Finite Group Databasebeta

← #21452111

Subgroups — #21452111

Order 7808.

804 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14519750C212yesmaximal, complemented
2C2×#14519749C212yesmaximal, complemented
3C2×#14519748C212yesmaximal, complemented
4#16839685C212yesmaximal, complemented
5#16839671C211yesmaximal, characteristic, complemented
6#16839670C212yesmaximal, complemented
7#16839636C212yesmaximal, complemented
8#16839634C211yesmaximal, characteristic, complemented
9C61×#392C211yesmaximal, characteristic, complemented
10C22×D244C2211yescharacteristic
11C22×C22⋊D61C2212yes—
12C22×C22⋊D61C2212yes—
13C22×C22⋊D61C2212yes—
14C23×Dic61C2211yescharacteristic
15C22×C4×D61C2212yes—
16#14519750—28no—
17#14519749—28no—
18#14519748—28no—
19C2×C4⋊Dic61C2212yes—
20C2×#11758196C2212yes—
21C2×#11758196C2212yes—
22C2×C22⋊Dic61C2212yes—
23C2×C22⋊Dic61C2212yes—
24C2×C22⋊Dic61C2211yescharacteristic
25C2×#11758198C2212yes—
26C2×#11758198C2212yes—
27#14519684C2211yescharacteristic
28#14519684C2212yes—
29C22×D4×C61C2212yes—
30C22×D4×C61C2211yescharacteristic
31C2×C22⋊C4×C61C2212yes—
32C2×C22⋊C4×C61C2211yescharacteristic
33#95×C61C2211yescharacteristic
34C24×C61C2312yes—
35C24×C61C2311yescharacteristic
36C23×D61C2312yes—
37C2×D4×C61—24no—
38C2×D4×C61—24no—
39C2×D4×C61—42no—
40C2×D4×C61—42no—
41C2×D4×C61—41no—
42C22×C4×C61C2311yescharacteristic
43C22×C4×C61C2311yescharacteristic
44C22×C4×C61C2312yes—
45C2×C22⋊D61—24no—
46C2×C22⋊D61—24no—
47C2×C22⋊D61—24no—
48C2×C22⋊D61—24no—
49C2×C22⋊D61—24no—
50C2×C22⋊D61—24no—
51C2×C22⋊D61—42no—
52C2×C22⋊D61—42no—
53C2×C22⋊D61—42no—
54C22×Dic61—24no—
55C22×Dic61—41no—
56C22×Dic61C2312yes—
57C22×Dic61C2312yes—
58C22×Dic61C2312yes—
59C2×D244—24no—
60C2×D244—41no—
61C2×C4×D61—24no—
62C2×C4×D61—24no—
63C2×C4×D61—42no—
64C22⋊C4×C61—24no—
65C22⋊C4×C61—41no—
66C22⋊Dic61—24no—
67C22⋊Dic61—24no—
68C22⋊Dic61D414yescomplemented
69#11758198—42no—
70#11758198D418yescomplemented
71C4⋊Dic61—24no—
72#11758196—24no—
73#11758196—24no—
74C23×C61—22no—
75C23×C61—21no—
76C23×C61—22no—
77C23×C61—21no—
78C23×C61—22no—
79C23×C61—22no—
80C23×C61—22no—
81C23×C61—22no—
82C23×C61—22no—
83C23×C61—22no—
84C23×C61—22no—
85C23×C61C2×D412yes—
86C23×C61C2411yescharacteristic
87C22×D61—22no—
88C22×D61—22no—
89C22×D61—22no—
90C22×D61—22no—
91C22×D61—22no—
92C22×D61—22no—
93C22×D61C2×D414yes—
94D4×C61—48no—
95D4×C61—44no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61C2×D414yes—
103C22⋊D61—48no—
104C22⋊D61—48no—
105C22⋊D61—48no—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—44no—
114C2×Dic61C2×D414yes—
115C2×Dic61C2×D414yes—
116D244—44no—
117C4×D61—48no—
118C22×C61—22no—
119C22×C61—24no—
120C22×C61—42no—
121C22×C61—41no—
122C22×C61—41no—
123C22×C61—42no—
124C22×C61—42no—
125C22×C61—42no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61C22×D412yes—
131C22×C61C22×D411yescharacteristic
132C22×C61H(5,𝔽2)11yescharacteristic
133C22×C61H(5,𝔽2)12yes—
134C22×C612−1+411yescharacteristic
135C2×D61—24no—
136C2×D61—24no—
137C2×D61—24no—
138C2×D61—42no—
139C2×D61—42no—
140C2×D61—42no—
141C2×D61—42no—
142C4×C61—24no—
143C4×C61—41no—
144C4×C61—41no—
145Dic61—24no—
146Dic61—24no—
147Dic61—42no—
148#2675—611nomaximal, Sylow
149C2×C61—42no—
150C2×C61—82no—
151C2×C61#53711yescharacteristic
152C2×C61#53712yes—
153C2×C61D4212yes—
154C2×C61D4211yescharacteristic
155C2×C61#56011yescharacteristic
156D61—44no—
157C2×#121—612no—
158C2×#121—612no—
159C2×#121—612no—
160#396—612no—
161#392—612no—
162#392—611no—
163#392D6111yescharacteristic, complemented
164#385—611no—
165#385—612no—
166C61#267511yescharacteristic, complemented, Sylow
167C22×D4—612no—
168C22×D4—612no—
169C22×D4—612no—
170C22×D4—611no—
171C22×D4C2×D6112yes—
172C22×D4C2×D6111yescharacteristic
173C23×C4—611no—
174C23×C4—612no—
175#121—1228no—
176#121—1228no—
177#121—1228no—
178C2×C4⋊C4—612no—
179C2×C4⋊C4—612no—
180C2×C4⋊C4—612no—
181C2×C22⋊C4—611no—
182C2×C22⋊C4—612no—
183C2×C22⋊C4—612no—
184C2×C22⋊C4—612no—
185C2×C22⋊C4—612no—
186C2×C22⋊C4C2×D6112yes—
187C2×C22⋊C4C2×D6111yescharacteristic
188#95—612no—
189#95—611no—
190#95C2×D6111yescharacteristic
191C24—612no—
192C24C22×D6112yes—
193C24C22×D6111yescharacteristic
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—1224no—
200C2×D4—1224no—
201C2×D4—1224no—
202C2×D4—1224no—
203C2×D4—1224no—
204C2×D4—1224no—
205C2×D4—1224no—
206C2×D4—2441no—
207C2×D4—2442no—
208C2×D4—2442no—
209C2×D4—2442no—
210C22×C4—612no—
211C22×C4—612no—
212C22×C4—612no—
213C22×C4—1224no—
214C22×C4—1224no—
215C22×C4—1224no—
216C22×C4—2442no—
217C22×C4—2441no—
218C22×C4C22×D6111yescharacteristic
219C22×C4C22×D6111yescharacteristic
220C22×C4C22×D6112yes—
221C4⋊C4—1224no—
222C4⋊C4—1224no—
223C4⋊C4—1224no—
224C22⋊C4—24no—
225C22⋊C4—41no—
226C22⋊C4—614no—
227C22⋊C4—618no—
228C22⋊C4—1224no—
229C22⋊C4—1224no—
230C22⋊C4—2442no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—614no—
243C23—1222no—
244C23—1222no—
245C23—1222no—
246C23—1222no—
247C23—1222no—
248C23—1222no—
249C23D4×D6112yes—
250C23C23×D6111yescharacteristic
251D4—48no—
252D4—44no—
253D4—2444no—
254D4—2448no—
255D4—2448no—
256D4—2448no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—614no—
264C2×C4—614no—
265C2×C4—1222no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—1222no—
270C2×C4—1222no—
271C2×C4—1222no—
272C2×C4—2448no—
273C2×C4—2444no—
274C2×C4D4×D6114yes—
275C22—22no—
276C22—24no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—1224no—
288C22—1224no—
289C22—1224no—
290C22—2442no—
291C22—2442no—
292C22—2442no—
293C22—2442no—
294C22C2×D4×D6111yescharacteristic
295C22C2×D4×D6112yes—
296C22#1451982211yescharacteristic
297C22#1451982212yes—
298C22#1451982411yescharacteristic
299C4—24no—
300C4—41no—
301C4—41no—
302C4—1224no—
303C4—1224no—
304C4—2442no—
305C2—42no—
306C2—82no—
307C2—2444no—
308C2#1684033911yescharacteristic
309C2#1684034112yes—
310C2#1684039412yes—
311C2#1684039711yescharacteristic
312C2#1684050111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC61×#392C2
hypercenter#392D61
hypocenterC61#2675
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#14519750C2semidirect
2. C2×#14519749C2semidirect
3. C2×#14519748C2semidirect
4. #16839685C2semidirect
5. #16839671C2semidirect
6. #16839670C2semidirect
7. #16839636C2semidirect
8. #16839634C2semidirect
9. C61×#392C2semidirect
68. C22⋊Dic61D4semidirect
70. #11758198D4semidirect
163. #392D61semidirect
166. C61#2675semidirect