Subgroups — #21423815

Order 7552.

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	C₂²×#11745664	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273686	C₂	1	1	yes	maximal, characteristic
3	#16837164	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	2	no	—
6	C₂×#11745664	—	2	2	no	—
7	C₂×#11745664	C₄	1	4	yes	complemented
8	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#14273686	—	2	2	no	—
10	#14273686	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
13	C₂×C₄×Dic₅₉	—	2	1	no	—
14	C₂³×C₄×C₅₉	—	2	1	no	—
15	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
18	C₂×C₄⋊C₄×C₅₉	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
20	C₂⁴×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	—	2	4	no	—
22	C₂²×C₄×C₅₉	—	2	2	no	—
23	C₂²×C₄×C₅₉	—	4	4	no	—
24	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	D₄	1	2	yes	—
29	C₂²×C₄×C₅₉	D₄	1	2	yes	—
30	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	2	no	—
33	C₄⋊C₄×C₅₉	—	2	4	no	—
34	C₄⋊C₄×C₅₉	—	2	2	no	—
35	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
36	C₂²⋊Dic₅₉	—	2	4	no	—
37	C₄⋊Dic₅₉	—	2	2	no	—
38	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	complemented
39	C₄×Dic₅₉	—	2	4	no	—
40	#11745664	—	2	8	no	—
41	C₂×C₅₉⋊C₈	—	2	2	no	—
42	C₂×C₅₉⋊C₈	—	2	2	no	—
43	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	complemented
44	C₂³×C₅₉	—	2	2	no	—
45	C₂³×C₅₉	—	2	2	no	—
46	C₂³×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	2	4	no	—
51	C₂×C₄×C₅₉	—	4	8	no	—
52	C₂×C₄×C₅₉	—	4	4	no	—
53	C₂×C₄×C₅₉	C₄²	1	4	yes	—
54	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×Dic₅₉	—	2	4	no	—
67	C₂×Dic₅₉	—	4	2	no	—
68	C₂×Dic₅₉	—	4	2	no	—
69	C₅₉⋊C₈	—	2	4	no	—
70	C₂²×C₅₉	—	2	4	no	—
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	—	2	4	no	—
74	C₂²×C₅₉	#95	1	4	yes	—
75	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂²×C₅₉	#137	1	2	yes	—
80	C₄×C₅₉	—	4	4	no	—
81	C₄×C₅₉	#95	1	4	yes	—
82	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
83	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	Dic₅₉	—	4	4	no	—
87	#1736	—	59	1	no	maximal, Sylow
88	C₂×C₅₉	—	2	4	no	—
89	C₂×C₅₉	C₂×#95	1	1	yes	characteristic
90	C₂×C₅₉	#417	1	2	yes	—
91	C₂×C₅₉	#418	1	1	yes	characteristic
92	C₂×C₅₉	#419	1	1	yes	characteristic
93	C₂×C₅₉	#428	1	2	yes	—
94	C₂²×M₁₆	—	59	1	no	—
95	C₂×#117	—	59	1	no	—
96	C₂²×C₄⋊C₄	D₅₉	1	1	yes	characteristic
97	#340	—	59	4	no	—
98	C₅₉	#1736	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	2	1	no	—
100	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
101	C₂×M₁₆	—	59	4	no	—
102	C₂×M₁₆	—	118	2	no	—
103	C₂×M₁₆	—	118	2	no	—
104	C₂²×C₈	—	59	2	no	—
105	#117	—	59	4	no	—
106	#117	—	118	2	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	59	2	no	—
110	C₂×C₄⋊C₄	Dic₅₉	1	4	yes	complemented
111	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
112	C₂×C₂²⋊C₄	—	118	1	no	—
113	C₂×C₄²	—	118	1	no	—
114	C₂⁴	C₂²⋊D₅₉	1	1	yes	characteristic
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	—	118	2	no	—
119	C₂²×C₄	Dic₁₁₈	1	2	yes	—
120	C₂²×C₄	C₄×D₅₉	1	2	yes	—
121	C₂²×C₄	C₄×D₅₉	1	2	yes	—
122	C₂²×C₄	D₂₃₆	1	2	yes	—
123	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
126	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
127	M₁₆	—	118	8	no	—
128	C₂×C₈	—	59	4	no	—
129	C₂×C₈	—	118	2	no	—
130	C₂×C₈	—	118	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	59	4	no	—
134	C₄⋊C₄	—	118	2	no	—
135	C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
136	C₂²⋊C₄	—	118	4	no	—
137	C₄²	—	118	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	Dic₅₉⋊C₄	1	2	yes	—
141	C₂³	(C₂×D₅₉)⋊C₄	1	2	yes	—
142	C₂³	C₂²⋊Dic₅₉	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	118	4	no	—
148	C₂×C₄	—	236	2	no	—
149	C₂×C₄	—	236	2	no	—
150	C₂×C₄	C₄×Dic₅₉	1	4	yes	—
151	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
152	C₂×C₄	Dic₅₉⋊C₄	1	4	yes	—
153	C₂×C₄	C₄⋊Dic₅₉	1	4	yes	—
154	C₂×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
155	C₂×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
156	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
157	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
163	C₈	—	118	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	C₅₉⋊#95	1	4	yes	—
169	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
170	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
171	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
172	C₂²	#14273753	1	2	yes	—
173	C₂²	#14273758	1	2	yes	—
174	C₄	—	4	4	no	—
175	C₄	—	236	4	no	—
176	C₄	C₅₉⋊#95	1	4	yes	—
177	C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
178	C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
179	C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
180	C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×C₅₉⋊#95	1	1	yes	characteristic
183	C₂	#16837401	1	2	yes	—
184	C₂	#16837407	1	1	yes	characteristic
185	C₂	#16837411	1	1	yes	characteristic
186	C₂	#16837442	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂²×C₄⋊C₄×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂²×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#1736
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #16837164	C₂	semidirect
7. C₂×#11745664	C₄	semidirect
10. #14273686	C₄	semidirect
18. C₂×C₄⋊C₄×C₅₉	C₄	semidirect
38. C₄⋊Dic₅₉	C₂×C₄	semidirect
43. C₂×C₅₉⋊C₈	C₂×C₄	semidirect
98. C₅₉	#1736	semidirect
110. C₂×C₄⋊C₄	Dic₅₉	semidirect