Finite Group Databasebeta

← #29581339

Subgroups — #29581339

Order 15616.

600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16840122C212yesmaximal, complemented
2#21453865C211yesmaximal, characteristic
3#21446638C211yesmaximal, characteristic, complemented
4C2×#16839501C212yesmaximal, complemented
5C61×#1295C211yesmaximal, characteristic, complemented
6C2×#14519777C2211yescharacteristic
7C2×#14519765C2212yes—
8#16840122—24no—
9#16840122—24no—
10C2×#14519702—22no—
11C2×#14519700—22no—
12C2×#14519699—22no—
13C2×#14519688C2211yescharacteristic
14#16839679—22no—
15#16839675—22no—
16#16839672—22no—
17#16839501—24no—
18C2×#97×C61C2212yes—
19C61×#382C2211yescharacteristic
20C22×Dic122—21no—
21C22×Dic122D412yes—
22C22×D244—22no—
23C22×D244D412yes—
24C22×#92418—21no—
25#14519777—22no—
26#14519777—21no—
27#14519777D412yescomplemented
28#14519765—22no—
29#14519765—22no—
30#14519765—22no—
31#14519765—22no—
32C2×C4⋊Dic61—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C2×#11758198—24no—
36#14519702—28no—
37#14519700—44no—
38#14519699—28no—
39#14519688—22no—
40#14519684—22no—
41C22×C8×C61—22no—
42C2×C4⋊C4×C61D412yes—
43C2×C42×C61C2311yescharacteristic
44#97×C61—22no—
45#97×C61D414yescomplemented
46#95×C61—22no—
47C23×D61—22no—
48C22×C4×C61—22no—
49C22×C4×C61C2×D412yes—
50C22×C4×C61C2×D411yescharacteristic
51C22×Dic61—22no—
52C22×Dic61—41no—
53C2×D244—22no—
54C2×D244—22no—
55C2×D244—44no—
56C2×D244—82no—
57C2×D244—84no—
58C2×Dic122—22no—
59C2×Dic122—22no—
60C2×Dic122—22no—
61C2×Dic122—44no—
62C2×Dic122—42no—
63C2×Dic122—42no—
64C2×Dic122—41no—
65C2×C8×C61—24no—
66C2×C8×C61—44no—
67C4⋊C4×C61—42no—
68C42×C61C2×D411yescharacteristic
69C42×C61C2×D412yes—
70C42×C61#4111yescharacteristic
71#11758198—42no—
72#11758198—42no—
73#11758198—42no—
74#11758198—42no—
75#11758198—84no—
76C4⋊Dic61—24no—
77C4⋊Dic61—42no—
78C2×#92418—22no—
79C2×#92418—41no—
80C2×#92418—41no—
81C23×C61#12011yescharacteristic
82C22×D61—42no—
83C22×D61—42no—
84C22×D61—42no—
85C22×D61—44no—
86C22×D61—44no—
87C2×C4×C61—21no—
88C2×C4×C61—21no—
89C2×C4×C61—24no—
90C2×C4×C61—42no—
91C2×C4×C61—44no—
92C2×C4×C61#12012yes—
93C2×C4×C61#12112yes—
94C2×C4×C61#12111yescharacteristic
95C2×C4×C61#12711yescharacteristic
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—44no—
99C2×Dic61—81no—
100C2×Dic61—81no—
101D244—44no—
102D244—82no—
103D244—88no—
104Dic122—44no—
105Dic122—44no—
106Dic122—82no—
107Dic122—82no—
108C8×C61—44no—
109#92418—42no—
110#10085—611nomaximal, Sylow
111C22×C61#39211yescharacteristic
112C22×C61#45311yescharacteristic
113C22×C61#45411yescharacteristic
114C22×C61#45514yes—
115C2×D61—84no—
116C2×D61—84no—
117C2×D61—82no—
118C2×D61—82no—
119C2×D61—82no—
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—82no—
125Dic61—82no—
126Dic61—82no—
127C2×#489—612no—
128#2819—611no—
129#1960—611no—
130C2×#329—612no—
131#1295D6111yescharacteristic, complemented
132C2×C61#165912yes—
133C2×C61#166212yes—
134C2×C61#200412yes—
135C2×C61#200611yescharacteristic
136D61—162no—
137C2×C4⋊Q8—611no—
138C2×#124—612no—
139#489—1224no—
140#489—1224no—
141C2×#103—1222no—
142C2×#103—1222no—
143C2×#102—1222no—
144C2×#97—611no—
145C2×#97C2×D6112yes—
146#394—1222no—
147#393—1222no—
148#392—1222no—
149#382C2×D6111yescharacteristic
150#329—1224no—
151C61#1008511yescharacteristic, complemented, Sylow
152C22×Q8—612no—
153C22×Q8—1221no—
154C22×D4—612no—
155C22×D4—1222no—
156C22×C8—22no—
157C22×C8—1221no—
158C4⋊Q8—612no—
159C4⋊Q8—1222no—
160C4⋊Q8—1221no—
161#124—1222no—
162#124—1222no—
163#124—1222no—
164#124—1222no—
165C2×C4⋊C4—1222no—
166C2×C4⋊C4D24412yes—
167C2×C22⋊C4—1224no—
168C2×C22⋊C4—1222no—
169C2×C22⋊C4—1222no—
170C2×C42C22×D6111yescharacteristic
171#103—1228no—
172#103—2444no—
173#102—1228no—
174#97—22no—
175#97—1222no—
176#97D24414yescomplemented
177#95—22no—
178#95—1222no—
179C24—1222no—
180C2×Q8—1222no—
181C2×Q8—1222no—
182C2×Q8—1222no—
183C2×Q8—2442no—
184C2×Q8—2442no—
185C2×Q8—2444no—
186C2×Q8—2441no—
187C2×D4—1222no—
188C2×D4—1222no—
189C2×D4—4884no—
190C2×D4—4882no—
191C2×D4—2444no—
192C22×C4—22no—
193C22×C4—1222no—
194C22×C4—2441no—
195C22×C4C2×D24411yescharacteristic
196C22×C4D4×D6112yes—
197C2×C8—24no—
198C2×C8—44no—
199C2×C8—1222no—
200C2×C8—2441no—
201C2×C8—2441no—
202C4⋊C4—42no—
203C4⋊C4—1224no—
204C4⋊C4—2442no—
205C22⋊C4—4884no—
206C22⋊C4—2442no—
207C22⋊C4—2442no—
208C22⋊C4—2442no—
209C22⋊C4—2442no—
210C42C2×D24412yes—
211C42D4×D6111yescharacteristic
212C42#1175822611yescharacteristic
213C23—2442no—
214C23—2444no—
215C23—2444no—
216C23—2442no—
217C23—2442no—
218C23#1451974711yescharacteristic
219Q8—4882no—
220Q8—4882no—
221Q8—2444no—
222Q8—2444no—
223D4—4882no—
224D4—4888no—
225D4—2444no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—4881no—
232C2×C4—4881no—
233C2×C4—2444no—
234C2×C4—2442no—
235C2×C4—2442no—
236C2×C4#1451974712yes—
237C2×C4#1451975212yes—
238C2×C4#1451975211yescharacteristic
239C2×C4#1451977411yescharacteristic
240C8—44no—
241C8—2442no—
242C22—4884no—
243C22—4884no—
244C22—4882no—
245C22—4882no—
246C22—4882no—
247C22#1683967211yescharacteristic
248C22#1683993211yescharacteristic
249C22#1683993711yescharacteristic
250C22#1683994114yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—4882no—
257C4—4882no—
258C2—9762no—
259C2#2144512612yes—
260C2#2144514312yes—
261C2#2144685112yes—
262C2#2144686311yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1295C2
socleC23×C61#120
prime radical#21453865C2
hypercenter#1295D61
hypocenterC61#10085
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
1. C2×#16840122C2semidirect
3. #21446638C2semidirect
4. C2×#16839501C2semidirect
5. C61×#1295C2semidirect
27. #14519777D4semidirect
45. #97×C61D4semidirect
131. #1295D61semidirect
151. C61#10085semidirect
176. #97D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168401222
#214538652
#214466382
C2×#168395012
C61×#12952
#1008561