Subgroups — #28742183

Order 15104.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433030	C₂	1	1	yes	maximal, characteristic
2	#21426004	C₂	1	4	yes	maximal, complemented
3	C₁₆×Dic₁₁₈	C₂	1	4	yes	maximal, complemented
4	#21425854	C₂	1	2	yes	maximal
5	#21425852	C₂	1	1	yes	maximal, characteristic
6	#21425692	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
8	#16837959	C₄	1	1	yes	characteristic
9	C₂×#14273667	C₂²	1	2	yes	—
10	#16837532	C₄	1	4	yes	—
11	C₈×Dic₁₁₈	C₂²	1	4	yes	—
12	#16837428	C₄	1	2	yes	—
13	#16837426	C₄	1	1	yes	characteristic
14	#16837357	C₂²	1	2	yes	—
15	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
16	Dic₅₉⋊C₁₆	C₂²	1	8	yes	—
17	#16837177	C₂²	1	4	yes	—
18	C₁₆×Dic₅₉	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
23	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
24	C₂×#11745664	C₂×C₄	1	2	yes	—
25	#14273727	C₈	1	2	yes	—
26	#14273719	C₈	1	2	yes	—
27	#14273706	C₈	1	4	yes	—
28	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
29	#14273686	C₂×C₄	1	2	yes	—
30	#14273667	—	2	4	no	—
31	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
32	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
33	Dic₅₉⋊C₈	C₂×C₄	1	8	yes	—
34	#14273637	C₂×C₄	1	4	yes	—
35	C₈×Dic₅₉	C₂³	1	4	yes	—
36	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₉	—	2	2	no	—
40	C₂×C₁₆×C₅₉	Q₈	1	4	yes	complemented
41	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
42	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₁₈	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₉	#41	1	2	yes	—
54	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
58	C₂²⋊Dic₅₉	C₂×C₈	1	4	yes	—
59	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
60	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
61	Dic₅₉⋊C₄	C₂×C₈	1	8	yes	—
62	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
63	#11745664	—	2	4	no	—
64	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
65	C₁₆×C₅₉	—	2	2	no	—
66	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
67	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
68	C₅₉⋊C₁₆	—	2	4	no	—
69	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	4	no	—
71	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₅₉	#131	1	2	yes	—
76	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₅₉	C₂²×C₈	1	8	yes	—
78	Dic₁₁₈	—	2	8	no	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	C₄×Q₈	1	2	yes	—
81	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
83	C₅₉⋊C₈	—	2	4	no	—
84	#33474	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
87	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	C₈×Q₈	1	2	yes	—
90	C₄×C₅₉	#504	1	4	yes	—
91	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
93	Dic₅₉	—	2	8	no	—
94	#2960	—	59	1	no	—
95	#2183	—	59	4	no	—
96	Q₈×C₁₆	—	59	4	no	—
97	#2151	—	59	2	no	—
98	#2150	—	59	1	no	—
99	#2108	—	59	2	no	—
100	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
101	C₂×C₅₉	—	2	2	no	—
102	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
103	C₂×C₅₉	C₂×#504	1	2	yes	—
104	#533	—	59	1	no	—
105	C₂×M₃₂	—	59	2	no	—
106	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
107	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
108	#446	—	59	4	no	—
109	C₈×Q₈	—	59	4	no	—
110	#424	—	59	2	no	—
111	#423	—	59	1	no	—
112	#405	—	59	2	no	—
113	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
114	C₄⋊C₁₆	—	59	8	no	—
115	C₄⋊C₁₆	—	59	4	no	—
116	#346	—	59	4	no	—
117	C₄×C₁₆	—	59	4	no	—
118	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
119	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
120	C₅₉	#33474	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	59	2	no	—
122	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	59	2	no	—
125	#125	—	59	2	no	—
126	C₂²⋊Q₈	—	59	4	no	—
127	C₄×Q₈	—	59	4	no	—
128	#117	—	59	2	no	—
129	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
130	M₃₂	—	118	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	59	4	no	—
133	C₂×C₁₆	Dic₁₁₈	1	4	yes	complemented
134	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
135	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
136	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
137	C₄⋊C₈	—	59	8	no	—
138	C₄⋊C₈	—	59	4	no	—
139	#97	—	59	4	no	—
140	C₄×C₈	—	59	4	no	—
141	C₄×C₈	C₄×D₅₉	1	2	yes	—
142	C₄×C₈	C₂²×D₅₉	1	2	yes	—
143	C₂×Q₈	—	59	4	no	—
144	C₂²×C₄	C₈×D₅₉	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
146	M₁₆	—	118	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	59	4	no	—
149	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
152	C₂×C₈	#11745685	1	2	yes	—
153	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
154	C₄⋊C₄	—	59	4	no	—
155	C₄⋊C₄	—	59	8	no	—
156	C₄⋊C₄	—	59	4	no	—
157	C₂²⋊C₄	—	59	4	no	—
158	C₄²	—	59	4	no	—
159	C₄²	C₈×D₅₉	1	1	yes	characteristic
160	C₄²	C₈×D₅₉	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₅₉	1	2	yes	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	118	4	no	—
164	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
165	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
166	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
167	Q₈	—	118	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	59	8	no	—
170	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
174	C₂×C₄	#14273733	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	118	4	no	—
178	C₈	C₄×Dic₁₁₈	1	2	yes	—
179	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
180	C₈	C₂×#11745685	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Dic₁₁₈	1	2	yes	—
183	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	118	8	no	—
186	C₄	C₈×Dic₁₁₈	1	2	yes	—
187	C₄	#16837818	1	4	yes	—
188	C₄	C₂×C₄×Dic₁₁₈	1	1	yes	characteristic
189	C₄	C₂×#14273733	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Dic₁₁₈	1	1	yes	characteristic
192	C₂	C₂×#16837818	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#33474
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21426004	C₂	semidirect
3. C₁₆×Dic₁₁₈	C₂	semidirect
40. C₂×C₁₆×C₅₉	Q₈	semidirect
120. C₅₉	#33474	semidirect
133. C₂×C₁₆	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21433030	2
#21426004	2
C₁₆×Dic₁₁₈	2
#21425854	2
#21425852	2
#21425692	2
C₂×C₄×C₁₆×C₅₉	2
#33474	59