Subgroups — #21433131

Order 7552.

464 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837480	C₂	1	2	yes	maximal, complemented
2	#16837474	C₂	1	4	yes	maximal, complemented
3	D₄×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837455	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837366	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#11745664	C₂²	1	4	yes	complemented
10	C₂²×C₅₉⋊C₈	C₂²	1	4	yes	complemented
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	complemented
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	complemented
13	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
16	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	complemented
17	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	complemented
18	#14273638	C₂²	1	2	yes	complemented
19	C₄×C₅₉⋊C₈	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₅₉	C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₉	C₂²	1	2	yes	—
22	C₄×D₄×C₅₉	C₄	1	2	yes	—
23	C₄×D₄×C₅₉	C₄	1	1	yes	characteristic
24	C₄×D₄×C₅₉	C₄	1	1	yes	characteristic
25	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₅₉	C₂²	1	2	yes	—
27	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₉	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₉	C₄	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
31	C₂⁴×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₅₉	—	2	4	no	—
33	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	—	4	4	no	—
37	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
45	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
46	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
49	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
50	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
52	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	#11745664	—	4	4	no	—
57	C₂×C₅₉⋊C₈	—	4	4	no	—
58	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
59	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
60	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
61	C₂³×C₅₉	—	2	4	no	—
62	C₂³×C₅₉	—	2	4	no	—
63	C₂³×C₅₉	—	4	2	no	—
64	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
65	C₂³×C₅₉	C₂²×C₄	1	2	yes	—
66	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	D₄×C₅₉	—	2	4	no	—
68	D₄×C₅₉	—	2	2	no	—
69	D₄×C₅₉	M₁₆	1	4	yes	complemented
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	2	4	no	—
73	C₂×C₄×C₅₉	—	2	4	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	4	2	no	—
76	C₂×C₄×C₅₉	—	4	2	no	—
77	C₂×C₄×C₅₉	—	4	2	no	—
78	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₅₉⋊C₈	—	2	2	no	—
87	C₅₉⋊C₈	—	2	4	no	—
88	C₅₉⋊C₈	—	2	2	no	—
89	C₂²×C₅₉	—	2	1	no	—
90	C₂²×C₅₉	—	2	2	no	—
91	C₂²×C₅₉	—	2	1	no	—
92	C₂²×C₅₉	—	2	2	no	—
93	C₂²×C₅₉	—	2	2	no	—
94	C₂²×C₅₉	—	2	2	no	—
95	C₂²×C₅₉	—	2	2	no	—
96	C₂²×C₅₉	—	4	8	no	—
97	C₂²×C₅₉	C₂×M₁₆	1	4	yes	—
98	C₂²×C₅₉	#131	1	2	yes	—
99	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
100	C₄×C₅₉	—	2	1	no	—
101	C₄×C₅₉	—	2	2	no	—
102	C₄×C₅₉	—	2	2	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	—	2	1	no	—
106	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
107	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
108	#2974	—	59	1	no	maximal, Sylow
109	C₂×C₅₉	—	2	2	no	—
110	C₂×C₅₉	—	2	4	no	—
111	C₂×C₅₉	—	4	2	no	—
112	C₂×C₅₉	#518	1	1	yes	characteristic
113	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
115	C₂×C₄×D₄	D₅₉	1	1	yes	characteristic
116	#436	—	59	2	no	—
117	#435	—	59	4	no	—
118	C₈×D₄	—	59	2	no	—
119	#432	—	59	1	no	—
120	#431	—	59	1	no	—
121	#407	—	59	2	no	—
122	C₂×C₂²⋊C₈	—	59	2	no	—
123	C₅₉	#2974	1	1	yes	characteristic, complemented, Sylow
124	C₂²×D₄	Dic₅₉	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₅₉	1	2	yes	—
126	C₂×M₁₆	—	59	4	no	—
127	C₂²×C₈	—	59	4	no	—
128	C₄×D₄	Dic₅₉	1	2	yes	—
129	C₄×D₄	Dic₅₉	1	1	yes	characteristic
130	C₄×D₄	Dic₅₉	1	1	yes	characteristic
131	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
132	C₄×D₄	C₂×D₅₉	1	2	yes	—
133	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
134	C₂×C₄⋊C₄	Dic₅₉	1	1	yes	characteristic
135	C₂×C₂²⋊C₄	Dic₅₉	1	2	yes	—
136	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
137	C₄⋊C₈	—	59	2	no	—
138	C₄⋊C₈	—	59	2	no	—
139	C₂²⋊C₈	—	59	1	no	—
140	C₂²⋊C₈	—	59	2	no	—
141	C₂²⋊C₈	—	59	1	no	—
142	C₂²⋊C₈	—	59	4	no	—
143	C₂²⋊C₈	—	59	4	no	—
144	#97	—	59	2	no	—
145	C₄×C₈	—	59	2	no	—
146	C₂⁴	C₂×Dic₅₉	1	2	yes	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
149	C₂×D₄	C₂×Dic₅₉	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₅₉	1	1	yes	characteristic
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
155	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
156	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
157	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
159	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
160	M₁₆	—	236	4	no	—
161	C₂×C₈	—	59	2	no	—
162	C₂×C₈	—	59	2	no	—
163	C₂×C₈	—	59	4	no	—
164	C₂×C₈	—	236	4	no	—
165	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
166	C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
167	C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
168	C₂²⋊C₄	C₂×Dic₅₉	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₅₉	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₅₉	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₅₉	1	2	yes	—
172	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
173	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
174	C₄²	C₂²×D₅₉	1	1	yes	characteristic
175	C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	C₂³	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	4	2	no	—
179	C₂³	C₂²×Dic₅₉	1	2	yes	—
180	C₂³	C₂²×Dic₅₉	1	2	yes	—
181	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
182	D₄	—	2	4	no	—
183	D₄	—	2	2	no	—
184	D₄	#11745664	1	4	yes	complemented
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
198	C₂×C₄	C₂²×Dic₅₉	1	2	yes	—
199	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
200	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
201	C₈	—	118	2	no	—
202	C₈	—	118	4	no	—
203	C₈	—	118	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	8	no	—
212	C₂²	C₂×#11745664	1	4	yes	—
213	C₂²	#14273734	1	2	yes	—
214	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	C₂×#11745664	1	2	yes	—
222	C₄	#14273772	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	4	no	—
225	C₂	—	4	2	no	—
226	C₂	#16837877	1	1	yes	characteristic
227	C₂	C₂²×#11745664	1	1	yes	characteristic
228	C₂	C₂×#14273734	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₂×C₄×D₄×C₅₉	C₂
hypercenter	C₂×C₄×D₄	D₅₉
hypocenter	C₅₉	#2974
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #16837480	C₂	semidirect
2. #16837474	C₂	semidirect
3. D₄×C₅₉⋊C₈	C₂	semidirect
4. #16837459	C₂	semidirect
5. #16837455	C₂	semidirect
6. #16837366	C₂	semidirect
7. C₂×C₂²⋊(C₅₉⋊C₈)	C₂	semidirect
9. C₂×#11745664	C₂²	semidirect
10. C₂²×C₅₉⋊C₈	C₂²	semidirect
11. C₄⋊(C₅₉⋊C₈)	C₂²	semidirect
12. C₄⋊(C₅₉⋊C₈)	C₂²	semidirect
16. C₂²⋊(C₅₉⋊C₈)	C₂²	semidirect
17. C₂²⋊(C₅₉⋊C₈)	C₂²	semidirect
18. #14273638	C₂²	semidirect
19. C₄×C₅₉⋊C₈	C₂²	semidirect
69. D₄×C₅₉	M₁₆	semidirect
123. C₅₉	#2974	semidirect
184. D₄	#11745664	semidirect