Subgroups — #21454726

Order 7808.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840560	C₂	1	1	yes	maximal, characteristic
2	#16840528	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839882	C₂	1	4	yes	maximal, complemented
4	#16839877	C₂	1	4	yes	maximal, complemented
5	#16839810	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14519704	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519703	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11758222	C₂²	1	1	yes	characteristic
11	#14519808	C₄	1	2	yes	complemented
12	#14519807	C₄	1	2	yes	complemented
13	#14519803	C₄	1	2	yes	complemented
14	#14519802	C₄	1	2	yes	complemented
15	#14519797	C₂²	1	4	yes	—
16	#14519785	—	2	2	no	—
17	#14519783	C₂²	1	4	yes	complemented
18	C₂×#11758194	C₂²	1	1	yes	characteristic
19	C₂×#11758194	C₂²	1	1	yes	characteristic
20	M₁₆×D₆₁	—	2	2	no	—
21	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
22	#14519712	C₂²	1	4	yes	complemented
23	#14519704	C₂²	1	2	yes	complemented
24	#14519704	C₂²	1	2	yes	complemented
25	#14519703	C₂²	1	2	yes	complemented
26	#14519703	C₂²	1	2	yes	complemented
27	#14519687	C₂²	1	2	yes	complemented
28	C₈×Dic₆₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
30	#131×C₆₁	—	2	2	no	—
31	#131×C₆₁	C₂²	1	4	yes	complemented
32	C₂×M₁₆×C₆₁	—	2	2	no	—
33	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
35	#41×C₆₁	—	2	1	no	—
36	#41×C₆₁	—	2	1	no	—
37	#41×C₆₁	C₂×C₄	1	2	yes	—
38	#41×C₆₁	C₂³	1	2	yes	—
39	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₆₁	—	2	1	no	—
41	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	—	2	1	no	—
43	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₆₁	—	2	1	no	—
45	C₂²×Dic₆₁	—	2	1	no	—
46	#11758222	—	2	1	no	—
47	#11758222	—	2	1	no	—
48	#11758222	C₂×C₄	1	2	yes	—
49	#11758222	C₂³	1	2	yes	—
50	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₆₁	—	2	1	no	—
52	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₆₁	—	2	1	no	—
54	M₁₆×C₆₁	—	2	4	no	—
55	M₁₆×C₆₁	D₄	1	2	yes	—
56	M₁₆×C₆₁	D₄	1	2	yes	—
57	M₁₆×C₆₁	C₂³	1	2	yes	—
58	C₂×C₈×C₆₁	—	2	1	no	—
59	C₂×C₈×C₆₁	—	2	4	no	—
60	C₂×C₈×C₆₁	C₂³	1	2	yes	—
61	#11758202	C₂×C₄	1	4	yes	complemented
62	#11758201	C₂×C₄	1	4	yes	complemented
63	#11758200	C₂×C₄	1	4	yes	complemented
64	#11758199	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₆₁	—	2	1	no	—
66	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
67	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
68	#11758194	—	2	1	no	—
69	#11758194	C₂×C₄	1	2	yes	—
70	#11758194	C₂×C₄	1	2	yes	—
71	#11758194	C₂³	1	2	yes	—
72	C₂×#92418	—	2	1	no	—
73	C₂×#92418	C₂³	1	2	yes	—
74	#11758189	—	2	4	no	—
75	C₈×D₆₁	—	2	4	no	—
76	C₂³×C₆₁	—	2	1	no	—
77	C₂³×C₆₁	#41	1	1	yes	characteristic
78	C₂²×D₆₁	—	2	1	no	—
79	Q₈×C₆₁	—	2	1	no	—
80	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
81	D₄×C₆₁	—	2	2	no	—
82	D₄×C₆₁	—	2	2	no	—
83	D₄×C₆₁	—	2	1	no	—
84	D₄×C₆₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₆₁	—	2	2	no	—
95	C₂²⋊D₆₁	—	2	2	no	—
96	C₂×Dic₆₁	—	2	1	no	—
97	C₂×Dic₆₁	—	2	1	no	—
98	C₂×Dic₆₁	—	2	1	no	—
99	C₂×Dic₆₁	—	4	2	no	—
100	D₂₄₄	—	2	1	no	—
101	D₂₄₄	C₂²×C₄	1	2	yes	—
102	C₄×D₆₁	—	2	2	no	—
103	C₄×D₆₁	—	2	2	no	—
104	Dic₁₂₂	—	2	1	no	—
105	Dic₁₂₂	C₂²×C₄	1	2	yes	—
106	C₈×C₆₁	—	2	2	no	—
107	C₈×C₆₁	C₂×D₄	1	4	yes	—
108	#92418	—	2	2	no	—
109	#92418	C₂²×C₄	1	4	yes	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	—	8	1	no	—
113	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
116	C₂×D₆₁	—	2	2	no	—
117	C₂×D₆₁	—	8	1	no	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
121	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	Dic₆₁	—	2	2	no	—
124	Dic₆₁	—	2	2	no	—
125	Dic₆₁	—	4	1	no	—
126	#2956	—	61	1	no	maximal, Sylow
127	C₂×C₆₁	—	2	1	no	—
128	C₂×C₆₁	—	2	1	no	—
129	C₂×C₆₁	—	2	1	no	—
130	C₂×C₆₁	—	4	2	no	—
131	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
132	D₆₁	—	4	2	no	—
133	#575	—	61	1	no	—
134	#568	—	61	1	no	—
135	#568	D₆₁	1	1	yes	characteristic, complemented
136	#444	—	61	4	no	—
137	#443	—	61	4	no	—
138	#430	—	61	1	no	—
139	C₂×#104	—	61	1	no	—
140	C₂×#104	—	61	1	no	—
141	C₄×M₁₆	—	61	1	no	—
142	C₆₁	#2956	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	61	1	no	—
144	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
145	#137	—	61	2	no	—
146	#137	—	61	2	no	—
147	AGL(1,ℤ/8)	—	61	2	no	—
148	AGL(1,ℤ/8)	—	61	2	no	—
149	#135	—	61	4	no	—
150	#131	—	2	2	no	—
151	#131	—	61	4	no	—
152	#131	—	122	2	no	—
153	#131	C₂×D₆₁	1	4	yes	complemented
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	61	1	no	—
156	C₂×M₁₆	—	61	1	no	—
157	C₂×M₁₆	—	122	2	no	—
158	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
159	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
160	C₂×C₄²	—	61	1	no	—
161	#108	—	61	4	no	—
162	#104	—	61	2	no	—
163	#104	—	61	2	no	—
164	#104	—	61	2	no	—
165	#104	—	61	2	no	—
166	#97	—	61	2	no	—
167	C₄×C₈	—	61	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	61	2	no	—
171	#41	—	61	2	no	—
172	#41	—	122	1	no	—
173	#41	—	122	1	no	—
174	#41	C₄×D₆₁	1	2	yes	—
175	#41	C₂²×D₆₁	1	2	yes	—
176	C₂×Q₈	—	61	1	no	—
177	C₂×Q₈	C₄×D₆₁	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	61	1	no	—
180	C₂×D₄	—	122	1	no	—
181	C₂×D₄	C₄×D₆₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	122	1	no	—
184	C₂²×C₄	—	122	1	no	—
185	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
186	Q₁₆	—	61	4	no	—
187	SD₁₆	—	61	4	no	—
188	SD₁₆	—	61	4	no	—
189	D₈	—	61	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	61	2	no	—
193	M₁₆	—	61	2	no	—
194	M₁₆	—	61	2	no	—
195	M₁₆	—	122	1	no	—
196	M₁₆	—	122	4	no	—
197	M₁₆	C₂²⋊D₆₁	1	2	yes	—
198	M₁₆	C₂²⋊D₆₁	1	2	yes	—
199	M₁₆	C₂²×D₆₁	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	61	2	no	—
203	C₂×C₈	—	122	1	no	—
204	C₂×C₈	—	122	4	no	—
205	C₂×C₈	C₂²×D₆₁	1	2	yes	—
206	C₄²	—	61	1	no	—
207	C₄²	—	61	1	no	—
208	C₄²	—	122	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	122	1	no	—
211	C₂³	#11758222	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	61	2	no	—
214	Q₈	—	122	1	no	—
215	Q₈	C₂×C₄×D₆₁	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	61	2	no	—
220	D₄	—	122	2	no	—
221	D₄	—	122	2	no	—
222	D₄	—	122	1	no	—
223	D₄	C₂×C₄×D₆₁	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	122	1	no	—
228	C₂×C₄	—	122	1	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	122	2	no	—
231	C₂×C₄	—	122	1	no	—
232	C₂×C₄	—	244	2	no	—
233	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
235	C₂×C₄	#11758222	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	61	4	no	—
241	C₈	—	122	2	no	—
242	C₈	C₂×C₂²⋊D₆₁	1	4	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	122	2	no	—
247	C₂²	—	488	1	no	—
248	C₂²	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
250	C₂²	C₂×#11758222	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	122	2	no	—
253	C₄	—	122	2	no	—
254	C₄	—	244	1	no	—
255	C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	244	2	no	—
264	C₂	C₂×C₄×C₂²⋊D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#568	C₂
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	#568	D₆₁
hypocenter	C₆₁	#2956
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #16840528	C₂	semidirect
3. #16839882	C₂	semidirect
4. #16839877	C₂	semidirect
5. #16839810	C₂	semidirect
6. C₂×#14519704	C₂	semidirect
7. C₂×#14519703	C₂	semidirect
8. M₁₆×Dic₆₁	C₂	semidirect
9. C₆₁×#568	C₂	semidirect
11. #14519808	C₄	semidirect
12. #14519807	C₄	semidirect
13. #14519803	C₄	semidirect
14. #14519802	C₄	semidirect
17. #14519783	C₂²	semidirect
22. #14519712	C₂²	semidirect
23. #14519704	C₂²	semidirect
24. #14519704	C₂²	semidirect
25. #14519703	C₂²	semidirect
26. #14519703	C₂²	semidirect
27. #14519687	C₂²	semidirect
28. C₈×Dic₆₁	C₂²	semidirect
31. #131×C₆₁	C₂²	semidirect
61. #11758202	C₂×C₄	semidirect
62. #11758201	C₂×C₄	semidirect
63. #11758200	C₂×C₄	semidirect
64. #11758199	C₂×C₄	semidirect
135. #568	D₆₁	semidirect
142. C₆₁	#2956	semidirect
153. #131	C₂×D₆₁	semidirect