Finite Group Databasebeta

← #21455196

Subgroups — #21455196

Order 7808.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 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#16840599C211yesmaximal, characteristic
2#137×D61C212yesmaximal, complemented
3#16840337C211yesmaximal, characteristic, complemented
4#16839947C211yesmaximal, characteristic, complemented
5#16839945C211yesmaximal, characteristic, complemented
6#16839944C211yesmaximal, characteristic, complemented
7#16839941C212yesmaximal, complemented
8#16839938C212yesmaximal, complemented
9#16839777C212yesmaximal, complemented
10#16839745C211yesmaximal, characteristic, complemented
11C61×#456C211yesmaximal, characteristic, complemented
12#14519825—22no—
13#14519825C2211yescharacteristic
14#14519823—22no—
15#14519823—22no—
16#14519823C2212yes—
17C2×#11758226—22no—
18C2×#11758226C2211yescharacteristic
19#41×D61—22no—
20#41×D61—22no—
21#41×D61C2212yes—
22C2×Q8×D61C2211yescharacteristic, complemented
23#14519810C2212yescomplemented
24#14519809—22no—
25#14519808C2212yescomplemented
26#14519807C2212yescomplemented
27#14519806—22no—
28Q16×D61—22no—
29SD16×D61—22no—
30M16×D61C2212yescomplemented
31#14519775C2211yescharacteristic, complemented
32#14519766C2211yescharacteristic, complemented
33#14519765C2211yescharacteristic, complemented
34#14519762—22no—
35#14519755—22no—
36#14519734C2211yescharacteristic, complemented
37#14519705C2212yescomplemented
38#14519704C2212yescomplemented
39#14519703C2212yescomplemented
40#14519696C2211yescharacteristic, complemented
41#14519693C2212yescomplemented
422−1+4×C61C2211yescharacteristic
43#137×C61C2212yescomplemented
44C4⋊Q8×C61C2211yescharacteristic, complemented
45#104×C61C2212yescomplemented
46#101×C61C2211yescharacteristic, complemented
47#41×C61—22no—
48#41×C61—22no—
49#41×C61C2312yes—
50C2×Q8×C61—22no—
51C2×Q8×C61C2311yescharacteristic
52C2×Q8×C61C2311yescharacteristic
53#11758226—22no—
54#11758226—24no—
55#11758226—22no—
56#11758226—22no—
57#11758226—22no—
58#11758226—22no—
59#11758226D412yes—
60Q8×D61—22no—
61Q8×D61—22no—
62Q8×D61—22no—
63Q8×D61—22no—
64Q8×D61D412yes—
65#11758224—22no—
66#11758224—22no—
67#11758224D412yes—
68D4×D61—24no—
69D4×D61—22no—
70D4×D61—24no—
71D4×D61—22no—
72D4×D61—22no—
73D4×D61D412yes—
74#11758222—24no—
75#11758222—24no—
76#11758222—22no—
77#11758222—22no—
78#11758222—22no—
79#11758222C2312yes—
80C2×D244—24no—
81C2×D244—22no—
82C2×D244D412yes—
83C2×D244C2311yescharacteristic
84C2×C4×D61—24no—
85C2×C4×D61—22no—
86C2×C4×D61—21no—
87C2×C4×D61D412yes—
88C2×C4×D61C2311yescharacteristic
89C2×Dic122—21no—
90C2×Dic122C2311yescharacteristic
91Q16×C61—22no—
92SD16×C61—22no—
93M16×C61C2312yes—
94C4⋊C4×C61—22no—
95C42×C61C2311yescharacteristic
96#11758202—22no—
97#11758202—22no—
98#11758201—22no—
99#11758200—22no—
100#11758198—21no—
101#11758198—22no—
102#11758198—22no—
103#11758196—22no—
104#11758196—21no—
105C4×Dic61C2311yescharacteristic
106#11758194C2312yes—
107Dic244—22no—
108#11758190—22no—
109#11758189—22no—
110C8×D61—22no—
111C22×D61—24no—
112C22×D61—22no—
113C22×D61C2×D412yes—
114C22×D61C2×D411yescharacteristic
115Q8×C61—22no—
116Q8×C61—22no—
117Q8×C61—22no—
118Q8×C61C2×D412yes—
119D4×C61—22no—
120D4×C61—22no—
121D4×C61C2×D412yes—
122C2×C4×C61—24no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61C2×D412yes—
127C2×C4×C61C2411yescharacteristic
128C22⋊D61—24no—
129C22⋊D61—22no—
130C22⋊D61—22no—
131C22⋊D61—22no—
132C2×Dic61—22no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61C2×D411yescharacteristic
136D244—24no—
137D244—22no—
138D244—22no—
139D244—24no—
140D244—22no—
141D244C2×D412yes—
142C4×D61—24no—
143C4×D61—24no—
144C4×D61—22no—
145C4×D61—22no—
146C4×D61—22no—
147C4×D61—22no—
148C4×D61—22no—
149C4×D61—42no—
150C4×D61C2×D412yes—
151C4×D61C2×D412yes—
152Dic122—22no—
153Dic122—22no—
154Dic122—22no—
155Dic122—41no—
156Dic122—41no—
157Dic122C2×D412yes—
158C8×C61—22no—
159#92418—22no—
160C22×C61—22no—
161C22×C61C22×D411yescharacteristic
162C2×D61—22no—
163C2×D61—22no—
164C2×D61—41no—
165C2×D61—42no—
166C2×D61—82no—
167C2×D61—84no—
168C2×D61#12012yes—
169C4×C61—22no—
170C4×C61—22no—
171C4×C61—41no—
172C4×C61—41no—
173C4×C61C22×D412yes—
174Dic61—22no—
175Dic61—41no—
176Dic61—41no—
177Dic61#12011yescharacteristic
178Dic61#12011yescharacteristic
179#3022—611nomaximal, Sylow
180C2×C61—21no—
181C2×C61—42no—
182C2×C61C2×#12011yescharacteristic
183D61—22no—
184D61—42no—
185D61—42no—
186#585—611no—
187C2×#137—612no—
188#536—611no—
189#456—611no—
190#456—611no—
191#456—611no—
192#456D6111yescharacteristic, complemented
193#455—612no—
194#455—612no—
195#421—612no—
196#413—611no—
197C61#302211yescharacteristic, complemented, Sylow
1982−1+4—611no—
1992−1+4—1222no—
2002−1+4C2×D6111yescharacteristic
201H(5,𝔽2)—612no—
202H(5,𝔽2)—1222no—
203H(5,𝔽2)—1222no—
204C2×#41—611no—
205C2×#41—612no—
206C2×#41—1222no—
207C2×#41—1222no—
208C2×#41—1222no—
209C22×Q8—611no—
210#137—612no—
211#137—612no—
212#137—612no—
213#137—1222no—
214#137—1222no—
215#137C2×D6112yescomplemented
216C2×Q16—1222no—
217C2×SD16—1222no—
218C2×M16—612no—
219C4⋊Q8—611no—
220C4⋊Q8C2×D6111yescharacteristic, complemented
221#124—611no—
222#124—611no—
223#123—1222no—
224C22⋊Q8—1222no—
225#117—611no—
226#104—612no—
227#104—612no—
228#104—612no—
229#104C2×D6112yescomplemented
230#101—611no—
231#101C2×D6111yescharacteristic, complemented
232#100—612no—
233#41—22no—
234#41—22no—
235#41—612no—
236#41—612no—
237#41—612no—
238#41—1222no—
239#41—1222no—
240#41—1224no—
241#41—1224no—
242#41—1222no—
243#41—1222no—
244#41—1222no—
245#41—1222no—
246#41—1222no—
247#41—1222no—
248#41—1224no—
249#41—1222no—
250#41—1222no—
251#41C22×D6112yes—
252C2×Q8—22no—
253C2×Q8—612no—
254C2×Q8—611no—
255C2×Q8—1221no—
256C2×Q8—1222no—
257C2×Q8—1222no—
258C2×Q8—1222no—
259C2×Q8—1222no—
260C2×Q8C22×D6111yescharacteristic
261C2×Q8C22×D6111yescharacteristic
262C2×D4—612no—
263C2×D4—612no—
264C2×D4—611no—
265C2×D4—1222no—
266C2×D4—1224no—
267C2×D4—1222no—
268C2×D4—1224no—
269C2×D4—1222no—
270C2×D4—1224no—
271C2×D4—1222no—
272C22×C4—611no—
273C22×C4—612no—
274C22×C4—1222no—
275C22×C4—1224no—
276C22×C4—1221no—
277Q16—22no—
278Q16—1222no—
279Q16—1222no—
280Q16—1222no—
281SD16—22no—
282SD16—1222no—
283SD16—1222no—
284SD16—1222no—
285M16—612no—
286M16—1222no—
287M16C22×D6112yes—
288C2×C8—1222no—
289C4⋊C4—22no—
290C4⋊C4—1221no—
291C4⋊C4—1222no—
292C22⋊C4—1221no—
293C22⋊C4—1222no—
294C22⋊C4—1222no—
295C42—611no—
296C42C22×D6111yescharacteristic
297C23—611no—
298C23—612no—
299C23—1222no—
300C23—1224no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—612no—
305Q8—1222no—
306Q8—1222no—
307Q8—1222no—
308Q8—2441no—
309Q8—2441no—
310Q8D4×D6112yes—
311D4—22no—
312D4—22no—
313D4—612no—
314D4—1224no—
315D4—1222no—
316D4—1224no—
317D4—1222no—
318D4—1222no—
319D4—1222no—
320D4—1224no—
321D4—1222no—
322D4—1222no—
323D4D4×D6112yes—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—611no—
329C2×C4—612no—
330C2×C4—612no—
331C2×C4—1222no—
332C2×C4—1222no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—1224no—
336C2×C4—1224no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—2442no—
342C2×C4D4×D6112yes—
343C2×C4C23×D6111yescharacteristic
344C8—22no—
345C8—1222no—
346C22—22no—
347C22—612no—
348C22—1222no—
349C22—1222no—
350C22—4882no—
351C22—4884no—
352C22—2441no—
353C22—2442no—
354C22C2×D4×D6111yescharacteristic
355C4—22no—
356C4—22no—
357C4—41no—
358C4—41no—
359C4—611no—
360C4—611no—
361C4—1222no—
362C4—2441no—
363C4—2441no—
364C4C2×D4×D6112yes—
365C2—21no—
366C2—42no—
367C2—1222no—
368C2—2442no—
369C2—2442no—
370C2#120×D6111yescharacteristic
subgroupG/H
centerC2#120×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#456C2
socleC2×C61C2×#120
hypercenter#456D61
hypocenterC61#3022
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D61C2semidirect
3. #16840337C2semidirect
4. #16839947C2semidirect
5. #16839945C2semidirect
6. #16839944C2semidirect
7. #16839941C2semidirect
8. #16839938C2semidirect
9. #16839777C2semidirect
10. #16839745C2semidirect
11. C61×#456C2semidirect
22. C2×Q8×D61C22semidirect
23. #14519810C22semidirect
25. #14519808C22semidirect
26. #14519807C22semidirect
30. M16×D61C22semidirect
31. #14519775C22semidirect
32. #14519766C22semidirect
33. #14519765C22semidirect
36. #14519734C22semidirect
37. #14519705C22semidirect
38. #14519704C22semidirect
39. #14519703C22semidirect
40. #14519696C22semidirect
41. #14519693C22semidirect
43. #137×C61C22semidirect
44. C4⋊Q8×C61C22semidirect
45. #104×C61C22semidirect
46. #101×C61C22semidirect
192. #456D61semidirect
197. C61#3022semidirect
215. #137C2×D61semidirect
220. C4⋊Q8C2×D61semidirect
229. #104C2×D61semidirect
231. #101C2×D61semidirect