Subgroups — #21432523

Order 7552.

556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273735	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273733	C₂	1	1	yes	maximal, characteristic
3	#16837870	C₂	1	1	yes	maximal, characteristic
4	#16837359	C₂	1	2	yes	maximal, complemented
5	#16837358	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837356	C₂	1	2	yes	maximal, complemented
7	M₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745659	C₂	1	2	yes	maximal, complemented
9	C₂×#14273637	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₅₉	C₂	1	2	yes	maximal, complemented
11	C₅₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11745685	C₂²	1	1	yes	characteristic
13	#14273735	C₄	1	8	yes	complemented
14	#14273733	C₄	1	8	yes	complemented
15	C₂×#11745664	C₂²	1	1	yes	characteristic
16	C₂×#11745659	C₂²	1	2	yes	—
17	C₂×#11745659	C₂²	1	2	yes	—
18	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	2	yes	—
20	C₂×C₈×D₅₉	C₂²	1	2	yes	—
21	#14273692	C₄	1	2	yes	—
22	#14273689	C₄	1	2	yes	—
23	(C₂²⋊D₅₉)⋊C₄	C₄	1	4	yes	—
24	#14273684	C₂²	1	2	yes	—
25	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
26	C₄²×D₅₉	C₂²	1	2	yes	—
27	#14273638	C₂²	1	2	yes	—
28	#14273637	C₂²	1	1	yes	characteristic
29	#14273637	C₂²	1	2	yes	—
30	#14273637	C₂²	1	1	yes	characteristic
31	#14273637	C₂²	1	1	yes	characteristic
32	#14273637	C₂²	1	1	yes	characteristic
33	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₅₉	C₂²	1	2	yes	—
35	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
37	#117×C₅₉	C₂²	1	1	yes	characteristic
38	#97×C₅₉	C₂²	1	2	yes	—
39	C₄×C₈×C₅₉	C₂²	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
42	C₂²×Dic₅₉	C₂×C₄	1	2	yes	—
43	#11745685	C₂×C₄	1	4	yes	—
44	#11745685	C₂×C₄	1	4	yes	—
45	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₉	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₅₉	C₂³	1	2	yes	—
48	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂³	1	2	yes	—
53	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
56	C₄²×C₅₉	C₂³	1	2	yes	—
57	(C₂×D₅₉)⋊C₄	C₂×C₄	1	4	yes	—
58	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₅₉	C₂×C₄	1	4	yes	—
60	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
61	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
62	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
63	#11745664	C₂×C₄	1	4	yes	complemented
64	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	complemented
65	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
66	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
68	#11745659	C₂×C₄	1	8	yes	complemented
69	#11745659	C₂×C₄	1	8	yes	complemented
70	C₈×D₅₉	C₂×C₄	1	8	yes	complemented
71	C₈×D₅₉	C₂×C₄	1	8	yes	complemented
72	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₂²⋊D₅₉	C₄²	1	8	yes	—
80	C₂×Dic₅₉	—	4	2	no	—
81	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
83	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
84	D₂₃₆	C₄²	1	4	yes	—
85	C₄×D₅₉	—	4	4	no	—
86	C₄×D₅₉	C₂²×C₄	1	4	yes	—
87	C₄×D₅₉	C₂²×C₄	1	4	yes	—
88	Dic₁₁₈	C₄²	1	4	yes	—
89	C₈×C₅₉	C₂²×C₄	1	2	yes	—
90	C₈×C₅₉	C₂²×C₄	1	4	yes	—
91	C₈×C₅₉	C₂²×C₄	1	2	yes	—
92	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
93	C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
94	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂²×C₅₉	—	2	2	no	—
96	C₂²×C₅₉	C₂×C₄²	1	2	yes	—
97	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₅₉	—	2	4	no	—
99	C₂×D₅₉	C₂×C₄²	1	4	yes	—
100	C₄×C₅₉	—	2	4	no	—
101	C₄×C₅₉	C₂×C₄²	1	2	yes	—
102	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
103	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
104	Dic₅₉	—	2	2	no	—
105	Dic₅₉	—	2	2	no	—
106	Dic₅₉	C₂×C₄²	1	4	yes	—
107	#2875	—	59	1	no	maximal, Sylow
108	C₂×C₅₉	—	2	2	no	—
109	C₂×C₅₉	C₂²×C₄²	1	1	yes	characteristic
110	C₂×C₅₉	#568	1	2	yes	—
111	D₅₉	—	2	4	no	—
112	C₂×#131	—	59	1	no	—
113	C₂×#131	—	59	1	no	—
114	C₄×#41	—	59	1	no	—
115	#405	—	59	1	no	—
116	#405	—	59	2	no	—
117	#405	—	59	2	no	—
118	#405	D₅₉	1	1	yes	characteristic, complemented
119	C₄×M₁₆	—	59	1	no	—
120	C₄×M₁₆	—	59	2	no	—
121	C₂×#97	—	59	1	no	—
122	C₂×#97	—	59	2	no	—
123	C₅₉	#2875	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	59	1	no	—
125	#131	—	59	8	no	—
126	#131	—	59	8	no	—
127	C₂×M₁₆	—	59	1	no	—
128	C₂×M₁₆	—	59	2	no	—
129	C₂×M₁₆	—	59	2	no	—
130	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
131	C₂²×C₈	—	59	1	no	—
132	C₂²×C₈	—	59	2	no	—
133	C₂²×C₈	—	59	2	no	—
134	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
135	C₄×Q₈	—	59	2	no	—
136	C₄×D₄	—	59	2	no	—
137	C₄×D₄	—	59	4	no	—
138	#117	—	59	2	no	—
139	#117	C₂×D₅₉	1	1	yes	characteristic
140	C₂×C₄²	—	59	1	no	—
141	C₂×C₄²	—	59	2	no	—
142	#97	—	59	1	no	—
143	#97	—	59	2	no	—
144	#97	—	59	1	no	—
145	#97	—	59	1	no	—
146	#97	—	59	1	no	—
147	#97	—	59	2	no	—
148	#97	C₂×D₅₉	1	2	yes	—
149	C₄×C₈	—	59	2	no	—
150	C₄×C₈	—	59	2	no	—
151	C₄×C₈	C₂×D₅₉	1	2	yes	—
152	#41	—	59	4	no	—
153	#41	—	59	4	no	—
154	C₂×Q₈	—	59	1	no	—
155	C₂×D₄	—	59	2	no	—
156	C₂×D₄	—	59	1	no	—
157	C₂²×C₄	—	59	2	no	—
158	C₂²×C₄	—	59	2	no	—
159	C₂²×C₄	—	59	4	no	—
160	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
161	M₁₆	—	59	4	no	—
162	M₁₆	—	59	8	no	—
163	M₁₆	—	59	8	no	—
164	M₁₆	C₄×D₅₉	1	4	yes	complemented
165	C₂×C₈	—	59	4	no	—
166	C₂×C₈	—	59	2	no	—
167	C₂×C₈	—	59	1	no	—
168	C₂×C₈	—	59	1	no	—
169	C₂×C₈	—	59	8	no	—
170	C₂×C₈	—	59	8	no	—
171	C₂×C₈	C₄×D₅₉	1	4	yes	complemented
172	C₂×C₈	C₂²×D₅₉	1	2	yes	—
173	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
175	C₄⋊C₄	—	59	4	no	—
176	C₄⋊C₄	C₄×D₅₉	1	2	yes	—
177	C₂²⋊C₄	—	59	4	no	—
178	C₂²⋊C₄	C₄×D₅₉	1	2	yes	—
179	C₄²	—	59	4	no	—
180	C₄²	—	59	2	no	—
181	C₄²	—	59	1	no	—
182	C₄²	—	59	1	no	—
183	C₄²	C₂²×D₅₉	1	2	yes	—
184	C₂³	—	59	2	no	—
185	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
186	Q₈	—	59	4	no	—
187	D₄	—	59	8	no	—
188	D₄	—	59	4	no	—
189	C₂×C₄	—	59	2	no	—
190	C₂×C₄	—	59	4	no	—
191	C₂×C₄	—	59	4	no	—
192	C₂×C₄	—	59	2	no	—
193	C₂×C₄	—	59	2	no	—
194	C₂×C₄	—	236	4	no	—
195	C₂×C₄	—	236	2	no	—
196	C₂×C₄	C₂×C₄×D₅₉	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
200	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
201	C₈	—	59	2	no	—
202	C₈	—	59	4	no	—
203	C₈	—	59	2	no	—
204	C₈	C₂×C₄×D₅₉	1	2	yes	—
205	C₈	C₂×C₄×D₅₉	1	4	yes	—
206	C₈	C₂×C₄×D₅₉	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	59	4	no	—
209	C₂²	—	118	4	no	—
210	C₂²	C₄²×D₅₉	1	2	yes	—
211	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	59	4	no	—
214	C₄	—	118	2	no	—
215	C₄	—	118	2	no	—
216	C₄	C₄²×D₅₉	1	2	yes	—
217	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
218	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	118	4	no	—
221	C₂	C₂×C₄²×D₅₉	1	1	yes	characteristic
222	C₂	#16838167	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄²
Fitting	C₅₉×#405	C₂
socle	C₂²×C₅₉	C₂³×C₄
hypercenter	#405	D₅₉
hypocenter	C₅₉	#2875
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #16837359	C₂	semidirect
5. #16837358	C₂	semidirect
6. #16837356	C₂	semidirect
7. M₁₆×Dic₅₉	C₂	semidirect
8. C₄×#11745659	C₂	semidirect
9. C₂×#14273637	C₂	semidirect
10. #97×D₅₉	C₂	semidirect
11. C₅₉×#405	C₂	semidirect
13. #14273735	C₄	semidirect
14. #14273733	C₄	semidirect
49. M₁₆×C₅₉	C₂×C₄	semidirect
50. C₂×C₈×C₅₉	C₂×C₄	semidirect
63. #11745664	C₂×C₄	semidirect
64. C₂×C₅₉⋊C₈	C₂×C₄	semidirect
68. #11745659	C₂×C₄	semidirect
69. #11745659	C₂×C₄	semidirect
70. C₈×D₅₉	C₂×C₄	semidirect
71. C₈×D₅₉	C₂×C₄	semidirect
118. #405	D₅₉	semidirect
123. C₅₉	#2875	semidirect
164. M₁₆	C₄×D₅₉	semidirect
171. C₂×C₈	C₄×D₅₉	semidirect