Subgroups — #28506749

Order 15104.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837228	C₂	1	1	yes	maximal, characteristic
3	#21424881	C₂	1	2	yes	maximal
4	#21422205	C₂	1	2	yes	maximal
5	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	C₂²	1	2	yes	—
7	#16837445	—	2	2	no	—
8	C₂×#14273661	C₂²	1	2	yes	—
9	#16837232	C₄	1	4	yes	complemented
10	#16837228	—	2	1	no	—
11	#16837228	—	2	1	no	—
12	#16837170	—	2	2	no	—
13	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#405	C₂²	1	1	yes	characteristic
16	C₅₉×#346	—	2	1	no	—
17	C₄×C₁₆×C₅₉	—	2	1	no	—
18	C₂×#11745664	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	#14273663	—	2	2	no	—
21	#14273663	C₂×C₄	1	4	yes	—
22	#14273661	—	2	2	no	—
23	#14273661	—	2	2	no	—
24	#14273660	—	4	2	no	—
25	#14273658	—	4	2	no	—
26	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	#117×C₅₉	D₄	1	1	yes	characteristic
29	M₃₂×C₅₉	—	2	1	no	—
30	M₃₂×C₅₉	Q₈	1	2	yes	—
31	C₂×C₁₆×C₅₉	—	2	1	no	—
32	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
36	#97×C₅₉	—	2	1	no	—
37	C₄×C₈×C₅₉	—	2	1	no	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₉	—	4	2	no	—
40	M₁₆×C₅₉	—	2	1	no	—
41	M₁₆×C₅₉	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
51	C₂²⋊C₄×C₅₉	—	2	1	no	—
52	C₄²×C₅₉	—	2	1	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	C₄⋊Dic₅₉	—	4	2	no	—
55	#11745664	—	2	4	no	—
56	#11745664	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	4	2	no	—
58	C₁₆×C₅₉	—	2	2	no	—
59	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
61	C₂³×C₅₉	#135	1	1	yes	characteristic
62	C₂×C₄×C₅₉	—	2	1	no	—
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	16	2	no	—
72	C₈×C₅₉	—	2	2	no	—
73	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₉	#125	1	1	yes	characteristic
77	C₅₉⋊C₈	—	4	4	no	—
78	#13171	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	—	2	1	no	—
80	C₂²×C₅₉	—	2	1	no	—
81	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
82	C₂²×C₅₉	#466	1	1	yes	characteristic
83	C₂²×C₅₉	#499	1	1	yes	characteristic
84	C₄×C₅₉	—	4	1	no	—
85	C₄×C₅₉	—	4	1	no	—
86	C₄×C₅₉	#389	1	1	yes	characteristic
87	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
88	C₄×C₅₉	#467	1	1	yes	characteristic
89	C₄×C₅₉	#500	1	1	yes	characteristic
90	Dic₅₉	—	16	2	no	—
91	C₂×#368	—	59	1	no	—
92	C₂×#366	—	59	1	no	—
93	#2108	D₅₉	1	1	yes	characteristic
94	#1952	—	59	2	no	—
95	#1387	—	59	2	no	—
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	#1942	1	1	yes	characteristic
99	C₂×C₅₉	#2221	1	1	yes	characteristic
100	C₂×C₅₉	#2222	1	1	yes	characteristic
101	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
103	C₂×#108	—	59	2	no	—
104	#428	—	118	2	no	—
105	C₂×#107	—	59	2	no	—
106	#405	C₂×D₅₉	1	1	yes	characteristic
107	#368	—	59	4	no	—
108	#366	—	118	1	no	—
109	#366	—	118	1	no	—
110	#346	—	2	1	no	—
111	C₄×C₁₆	—	2	1	no	—
112	#343	—	118	2	no	—
113	C₅₉	#13171	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	118	2	no	—
115	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	#117	D₂₃₆	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	118	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	Dic₁₁₈	1	2	yes	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
123	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
126	#108	—	59	4	no	—
127	#108	—	118	2	no	—
128	#107	—	118	2	no	—
129	#107	—	118	2	no	—
130	#107	—	236	2	no	—
131	#106	—	236	2	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂²×C₄	—	236	2	no	—
135	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	118	4	no	—
138	M₁₆	—	236	2	no	—
139	M₁₆	Dic₂₃₆	1	2	yes	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	236	2	no	—
142	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
145	C₂×C₈	#11745685	1	1	yes	characteristic
146	C₂×C₈	#11745685	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
148	C₄⋊C₄	—	236	2	no	—
149	C₄⋊C₄	—	236	2	no	—
150	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
151	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	Dic₅₉⋊C₄	1	2	yes	—
156	C₁₆	Dic₅₉⋊C₄	1	2	yes	—
157	C₂³	#14273749	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	944	2	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
163	C₂×C₄	#14273701	1	1	yes	characteristic
164	C₂×C₄	#14273727	1	1	yes	characteristic
165	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
167	C₂×C₄	#14273749	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	236	4	no	—
170	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
171	C₈	C₄×Dic₁₁₈	1	1	yes	characteristic
172	C₈	#14273706	1	1	yes	characteristic
173	C₈	#14273719	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×Dic₂₃₆	1	1	yes	characteristic
177	C₂²	#16837638	1	1	yes	characteristic
178	C₂²	#16837804	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	944	2	no	—
182	C₄	#16837300	1	1	yes	characteristic
183	C₄	C₄×D₄₇₂	1	1	yes	characteristic
184	C₄	#16837643	1	1	yes	characteristic
185	C₄	#16837807	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#21424829	1	1	yes	characteristic
189	C₂	#21426216	1	1	yes	characteristic
190	C₂	#21426221	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837638
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂³×C₅₉	#135
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#13171
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
9. #16837232	C₄	semidirect
113. C₅₉	#13171	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837232	2
C₂×#16837228	2
#21424881	2
#21422205	2
C₅₉×#2108	2
#13171	59