Subgroups — #21432521

Order 7552.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273733	C₂	1	2	yes	maximal
2	C₄×#11745685	C₂	1	1	yes	maximal, characteristic
3	#16837359	C₂	1	4	yes	maximal, complemented
4	#16837357	C₂	1	2	yes	maximal, complemented
5	C₄×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837354	C₂	1	2	yes	maximal, complemented
7	#97×D₅₉	C₂	1	2	yes	maximal, complemented
8	C₂×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11745685	C₂²	1	1	yes	characteristic
10	#14273733	C₄	1	16	yes	complemented
11	C₂×#11745664	C₂²	1	2	yes	—
12	C₂×#11745659	C₂²	1	4	yes	—
13	C₂×C₈×D₅₉	C₂²	1	4	yes	—
14	C₄×Dic₁₁₈	C₄	1	2	yes	—
15	C₄×D₂₃₆	C₄	1	2	yes	—
16	C₄×C₂²⋊D₅₉	C₄	1	4	yes	—
17	#14273686	C₂²	1	1	yes	characteristic
18	#14273684	C₂²	1	2	yes	—
19	C₄²×D₅₉	C₂²	1	2	yes	—
20	#14273638	C₂²	1	2	yes	—
21	#14273637	C₂²	1	4	yes	—
22	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
23	C₈×Dic₅₉	C₂²	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
26	#97×C₅₉	C₂²	1	1	yes	characteristic
27	#97×C₅₉	C₂²	1	1	yes	characteristic
28	#97×C₅₉	C₂²	1	2	yes	—
29	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
32	#11745685	C₂×C₄	1	8	yes	—
33	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₉	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₅₉	C₂³	1	2	yes	—
36	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₅₉	C₂³	1	2	yes	—
39	C₂×C₈×C₅₉	C₂³	1	2	yes	—
40	C₄²×C₅₉	C₂×C₄	1	2	yes	—
41	C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
44	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
46	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
47	C₄×Dic₅₉	C₂×C₄	1	4	yes	—
48	C₄×Dic₅₉	C₂³	1	2	yes	—
49	#11745664	C₂×C₄	1	8	yes	complemented
50	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
51	#11745659	C₂×C₄	1	16	yes	complemented
52	C₈×D₅₉	C₂×C₄	1	16	yes	complemented
53	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₅₉	—	4	2	no	—
56	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₅₉	C₄²	1	8	yes	—
62	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
63	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
64	D₂₃₆	C₄²	1	4	yes	—
65	C₄×D₅₉	—	4	4	no	—
66	C₄×D₅₉	C₂²×C₄	1	8	yes	—
67	Dic₁₁₈	C₄²	1	4	yes	—
68	C₈×C₅₉	C₂²×C₄	1	4	yes	—
69	C₈×C₅₉	C₂²×C₄	1	4	yes	—
70	C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂²×C₅₉	—	2	2	no	—
72	C₂²×C₅₉	C₂×C₄²	1	2	yes	—
73	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₅₉	—	2	4	no	—
75	C₂×D₅₉	C₂×C₄²	1	4	yes	—
76	C₄×C₅₉	—	2	2	no	—
77	C₄×C₅₉	—	2	2	no	—
78	C₄×C₅₉	C₂×C₄²	1	2	yes	—
79	C₄×C₅₉	C₂³×C₄	1	2	yes	—
80	Dic₅₉	—	2	4	no	—
81	Dic₅₉	C₂×C₄²	1	4	yes	—
82	#2875	—	59	1	no	maximal, Sylow
83	C₂×C₅₉	—	2	2	no	—
84	C₂×C₅₉	C₂²×C₄²	1	1	yes	characteristic
85	C₂×C₅₉	#568	1	2	yes	—
86	D₅₉	—	2	4	no	—
87	C₂×#131	—	59	2	no	—
88	C₄×#41	—	59	1	no	—
89	#405	—	59	2	no	—
90	#405	—	59	4	no	—
91	C₄×M₁₆	—	59	1	no	—
92	C₄×M₁₆	—	59	2	no	—
93	C₂×#97	—	59	2	no	—
94	C₂×#97	D₅₉	1	1	yes	characteristic, complemented
95	C₅₉	#2875	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	59	1	no	—
97	#131	—	59	16	no	—
98	C₂×M₁₆	—	59	2	no	—
99	C₂×M₁₆	—	59	4	no	—
100	C₂²×C₈	—	59	4	no	—
101	C₂²×C₈	C₂×D₅₉	1	2	yes	—
102	C₄×Q₈	—	59	2	no	—
103	C₄×D₄	—	59	4	no	—
104	C₄×D₄	—	59	2	no	—
105	#117	—	59	1	no	—
106	#117	—	59	2	no	—
107	C₂×C₄²	—	59	2	no	—
108	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
109	#97	—	59	4	no	—
110	#97	—	59	2	no	—
111	#97	C₂×D₅₉	1	1	yes	characteristic
112	#97	C₂×D₅₉	1	1	yes	characteristic
113	#97	C₂×D₅₉	1	2	yes	—
114	C₄×C₈	—	59	4	no	—
115	C₄×C₈	—	59	2	no	—
116	#41	—	59	8	no	—
117	C₂×Q₈	—	59	1	no	—
118	C₂×D₄	—	59	2	no	—
119	C₂×D₄	—	59	1	no	—
120	C₂²×C₄	—	59	2	no	—
121	C₂²×C₄	—	59	4	no	—
122	C₂²×C₄	C₄×D₅₉	1	2	yes	—
123	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
124	M₁₆	—	59	8	no	—
125	M₁₆	—	59	16	no	—
126	C₂×C₈	—	59	4	no	—
127	C₂×C₈	—	59	16	no	—
128	C₂×C₈	C₄×D₅₉	1	8	yes	complemented
129	C₂×C₈	C₂²×D₅₉	1	2	yes	—
130	C₂×C₈	C₂²×D₅₉	1	2	yes	—
131	C₄⋊C₄	—	59	4	no	—
132	C₄⋊C₄	—	59	2	no	—
133	C₂²⋊C₄	—	59	2	no	—
134	C₂²⋊C₄	—	59	4	no	—
135	C₄²	—	59	4	no	—
136	C₄²	—	59	2	no	—
137	C₄²	C₄×D₅₉	1	2	yes	—
138	C₄²	C₂²×D₅₉	1	1	yes	characteristic
139	C₄²	C₂²×D₅₉	1	1	yes	characteristic
140	C₂³	—	59	2	no	—
141	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
142	Q₈	—	59	4	no	—
143	D₄	—	59	8	no	—
144	D₄	—	59	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	59	2	no	—
147	C₂×C₄	—	59	8	no	—
148	C₂×C₄	—	59	4	no	—
149	C₂×C₄	—	236	4	no	—
150	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₅₉	1	4	yes	—
154	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
155	C₈	—	59	8	no	—
156	C₈	C₂×C₄×D₅₉	1	4	yes	—
157	C₈	C₂×C₄×D₅₉	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	59	4	no	—
160	C₂²	—	118	4	no	—
161	C₂²	C₄²×D₅₉	1	2	yes	—
162	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	59	4	no	—
166	C₄	—	118	4	no	—
167	C₄	C₄²×D₅₉	1	2	yes	—
168	C₄	C₂²×C₄×D₅₉	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	118	4	no	—
171	C₂	C₂×C₄²×D₅₉	1	1	yes	characteristic
172	C₂	#16838168	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄²
Fitting	C₂×#97×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×C₄
hypercenter	C₂×#97	D₅₉
hypocenter	C₅₉	#2875
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. #16837359	C₂	semidirect
4. #16837357	C₂	semidirect
5. C₄×#11745664	C₂	semidirect
6. #16837354	C₂	semidirect
7. #97×D₅₉	C₂	semidirect
8. C₂×#97×C₅₉	C₂	semidirect
10. #14273733	C₄	semidirect
37. C₂×C₈×C₅₉	C₂×C₄	semidirect
49. #11745664	C₂×C₄	semidirect
51. #11745659	C₂×C₄	semidirect
52. C₈×D₅₉	C₂×C₄	semidirect
94. C₂×#97	D₅₉	semidirect
95. C₅₉	#2875	semidirect
128. C₂×C₈	C₄×D₅₉	semidirect