Subgroups — #28504011

Order 15104.

358 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₄×#14273667	C₂	1	2	yes	maximal
3	#21422178	C₂	1	2	yes	maximal
4	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	4	yes	—
6	#16837223	C₄	1	8	yes	complemented
7	#16837178	—	2	2	no	—
8	C₄×C₅₉⋊C₁₆	—	2	2	no	—
9	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
10	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
11	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#376	C₄	1	1	yes	characteristic
16	#14273667	—	2	4	no	—
17	#14273667	C₂×C₄	1	8	yes	—
18	C₂×C₅₉⋊C₁₆	—	2	4	no	—
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	C₂³	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	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	#97×C₅₉	—	2	2	no	—
27	#97×C₅₉	C₈	1	4	yes	—
28	#97×C₅₉	C₈	1	4	yes	—
29	C₄×C₈×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₉	D₄	1	2	yes	—
32	C₄×C₈×C₅₉	Q₈	1	2	yes	—
33	#95×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	—	2	1	no	—
35	C₂²×C₄×C₅₉	—	2	1	no	—
36	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	C₄²	1	2	yes	—
41	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₉	#41	1	2	yes	—
51	C₄²×C₅₉	—	2	1	no	—
52	C₄²×C₅₉	—	2	1	no	—
53	C₄²×C₅₉	C₄²	1	2	yes	—
54	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅₉	C₂×C₈	1	2	yes	—
56	C₄²×C₅₉	C₂×C₈	1	2	yes	—
57	C₅₉⋊C₁₆	—	2	8	no	—
58	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	4	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	4	2	no	—
64	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
65	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₅₉	—	2	2	no	—
72	C₈×C₅₉	C₄×C₈	1	4	yes	—
73	C₈×C₅₉	C₄⋊C₈	1	4	yes	—
74	C₈×C₅₉	C₄×D₄	1	2	yes	—
75	C₈×C₅₉	C₄×Q₈	1	2	yes	—
76	#12656	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	2	no	—
78	C₂²×C₅₉	C₄×M₁₆	1	1	yes	characteristic
79	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₉	#431	1	1	yes	characteristic
81	C₄×C₅₉	—	2	4	no	—
82	C₄×C₅₉	—	4	2	no	—
83	C₄×C₅₉	—	4	2	no	—
84	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₅₉	#431	1	1	yes	characteristic
88	C₂×#364	—	59	2	no	—
89	C₄×M₃₂	—	59	2	no	—
90	#1726	D₅₉	1	1	yes	characteristic
91	#1376	—	59	2	no	—
92	C₂×C₅₉	—	2	1	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₅₉	#2153	1	2	yes	—
96	C₂×M₃₂	—	59	4	no	—
97	C₂×#97	Dic₅₉	1	1	yes	characteristic
98	C₂×#97	Dic₅₉	1	1	yes	characteristic
99	C₂×#97	Dic₅₉	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
103	#376	Dic₅₉	1	1	yes	characteristic
104	#364	—	59	8	no	—
105	#346	—	118	2	no	—
106	C₄×C₁₆	—	118	2	no	—
107	C₅₉	#12656	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
115	M₃₂	—	59	8	no	—
116	M₃₂	—	118	4	no	—
117	C₂×C₁₆	—	118	4	no	—
118	#97	—	2	2	no	—
119	#97	C₅₉⋊C₈	1	4	yes	—
120	#97	C₅₉⋊C₈	1	4	yes	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	Dic₁₁₈	1	2	yes	—
123	C₄×C₈	C₄×D₅₉	1	4	yes	—
124	C₄×C₈	D₂₃₆	1	2	yes	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	#11745664	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
133	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
134	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
135	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
136	C₂×C₈	#11745664	1	2	yes	—
137	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
138	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
139	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
141	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
142	C₂×C₈	#11745685	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
146	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
147	C₄²	C₄×Dic₅₉	1	2	yes	—
148	C₄²	C₄⋊Dic₅₉	1	2	yes	—
149	C₁₆	—	118	8	no	—
150	C₂³	C₂×#11745664	1	1	yes	characteristic
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	—	4	2	no	—
156	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
157	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
159	C₂×C₄	#14273686	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	C₄×C₅₉⋊C₈	1	4	yes	—
165	C₈	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
166	C₈	C₄×D₂₃₆	1	2	yes	—
167	C₈	C₄×Dic₁₁₈	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	C₄×#11745664	1	1	yes	characteristic
170	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
171	C₂²	#16837455	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
176	C₄	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
177	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
178	C₄	#16837455	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
182	C₂	#21425862	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1726	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#12656
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #16837223	C₄	semidirect
107. C₅₉	#12656	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
C₄×#14273667	2
#21422178	2
C₅₉×#1726	2
#12656	59