Subgroups — #28505047

Order 15104.

314 conjugacy classes of subgroups.

The subgroup lattice has 182 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	C₄×#97×C₅₉	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	C₄	1	1	yes	characteristic
9	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×#97×C₅₉	C₂²	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	C₄³×C₅₉	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	—	2	2	no	—
22	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	#97×C₅₉	—	2	2	no	—
26	#97×C₅₉	—	2	2	no	—
27	#97×C₅₉	—	2	2	no	—
28	#97×C₅₉	C₈	1	4	yes	—
29	C₄×C₈×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	D₄	1	2	yes	—
31	C₄×C₈×C₅₉	Q₈	1	2	yes	—
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	—	4	4	no	—
54	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅₉	C₂×C₈	1	2	yes	—
56	C₅₉⋊C₁₆	—	4	4	no	—
57	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	4	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	4	2	no	—
64	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	—	2	2	no	—
72	C₈×C₅₉	C₄⋊C₈	1	4	yes	—
73	C₈×C₅₉	#124	1	2	yes	—
74	C₈×C₅₉	#125	1	2	yes	—
75	#12882	—	59	1	no	maximal, Sylow
76	C₂²×C₅₉	—	2	2	no	—
77	C₂²×C₅₉	#423	1	1	yes	characteristic
78	C₂²×C₅₉	#431	1	1	yes	characteristic
79	C₂²×C₅₉	#432	1	1	yes	characteristic
80	C₄×C₅₉	—	2	4	no	—
81	C₄×C₅₉	—	4	2	no	—
82	C₄×C₅₉	—	4	2	no	—
83	C₄×C₅₉	#382	1	1	yes	characteristic
84	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₅₉	#431	1	1	yes	characteristic
86	C₄×C₅₉	#432	1	1	yes	characteristic
87	C₂×#364	—	59	2	no	—
88	C₄×#97	D₅₉	1	1	yes	characteristic
89	#1377	—	59	2	no	—
90	#1375	—	59	2	no	—
91	C₂×C₅₉	—	2	1	no	—
92	C₂×C₅₉	—	2	1	no	—
93	C₂×C₅₉	#1831	1	1	yes	characteristic
94	C₂×C₅₉	#2165	1	2	yes	—
95	C₂×M₃₂	—	59	4	no	—
96	C₂×#97	Dic₅₉	1	1	yes	characteristic
97	C₂×#97	Dic₅₉	1	1	yes	characteristic
98	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
99	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
100	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₄³	Dic₅₉	1	1	yes	characteristic
103	#364	—	118	4	no	—
104	C₅₉	#12882	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
106	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
113	M₃₂	—	118	4	no	—
114	M₃₂	—	118	4	no	—
115	C₂×C₁₆	—	118	4	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	—	2	2	no	—
119	#97	C₅₉⋊C₈	1	4	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	Dic₁₁₈	1	2	yes	—
122	C₄×C₈	D₂₃₆	1	2	yes	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	#11745664	1	2	yes	—
126	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
132	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
133	C₂×C₈	#11745664	1	2	yes	—
134	C₂×C₈	#11745664	1	2	yes	—
135	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
137	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
138	C₂×C₈	#11745685	1	2	yes	—
139	C₂×C₈	#11745685	1	2	yes	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
146	C₄²	C₄⋊Dic₅₉	1	2	yes	—
147	C₁₆	—	236	4	no	—
148	C₂³	C₂×#11745664	1	1	yes	characteristic
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
156	C₂×C₄	#14273686	1	1	yes	characteristic
157	C₂×C₄	#14273686	1	1	yes	characteristic
158	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
159	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
160	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
164	C₈	#14273715	1	2	yes	—
165	C₈	#14273719	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#16837426	1	1	yes	characteristic
168	C₂²	#16837455	1	1	yes	characteristic
169	C₂²	#16837459	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	#16837268	1	1	yes	characteristic
174	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
175	C₄	#16837455	1	1	yes	characteristic
176	C₄	#16837459	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#21424213	1	1	yes	characteristic
180	C₂	#21425908	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14273686
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#97×C₅₉	C₂
socle	C₂²×C₅₉	#432
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#97	D₅₉
hypocenter	C₅₉	#12882
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
104. C₅₉	#12882	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
C₄×#97×C₅₉	2
#12882	59