4—22no—
242D4—2444no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—1222no—
246C2×C4—1221no—
247C2×C4—1222no—
248C2×C4—2441no—
249C2×C4—2442no—
250C2×C4—2441no—
251C2×C4#1451968912yes—
252C2×C4C2×#1175819811yescharacteristic
253C2×C4C4×C22⋊D6112yes—
254C2×C4C2×#1175818911yescharacteristic
255C2×C4#1451978312yes—
256C8—22no—
257C8—2441no—
258C8—2441no—
259C8#1451974611yescharacteristic
260C8#1451975112yes—
261C8#1451976011yescharacteristic
262C22—22no—
263C22—41no—
264C22—42no—
265C22—1222no—
266C22—1224no—
267C22—2441no—
268C22—2442no—
269C22—2441no—
270C22C2×#1451968911yescharacteristic
271C22C8×C22⋊D6112yes—
272C4—22no—
273C4—2442no—
274C4—2441no—
275C4#1683964111yescharacteristic
276C4#1683972811yescharacteristic
277C4#1683983312yes—
278C2—21no—
279C2—22no—
280C2—42no—
281C2—2442no—
282C2#2144611111yescharacteristic
subgroupG/H
centerC8#14519746
commutatorC23×C61C22×C8
FrattiniC22×C8C22×D61
FittingC61×#3407C2
socleC2×C61#1869
prime radical#16840269C4
cosocleC22×C8×C61C23
hypercenter#3407D61
hypocenterC61#12981
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. #21447440C2semidirect
3. #21447429C2semidirect
4. #21443904C2semidirect
5. C61×#3407C2semidirect
53. C2×C16×C61D4semidirect
137. #3407D61semidirect
162. C61#12981semidirect
190. C2×C16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214545112
#214474402
#214474292
#214439042
C61×#34072
#1298161