Subgroups — #28500297

Order 15104.

220 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	#21425899	C₂	1	1	yes	maximal, characteristic
2	#21425895	C₂	1	1	yes	maximal, characteristic
3	#21422166	C₂	1	2	yes	maximal, complemented
4	#21422160	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
7	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
8	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
10	#16837178	—	2	1	no	—
11	C₄×C₅₉⋊C₁₆	—	2	1	no	—
12	C₅₉×#431	C₄	1	1	yes	characteristic
13	C₅₉×#431	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#322	C₄	1	1	yes	characteristic
16	C₅₉×#322	C₄	1	1	yes	characteristic
17	C₈²×C₅₉	C₂²	1	1	yes	characteristic
18	C₈²×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₅₉	—	2	1	no	—
25	C₄⋊C₈×C₅₉	C₈	1	2	yes	—
26	C₂²⋊C₈×C₅₉	—	2	1	no	—
27	C₂²⋊C₈×C₅₉	C₈	1	2	yes	—
28	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	2	1	no	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
52	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₅₉⋊C₁₆	—	4	2	no	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	C₂³×C₅₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	1	no	—
57	C₂×C₄×C₅₉	—	2	1	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₅₉	—	2	1	no	—
67	C₈×C₅₉	—	2	1	no	—
68	C₈×C₅₉	—	2	2	no	—
69	C₈×C₅₉	—	2	2	no	—
70	C₈×C₅₉	—	4	1	no	—
71	C₈×C₅₉	—	4	1	no	—
72	C₈×C₅₉	#108	1	2	yes	—
73	C₈×C₅₉	#135	1	2	yes	—
74	#12014	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	—	2	1	no	—
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₅₉	—	2	1	no	—
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	#427	1	1	yes	characteristic
84	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
85	C₄×C₅₉	#504	1	2	yes	—
86	#2163	—	59	1	no	—
87	#2162	—	59	1	no	—
88	#1450	D₅₉	1	1	yes	characteristic
89	#1370	—	59	2	no	—
90	#1367	—	59	2	no	—
91	C₂×C₅₉	—	4	1	no	—
92	C₂×C₅₉	#1564	1	1	yes	characteristic
93	C₂×C₅₉	#2151	1	1	yes	characteristic
94	C₂×C₅₉	#2153	1	1	yes	characteristic
95	#431	Dic₅₉	1	1	yes	characteristic
96	#431	Dic₅₉	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	C₄⋊C₁₆	—	59	2	no	—
99	C₄⋊C₁₆	—	59	2	no	—
100	C₂²⋊C₁₆	—	118	1	no	—
101	C₂²⋊C₁₆	—	118	1	no	—
102	#346	—	118	1	no	—
103	C₄×C₁₆	—	118	1	no	—
104	#322	Dic₅₉	1	1	yes	characteristic
105	#322	Dic₅₉	1	1	yes	characteristic
106	C₈²	C₂×D₅₉	1	1	yes	characteristic
107	C₈²	C₂×D₅₉	1	1	yes	characteristic
108	C₅₉	#12014	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
110	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
112	C₂×C₁₆	—	118	2	no	—
113	C₂×C₁₆	—	118	2	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	C₅₉⋊C₈	1	2	yes	—
116	C₂²⋊C₈	—	2	1	no	—
117	C₂²⋊C₈	C₅₉⋊C₈	1	2	yes	—
118	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
119	C₄×C₈	D₂₃₆	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
123	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
124	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	#11745664	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
134	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
135	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
136	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
137	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
138	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
139	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
140	C₄²	#11745664	1	1	yes	characteristic
141	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
142	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
143	C₁₆	—	236	2	no	—
144	C₁₆	—	236	2	no	—
145	C₂³	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
152	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
154	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
155	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	#14273663	1	2	yes	—
163	C₈	#14273749	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	#16837440	1	1	yes	characteristic
173	C₄	C₂×#14273663	1	1	yes	characteristic
174	C₄	#16837819	1	2	yes	—
175	C₂	—	4	1	no	—
176	C₂	#21422844	1	1	yes	characteristic
177	C₂	#21425855	1	1	yes	characteristic
178	C₂	#21425862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₄⋊C₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#12014
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #21422166	C₂	semidirect
4. #21422160	C₂	semidirect
108. C₅₉	#12014	semidirect

Maximal subgroups

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

subgroup	index
#21425899	2
#21425895	2
#21422166	2
#21422160	2
C₅₉×#1450	2
#12014	59