Subgroups — #21421980

Order 7552.

188 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#16837272	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₈	—	2	1	no	—
5	C₂×C₄⋊Dic₅₉	—	2	1	no	—
6	C₂×C₄⋊Dic₅₉	—	2	1	no	—
7	C₂×C₄⋊Dic₅₉	C₄	1	2	yes	—
8	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	complemented
10	C₂²×C₈×C₅₉	—	2	1	no	—
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	—	2	1	no	—
13	C₄⋊C₈×C₅₉	C₄	1	2	yes	complemented
14	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	—	2	2	no	—
19	C₂×C₈×C₅₉	—	4	1	no	—
20	C₂×C₈×C₅₉	—	4	1	no	—
21	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	Q₈	1	1	yes	characteristic
25	C₄⋊Dic₅₉	—	4	1	no	—
26	C₄⋊Dic₅₉	—	4	1	no	—
27	C₄⋊Dic₅₉	—	4	1	no	—
28	C₄⋊Dic₅₉	—	4	1	no	—
29	C₄⋊Dic₅₉	—	4	2	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	—	4	1	no	—
32	C₂×C₅₉⋊C₈	—	4	1	no	—
33	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
43	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
44	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
50	C₂×Dic₅₉	—	4	2	no	—
51	C₂×Dic₅₉	—	4	2	no	—
52	C₂×Dic₅₉	—	4	2	no	—
53	C₈×C₅₉	—	4	2	no	—
54	C₅₉⋊C₈	—	4	2	no	—
55	C₂²×C₅₉	#95	1	1	yes	characteristic
56	C₂²×C₅₉	#100	1	1	yes	characteristic
57	C₂²×C₅₉	#101	1	1	yes	characteristic
58	C₂²×C₅₉	#102	1	1	yes	characteristic
59	C₂²×C₅₉	#102	1	1	yes	characteristic
60	C₂²×C₅₉	#103	1	1	yes	characteristic
61	C₂²×C₅₉	#103	1	1	yes	characteristic
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	2	1	no	—
64	C₄×C₅₉	#102	1	1	yes	characteristic
65	C₄×C₅₉	#102	1	1	yes	characteristic
66	C₄×C₅₉	#103	1	1	yes	characteristic
67	C₄×C₅₉	#103	1	1	yes	characteristic
68	C₄×C₅₉	#106	1	1	yes	characteristic
69	C₄×C₅₉	#106	1	1	yes	characteristic
70	C₄×C₅₉	#107	1	1	yes	characteristic
71	C₄×C₅₉	#107	1	1	yes	characteristic
72	Dic₅₉	—	8	2	no	—
73	#1296	—	59	1	no	maximal, Sylow
74	C₂×C₅₉	#331	1	1	yes	characteristic
75	C₂×C₅₉	#332	1	1	yes	characteristic
76	C₂×C₅₉	#332	1	1	yes	characteristic
77	C₂×C₅₉	#333	1	1	yes	characteristic
78	C₂×C₅₉	#340	1	1	yes	characteristic
79	C₂×C₅₉	#340	1	1	yes	characteristic
80	C₂×C₅₉	#343	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	59	1	no	—
82	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
83	#384	—	59	1	no	—
84	C₅₉	#1296	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	118	1	no	—
87	C₂×C₄⋊C₄	—	59	2	no	—
88	C₂×C₄⋊C₄	—	118	1	no	—
89	C₂×C₄⋊C₄	—	118	1	no	—
90	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	59	2	no	—
93	C₄⋊C₈	—	118	1	no	—
94	C₄⋊C₈	Dic₅₉	1	2	yes	complemented
95	C₂²×C₄	—	118	2	no	—
96	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
97	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	118	2	no	—
103	C₂×C₈	—	236	1	no	—
104	C₂×C₈	—	236	1	no	—
105	C₄⋊C₄	—	236	2	no	—
106	C₄⋊C₄	—	236	1	no	—
107	C₄⋊C₄	—	236	1	no	—
108	C₄⋊C₄	—	236	1	no	—
109	C₄⋊C₄	—	236	1	no	—
110	C₄²	Dic₁₁₈	1	1	yes	characteristic
111	C₄²	C₄×D₅₉	1	1	yes	characteristic
112	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
113	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
114	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	236	2	no	—
118	C₂×C₄	—	236	2	no	—
119	C₂×C₄	—	236	2	no	—
120	C₂×C₄	#11745660	1	1	yes	characteristic
121	C₂×C₄	#11745660	1	1	yes	characteristic
122	C₂×C₄	D₄₇₂	1	1	yes	characteristic
123	C₂×C₄	Dic₂₃₆	1	1	yes	characteristic
124	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
125	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
126	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	#11745669	1	1	yes	characteristic
130	C₂×C₄	#11745670	1	1	yes	characteristic
131	C₂×C₄	#11745671	1	1	yes	characteristic
132	C₂×C₄	#11745672	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	236	2	no	—
136	C₂²	C₅₉⋊#95	1	1	yes	characteristic
137	C₂²	#14273643	1	1	yes	characteristic
138	C₂²	#14273645	1	1	yes	characteristic
139	C₂²	#14273647	1	1	yes	characteristic
140	C₂²	#14273649	1	1	yes	characteristic
141	C₂²	#14273650	1	1	yes	characteristic
142	C₂²	#14273652	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	472	2	no	—
146	C₄	#14273648	1	1	yes	characteristic
147	C₄	#14273649	1	1	yes	characteristic
148	C₄	#14273651	1	1	yes	characteristic
149	C₄	#14273652	1	1	yes	characteristic
150	C₄	#14273658	1	1	yes	characteristic
151	C₄	#14273659	1	1	yes	characteristic
152	C₄	#14273660	1	1	yes	characteristic
153	C₄	#14273661	1	1	yes	characteristic
154	C₂	#16837145	1	1	yes	characteristic
155	C₂	#16837147	1	1	yes	characteristic
156	C₂	#16837149	1	1	yes	characteristic
157	C₂	#16837150	1	1	yes	characteristic
158	C₂	#16837164	1	1	yes	characteristic
159	C₂	#16837165	1	1	yes	characteristic
160	C₂	#16837170	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₄²
Frattini	C₂×C₄²	C₂×D₅₉
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#1296
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₅₉⋊C₈)	C₄	semidirect
13. C₄⋊C₈×C₅₉	C₄	semidirect
84. C₅₉	#1296	semidirect
94. C₄⋊C₈	Dic₅₉	semidirect