Subgroups — #28475871

Order 15104.

208 conjugacy classes of subgroups.

The subgroup lattice has 178 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	1	yes	maximal, characteristic
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273667	—	2	1	no	—
5	C₂×#14273663	—	2	1	no	—
6	C₂×#14273663	—	2	1	no	—
7	#16837426	C₄	1	2	yes	—
8	#16837223	—	2	1	no	—
9	#16837223	C₄	1	2	yes	—
10	C₂×M₃₂×C₅₉	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#364	—	2	1	no	—
13	C₅₉×#364	C₄	1	2	yes	—
14	C₂×#11745664	—	2	2	no	—
15	#14273667	—	2	2	no	—
16	#14273667	—	4	1	no	—
17	C₂×C₅₉⋊C₁₆	—	4	1	no	—
18	#14273663	—	4	1	no	—
19	#14273663	—	4	1	no	—
20	#14273663	—	4	1	no	—
21	#14273663	—	4	1	no	—
22	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₅₉	—	2	2	no	—
27	M₃₂×C₅₉	—	4	1	no	—
28	C₂×C₁₆×C₅₉	—	4	1	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	—	2	1	no	—
34	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
42	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₉	—	2	1	no	—
48	C₄²×C₅₉	C₄²	1	1	yes	characteristic
49	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	#11745664	—	4	2	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₁₆×C₅₉	—	4	2	no	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	C₂³×C₅₉	#101	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	1	no	—
57	C₂×C₄×C₅₉	—	2	1	no	—
58	C₂×C₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
65	C₈×C₅₉	#102	1	1	yes	characteristic
66	C₈×C₅₉	#102	1	1	yes	characteristic
67	C₈×C₅₉	#103	1	1	yes	characteristic
68	C₈×C₅₉	#103	1	1	yes	characteristic
69	C₈×C₅₉	#106	1	1	yes	characteristic
70	C₈×C₅₉	#106	1	1	yes	characteristic
71	C₈×C₅₉	#107	1	1	yes	characteristic
72	C₈×C₅₉	#107	1	1	yes	characteristic
73	C₅₉⋊C₈	—	8	2	no	—
74	#7272	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	4	1	no	—
76	C₂²×C₅₉	#332	1	1	yes	characteristic
77	C₂²×C₅₉	#332	1	1	yes	characteristic
78	C₂²×C₅₉	#343	1	1	yes	characteristic
79	C₄×C₅₉	—	2	2	no	—
80	C₄×C₅₉	—	4	1	no	—
81	C₄×C₅₉	#331	1	1	yes	characteristic
82	C₄×C₅₉	#333	1	1	yes	characteristic
83	C₄×C₅₉	#340	1	1	yes	characteristic
84	C₄×C₅₉	#340	1	1	yes	characteristic
85	C₂×#364	—	59	1	no	—
86	C₂×#364	D₅₉	1	1	yes	characteristic
87	#1848	—	59	1	no	—
88	C₂×C₅₉	—	2	1	no	—
89	C₂×C₅₉	—	2	1	no	—
90	C₂×C₅₉	—	2	1	no	—
91	C₂×C₅₉	#1296	1	1	yes	characteristic
92	C₂×M₃₂	—	2	1	no	—
93	C₂×M₃₂	—	118	1	no	—
94	C₂×#108	—	118	1	no	—
95	C₂×#108	—	118	1	no	—
96	#423	—	59	2	no	—
97	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	#364	—	2	1	no	—
99	#364	—	59	2	no	—
100	#364	—	118	1	no	—
101	#364	Dic₅₉	1	2	yes	—
102	C₅₉	#7272	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	118	2	no	—
104	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	4	1	no	—
109	M₃₂	—	118	2	no	—
110	M₃₂	—	236	1	no	—
111	C₂×C₁₆	—	4	1	no	—
112	C₂×C₁₆	—	236	1	no	—
113	#108	—	236	1	no	—
114	#108	—	236	1	no	—
115	#108	—	236	1	no	—
116	#108	—	236	1	no	—
117	C₄⋊C₈	—	236	2	no	—
118	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
124	M₁₆	—	236	2	no	—
125	M₁₆	—	236	2	no	—
126	C₂×C₈	—	236	2	no	—
127	C₂×C₈	#11745660	1	1	yes	characteristic
128	C₂×C₈	#11745660	1	1	yes	characteristic
129	C₂×C₈	D₄₇₂	1	1	yes	characteristic
130	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
131	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
132	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
133	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
134	C₂×C₈	#11745669	1	1	yes	characteristic
135	C₂×C₈	#11745670	1	1	yes	characteristic
136	C₂×C₈	#11745671	1	1	yes	characteristic
137	C₂×C₈	#11745672	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
139	C₄²	—	2	1	no	—
140	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
141	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
142	C₁₆	—	4	2	no	—
143	C₁₆	—	236	2	no	—
144	C₂³	#14273645	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
149	C₂×C₄	#14273643	1	1	yes	characteristic
150	C₂×C₄	#14273647	1	1	yes	characteristic
151	C₂×C₄	#14273649	1	1	yes	characteristic
152	C₂×C₄	#14273650	1	1	yes	characteristic
153	C₂×C₄	#14273652	1	1	yes	characteristic
154	C₈	—	472	2	no	—
155	C₈	#14273648	1	1	yes	characteristic
156	C₈	#14273649	1	1	yes	characteristic
157	C₈	#14273651	1	1	yes	characteristic
158	C₈	#14273652	1	1	yes	characteristic
159	C₈	#14273658	1	1	yes	characteristic
160	C₈	#14273659	1	1	yes	characteristic
161	C₈	#14273660	1	1	yes	characteristic
162	C₈	#14273661	1	1	yes	characteristic
163	C₂²	—	4	1	no	—
164	C₂²	#16837147	1	1	yes	characteristic
165	C₂²	#16837149	1	1	yes	characteristic
166	C₂²	#16837170	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	#16837145	1	1	yes	characteristic
170	C₄	#16837150	1	1	yes	characteristic
171	C₄	#16837164	1	1	yes	characteristic
172	C₄	#16837165	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#21421980	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837150
commutator	C₄²×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂×C₅₉	#1296
prime radical	C₂³×C₅₉	#101
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#7272
LCS	G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
102. C₅₉	#7272	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
#21424268	2
C₂×C₅₉×#364	2
#7272	59