Subgroups — #28505043

Order 15104.

298 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837223	C₂	1	2	yes	maximal
2	#21422180	C₂	1	2	yes	maximal
3	#21422176	C₂	1	2	yes	maximal
4	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	4	yes	—
6	#16837223	—	2	4	no	—
7	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
8	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
9	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#376	C₄	1	1	yes	characteristic
14	#14273667	—	2	4	no	—
15	#14273667	—	2	4	no	—
16	C₂×C₅₉⋊C₁₆	—	2	4	no	—
17	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
24	#97×C₅₉	—	2	2	no	—
25	#97×C₅₉	—	2	2	no	—
26	#97×C₅₉	—	2	2	no	—
27	C₄×C₈×C₅₉	—	2	2	no	—
28	C₄×C₈×C₅₉	C₈	1	4	yes	—
29	C₄×C₈×C₅₉	D₄	1	2	yes	—
30	C₄×C₈×C₅₉	Q₈	1	2	yes	—
31	#95×C₅₉	—	2	2	no	—
32	C₂²×C₄×C₅₉	—	2	1	no	—
33	C₂²×C₄×C₅₉	—	2	1	no	—
34	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₉	#41	1	2	yes	—
48	C₂×C₈×C₅₉	#41	1	2	yes	—
49	C₄²×C₅₉	—	2	1	no	—
50	C₄²×C₅₉	—	2	1	no	—
51	C₄²×C₅₉	—	2	1	no	—
52	C₄²×C₅₉	—	2	1	no	—
53	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₉	C₂×C₈	1	2	yes	—
55	C₅₉⋊C₁₆	—	4	4	no	—
56	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	4	2	no	—
61	C₂×C₄×C₅₉	—	4	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₅₉	—	2	2	no	—
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	C₄⋊C₈	1	4	yes	—
72	C₈×C₅₉	#124	1	2	yes	—
73	C₈×C₅₉	#125	1	2	yes	—
74	#12881	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	2	no	—
76	C₂²×C₅₉	#423	1	1	yes	characteristic
77	C₂²×C₅₉	#431	1	1	yes	characteristic
78	C₂²×C₅₉	#432	1	1	yes	characteristic
79	C₄×C₅₉	—	4	2	no	—
80	C₄×C₅₉	—	4	2	no	—
81	C₄×C₅₉	—	4	2	no	—
82	C₄×C₅₉	#382	1	1	yes	characteristic
83	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₅₉	#431	1	1	yes	characteristic
85	C₄×C₅₉	#432	1	1	yes	characteristic
86	C₂×#364	—	59	2	no	—
87	#1726	D₅₉	1	1	yes	characteristic
88	#1377	—	59	2	no	—
89	#1375	—	59	2	no	—
90	C₂×C₅₉	—	2	1	no	—
91	C₂×C₅₉	—	2	1	no	—
92	C₂×C₅₉	#1831	1	1	yes	characteristic
93	C₂×C₅₉	#2166	1	2	yes	—
94	C₂×M₃₂	—	59	4	no	—
95	C₂×#97	Dic₅₉	1	1	yes	characteristic
96	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
97	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
98	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
99	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	#376	Dic₅₉	1	1	yes	characteristic
102	#364	—	118	4	no	—
103	C₅₉	#12881	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
109	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
111	M₃₂	—	118	4	no	—
112	M₃₂	—	118	4	no	—
113	C₂×C₁₆	—	118	4	no	—
114	#97	—	2	2	no	—
115	#97	—	2	2	no	—
116	#97	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	C₅₉⋊C₈	1	4	yes	—
119	C₄×C₈	Dic₁₁₈	1	2	yes	—
120	C₄×C₈	D₂₃₆	1	2	yes	—
121	#95	—	2	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	#11745664	1	2	yes	—
125	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
131	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
132	C₂×C₈	#11745664	1	2	yes	—
133	C₂×C₈	#11745664	1	2	yes	—
134	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
136	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂×C₈	#11745685	1	2	yes	—
138	C₂×C₈	#11745685	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
144	C₄²	C₄⋊Dic₅₉	1	2	yes	—
145	C₁₆	—	236	4	no	—
146	C₂³	C₂×#11745664	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
154	C₂×C₄	#14273686	1	1	yes	characteristic
155	C₂×C₄	#14273686	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
157	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
158	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
162	C₈	#14273715	1	2	yes	—
163	C₈	#14273719	1	2	yes	—
164	C₂²	—	2	2	no	—
165	C₂²	#16837426	1	1	yes	characteristic
166	C₂²	#16837455	1	1	yes	characteristic
167	C₂²	#16837459	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	#16837268	1	1	yes	characteristic
172	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
173	C₄	#16837455	1	1	yes	characteristic
174	C₄	#16837459	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#21424213	1	1	yes	characteristic
178	C₂	#21425912	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14273686
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1726	C₂
socle	C₂²×C₅₉	#432
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#12881
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
103. C₅₉	#12881	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837223	2
#21422180	2
#21422176	2
C₅₉×#1726	2
#12881	59