Subgroups — #21423834

Order 7552.

388 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11745664	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273686	C₂	1	1	yes	maximal, characteristic
3	#16837170	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	2	no	—
6	C₂×#11745664	—	2	1	no	—
7	C₂×#11745664	C₄	1	4	yes	complemented
8	C₂×#11745664	C₂²	1	2	yes	—
9	C₂²×C₅₉⋊C₈	—	2	1	no	—
10	#14273686	—	2	1	no	—
11	#14273686	—	2	1	no	—
12	#14273686	—	2	1	no	—
13	#14273686	—	2	1	no	—
14	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₅₉	C₄	1	2	yes	—
16	C₂×C₄×Dic₅₉	C₄	1	2	yes	—
17	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₉	—	2	2	no	—
19	C₂×M₁₆×C₅₉	—	2	1	no	—
20	C₂×M₁₆×C₅₉	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
22	C₂²×C₈×C₅₉	—	2	1	no	—
23	C₂⁴×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₉	D₄	1	2	yes	—
28	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
33	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₉	C₂×C₄	1	4	yes	—
35	M₁₆×C₅₉	—	2	4	no	—
36	M₁₆×C₅₉	—	2	2	no	—
37	M₁₆×C₅₉	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₅₉	—	2	4	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂²⋊Dic₅₉	—	2	2	no	—
41	C₂²⋊Dic₅₉	—	2	2	no	—
42	C₄⋊Dic₅₉	—	2	2	no	—
43	C₄⋊Dic₅₉	—	2	2	no	—
44	C₄×Dic₅₉	—	2	2	no	—
45	C₄×Dic₅₉	—	2	2	no	—
46	#11745664	—	2	4	no	—
47	#11745664	—	2	2	no	—
48	#11745664	C₂×C₄	1	4	yes	—
49	C₂×C₅₉⋊C₈	—	2	4	no	—
50	C₂×C₅₉⋊C₈	—	2	2	no	—
51	C₂³×C₅₉	—	4	2	no	—
52	C₂³×C₅₉	C₄²	1	2	yes	—
53	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
56	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₄²	1	2	yes	—
58	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×Dic₅₉	—	2	4	no	—
76	C₂×Dic₅₉	—	4	4	no	—
77	C₈×C₅₉	—	2	4	no	—
78	C₅₉⋊C₈	—	2	4	no	—
79	C₂²×C₅₉	—	2	2	no	—
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	—	2	2	no	—
82	C₂²×C₅₉	—	4	2	no	—
83	C₂²×C₅₉	—	4	2	no	—
84	C₂²×C₅₉	#95	1	2	yes	—
85	C₂²×C₅₉	#95	1	2	yes	—
86	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
87	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₅₉	#95	1	2	yes	—
90	C₄×C₅₉	#95	1	2	yes	—
91	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	Dic₅₉	—	4	4	no	—
96	#1742	—	59	1	no	maximal, Sylow
97	C₂×C₅₉	—	2	2	no	—
98	C₂×C₅₉	—	2	2	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	—	2	1	no	—
101	C₂×C₅₉	C₂×#95	1	1	yes	characteristic
102	C₂×C₅₉	#413	1	2	yes	—
103	C₂²×M₁₆	—	59	1	no	—
104	C₂²×M₁₆	D₅₉	1	1	yes	characteristic
105	C₂×#117	—	59	1	no	—
106	#343	—	59	4	no	—
107	C₅₉	#1742	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂×M₁₆	—	59	4	no	—
112	C₂×M₁₆	—	59	2	no	—
113	C₂×M₁₆	—	118	1	no	—
114	C₂×M₁₆	—	118	2	no	—
115	C₂×M₁₆	Dic₅₉	1	4	yes	complemented
116	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	118	1	no	—
119	#117	—	118	1	no	—
120	#117	—	118	1	no	—
121	#117	—	118	1	no	—
122	#117	—	118	1	no	—
123	C₂×C₄⋊C₄	—	59	2	no	—
124	C₂×C₂²⋊C₄	—	59	2	no	—
125	C₂×C₄²	—	59	2	no	—
126	C₂⁴	C₄×D₅₉	1	1	yes	characteristic
127	C₂²×C₄	—	59	4	no	—
128	C₂²×C₄	Dic₁₁₈	1	2	yes	—
129	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	C₄×D₅₉	1	2	yes	—
131	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
132	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
134	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
137	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
138	M₁₆	—	2	4	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	59	4	no	—
141	M₁₆	—	118	4	no	—
142	M₁₆	—	118	2	no	—
143	M₁₆	C₂×Dic₅₉	1	4	yes	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	118	4	no	—
147	C₂×C₈	—	118	2	no	—
148	C₄⋊C₄	—	118	2	no	—
149	C₄⋊C₄	—	118	2	no	—
150	C₂²⋊C₄	—	118	2	no	—
151	C₂²⋊C₄	—	118	2	no	—
152	C₄²	—	118	2	no	—
153	C₄²	—	118	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₄×Dic₅₉	1	2	yes	—
156	C₂³	Dic₅₉⋊C₄	1	2	yes	—
157	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
158	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
159	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₄	—	118	4	no	—
161	C₂×C₄	—	236	4	no	—
162	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
163	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
164	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
165	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
166	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
167	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
168	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
170	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
171	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
173	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
174	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	118	4	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	C₅₉⋊#95	1	2	yes	—
188	C₂²	C₅₉⋊#95	1	2	yes	—
189	C₂²	C₂×C₄×Dic₅₉	1	1	yes	characteristic
190	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
191	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
192	C₄	—	236	4	no	—
193	C₄	C₅₉⋊#95	1	2	yes	—
194	C₄	C₅₉⋊#95	1	2	yes	—
195	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
196	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
197	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
198	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	—	2	2	no	—
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	C₂×C₅₉⋊#95	1	1	yes	characteristic
204	C₂	#16837384	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	(C₂×D₅₉)⋊C₄
commutator	C₂²×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂²×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	C₂²×M₁₆	D₅₉
hypocenter	C₅₉	#1742
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16837170	C₂	semidirect
7. C₂×#11745664	C₄	semidirect
20. C₂×M₁₆×C₅₉	C₄	semidirect
107. C₅₉	#1742	semidirect
115. C₂×M₁₆	Dic₅₉	semidirect