Subgroups — #28554299

Order 15104.

826 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	#41×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#41×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#21422388	C₂	1	3	yes	maximal, complemented
4	#21422383	C₂	1	3	yes	maximal, complemented
5	#21422383	C₂	1	3	yes	maximal, complemented
6	C₂×#16837124	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₅₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₅₉⋊C₈	C₄	1	2	yes	—
9	Q₈×C₅₉⋊C₈	C₄	1	2	yes	—
10	D₄×C₅₉⋊C₈	C₄	1	6	yes	—
11	D₄×C₅₉⋊C₈	C₄	1	6	yes	—
12	#16837455	C₂²	1	3	yes	—
13	#16837455	C₂²	1	3	yes	—
14	C₂×C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
16	#16837124	C₂²	1	1	yes	characteristic
17	#16837124	C₂²	1	3	yes	—
18	#16837124	C₂²	1	3	yes	—
19	#16837124	C₂²	1	3	yes	—
20	#16837124	C₂²	1	6	yes	—
21	C₂×#131×C₅₉	C₄	1	2	yes	—
22	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#405	C₄	1	6	yes	—
24	C₄×M₁₆×C₅₉	C₂²	1	3	yes	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
26	C₂²×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
27	C₂²×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
28	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	6	yes	—
29	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	6	yes	—
30	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	6	yes	—
31	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	6	yes	—
32	C₄×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
33	C₄×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
34	C₄×C₅₉⋊C₈	C₂³	1	3	yes	—
35	C₄×C₅₉⋊C₈	C₂³	1	3	yes	—
36	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
38	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
39	#131×C₅₉	—	2	8	no	—
40	C₂×M₁₆×C₅₉	C₂×C₄	1	6	yes	—
41	C₂²×C₈×C₅₉	C₂×C₄	1	6	yes	—
42	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₅₉	C₂×C₄	1	3	yes	—
45	C₄×D₄×C₅₉	C₂×C₄	1	3	yes	—
46	#117×C₅₉	C₂×C₄	1	3	yes	—
47	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
48	#97×C₅₉	C₂×C₄	1	6	yes	—
49	C₄×C₈×C₅₉	C₂×C₄	1	6	yes	—
50	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₉	C₂³	1	3	yes	—
52	#41×C₅₉	M₁₆	1	8	yes	—
53	C₂×Q₈×C₅₉	C₄²	1	1	yes	characteristic
54	C₂×D₄×C₅₉	C₄²	1	3	yes	—
55	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
58	M₁₆×C₅₉	—	2	12	no	—
59	C₂×C₈×C₅₉	—	2	12	no	—
60	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
62	C₄⋊C₄×C₅₉	C₄²	1	3	yes	—
63	C₄⋊C₄×C₅₉	C₄²	1	3	yes	—
64	C₂²⋊C₄×C₅₉	C₄²	1	6	yes	—
65	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
66	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
67	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
68	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₅₉⋊C₈	—	2	12	no	—
70	C₂×C₅₉⋊C₈	—	4	6	no	—
71	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₅₉⋊C₈	C₂²×C₄	1	6	yes	—
73	C₂×C₅₉⋊C₈	C₂²×C₄	1	6	yes	—
74	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
75	C₂³×C₅₉	C₂×C₄²	1	3	yes	—
76	Q₈×C₅₉	#97	1	4	yes	—
77	D₄×C₅₉	#97	1	12	yes	—
78	C₂×C₄×C₅₉	—	2	6	no	—
79	C₂×C₄×C₅₉	—	2	6	no	—
80	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
81	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
82	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
83	C₂×C₄×C₅₉	C₂×M₁₆	1	12	yes	—
84	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₅₉	—	2	2	no	—
88	C₈×C₅₉	—	2	6	no	—
89	C₅₉⋊C₈	—	2	6	no	—
90	C₅₉⋊C₈	—	2	2	no	—
91	C₅₉⋊C₈	—	2	6	no	—
92	C₅₉⋊C₈	#131	1	4	yes	complemented
93	#17320	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	—	2	3	no	—
95	C₂²×C₅₉	—	2	3	no	—
96	C₂²×C₅₉	C₂×#97	1	6	yes	—
97	C₂²×C₅₉	C₂²×C₄²	1	1	yes	characteristic
98	C₄×C₅₉	—	2	3	no	—
99	C₄×C₅₉	—	2	3	no	—
100	C₄×C₅₉	C₂×#97	1	6	yes	—
101	C₄×C₅₉	C₂²×M₁₆	1	2	yes	—
102	C₄×C₅₉	C₂×#131	1	2	yes	—
103	C₄×C₅₉	#568	1	2	yes	—
104	C₈×#41	—	59	1	no	—
105	C₈×#41	—	59	1	no	—
106	C₄×#131	D₅₉	1	1	yes	characteristic
107	#1452	—	59	3	no	—
108	#1451	—	59	3	no	—
109	#1451	—	59	3	no	—
110	C₂×#322	—	59	3	no	—
111	C₂×C₅₉	—	2	6	no	—
112	C₂×C₅₉	C₂²×#97	1	1	yes	characteristic
113	C₂×C₅₉	C₄×#131	1	1	yes	characteristic
114	C₂×C₅₉	#2875	1	1	yes	characteristic
115	C₂×#131	Dic₅₉	1	2	yes	—
116	C₄×#41	C₂×D₅₉	1	1	yes	characteristic
117	C₈×Q₈	—	59	2	no	—
118	C₈×Q₈	—	59	2	no	—
119	C₈×D₄	—	59	6	no	—
120	C₈×D₄	—	59	6	no	—
121	#431	—	59	3	no	—
122	#431	—	59	3	no	—
123	#405	Dic₅₉	1	6	yes	—
124	C₄×M₁₆	C₂×D₅₉	1	3	yes	—
125	C₂×C₄×C₈	—	59	3	no	—
126	C₂×C₄×C₈	—	59	3	no	—
127	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
128	#322	—	59	1	no	—
129	#322	—	59	3	no	—
130	#322	—	59	3	no	—
131	#322	—	59	3	no	—
132	#322	—	59	6	no	—
133	C₅₉	#17320	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	C₂×Dic₅₉	1	1	yes	characteristic
135	#131	—	2	8	no	—
136	C₂×M₁₆	C₂×Dic₅₉	1	6	yes	—
137	C₂²×C₈	—	59	6	no	—
138	C₂²×C₈	—	59	6	no	—
139	C₂²×C₈	C₂×Dic₅₉	1	6	yes	—
140	C₄×Q₈	C₄×D₅₉	1	1	yes	characteristic
141	C₄×Q₈	C₄×D₅₉	1	1	yes	characteristic
142	C₄×D₄	C₄×D₅₉	1	3	yes	—
143	C₄×D₄	C₄×D₅₉	1	3	yes	—
144	#117	C₂×Dic₅₉	1	3	yes	—
145	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
146	C₄⋊C₈	—	59	6	no	—
147	C₄⋊C₈	—	59	6	no	—
148	C₂²⋊C₈	—	59	6	no	—
149	C₂²⋊C₈	—	59	6	no	—
150	#97	C₂×Dic₅₉	1	6	yes	—
151	C₄×C₈	—	59	6	no	—
152	C₄×C₈	—	59	6	no	—
153	C₄×C₈	—	59	1	no	—
154	C₄×C₈	—	59	1	no	—
155	C₄×C₈	—	59	3	no	—
156	C₄×C₈	—	59	3	no	—
157	C₄×C₈	C₂×Dic₅₉	1	6	yes	—
158	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₅₉	1	3	yes	—
160	#41	#11745664	1	8	yes	—
161	C₂×Q₈	C₄×Dic₅₉	1	1	yes	characteristic
162	C₂×D₄	C₄×Dic₅₉	1	3	yes	—
163	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
164	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
165	C₂²×C₄	C₂²×Dic₅₉	1	3	yes	—
166	M₁₆	—	2	12	no	—
167	C₂×C₈	—	2	12	no	—
168	C₂×C₈	—	59	2	no	—
169	C₂×C₈	—	59	2	no	—
170	C₂×C₈	—	59	6	no	—
171	C₂×C₈	—	59	6	no	—
172	C₂×C₈	—	118	12	no	—
173	C₂×C₈	—	236	6	no	—
174	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
175	C₂×C₈	C₂²×Dic₅₉	1	6	yes	—
176	C₄⋊C₄	C₄×Dic₅₉	1	3	yes	—
177	C₄⋊C₄	C₄×Dic₅₉	1	3	yes	—
178	C₂²⋊C₄	C₄×Dic₅₉	1	6	yes	—
179	C₄²	C₂×C₄×D₅₉	1	3	yes	—
180	C₄²	C₂×C₄×D₅₉	1	3	yes	—
181	C₄²	C₂²×Dic₅₉	1	3	yes	—
182	C₄²	C₂³×D₅₉	1	1	yes	characteristic
183	C₂³	C₂×C₄×Dic₅₉	1	3	yes	—
184	Q₈	#14273638	1	4	yes	—
185	D₄	#14273638	1	12	yes	—
186	C₂×C₄	—	2	6	no	—
187	C₂×C₄	—	2	6	no	—
188	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
189	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
190	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
191	C₂×C₄	C₂×#11745664	1	12	yes	—
192	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
194	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	6	no	—
197	C₈	—	59	4	no	—
198	C₈	—	118	6	no	—
199	C₈	—	118	6	no	—
200	C₈	—	118	2	no	—
201	C₂²	—	2	3	no	—
202	C₂²	—	2	3	no	—
203	C₂²	C₂×#14273638	1	6	yes	—
204	C₂²	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
205	C₄	—	2	3	no	—
206	C₄	—	2	3	no	—
207	C₄	C₂×#14273638	1	6	yes	—
208	C₄	C₂²×#11745664	1	2	yes	—
209	C₄	#131×D₅₉	1	2	yes	—
210	C₄	#16838167	1	2	yes	—
211	C₂	—	2	6	no	—
212	C₂	C₂²×#14273638	1	1	yes	characteristic
213	C₂	#131×Dic₅₉	1	1	yes	characteristic
214	C₂	#21432522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₄²
Fitting	C₄×#131×C₅₉	C₂
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₄×#131	D₅₉
hypocenter	C₅₉	#17320
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #21422388	C₂	semidirect
4. #21422383	C₂	semidirect
5. #21422383	C₂	semidirect
6. C₂×#16837124	C₂	semidirect
92. C₅₉⋊C₈	#131	semidirect
133. C₅₉	#17320	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#41×C₅₉⋊C₈	2
#21422388	2
#21422383	2
C₂×#16837124	2
C₄×#131×C₅₉	2
#17320	59