Finite Group Databasebeta

← #21452656

Subgroups — #21452656

Order 7808.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14519764C211yesmaximal, characteristic
2#16839690C212yesmaximal, complemented
3#16839690C212yesmaximal, complemented
4#16839682C212yesmaximal, complemented
5#16839643C212yesmaximal, complemented
6#16839642C211yesmaximal, characteristic
7#16839635C212yesmaximal, complemented
8#16839620C212yesmaximal, complemented
9C2×#123×C61C211yesmaximal, characteristic
10C22×Dic122C2211yescharacteristic
11C23×Dic61C2211yescharacteristic
12#14519764—24no—
13C2×#11758196C2212yes—
14C2×#11758196C2212yes—
15C2×#11758196C2212yes—
16C2×C22⋊Dic61C2212yes—
17C2×C22⋊Dic61C2212yes—
18C2×C22⋊Dic61C2212yes—
19C2×C22⋊Dic61C2211yescharacteristic
20C2×C4×Dic61C2211yescharacteristic
21#14519684C2212yes—
22#14519684C2212yes—
23#14519684C2212yes—
24#14519684C2212yes—
25#14519684C2212yes—
26#14519684C2212yes—
27C22×D4×C61C2211yescharacteristic
28C23×C4×C61C2211yescharacteristic
29#123×C61—24no—
30C2×C4⋊C4×C61C2212yes—
31C2×C22⋊C4×C61C2211yescharacteristic
32C2×C22⋊C4×C61C2212yes—
33C24×C61C2311yescharacteristic
34C24×C61C2311yescharacteristic
35C2×D4×C61—41no—
36C2×D4×C61—41no—
37C2×D4×C61D414yes—
38C22×C4×C61—22no—
39C22×C4×C61—41no—
40C22×C4×C61—41no—
41C22×C4×C61C2312yes—
42C22×C4×C61C2312yes—
43C22×C4×C61C2311yescharacteristic
44C22×Dic61—24no—
45C22×Dic61—41no—
46C22×Dic61C2312yes—
47C22×Dic61C2312yes—
48C22×Dic61C2312yes—
49C22×Dic61C2312yes—
50C2×Dic122—22no—
51C2×Dic122—41no—
52C2×Dic122—41no—
53C4⋊C4×C61—24no—
54C22⋊C4×C61—22no—
55C22⋊C4×C61—24no—
56C22⋊Dic61—24no—
57C22⋊Dic61—24no—
58C22⋊Dic61—24no—
59C22⋊Dic61—41no—
60#11758196—24no—
61#11758196—42no—
62#11758196—42no—
63C4×Dic61—22no—
64C23×C61—22no—
65C23×C61—22no—
66C23×C61—21no—
67C23×C61—21no—
68C23×C61—21no—
69C23×C61—21no—
70C23×C61—22no—
71C23×C61—22no—
72C23×C61C2×D412yes—
73C23×C61C2×D412yes—
74C23×C61C2411yescharacteristic
75D4×C61—44no—
76C2×C4×C61—21no—
77C2×C4×C61—21no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—22no—
84C2×C4×C61—44no—
85C2×C4×C61C2×D412yes—
86C2×Dic61—22no—
87C2×Dic61—22no—
88C2×Dic61—22no—
89C2×Dic61—22no—
90C2×Dic61—22no—
91C2×Dic61—22no—
92C2×Dic61—22no—
93C2×Dic61—22no—
94C2×Dic61—22no—
95C2×Dic61—22no—
96C2×Dic61—22no—
97C2×Dic61—44no—
98C2×Dic61#4114yes—
99Dic122—44no—
100C22×C61—22no—
101C22×C61—22no—
102C22×C61—24no—
103C22×C61—24no—
104C22×C61—42no—
105C22×C61—42no—
106C22×C61—41no—
107C22×C61—41no—
108C22×C61—41no—
109C22×C61—41no—
110C22×C61C22×D411yescharacteristic
111C22×C61C2×#4112yes—
112C22×C61H(5,𝔽2)11yescharacteristic
113C22×C61H(5,𝔽2)11yescharacteristic
114C22×C612−1+412yes—
115C4×C61—22no—
116C4×C61—42no—
117C4×C61—42no—
118Dic61—24no—
119Dic61—42no—
120Dic61—42no—
121Dic61—42no—
122#2717—611nomaximal, Sylow
123C2×C61—42no—
124C2×C61—42no—
125C2×C61#53411yescharacteristic
126C2×C61#54212yes—
127C2×C61#54712yes—
128C2×C61#55111yescharacteristic
129C2×C61#56311yescharacteristic
130C2×#124—611no—
131C2×#123D6111yescharacteristic
132#397—612no—
133#397—612no—
134#395—612no—
135#387—612no—
136#386—611no—
137#385—612no—
138#380—612no—
139C61#271711yescharacteristic, complemented, Sylow
140C22×Q8—611no—
141C22×D4C2×D6111yescharacteristic
142C23×C4—611no—
143C23×C4C2×D6111yescharacteristic
144#124—1224no—
145#123—24no—
146C2×C4⋊C4—612no—
147C2×C4⋊C4—612no—
148C2×C4⋊C4—612no—
149C2×C4⋊C4C2×D6112yes—
150C2×C22⋊C4—612no—
151C2×C22⋊C4—612no—
152C2×C22⋊C4—612no—
153C2×C22⋊C4—611no—
154C2×C22⋊C4C2×D6112yes—
155C2×C22⋊C4C2×D6111yescharacteristic
156C2×C42—611no—
157#95—612no—
158#95—612no—
159#95—612no—
160#95—612no—
161#95—612no—
162#95—612no—
163C24C22×D6111yescharacteristic
164C24C22×D6111yescharacteristic
165C2×Q8—1222no—
166C2×Q8—2441no—
167C2×Q8—2441no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4C22⋊D6114yes—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—612no—
175C22×C4—612no—
176C22×C4—612no—
177C22×C4—612no—
178C22×C4—1224no—
179C22×C4—2441no—
180C22×C4C22×D6112yes—
181C22×C4C22×D6112yes—
182C22×C4C22×D6111yescharacteristic
183C4⋊C4—24no—
184C4⋊C4—1224no—
185C4⋊C4—2442no—
186C4⋊C4—2442no—
187C22⋊C4—22no—
188C22⋊C4—24no—
189C22⋊C4—1224no—
190C22⋊C4—1224no—
191C22⋊C4—1224no—
192C22⋊C4—2441no—
193C42—1222no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×C22⋊D6112yes—
203C23C2×C22⋊D6112yes—
204C23C23×D6111yescharacteristic
205Q8—2444no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—614no—
217C2×C4—1222no—
218C2×C4—1222no—
219C2×C4—1222no—
220C2×C4—1222no—
221C2×C4—1222no—
222C2×C4—1222no—
223C2×C4—1222no—
224C2×C4—1222no—
225C2×C4—1222no—
226C2×C4—1222no—
227C2×C4—1222no—
228C2×C4—2444no—
229C2×C4C2×C22⋊D6112yes—
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22C22×C22⋊D6111yescharacteristic
241C22#41×D6112yes—
242C22#1451982211yescharacteristic
243C22#1451982211yescharacteristic
244C22#1451982412yes—
245C4—22no—
246C4—42no—
247C4—42no—
248C4—1224no—
249C4—2442no—
250C4—2442no—
251C4—2442no—
252C2—42no—
253C2—42no—
254C2#1684032111yescharacteristic
255C2#1684038012yes—
256C2#1684041112yes—
257C2#1684044011yescharacteristic
258C2#1684051511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×#123×C61C2
hypercenterC2×#123D61
hypocenterC61#2717
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #16839690C2semidirect
3. #16839690C2semidirect
4. #16839682C2semidirect
5. #16839643C2semidirect
7. #16839635C2semidirect
8. #16839620C2semidirect
139. C61#2717semidirect