Subgroups — #21433209

Order 7552.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11745685	C₂	1	1	yes	maximal, characteristic
2	#16837532	C₂	1	2	yes	maximal, complemented
3	#16837481	C₂	1	2	yes	maximal, complemented
4	#16837470	C₂	1	4	yes	maximal, complemented
5	#16837458	C₂	1	2	yes	maximal, complemented
6	#16837424	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11745685	C₄	1	1	yes	characteristic
10	C₂×#11745659	C₂²	1	4	yes	—
11	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
12	C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
13	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
14	C₄×D₂₃₆	C₄	1	1	yes	characteristic
15	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₅₉	C₄	1	2	yes	—
17	C₄×C₂²⋊D₅₉	C₂²	1	2	yes	—
18	#14273686	C₄	1	1	yes	characteristic
19	#14273684	C₄	1	2	yes	—
20	C₄²×D₅₉	C₂²	1	2	yes	—
21	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
23	Dic₅₉⋊C₈	C₂²	1	4	yes	—
24	(C₂×D₅₉)⋊C₈	C₂²	1	4	yes	—
25	#14273637	C₂²	1	4	yes	—
26	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
30	C₄×C₈×C₅₉	C₂²	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
35	#11745685	—	2	4	no	—
36	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₉	C₂³	1	2	yes	—
40	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂³	1	2	yes	—
42	C₂×C₈×C₅₉	C₂³	1	2	yes	—
43	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
48	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
49	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
52	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
56	C₄×Dic₅₉	C₂³	1	2	yes	—
57	C₂×C₅₉⋊C₈	—	4	2	no	—
58	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
59	#11745659	—	4	4	no	—
60	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₅₉	—	2	1	no	—
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	4	1	no	—
65	C₂×C₄×C₅₉	M₁₆	1	4	yes	complemented
66	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₅₉	—	2	4	no	—
73	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
76	D₂₃₆	—	2	2	no	—
77	C₄×D₅₉	—	2	4	no	—
78	C₄×D₅₉	—	2	2	no	—
79	C₄×D₅₉	—	2	2	no	—
80	C₄×D₅₉	—	4	2	no	—
81	Dic₁₁₈	—	2	2	no	—
82	C₈×C₅₉	—	2	2	no	—
83	C₈×C₅₉	—	2	2	no	—
84	C₅₉⋊C₈	—	2	4	no	—
85	C₂²×C₅₉	—	2	1	no	—
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	C₂×M₁₆	1	2	yes	—
88	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₅₉	—	2	2	no	—
90	C₂×D₅₉	—	2	2	no	—
91	C₂×D₅₉	—	2	2	no	—
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	—	2	1	no	—
95	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
96	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
97	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₅₉	—	2	2	no	—
100	Dic₅₉	—	2	2	no	—
101	Dic₅₉	—	2	2	no	—
102	#2989	—	59	1	no	maximal, Sylow
103	C₂×C₅₉	—	2	2	no	—
104	C₂×C₅₉	#520	1	1	yes	characteristic
105	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₅₉	#568	1	1	yes	characteristic
107	D₅₉	—	4	2	no	—
108	C₄×#41	—	59	1	no	—
109	#446	—	59	2	no	—
110	#436	—	59	2	no	—
111	#435	—	59	4	no	—
112	#432	—	59	2	no	—
113	#431	D₅₉	1	1	yes	characteristic, complemented
114	#423	—	59	2	no	—
115	C₂×C₄⋊C₈	—	59	1	no	—
116	C₅₉	#2989	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	—	59	1	no	—
118	C₂×M₁₆	—	59	4	no	—
119	C₂²×C₈	—	59	2	no	—
120	C₄×Q₈	—	59	1	no	—
121	C₄×Q₈	—	59	1	no	—
122	C₄×D₄	—	59	2	no	—
123	C₄×D₄	—	59	2	no	—
124	C₄×D₄	—	59	1	no	—
125	C₄×D₄	—	59	1	no	—
126	#117	—	59	1	no	—
127	#117	—	59	2	no	—
128	C₂×C₄²	—	59	2	no	—
129	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
130	C₄⋊C₈	—	59	4	no	—
131	C₄⋊C₈	—	59	2	no	—
132	C₄⋊C₈	—	59	2	no	—
133	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
135	C₂²⋊C₈	—	59	4	no	—
136	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
137	#97	—	59	4	no	—
138	C₄×C₈	C₂×D₅₉	1	2	yes	—
139	#41	—	118	4	no	—
140	C₂×Q₈	—	59	1	no	—
141	C₂×D₄	—	59	2	no	—
142	C₂×D₄	—	59	1	no	—
143	C₂²×C₄	—	59	2	no	—
144	C₂²×C₄	—	59	2	no	—
145	C₂²×C₄	—	59	2	no	—
146	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
149	M₁₆	—	236	4	no	—
150	C₂×C₈	—	59	4	no	—
151	C₂×C₈	—	236	2	no	—
152	C₂×C₈	C₂²×D₅₉	1	2	yes	—
153	C₂×C₈	C₂²×D₅₉	1	2	yes	—
154	C₄⋊C₄	—	59	2	no	—
155	C₄⋊C₄	—	59	2	no	—
156	C₄⋊C₄	—	59	1	no	—
157	C₄⋊C₄	—	59	1	no	—
158	C₂²⋊C₄	—	59	2	no	—
159	C₂²⋊C₄	—	59	2	no	—
160	C₂²⋊C₄	—	59	2	no	—
161	C₄²	—	59	2	no	—
162	C₄²	—	59	2	no	—
163	C₄²	—	59	2	no	—
164	C₄²	C₄×D₅₉	1	1	yes	characteristic
165	C₄²	C₄×D₅₉	1	1	yes	characteristic
166	C₄²	C₂²×D₅₉	1	1	yes	characteristic
167	C₄²	C₂²×D₅₉	1	1	yes	characteristic
168	C₂³	—	59	2	no	—
169	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
170	Q₈	—	118	2	no	—
171	D₄	—	118	4	no	—
172	D₄	—	118	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	59	2	no	—
177	C₂×C₄	—	59	2	no	—
178	C₂×C₄	—	59	2	no	—
179	C₂×C₄	—	118	4	no	—
180	C₂×C₄	—	118	2	no	—
181	C₂×C₄	—	118	2	no	—
182	C₂×C₄	—	236	2	no	—
183	C₂×C₄	#11745659	1	4	yes	complemented
184	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	118	4	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	118	2	no	—
196	C₂²	—	118	2	no	—
197	C₂²	—	118	2	no	—
198	C₂²	C₂×#11745659	1	2	yes	—
199	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	118	2	no	—
204	C₄	—	118	2	no	—
205	C₄	—	118	2	no	—
206	C₄	C₂×#11745659	1	2	yes	—
207	C₄	C₂×#11745659	1	2	yes	—
208	C₄	#14273773	1	1	yes	characteristic
209	C₄	#14273774	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	236	2	no	—
212	C₂	#16837882	1	1	yes	characteristic
213	C₂	C₂²×#11745659	1	1	yes	characteristic
214	C₂	#16838168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#431	C₂
hypercenter	#431	D₅₉
hypocenter	C₅₉	#2989
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16837532	C₂	semidirect
3. #16837481	C₂	semidirect
4. #16837470	C₂	semidirect
5. #16837458	C₂	semidirect
6. #16837424	C₂	semidirect
7. C₂×C₄⋊(C₅₉⋊C₈)	C₂	semidirect
8. C₅₉×#431	C₂	semidirect
65. C₂×C₄×C₅₉	M₁₆	semidirect
113. #431	D₅₉	semidirect
116. C₅₉	#2989	semidirect
183. C₂×C₄	#11745659	semidirect