Finite Group Databasebeta

← #21456641

Subgroups — #21456641

Order 7808.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14519753C211yesmaximal, characteristic
2C22⋊Q8×D61C211yesmaximal, characteristic
3#16840093C211yesmaximal, characteristic, complemented
4#16840091C212yesmaximal, complemented
5#16840090C211yesmaximal, characteristic, complemented
6#16840019C212yesmaximal, complemented
7#16840017C212yesmaximal, complemented
8#16839918C211yesmaximal, characteristic, complemented
9#16839916C212yesmaximal, complemented
10#16839724C211yesmaximal, characteristic
11C61×#451C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2211yescharacteristic
13C22×C4×D61C2211yescharacteristic
14C2×#11758202C2212yescomplemented
15C2×#11758189C2212yes—
16#14519757C2211yescharacteristic
17#14519756C2211yescharacteristic
18#14519755—21no—
19#14519754—21no—
20#14519753—21no—
21#14519753—21no—
22#14519753C2211yescharacteristic
23#14519753C2212yes—
24#14519753C2212yes—
25C2×C4⋊Dic61C2211yescharacteristic
26C2×#11758196—21no—
27C4⋊C4×D61—21no—
28C4⋊C4×D61C2211yescharacteristic
29C2×#11758198—21no—
30C22⋊C4×D61—21no—
31#14519709C2212yescomplemented
32#14519708C2212yescomplemented
33#14519702C2212yescomplemented
34#14519701C2212yescomplemented
35#14519701C2212yescomplemented
36C22⋊#92418C2211yescharacteristic
37#14519689C2212yes—
38C22×Q8×C61C2211yescharacteristic
39C2×Q16×C61C2212yescomplemented
40C22⋊Q8×C61C2211yescharacteristic
41#103×C61C2212yescomplemented
42C22⋊C8×C61C2211yescharacteristic
43C23×D61D411yescharacteristic
44C2×Q8×C61—21no—
45C2×Q8×C61—42no—
46C2×Q8×C61C2311yescharacteristic
47C2×Q8×C61C2312yes—
48C22×C4×C61—21no—
49C22×C4×C61C2311yescharacteristic
50C22×Dic61—21no—
51C22×Dic61D411yescharacteristic
52Q8×D61—42no—
53C2×C4×D61—21no—
54C2×C4×D61—21no—
55C2×C4×D61—42no—
56C2×C4×D61D411yescharacteristic
57C2×C4×D61D411yescharacteristic
58C2×C4×D61C2311yescharacteristic
59C2×C4×D61C2311yescharacteristic
60C2×Dic122—21no—
61C2×Dic122C2311yescharacteristic
62Q16×C61—42no—
63C2×C8×C61C2312yes—
64C4⋊C4×C61—21no—
65C4⋊C4×C61C2311yescharacteristic
66C22⋊C4×C61—21no—
67C22⋊Dic61—21no—
68#11758202—42no—
69#11758198—21no—
70#11758198—21no—
71#11758198—22no—
72#11758198—22no—
73C4⋊Dic61—21no—
74C4⋊Dic61—21no—
75C4⋊Dic61C2311yescharacteristic
76C4⋊Dic61C2312yes—
77#11758196—21no—
78#11758196—21no—
79#11758196—21no—
80#11758196—22no—
81#11758196—22no—
82C2×#92418C2312yes—
83#11758189—42no—
84C23×C61C2×D411yescharacteristic
85C22×D61—22no—
86C22×D61—22no—
87C22×D61—41no—
88C22×D61C2×D411yescharacteristic
89C22×D61C2×D411yescharacteristic
90Q8×C61—22no—
91Q8×C61—22no—
92Q8×C61—22no—
93Q8×C61—42no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—42no—
99C2×C4×C61C2×D411yescharacteristic
100C2×C4×C61C2411yescharacteristic
101C2×Dic61—21no—
102C2×Dic61—22no—
103C2×Dic61—21no—
104C2×Dic61—41no—
105C2×Dic61—42no—
106C2×Dic61C2×D411yescharacteristic
107C2×Dic61C2×D411yescharacteristic
108C4×D61—22no—
109C4×D61—22no—
110C4×D61—41no—
111C4×D61—41no—
112C4×D61—42no—
113C4×D61—42no—
114Dic122—22no—
115Dic122—42no—
116C8×C61—22no—
117#92418—22no—
118C22×C61—21no—
119C22×C61—21no—
120C22×C61#13712yes—
121C22×C61C22×D411yescharacteristic
122C2×D61—21no—
123C2×D61—21no—
124C2×D61—21no—
125C2×D61—21no—
126C2×D61—21no—
127C2×D61—44no—
128C2×D61#13712yes—
129C4×C61—21no—
130C4×C61—41no—
131C4×C61—42no—
132C4×C61—41no—
133C4×C61H(5,𝔽2)12yes—
134Dic61—21no—
135Dic61—21no—
136Dic61—41no—
137Dic61—41no—
138Dic61—42no—
139#3195—611nomaximal, Sylow
140C2×C61—22no—
141C2×C61#53411yescharacteristic
142C2×C61C2×#13711yescharacteristic
143C2×C61C2×#13711yescharacteristic
144D61—22no—
145D61—41no—
146C2×C22⋊Q8—611no—
147C2×C22⋊Q8—611no—
148#483—611no—
149#483—612no—
150#483—611no—
151#470—612no—
152#470—612no—
153#451—611no—
154#451—612no—
155#451D6111yescharacteristic, complemented
156#407—611no—
157C61#319511yescharacteristic, complemented, Sylow
158C22×Q8—611no—
159C22×Q8C2×D6111yescharacteristic
160C23×C4—611no—
161C2×Q16—612no—
162C2×Q16C2×D6112yescomplemented
163C2×M16—612no—
164C22⋊Q8—612no—
165C22⋊Q8—611no—
166C22⋊Q8—612no—
167C22⋊Q8—611no—
168C22⋊Q8—611no—
169C22⋊Q8—1221no—
170C22⋊Q8—1221no—
171C22⋊Q8—1221no—
172C22⋊Q8—1221no—
173C22⋊Q8C2×D6111yescharacteristic
174C2×C4⋊C4—611no—
175C2×C4⋊C4—611no—
176C2×C4⋊C4—1221no—
177C2×C4⋊C4—1221no—
178C2×C22⋊C4—1221no—
179C2×C22⋊C4—1221no—
180#106—612no—
181#106—612no—
182#103—612no—
183#103—612no—
184#103—612no—
185#103C2×D6112yescomplemented
186C22⋊C8—611no—
187C22⋊C8—612no—
188C22⋊C8C2×D6111yescharacteristic
189C24—611no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—611no—
193C2×Q8—1221no—
194C2×Q8—2442no—
195C2×Q8C22×D6111yescharacteristic
196C2×Q8C22×D6112yes—
197C22×C4—21no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—611no—
201C22×C4—611no—
202C22×C4—611no—
203C22×C4—1221no—
204C22×C4—1221no—
205C22×C4—1221no—
206C22×C4—2442no—
207C22×C4C22×D6111yescharacteristic
208Q16—42no—
209Q16—2442no—
210M16—2442no—
211C2×C8—612no—
212C2×C8C22×D6112yes—
213C4⋊C4—21no—
214C4⋊C4—611no—
215C4⋊C4—612no—
216C4⋊C4—1221no—
217C4⋊C4—1221no—
218C4⋊C4—1222no—
219C4⋊C4—1222no—
220C4⋊C4—1221no—
221C4⋊C4—1221no—
222C4⋊C4—1221no—
223C4⋊C4C22×D6111yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1221no—
226C22⋊C4—1221no—
227C22⋊C4—1222no—
228C22⋊C4—1222no—
229C22⋊C4—1221no—
230C23—611no—
231C23—611no—
232C23—1222no—
233C23—1222no—
234C23—2441no—
235C23D4×D6111yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—1222no—
241Q8—2442no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—611no—
248C2×C4—611no—
249C2×C4—1222no—
250C2×C4—1222no—
251C2×C4—1221no—
252C2×C4—1222no—
253C2×C4—1221no—
254C2×C4—2441no—
255C2×C4—2441no—
256C2×C4—2442no—
257C2×C4—2442no—
258C2×C4—2441no—
259C2×C4—2442no—
260C2×C4D4×D6111yescharacteristic
261C2×C4C23×D6111yescharacteristic
262C8—22no—
263C8—1222no—
264C22—21no—
265C22—21no—
266C22—612no—
267C22—1221no—
268C22—1221no—
269C22—1221no—
270C22—1221no—
271C22—1221no—
272C22—2444no—
273C22#1451980912yes—
274C22C2×D4×D6111yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—1221no—
280C4—1221no—
281C4—2441no—
282C4—2442no—
283C4—2441no—
284C4#1451982212yes—
285C2—22no—
286C2—1222no—
287C2—2441no—
288C2#1684032211yescharacteristic
289C2#137×D6111yescharacteristic
290C2C2×#1451980911yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#451C2
socleC22×C61C22×D4
hypercenter#451D61
hypocenterC61#3195
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #16840093C2semidirect
4. #16840091C2semidirect
5. #16840090C2semidirect
6. #16840019C2semidirect
7. #16840017C2semidirect
8. #16839918C2semidirect
9. #16839916C2semidirect
11. C61×#451C2semidirect
14. C2×#11758202C22semidirect
31. #14519709C22semidirect
32. #14519708C22semidirect
33. #14519702C22semidirect
34. #14519701C22semidirect
35. #14519701C22semidirect
39. C2×Q16×C61C22semidirect
41. #103×C61C22semidirect
155. #451D61semidirect
157. C61#3195semidirect
162. C2×Q16C2×D61semidirect
185. #103C2×D61semidirect