Finite Group Databasebeta

← #29573425

Subgroups — #29573425

Order 15616.

526 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 107 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#21448225C211yesmaximal, characteristic
2#21445582C212yesmaximal
3#21443701C212yesmaximal, complemented
4#21443700C211yesmaximal, characteristic, complemented
5C61×#1295C211yesmaximal, characteristic, complemented
6C2×#14519735C412yes—
7C2×C42×D61C2211yescharacteristic
8C2×#14519689—22no—
9C2×#14519688C2212yes—
10#16839631C412yes—
11#16839628C2211yescharacteristic
12#16839621—22no—
13#16839613—22no—
14#16839516—22no—
15C2×#97×C61C2212yes—
16C61×#382C2211yescharacteristic
17C22×C4×D61C2×C411yescharacteristic
18C22×C4×D61D412yes—
19C22×#92418—22no—
20#14519735—44no—
21C2×C4⋊Dic61C2×C412yes—
22C2×#11758196—22no—
23C2×#11758196—22no—
24C2×#11758198—22no—
25C2×C4×Dic61—22no—
26C2×C4×Dic61C2×C411yescharacteristic
27C2×C4×Dic61D412yes—
28C42×D61—22no—
29C42×D61—21no—
30C42×D61—21no—
31#14519689—28no—
32#14519688—22no—
33#14519688—22no—
34#14519684—22no—
35C22×C8×C61—22no—
36C2×C4⋊C4×C61C2×C412yes—
37C2×C42×C61C2311yescharacteristic
38#97×C61—22no—
39#97×C61—22no—
40#95×C61—22no—
41C23×D61C22⋊C411yescharacteristic
42C22×C4×C61—22no—
43C22×C4×C61C22×C411yescharacteristic
44C22×C4×C61C2×D412yes—
45C22×Dic61—22no—
46C22×Dic61C22⋊C411yescharacteristic
47C22×Dic61C22⋊C412yes—
48C2×C4×D61—22no—
49C2×C4×D61—21no—
50C2×C4×D61—21no—
51C2×C4×D61—22no—
52C2×C4×D61—22no—
53C2×C4×D61—22no—
54C2×C4×D61—24no—
55C2×C4×D61—42no—
56C2×C8×C61—24no—
57C2×C8×C61—42no—
58C2×C8×C61—42no—
59C4⋊C4×C61—42no—
60C42×C61#4112yes—
61C42×C61#4111yescharacteristic
62C42×C61#4111yescharacteristic
63#11758198—44no—
64C4⋊Dic61—42no—
65#11758196—44no—
66#11758196—44no—
67C4×Dic61—21no—
68C4×Dic61—21no—
69C4×Dic61—22no—
70C4×Dic61—22no—
71C4×Dic61—44no—
72C2×#92418—24no—
73C2×#92418—42no—
74C2×#92418—42no—
75C23×C61C2×C22⋊C411yescharacteristic
76C22×D61—21no—
77C22×D61—21no—
78C22×D61—22no—
79C22×D61—22no—
80C22×D61#10012yes—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—22no—
84C2×C4×C61—22no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61#11711yescharacteristic
89C2×C4×C61#11711yescharacteristic
90C2×C4×C61#12312yes—
91C2×C4×C61#12312yes—
92C2×Dic61—21no—
93C2×Dic61—22no—
94C2×Dic61—21no—
95C2×Dic61—22no—
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—42no—
99C2×Dic61#10112yes—
100C2×Dic61#10414yes—
101C4×D61—24no—
102C4×D61—24no—
103C4×D61—44no—
104C4×D61—44no—
105C8×C61—44no—
106#92418—44no—
107#8722—611nomaximal, Sylow
108C22×C61#38111yescharacteristic
109C22×C61C2×#10011yescharacteristic
110C22×C61C2×#10111yescharacteristic
111C22×C61C2×#10412yes—
112C22×C61#42112yes—
113C2×D61—22no—
114C2×D61—24no—
115C2×D61—41no—
116C2×D61—42no—
117C2×D61—41no—
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—82no—
123Dic61—22no—
124Dic61—24no—
125Dic61—82no—
126#2285—611no—
127#1749—612no—
128#1295—612no—
129#1295—611no—
130#1295D6111yescharacteristic, complemented
131C2×C61#152611yescharacteristic
132C2×C61#153012yes—
133C2×C61#153711yescharacteristic
134C2×C61#182111yescharacteristic
135C2×C61#182612yes—
136D61—42no—
137C2×#117—612no—
138C22×C42—611no—
139C2×C22⋊C8—1222no—
140C2×#97—612no—
141C2×#97C2×D6112yes—
142#384—612no—
143#382—611no—
144#382C2×D6111yescharacteristic
145#380—1222no—
146C4×C4⋊C4—1222no—
147#336—1222no—
148C61#872211yescharacteristic, complemented, Sylow
149C23×C4—611no—
150C23×C4—612no—
151C22×C8—22no—
152C22×C8—1222no—
153#117—2444no—
154C2×C4⋊C4—612no—
155C2×C4⋊C4—1222no—
156C2×C4⋊C4—1222no—
157C2×C4⋊C4C4×D6112yes—
158C2×C22⋊C4—1222no—
159C2×C42—611no—
160C2×C42—612no—
161C2×C42—1222no—
162C2×C42—1221no—
163C2×C42—1221no—
164C2×C42—1222no—
165C2×C42C22×D6111yescharacteristic
166C22⋊C8—1228no—
167#97—22no—
168#97—22no—
169#97—1222no—
170#97—1222no—
171#95—22no—
172#95—1222no—
173C24—611no—
174C22×C4—22no—
175C22×C4—611no—
176C22×C4—612no—
177C22×C4—1222no—
178C22×C4—1224no—
179C22×C4—1222no—
180C22×C4—1221no—
181C22×C4—1221no—
182C22×C4—1222no—
183C22×C4—1222no—
184C22×C4—1222no—
185C22×C4—2442no—
186C22×C4C2×C4×D6111yescharacteristic
187C22×C4D4×D6112yes—
188C2×C8—24no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—1224no—
192C2×C8—2442no—
193C2×C8—2442no—
194C4⋊C4—42no—
195C4⋊C4—2442no—
196C4⋊C4—2444no—
197C4⋊C4—2444no—
198C22⋊C4—2444no—
199C42—1221no—
200C42—1221no—
201C42—1222no—
202C42—1222no—
203C42—2444no—
204C42#1175822212yes—
205C42#1175822411yescharacteristic
206C42#1175822611yescharacteristic
207C23—612no—
208C23—1221no—
209C23—1222no—
210C23—1222no—
211C23—1221no—
212C23C22⋊C4×D6111yescharacteristic
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—612no—
221C2×C4—614no—
222C2×C4—1224no—
223C2×C4—1224no—
224C2×C4—1221no—
225C2×C4—1222no—
226C2×C4—1221no—
227C2×C4—1222no—
228C2×C4—2444no—
229C2×C4—2444no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4#1451973411yescharacteristic
234C2×C4#1451973511yescharacteristic
235C2×C4#1451975912yes—
236C2×C4#1451976212yes—
237C8—44no—
238C8—2444no—
239C22—1222no—
240C22—1224no—
241C22—2442no—
242C22—2441no—
243C22—2441no—
244C22#1683962411yescharacteristic
245C22#100×D6111yescharacteristic
246C22#101×D6111yescharacteristic
247C22#104×D6112yes—
248C22#1683977712yes—
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—82no—
254C4—1222no—
255C4—1224no—
256C4—4882no—
257C2—2442no—
258C2#2144441211yescharacteristic
259C2#2144442812yes—
260C2#2144445511yescharacteristic
261C2#2144589811yescharacteristic
262C2#2144591712yes—
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1295C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1295D61
hypocenterC61#8722
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #21443701C2semidirect
4. #21443700C2semidirect
5. C61×#1295C2semidirect
130. #1295D61semidirect
148. C61#8722semidirect

Maximal subgroups

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

subgroupindex
#214482252
#214455822
#214437012
#214437002
C61×#12952
#872261