Subgroups — #28504019

Order 15104.

350 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	#21425852	C₂	1	2	yes	maximal
2	C₂×#16837180	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	C₄⋊(C₅₉⋊C₁₆)	C₄	1	8	yes	complemented
7	#16837180	—	2	4	no	—
8	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
9	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
10	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄³×C₅₉	C₄	1	1	yes	characteristic
15	#14273667	—	2	4	no	—
16	#14273667	—	2	4	no	—
17	C₂×C₅₉⋊C₁₆	C₂×C₄	1	8	yes	—
18	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅₉	—	2	2	no	—
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	—	2	2	no	—
28	#97×C₅₉	—	2	2	no	—
29	#97×C₅₉	D₄	1	2	yes	—
30	#97×C₅₉	Q₈	1	2	yes	—
31	C₄×C₈×C₅₉	—	2	1	no	—
32	C₄×C₈×C₅₉	—	2	1	no	—
33	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
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	—	2	1	no	—
41	C₂×C₈×C₅₉	—	2	1	no	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₂×C₈×C₅₉	C₄²	1	2	yes	—
45	C₂×C₈×C₅₉	C₄⋊C₄	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	—	4	4	no	—
54	C₄²×C₅₉	C₄²	1	2	yes	—
55	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
56	C₄²×C₅₉	M₁₆	1	2	yes	—
57	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
58	C₄²×C₅₉	M₁₆	1	1	yes	characteristic
59	C₅₉⋊C₁₆	—	2	8	no	—
60	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₉	—	2	4	no	—
62	C₂×C₄×C₅₉	—	2	2	no	—
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	—	4	2	no	—
66	C₂×C₄×C₅₉	#97	1	2	yes	—
67	C₂×C₄×C₅₉	#97	1	2	yes	—
68	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₅₉	—	2	2	no	—
75	C₈×C₅₉	—	2	2	no	—
76	C₈×C₅₉	—	2	2	no	—
77	C₈×C₅₉	C₄×D₄	1	2	yes	—
78	C₈×C₅₉	C₄×Q₈	1	2	yes	—
79	#12658	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
82	C₂²×C₅₉	#423	1	1	yes	characteristic
83	C₂²×C₅₉	#432	1	1	yes	characteristic
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	4	2	no	—
86	C₄×C₅₉	—	4	2	no	—
87	C₄×C₅₉	#347	1	4	yes	complemented
88	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
90	C₄×C₅₉	#423	1	1	yes	characteristic
91	C₄×C₅₉	#432	1	1	yes	characteristic
92	#2150	—	59	2	no	—
93	C₂×#347	—	59	2	no	—
94	C₄×#97	D₅₉	1	1	yes	characteristic
95	#1375	—	59	2	no	—
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	#1767	1	1	yes	characteristic
99	C₂×C₅₉	C₂×#347	1	2	yes	—
100	C₂×M₃₂	—	59	4	no	—
101	C₂×#97	Dic₅₉	1	1	yes	characteristic
102	C₂×#97	Dic₅₉	1	1	yes	characteristic
103	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
104	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
105	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₄³	Dic₅₉	1	1	yes	characteristic
108	C₄⋊C₁₆	—	59	8	no	—
109	#347	—	118	4	no	—
110	C₅₉	#12658	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
115	C₂×C₄²	—	2	2	no	—
116	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
119	M₃₂	—	118	4	no	—
120	M₃₂	—	118	4	no	—
121	C₂×C₁₆	—	59	8	no	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	Dic₁₁₈	1	2	yes	—
126	#97	D₂₃₆	1	2	yes	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₄×D₅₉	1	4	yes	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	#11745664	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	#11745664	1	2	yes	—
141	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
142	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
145	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
146	C₂×C₈	#11745685	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	#11745664	1	2	yes	—
151	C₄²	#11745664	1	1	yes	characteristic
152	C₄²	#11745664	1	1	yes	characteristic
153	C₄²	C₄×Dic₅₉	1	2	yes	—
154	C₄²	C₄⋊Dic₅₉	1	2	yes	—
155	C₁₆	—	118	8	no	—
156	C₂³	C₂×#11745664	1	1	yes	characteristic
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	#14273638	1	2	yes	—
163	C₂×C₄	#14273638	1	2	yes	—
164	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
166	C₂×C₄	#14273686	1	1	yes	characteristic
167	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
168	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
169	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	C₄×D₂₃₆	1	2	yes	—
174	C₈	C₄×Dic₁₁₈	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	C₂×#14273638	1	1	yes	characteristic
177	C₂²	#16837426	1	1	yes	characteristic
178	C₂²	#16837459	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	#16837180	1	4	yes	complemented
183	C₄	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
184	C₄	C₄×#11745664	1	1	yes	characteristic
185	C₄	#16837426	1	1	yes	characteristic
186	C₄	#16837459	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#21423949	1	1	yes	characteristic
190	C₂	C₂×#16837180	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#97×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#97
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#97	D₅₉
hypocenter	C₅₉	#12658
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
6. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
87. C₄×C₅₉	#347	semidirect
110. C₅₉	#12658	semidirect
182. C₄	#16837180	semidirect

Maximal subgroups

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

subgroup	index
#21425852	2
C₂×#16837180	2
#21422176	2
C₄×#97×C₅₉	2
#12658	59