Subgroups — #21454666

Order 7808.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#16840234	C₂	1	2	yes	maximal, complemented
3	#16839872	C₂	1	2	yes	maximal, complemented
4	#16839871	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839844	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839799	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839765	C₂	1	2	yes	maximal, complemented
10	#16839716	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₆₁	C₂²	1	2	yes	—
13	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
14	C₂×#11758199	C₂²	1	2	yes	—
15	D₈×D₆₁	C₄	1	8	yes	complemented
16	C₂×#11758189	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
18	C₄×D₂₄₄	C₂²	1	2	yes	complemented
19	D₄×Dic₆₁	C₂²	1	2	yes	complemented
20	#14519739	C₂²	1	2	yes	complemented
21	C₄×C₂²⋊D₆₁	—	2	2	no	—
22	#14519737	—	2	2	no	—
23	#14519735	C₂²	1	2	yes	—
24	#14519734	C₂²	1	1	yes	characteristic
25	C₂×#11758198	—	2	2	no	—
26	C₂²⋊C₄×D₆₁	—	2	2	no	—
27	#14519711	C₂²	1	1	yes	characteristic, complemented
28	#14519708	C₂²	1	2	yes	complemented
29	#14519699	C₂²	1	2	yes	complemented
30	#14519698	C₂²	1	2	yes	complemented
31	#14519697	C₂²	1	2	yes	complemented
32	#14519688	C₂²	1	1	yes	characteristic, complemented
33	#14519687	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₆₁	C₂²	1	2	yes	complemented
37	#107×C₆₁	C₂²	1	1	yes	characteristic, complemented
38	#102×C₆₁	C₂²	1	2	yes	complemented
39	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₆₁	—	2	2	no	—
41	C₂×D₄×C₆₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₆₁	—	2	2	no	—
43	C₂×C₂²⋊D₆₁	—	2	2	no	—
44	C₂²×Dic₆₁	—	2	2	no	—
45	D₄×D₆₁	—	2	4	no	—
46	D₄×D₆₁	C₂×C₄	1	8	yes	complemented
47	C₂×D₂₄₄	C₂³	1	2	yes	—
48	C₂×C₄×D₆₁	—	2	2	no	—
49	C₂×C₄×D₆₁	—	2	2	no	—
50	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
51	D₈×C₆₁	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₆₁	—	2	2	no	—
56	C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₆₁	—	2	2	no	—
58	#11758199	C₂×C₄	1	8	yes	complemented
59	#11758198	—	2	2	no	—
60	#11758198	—	2	2	no	—
61	#11758198	—	2	2	no	—
62	#11758198	D₄	1	1	yes	characteristic
63	#11758198	D₄	1	1	yes	characteristic
64	C₄⋊Dic₆₁	C₂³	1	2	yes	—
65	#11758196	D₄	1	1	yes	characteristic
66	#11758196	D₄	1	1	yes	characteristic
67	C₄×Dic₆₁	—	2	2	no	—
68	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
69	C₂×#92418	C₂³	1	1	yes	characteristic
70	C₂×#92418	C₂³	1	1	yes	characteristic
71	D₄₈₈	C₂×C₄	1	4	yes	complemented
72	#11758189	—	2	2	no	—
73	C₈×D₆₁	C₂×C₄	1	4	yes	complemented
74	C₂³×C₆₁	—	2	2	no	—
75	C₂²×D₆₁	—	2	2	no	—
76	C₂²×D₆₁	—	2	8	no	—
77	C₂²×D₆₁	—	8	2	no	—
78	C₂²×D₆₁	—	8	2	no	—
79	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
80	D₄×C₆₁	—	2	2	no	—
81	D₄×C₆₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₆₁	—	2	2	no	—
83	C₂×C₄×C₆₁	—	8	2	no	—
84	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₆₁	—	2	8	no	—
88	C₂×Dic₆₁	—	2	2	no	—
89	C₂×Dic₆₁	—	8	2	no	—
90	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
93	D₂₄₄	—	2	2	no	—
94	D₂₄₄	C₂²×C₄	1	4	yes	—
95	C₄×D₆₁	—	8	2	no	—
96	C₄×D₆₁	—	8	2	no	—
97	C₄×D₆₁	C₂²×C₄	1	2	yes	—
98	C₄×D₆₁	#41	1	2	yes	—
99	C₈×C₆₁	—	2	1	no	—
100	C₈×C₆₁	C₂²×C₄	1	2	yes	—
101	#92418	—	2	1	no	—
102	#92418	C₂²×C₄	1	2	yes	—
103	C₂²×C₆₁	—	2	4	no	—
104	C₂²×C₆₁	—	4	2	no	—
105	C₂²×C₆₁	—	4	2	no	—
106	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₆₁	—	2	1	no	—
108	C₂×D₆₁	—	2	1	no	—
109	C₂×D₆₁	—	2	4	no	—
110	C₂×D₆₁	—	4	2	no	—
111	C₂×D₆₁	—	4	2	no	—
112	C₂×D₆₁	—	8	8	no	—
113	C₂×D₆₁	C₄×D₄	1	2	yes	—
114	C₄×C₆₁	—	2	1	no	—
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	4	2	no	—
117	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
119	Dic₆₁	—	2	1	no	—
120	Dic₆₁	—	2	1	no	—
121	Dic₆₁	—	4	2	no	—
122	Dic₆₁	C₄×D₄	1	2	yes	—
123	#2948	—	61	1	no	maximal, Sylow
124	C₂×C₆₁	—	4	4	no	—
125	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×C₆₁	#576	1	1	yes	characteristic
127	C₂×C₆₁	#576	1	1	yes	characteristic
128	D₆₁	—	2	2	no	—
129	D₆₁	—	4	4	no	—
130	C₂²×D₈	—	61	1	no	—
131	#518	—	61	2	no	—
132	#442	—	61	2	no	—
133	#442	—	61	2	no	—
134	C₄×D₈	—	61	1	no	—
135	C₄×D₈	—	61	1	no	—
136	C₄×D₈	—	61	1	no	—
137	C₄×D₈	D₆₁	1	1	yes	characteristic, complemented
138	#427	—	61	1	no	—
139	#418	—	61	2	no	—
140	#405	—	61	1	no	—
141	C₆₁	#2948	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	—	61	2	no	—
143	C₂×D₈	—	61	2	no	—
144	C₂×D₈	—	61	8	no	—
145	C₂×D₈	—	61	1	no	—
146	C₂×D₈	C₂×D₆₁	1	1	yes	characteristic
147	C₂×M₁₆	—	61	1	no	—
148	C₂²×C₈	—	61	1	no	—
149	C₄×D₄	—	61	2	no	—
150	C₄×D₄	—	61	2	no	—
151	C₄×D₄	—	61	2	no	—
152	C₄×D₄	—	122	2	no	—
153	C₄×D₄	—	122	2	no	—
154	C₄×D₄	C₂×D₆₁	1	2	yes	complemented
155	#117	—	61	1	no	—
156	#117	—	61	2	no	—
157	C₂×C₂²⋊C₄	—	122	2	no	—
158	C₂×C₂²⋊C₄	—	122	2	no	—
159	#107	—	61	1	no	—
160	#107	C₂×D₆₁	1	1	yes	characteristic, complemented
161	#106	—	61	2	no	—
162	#102	—	61	2	no	—
163	#102	—	61	2	no	—
164	#102	—	61	2	no	—
165	#102	C₂×D₆₁	1	2	yes	complemented
166	#97	—	61	1	no	—
167	#97	—	61	1	no	—
168	C₄×C₈	—	61	1	no	—
169	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic, complemented
170	C₂⁴	—	122	2	no	—
171	C₂×D₄	—	61	8	no	—
172	C₂×D₄	—	61	2	no	—
173	C₂×D₄	—	122	4	no	—
174	C₂×D₄	—	122	2	no	—
175	C₂×D₄	C₂²×D₆₁	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	61	1	no	—
178	C₂²×C₄	—	122	2	no	—
179	C₂²×C₄	—	122	2	no	—
180	C₂²×C₄	—	122	2	no	—
181	D₈	—	61	8	no	—
182	D₈	—	61	4	no	—
183	D₈	C₄×D₆₁	1	4	yes	complemented
184	M₁₆	—	122	2	no	—
185	C₂×C₈	—	61	1	no	—
186	C₂×C₈	—	61	4	no	—
187	C₂×C₈	—	61	1	no	—
188	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
190	C₄⋊C₄	—	61	1	no	—
191	C₄⋊C₄	—	61	1	no	—
192	C₄⋊C₄	—	61	2	no	—
193	C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	61	1	no	—
196	C₂²⋊C₄	—	61	1	no	—
197	C₂²⋊C₄	—	122	2	no	—
198	C₂²⋊C₄	—	122	2	no	—
199	C₂²⋊C₄	—	122	2	no	—
200	C₂²⋊C₄	—	122	2	no	—
201	C₄²	—	61	1	no	—
202	C₄²	—	122	2	no	—
203	C₄²	C₂²×D₆₁	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	61	1	no	—
206	C₂³	—	122	8	no	—
207	C₂³	—	122	2	no	—
208	C₂³	—	488	2	no	—
209	C₂³	—	488	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	61	4	no	—
212	D₄	—	122	8	no	—
213	D₄	—	122	2	no	—
214	D₄	C₂×C₄×D₆₁	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	61	1	no	—
218	C₂×C₄	—	61	2	no	—
219	C₂×C₄	—	61	2	no	—
220	C₂×C₄	—	61	1	no	—
221	C₂×C₄	—	61	1	no	—
222	C₂×C₄	—	122	2	no	—
223	C₂×C₄	—	488	2	no	—
224	C₂×C₄	—	488	2	no	—
225	C₂×C₄	—	488	2	no	—
226	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
227	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	61	2	no	—
231	C₈	—	122	1	no	—
232	C₈	C₂×C₄×D₆₁	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	61	2	no	—
237	C₂²	—	122	1	no	—
238	C₂²	—	122	1	no	—
239	C₂²	—	122	4	no	—
240	C₂²	—	488	8	no	—
241	C₂²	—	244	2	no	—
242	C₂²	—	244	2	no	—
243	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	61	2	no	—
248	C₄	—	122	1	no	—
249	C₄	—	122	1	no	—
250	C₄	—	244	2	no	—
251	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
252	C₄	#41×D₆₁	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	122	2	no	—
255	C₂	—	244	4	no	—
256	C₂	C₄×D₄×D₆₁	1	1	yes	characteristic
257	C₂	#16840563	1	1	yes	characteristic
258	C₂	#16840564	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₄×D₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₄×D₈	D₆₁
hypocenter	C₆₁	#2948
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #16840234	C₂	semidirect
3. #16839872	C₂	semidirect
4. #16839871	C₂	semidirect
5. C₄×D₄₈₈	C₂	semidirect
6. D₈×Dic₆₁	C₂	semidirect
7. #16839844	C₂	semidirect
8. #16839799	C₂	semidirect
9. #16839765	C₂	semidirect
10. #16839716	C₂	semidirect
11. C₄×D₈×C₆₁	C₂	semidirect
15. D₈×D₆₁	C₄	semidirect
18. C₄×D₂₄₄	C₂²	semidirect
19. D₄×Dic₆₁	C₂²	semidirect
20. #14519739	C₂²	semidirect
27. #14519711	C₂²	semidirect
28. #14519708	C₂²	semidirect
29. #14519699	C₂²	semidirect
30. #14519698	C₂²	semidirect
31. #14519697	C₂²	semidirect
32. #14519688	C₂²	semidirect
33. #14519687	C₂²	semidirect
34. C₈×Dic₆₁	C₂²	semidirect
36. C₄×D₄×C₆₁	C₂²	semidirect
37. #107×C₆₁	C₂²	semidirect
38. #102×C₆₁	C₂²	semidirect
39. C₄×C₈×C₆₁	C₂²	semidirect
46. D₄×D₆₁	C₂×C₄	semidirect
51. D₈×C₆₁	C₂×C₄	semidirect
58. #11758199	C₂×C₄	semidirect
71. D₄₈₈	C₂×C₄	semidirect
73. C₈×D₆₁	C₂×C₄	semidirect
137. C₄×D₈	D₆₁	semidirect
141. C₆₁	#2948	semidirect
154. C₄×D₄	C₂×D₆₁	semidirect
160. #107	C₂×D₆₁	semidirect
165. #102	C₂×D₆₁	semidirect
169. C₄×C₈	C₂×D₆₁	semidirect
183. D₈	C₄×D₆₁	semidirect