Subgroups — #21454694

Order 7808.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519800	C₂	1	1	yes	maximal, characteristic
2	#16840244	C₂	1	2	yes	maximal, complemented
3	#16839865	C₂	1	2	yes	maximal, complemented
4	#16839860	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839844	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839762	C₂	1	2	yes	maximal, complemented
10	#16839716	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11758226	C₂²	1	2	yes	—
13	#14519800	C₄	1	8	yes	complemented
14	C₂×#11758201	C₂²	1	2	yes	—
15	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
16	C₂×#11758189	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
18	Q₈×Dic₆₁	C₂²	1	2	yes	—
19	C₄×D₂₄₄	—	2	2	no	—
20	C₄×D₂₄₄	C₂²	1	2	yes	—
21	#14519739	—	2	2	no	—
22	#14519739	C₂²	1	2	yes	—
23	#14519734	—	2	2	no	—
24	#14519734	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₆₁	—	2	2	no	—
26	C₄⋊C₄×D₆₁	C₂²	1	2	yes	—
27	#14519711	C₂²	1	1	yes	characteristic
28	#14519710	C₂²	1	2	yes	—
29	#14519701	C₂²	1	2	yes	—
30	#14519699	C₂²	1	2	yes	—
31	#14519697	C₂²	1	2	yes	—
32	#14519688	C₂²	1	1	yes	characteristic
33	#14519687	C₂²	1	1	yes	characteristic
34	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
37	#107×C₆₁	C₂²	1	1	yes	characteristic
38	#103×C₆₁	C₂²	1	2	yes	—
39	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
41	#11758226	—	2	4	no	—
42	#11758226	C₂×C₄	1	8	yes	complemented
43	C₂×D₂₄₄	—	2	2	no	—
44	C₂×D₂₄₄	C₂³	1	2	yes	—
45	C₂×C₄×D₆₁	—	2	2	no	—
46	C₂×C₄×D₆₁	—	2	2	no	—
47	C₂×C₄×D₆₁	—	2	2	no	—
48	C₂×C₄×D₆₁	—	2	2	no	—
49	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
50	Q₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
55	C₄²×C₆₁	—	2	2	no	—
56	C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	#11758201	C₂×C₄	1	8	yes	complemented
58	#11758198	—	2	2	no	—
59	#11758198	—	2	2	no	—
60	#11758198	D₄	1	1	yes	characteristic
61	#11758198	D₄	1	1	yes	characteristic
62	C₄⋊Dic₆₁	—	2	2	no	—
63	C₄⋊Dic₆₁	C₂³	1	2	yes	—
64	#11758196	—	2	2	no	—
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	C₂²×D₆₁	—	2	2	no	—
75	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₆₁	—	2	2	no	—
77	Q₈×C₆₁	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
83	C₂×Dic₆₁	—	2	2	no	—
84	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
87	D₂₄₄	—	2	8	no	—
88	D₂₄₄	—	2	2	no	—
89	D₂₄₄	C₂²×C₄	1	4	yes	—
90	C₄×D₆₁	—	2	8	no	—
91	C₄×D₆₁	—	8	2	no	—
92	C₄×D₆₁	—	8	2	no	—
93	C₄×D₆₁	—	8	2	no	—
94	C₄×D₆₁	C₂²×C₄	1	2	yes	—
95	C₄×D₆₁	#41	1	2	yes	—
96	C₈×C₆₁	—	2	1	no	—
97	C₈×C₆₁	C₂²×C₄	1	2	yes	—
98	#92418	—	2	1	no	—
99	#92418	C₂²×C₄	1	2	yes	—
100	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₆₁	—	2	1	no	—
102	C₂×D₆₁	—	2	1	no	—
103	C₂×D₆₁	—	2	4	no	—
104	C₂×D₆₁	—	4	2	no	—
105	C₂×D₆₁	—	4	2	no	—
106	C₂×D₆₁	C₄×D₄	1	2	yes	—
107	C₄×C₆₁	—	2	4	no	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	—	2	1	no	—
110	C₄×C₆₁	—	4	2	no	—
111	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
113	Dic₆₁	—	2	1	no	—
114	Dic₆₁	—	2	1	no	—
115	Dic₆₁	—	4	2	no	—
116	Dic₆₁	C₄×D₄	1	2	yes	—
117	#2951	—	61	1	no	maximal, Sylow
118	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₆₁	#576	1	1	yes	characteristic
120	C₂×C₆₁	#578	1	1	yes	characteristic
121	D₆₁	—	2	2	no	—
122	D₆₁	—	4	4	no	—
123	C₂×#135	—	61	1	no	—
124	#520	—	61	2	no	—
125	#440	—	61	2	no	—
126	#440	—	61	2	no	—
127	C₄×Q₁₆	—	61	1	no	—
128	C₄×Q₁₆	D₆₁	1	1	yes	characteristic, complemented
129	C₄×D₈	—	61	1	no	—
130	C₄×D₈	—	61	1	no	—
131	C₂×#107	—	61	1	no	—
132	#417	—	61	2	no	—
133	#405	—	61	1	no	—
134	C₆₁	#2951	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	61	2	no	—
136	#135	—	61	8	no	—
137	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
138	C₂×SD₁₆	—	61	2	no	—
139	C₂×D₈	—	61	1	no	—
140	C₂×M₁₆	—	61	1	no	—
141	C₂²×C₈	—	61	1	no	—
142	C₄×Q₈	—	61	2	no	—
143	C₄×Q₈	C₂×D₆₁	1	2	yes	—
144	C₄×D₄	—	61	2	no	—
145	C₄×D₄	—	61	2	no	—
146	C₄×D₄	—	122	2	no	—
147	C₄×D₄	—	122	2	no	—
148	#117	—	61	1	no	—
149	#117	—	122	2	no	—
150	C₂×C₄⋊C₄	—	61	2	no	—
151	C₂×C₄⋊C₄	—	122	2	no	—
152	#107	—	61	1	no	—
153	#107	—	61	2	no	—
154	#107	C₂×D₆₁	1	1	yes	characteristic
155	#103	—	61	2	no	—
156	#103	C₂×D₆₁	1	2	yes	—
157	#102	—	61	2	no	—
158	#102	—	61	2	no	—
159	#97	—	61	1	no	—
160	#97	—	61	1	no	—
161	C₄×C₈	—	61	1	no	—
162	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
163	#41	—	61	8	no	—
164	#41	—	122	4	no	—
165	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
166	C₂×D₄	—	61	2	no	—
167	C₂×D₄	—	122	2	no	—
168	C₂²×C₄	—	61	1	no	—
169	C₂²×C₄	—	122	2	no	—
170	C₂²×C₄	—	122	2	no	—
171	C₂²×C₄	—	122	2	no	—
172	C₂²×C₄	—	122	2	no	—
173	Q₁₆	C₄×D₆₁	1	4	yes	complemented
174	SD₁₆	—	61	8	no	—
175	D₈	—	61	4	no	—
176	M₁₆	—	122	2	no	—
177	C₂×C₈	—	61	1	no	—
178	C₂×C₈	—	61	4	no	—
179	C₂×C₈	—	61	1	no	—
180	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	61	1	no	—
184	C₄⋊C₄	—	61	1	no	—
185	C₄⋊C₄	—	61	2	no	—
186	C₄⋊C₄	—	122	2	no	—
187	C₄⋊C₄	—	122	2	no	—
188	C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
189	C₂²⋊C₄	—	61	1	no	—
190	C₂²⋊C₄	—	61	1	no	—
191	C₂²⋊C₄	—	122	2	no	—
192	C₂²⋊C₄	—	122	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	61	1	no	—
195	C₄²	—	122	2	no	—
196	C₄²	C₂²×D₆₁	1	1	yes	characteristic
197	C₂³	—	61	1	no	—
198	C₂³	—	122	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	C₂×C₄×D₆₁	1	4	yes	—
201	D₄	—	61	4	no	—
202	D₄	—	122	8	no	—
203	D₄	—	122	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	61	1	no	—
207	C₂×C₄	—	61	2	no	—
208	C₂×C₄	—	61	2	no	—
209	C₂×C₄	—	61	1	no	—
210	C₂×C₄	—	61	1	no	—
211	C₂×C₄	—	122	8	no	—
212	C₂×C₄	—	122	2	no	—
213	C₂×C₄	—	488	2	no	—
214	C₂×C₄	—	488	2	no	—
215	C₂×C₄	—	488	2	no	—
216	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
217	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	61	2	no	—
221	C₈	—	122	1	no	—
222	C₈	C₂×C₄×D₆₁	1	2	yes	—
223	C₂²	—	61	2	no	—
224	C₂²	—	122	1	no	—
225	C₂²	—	122	1	no	—
226	C₂²	—	122	4	no	—
227	C₂²	—	244	2	no	—
228	C₂²	—	244	2	no	—
229	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
230	C₄	—	2	4	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	61	2	no	—
235	C₄	—	122	1	no	—
236	C₄	—	122	1	no	—
237	C₄	—	244	2	no	—
238	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
239	C₄	#41×D₆₁	1	1	yes	characteristic
240	C₂	—	122	2	no	—
241	C₂	—	244	4	no	—
242	C₂	C₄×D₄×D₆₁	1	1	yes	characteristic
243	C₂	#16840564	1	1	yes	characteristic
244	C₂	#16840575	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₄×Q₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₄×Q₁₆	D₆₁
hypocenter	C₆₁	#2951
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #16840244	C₂	semidirect
3. #16839865	C₂	semidirect
4. #16839860	C₂	semidirect
5. Q₁₆×Dic₆₁	C₂	semidirect
6. C₄×D₄₈₈	C₂	semidirect
7. #16839844	C₂	semidirect
8. #107×D₆₁	C₂	semidirect
9. #16839762	C₂	semidirect
10. #16839716	C₂	semidirect
11. C₄×Q₁₆×C₆₁	C₂	semidirect
13. #14519800	C₄	semidirect
42. #11758226	C₂×C₄	semidirect
50. Q₁₆×C₆₁	C₂×C₄	semidirect
57. #11758201	C₂×C₄	semidirect
71. D₄₈₈	C₂×C₄	semidirect
73. C₈×D₆₁	C₂×C₄	semidirect
128. C₄×Q₁₆	D₆₁	semidirect
134. C₆₁	#2951	semidirect
173. Q₁₆	C₄×D₆₁	semidirect