Finite Group Databasebeta

← #29600362

Subgroups — #29600362

Order 15616.

388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 59 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#21456192C211yesmaximal, characteristic
2#353×D61C212yesmaximal
3#21443990C211yesmaximal, characteristic, complemented
4#21443989C212yesmaximal, complemented
5C61×#1412C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic, complemented
7#127×D61C2211yescharacteristic
8#16840128C411yescharacteristic, complemented
9#16840127C411yescharacteristic, complemented
10#16840126C2211yescharacteristic
11#16840107—22no—
12#16839765—22no—
13#99×D61—22no—
14#16839714C411yescharacteristic, complemented
15#16839555C2212yes—
16#16839554—22no—
17C61×#490C2211yescharacteristic
18C61×#353C2212yes—
19C2×D4×D61—41no—
20C2×D4×D61D412yes—
21C2×#11758189—21no—
22#14519776—21no—
23#14519774C2×C411yescharacteristic
24#14519773—41no—
25#14519773D412yes—
26#14519769C2×C411yescharacteristic, complemented
27#14519755—41no—
28#14519742—41no—
29#14519735—21no—
30C22⋊C4×D61—42no—
31C42×D61C2×C411yescharacteristic
32#14519699—22no—
33#14519698—22no—
34#14519697—22no—
35#14519692—22no—
36#14519691—42no—
37#14519688C2×C411yescharacteristic, complemented
38C8×Dic61—21no—
39#127×C61C2311yescharacteristic
40#125×C61C2×C411yescharacteristic, complemented
41#102×C61—22no—
42#99×C61—22no—
43#97×C61C2×C411yescharacteristic, complemented
44C23×D61—22no—
45C2×D4×C61—41no—
46C2×D4×C61C2×D412yes—
47C2×C22⋊D61—22no—
48C2×C22⋊D61—42no—
49D4×D61—42no—
50D4×D61—42no—
51D4×D61—84no—
52C2×D244—41no—
53C2×D244C22⋊C412yes—
54C2×C4×D61—21no—
55C2×C4×D61—82no—
56C2×C4×D61C22⋊C411yescharacteristic
57C2×Dic122—41no—
58C2×C8×C61—21no—
59C4⋊C4×C61—21no—
60C4⋊C4×C61—41no—
61C22⋊C4×C61—42no—
62C42×C61C22×C411yescharacteristic
63C22⋊Dic61—42no—
64#11758198—41no—
65#11758198—82no—
66C4⋊Dic61—21no—
67#11758196—41no—
68#11758196—41no—
69C4×Dic61—21no—
70C4×Dic61—41no—
71C4×Dic61C22⋊C411yescharacteristic
72C2×#92418—21no—
73#11758189—42no—
74C23×C61—22no—
75C22×D61—22no—
76C22×D61—42no—
77C22×D61—42no—
78C22×D61—84no—
79C22×D61#9911yescharacteristic
80D4×C61—42no—
81D4×C61—82no—
82C2×C4×C61—21no—
83C2×C4×C61—41no—
84C2×C4×C61—82no—
85C2×C4×C61C2×C22⋊C411yescharacteristic
86C22⋊D61—82no—
87C22⋊D61—84no—
88C2×Dic61—21no—
89C2×Dic61—41no—
90C2×Dic61—82no—
91C2×Dic61#9911yescharacteristic
92D244—42no—
93D244—82no—
94C4×D61—22no—
95C4×D61—42no—
96C4×D61—164no—
97Dic122—82no—
98C8×C61—41no—
99#92418—41no—
100#13595—611nomaximal, Sylow
101C22×C61—42no—
102C22×C61—82no—
103C22×C61C2×#9911yescharacteristic
104C2×D61—41no—
105C2×D61—42no—
106C2×D61—84no—
107C2×D61—82no—
108C2×D61—162no—
109C2×D61#35412yes—
110C4×C61—21no—
111C4×C61—41no—
112C4×C61—81no—
113C4×C61—162no—
114Dic61—22no—
115Dic61—21no—
116Dic61—81no—
117Dic61—162no—
118#3138—611no—
119C2×#353—612no—
120#1412—611no—
121#1412—612no—
122#1412D6111yescharacteristic, complemented
123C2×C61—21no—
124C2×C61—82no—
125C2×C61C2×#35411yescharacteristic
126D61—22no—
127D61—82no—
128#533—611no—
129C2×#127—611no—
130#490—611no—
131#490—611no—
132#490—611no—
133#490C2×D6111yescharacteristic
134#486—1222no—
135#418—1222no—
136C2×#99—1222no—
137C4×M16—611no—
138#353—612no—
139#353—1222no—
140#353C2×D6112yes—
141C61#1359511yescharacteristic, complemented, Sylow
142C22×D4—612no—
143C22×D4—2441no—
144C2×M16—1221no—
145C4⋊Q8—1221no—
146#127—612no—
147#127—611no—
148#127—2441no—
149#127C22×D6111yescharacteristic
150#125—611no—
151#125C4×D6111yescharacteristic, complemented
152C22⋊Q8—2441no—
153C4×Q8—2441no—
154#117—1221no—
155C2×C22⋊C4—2442no—
156C2×C42—611no—
157#102—22no—
158#102—1222no—
159#102—1222no—
160#102—1222no—
161#99—22no—
162#99—1222no—
163#99—2442no—
164#97—611no—
165#97C4×D6111yescharacteristic, complemented
166C4×C8—1221no—
167C24—1222no—
168C2×Q8—2441no—
169C2×D4—41no—
170C2×D4—612no—
171C2×D4—1222no—
172C2×D4—4884no—
173C2×D4—2441no—
174C2×D4—2442no—
175C2×D4—2442no—
176C2×D4—2442no—
177C2×D4D4×D6112yes—
178C22×C4—611no—
179C22×C4—1221no—
180C22×C4—4882no—
181M16—2442no—
182C2×C8—21no—
183C2×C8—1221no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—1221no—
187C4⋊C4—2441no—
188C4⋊C4—2441no—
189C22⋊C4—42no—
190C22⋊C4—4882no—
191C22⋊C4—2441no—
192C22⋊C4—2442no—
193C42—611no—
194C42—1221no—
195C42—2441no—
196C42C2×C4×D6111yescharacteristic
197C23—22no—
198C23—611no—
199C23—1222no—
200C23—4884no—
201C23—2442no—
202C23—2442no—
203Q8—4882no—
204D4—42no—
205D4—82no—
206D4—4882no—
207D4—4882no—
208D4—4884no—
209D4—2442no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—82no—
213C2×C4—611no—
214C2×C4—1222no—
215C2×C4—1221no—
216C2×C4—9764no—
217C2×C4—4882no—
218C2×C4—2442no—
219C2×C4—2441no—
220C2×C4C22⋊C4×D6111yescharacteristic
221C8—41no—
222C8—2441no—
223C22—42no—
224C22—82no—
225C22—612no—
226C22—9762no—
227C22—4882no—
228C22—4884no—
229C22—2441no—
230C22—2442no—
231C22#99×D6111yescharacteristic
232C4—21no—
233C4—41no—
234C4—81no—
235C4—162no—
236C4—1222no—
237C4—1221no—
238C4—9762no—
239C4—4881no—
240C2—21no—
241C2—82no—
242C2—1222no—
243C2—4882no—
244C2#354×D6111yescharacteristic
subgroupG/H
centerC2#354×D61
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1412C2
socleC2×C61C2×#354
prime radical#127×D61C22
cosocle#127×C61C23
hypercenter#1412D61
hypocenterC61#13595
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
3. #21443990C2semidirect
4. #21443989C2semidirect
5. C61×#1412C2semidirect
6. #16840318C4semidirect
8. #16840128C4semidirect
9. #16840127C4semidirect
14. #16839714C4semidirect
26. #14519769C2×C4semidirect
37. #14519688C2×C4semidirect
40. #125×C61C2×C4semidirect
43. #97×C61C2×C4semidirect
122. #1412D61semidirect
141. C61#13595semidirect
151. #125C4×D61semidirect
165. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214561922
#353×D612
#214439902
#214439892
C61×#14122
#1359561