Subgroups — #21445539

Order 7808.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519736	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839525	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₆₁	—	2	1	no	—
6	#14519736	—	2	2	no	—
7	#14519736	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₆₁	—	2	1	no	—
9	C₂×C₄⋊Dic₆₁	—	2	2	no	—
10	C₂×C₄⋊Dic₆₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₆₁	—	2	1	no	—
14	C₂×C₄×Dic₆₁	—	2	1	no	—
15	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₆₁	—	2	2	no	—
17	C₂×M₁₆×C₆₁	—	2	2	no	—
18	C₂×M₁₆×C₆₁	C₄	1	4	yes	complemented
19	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
20	C₂⁴×C₆₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	D₄	1	2	yes	—
26	C₂²×C₄×C₆₁	D₄	1	2	yes	—
27	C₂²×C₄×C₆₁	Q₈	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₆₁	—	2	4	no	—
30	C₂²×Dic₆₁	—	2	2	no	—
31	C₂²×Dic₆₁	—	2	2	no	—
32	C₂²×Dic₆₁	—	4	4	no	—
33	M₁₆×C₆₁	—	2	8	no	—
34	C₂×C₈×C₆₁	—	2	2	no	—
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	complemented
37	C₂²⋊Dic₆₁	—	2	4	no	—
38	C₄⋊Dic₆₁	—	2	4	no	—
39	C₄⋊Dic₆₁	—	2	2	no	—
40	C₄⋊Dic₆₁	—	2	2	no	—
41	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	complemented
43	C₄×Dic₆₁	—	2	4	no	—
44	C₂³×C₆₁	—	2	2	no	—
45	C₂³×C₆₁	—	2	2	no	—
46	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
49	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₄²	1	4	yes	—
51	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
54	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
57	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×Dic₆₁	—	2	4	no	—
64	C₂×Dic₆₁	—	2	4	no	—
65	C₂×Dic₆₁	—	4	8	no	—
66	C₂×Dic₆₁	—	4	2	no	—
67	C₂×Dic₆₁	—	4	4	no	—
68	C₂×Dic₆₁	—	4	2	no	—
69	C₈×C₆₁	—	2	4	no	—
70	C₂²×C₆₁	—	2	4	no	—
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	—	2	2	no	—
73	C₂²×C₆₁	—	2	4	no	—
74	C₂²×C₆₁	#95	1	4	yes	—
75	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²×C₆₁	#137	1	2	yes	—
80	C₄×C₆₁	#95	1	4	yes	—
81	C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
82	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	Dic₆₁	—	4	4	no	—
86	Dic₆₁	—	4	4	no	—
87	#1736	—	61	1	no	maximal, Sylow
88	C₂×C₆₁	—	2	4	no	—
89	C₂×C₆₁	C₂×#95	1	1	yes	characteristic
90	C₂×C₆₁	#417	1	2	yes	—
91	C₂×C₆₁	#418	1	1	yes	characteristic
92	C₂×C₆₁	#419	1	1	yes	characteristic
93	C₂×C₆₁	#428	1	2	yes	—
94	C₂²×M₁₆	D₆₁	1	1	yes	characteristic
95	C₂×#117	—	61	1	no	—
96	C₂²×C₄⋊C₄	—	61	1	no	—
97	#340	—	61	4	no	—
98	C₆₁	#1736	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	122	1	no	—
100	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	Dic₆₁	1	4	yes	complemented
104	C₂²×C₈	C₂×D₆₁	1	2	yes	—
105	#117	—	61	4	no	—
106	#117	—	122	2	no	—
107	C₂×C₄⋊C₄	—	61	4	no	—
108	C₂×C₄⋊C₄	—	61	2	no	—
109	C₂×C₄⋊C₄	—	61	2	no	—
110	C₂×C₄⋊C₄	—	122	1	no	—
111	C₂×C₄⋊C₄	—	122	2	no	—
112	C₂×C₂²⋊C₄	—	122	1	no	—
113	C₂×C₄²	—	122	1	no	—
114	C₂⁴	D₂₄₄	1	1	yes	characteristic
115	C₂²×C₄	—	122	4	no	—
116	C₂²×C₄	—	122	2	no	—
117	C₂²×C₄	—	122	2	no	—
118	C₂²×C₄	—	244	4	no	—
119	C₂²×C₄	Dic₁₂₂	1	2	yes	—
120	C₂²×C₄	C₄×D₆₁	1	2	yes	—
121	C₂²×C₄	C₄×D₆₁	1	2	yes	—
122	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
123	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
125	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₂×Dic₆₁	1	4	yes	complemented
131	C₄⋊C₄	—	61	4	no	—
132	C₄⋊C₄	—	61	4	no	—
133	C₄⋊C₄	—	122	4	no	—
134	C₄⋊C₄	—	122	2	no	—
135	C₄⋊C₄	—	122	2	no	—
136	C₂²⋊C₄	—	122	4	no	—
137	C₄²	—	122	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₄⋊Dic₆₁	1	2	yes	—
141	C₂³	#11758198	1	2	yes	—
142	C₂³	#11758198	1	2	yes	—
143	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
144	C₂×C₄	—	122	4	no	—
145	C₂×C₄	—	122	4	no	—
146	C₂×C₄	—	244	8	no	—
147	C₂×C₄	—	244	2	no	—
148	C₂×C₄	—	244	2	no	—
149	C₂×C₄	—	244	4	no	—
150	C₂×C₄	C₄×Dic₆₁	1	4	yes	—
151	C₂×C₄	#11758196	1	4	yes	—
152	C₂×C₄	#11758196	1	4	yes	—
153	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
154	C₂×C₄	#11758198	1	2	yes	—
155	C₂×C₄	#11758198	1	2	yes	—
156	C₂×C₄	C₂²⋊Dic₆₁	1	4	yes	—
157	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	#14519684	1	4	yes	—
169	C₂²	C₂×#11758198	1	1	yes	characteristic
170	C₂²	C₂×#11758198	1	1	yes	characteristic
171	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
172	C₂²	#14519805	1	2	yes	—
173	C₂²	#14519810	1	2	yes	—
174	C₄	—	244	4	no	—
175	C₄	—	244	4	no	—
176	C₄	#14519684	1	4	yes	—
177	C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
178	C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
179	C₄	C₂×#11758196	1	1	yes	characteristic
180	C₄	C₂×#11758196	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×#14519684	1	1	yes	characteristic
183	C₂	#16839764	1	2	yes	—
184	C₂	#16839768	1	1	yes	characteristic
185	C₂	#16839772	1	1	yes	characteristic
186	C₂	#16839804	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂²×M₁₆×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂²×M₁₆	D₆₁
hypocenter	C₆₁	#1736
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16839525	C₂	semidirect
7. #14519736	C₄	semidirect
10. C₂×C₄⋊Dic₆₁	C₄	semidirect
18. C₂×M₁₆×C₆₁	C₄	semidirect
36. C₂×C₈×C₆₁	C₂×C₄	semidirect
42. C₄⋊Dic₆₁	C₂×C₄	semidirect
98. C₆₁	#1736	semidirect
103. C₂×M₁₆	Dic₆₁	semidirect
130. C₂×C₈	C₂×Dic₆₁	semidirect