Subgroups — C₄×#16837165 #28493870

Order 15104.

1182 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₅₉	C₂	1	2	yes	maximal
2	C₂×#16837165	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₉	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₅₉	—	2	2	no	—
6	C₄×C₄⋊Dic₅₉	—	2	4	no	—
7	C₄×C₄⋊Dic₅₉	C₄	1	8	yes	complemented
8	C₂×C₅₉⋊#95	—	2	2	no	—
9	#16837165	C₄	1	16	yes	direct product factor
10	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	—	2	4	no	—
13	C₂×C₄×C₈×C₅₉	C₄	1	4	yes	complemented
14	C₂³×Dic₅₉	—	2	4	no	—
15	C₂×C₄⋊Dic₅₉	—	2	4	no	—
16	C₂×C₄⋊Dic₅₉	—	2	8	no	—
17	C₂×C₄⋊Dic₅₉	C₂×C₄	1	16	yes	—
18	C₂×C₄⋊Dic₅₉	C₂×C₄	1	8	yes	—
19	C₂×C₄×Dic₅₉	—	2	8	no	—
20	C₂×C₄×Dic₅₉	—	4	4	no	—
21	C₂×C₄×Dic₅₉	—	4	4	no	—
22	C₅₉⋊#95	—	2	8	no	—
23	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	—	2	4	no	—
27	C₂²×C₈×C₅₉	—	2	4	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₉	D₄	1	4	yes	—
33	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
34	C₄×C₈×C₅₉	—	2	8	no	—
35	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	—	2	4	no	—
37	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₉	D₈	1	2	yes	—
42	C₂²×C₄×C₅₉	SD₁₆	1	4	yes	—
43	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	#41	1	4	yes	—
50	C₂²×Dic₅₉	—	2	8	no	—
51	C₂²×Dic₅₉	—	2	16	no	—
52	C₂²×Dic₅₉	—	4	4	no	—
53	C₂²×Dic₅₉	—	4	4	no	—
54	C₂²×Dic₅₉	—	4	8	no	—
55	C₂×C₈×C₅₉	—	2	4	no	—
56	C₂×C₈×C₅₉	—	2	4	no	—
57	C₂×C₈×C₅₉	—	2	16	no	—
58	C₂×C₈×C₅₉	C₄²	1	8	yes	complemented
59	C₄²×C₅₉	C₄²	1	4	yes	—
60	C₄²×C₅₉	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₅₉	C₄⋊C₄	1	8	yes	—
62	C₄⋊Dic₅₉	—	2	8	no	—
63	C₄⋊Dic₅₉	—	2	16	no	—
64	C₄⋊Dic₅₉	C₄²	1	16	yes	complemented
65	C₄×Dic₅₉	—	4	16	no	—
66	C₂³×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
67	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₅₉	C₄×D₄	1	4	yes	—
69	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
70	C₂³×C₅₉	C₂×SD₁₆	1	2	yes	—
71	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂³×C₅₉	#135	1	4	yes	—
73	C₂×C₄×C₅₉	—	2	8	no	—
74	C₂×C₄×C₅₉	—	2	4	no	—
75	C₂×C₄×C₅₉	#95	1	4	yes	—
76	C₂×C₄×C₅₉	#95	1	4	yes	—
77	C₂×C₄×C₅₉	#102	1	8	yes	—
78	C₂×C₄×C₅₉	#103	1	8	yes	—
79	C₂×C₄×C₅₉	#106	1	4	yes	—
80	C₂×C₄×C₅₉	#107	1	4	yes	—
81	C₂×C₄×C₅₉	C₂×C₄²	1	4	yes	—
82	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₅₉	#117	1	4	yes	—
87	C₂×C₄×C₅₉	#117	1	2	yes	—
88	C₂×C₄×C₅₉	C₄×D₄	1	8	yes	—
89	C₂×C₄×C₅₉	C₄×Q₈	1	4	yes	—
90	C₂×Dic₅₉	—	2	16	no	—
91	C₂×Dic₅₉	—	4	8	no	—
92	C₂×Dic₅₉	—	4	8	no	—
93	C₂×Dic₅₉	—	4	16	no	—
94	C₂×Dic₅₉	—	4	16	no	—
95	C₈×C₅₉	—	2	8	no	—
96	C₄×#340	—	59	1	no	maximal, Sylow
97	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
98	C₂²×C₅₉	C₄×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
100	C₂²×C₅₉	C₂×#102	1	2	yes	—
101	C₂²×C₅₉	C₂×#103	1	2	yes	—
102	C₂²×C₅₉	#416	1	4	yes	—
103	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
104	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
105	C₂²×C₅₉	#427	1	2	yes	—
106	C₂²×C₅₉	C₄×D₈	1	4	yes	—
107	C₂²×C₅₉	C₄×SD₁₆	1	8	yes	—
108	C₂²×C₅₉	C₄×Q₁₆	1	4	yes	—
109	C₄×C₅₉	—	2	4	no	—
110	C₄×C₅₉	#340	1	8	yes	complemented
111	C₄×C₅₉	C₄³	1	1	yes	characteristic
112	C₄×C₅₉	#376	1	1	yes	characteristic
113	C₄×C₅₉	C₄×C₂²⋊C₄	1	2	yes	—
114	C₄×C₅₉	C₄×C₄⋊C₄	1	4	yes	—
115	Dic₅₉	—	4	16	no	—
116	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
117	C₂×C₄×C₄⋊C₄	—	59	2	no	—
118	C₂×#340	—	59	4	no	—
119	C₂×C₅₉	C₄×#95	1	1	yes	characteristic
120	C₂×C₅₉	C₂×#340	1	1	yes	characteristic
121	C₂×C₅₉	#1735	1	1	yes	characteristic
122	C₂×C₅₉	C₄×#102	1	4	yes	—
123	C₂×C₅₉	C₄×#103	1	4	yes	—
124	C₂×C₅₉	C₄×#106	1	2	yes	—
125	C₂×C₅₉	C₄×#107	1	2	yes	—
126	C₂³×C₈	C₂×D₅₉	1	2	yes	—
127	C₂²×C₄⋊C₄	—	59	4	no	—
128	C₂²×C₄²	—	118	2	no	—
129	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	4	no	—
131	C₂×C₄×C₈	Dic₅₉	1	4	yes	complemented
132	C₄×C₄⋊C₄	—	59	8	no	—
133	C₄×C₄⋊C₄	—	118	4	no	—
134	C₂×#95	—	118	2	no	—
135	#340	—	59	16	no	—
136	C₅₉	C₄×#340	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	118	4	no	—
138	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
139	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₄×D₅₉	1	8	yes	—
144	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
145	C₂×C₄⋊C₄	—	59	16	no	—
146	C₂×C₄⋊C₄	—	59	8	no	—
147	C₂×C₄⋊C₄	—	118	8	no	—
148	C₂×C₄⋊C₄	—	118	4	no	—
149	C₂×C₄²	—	118	8	no	—
150	C₂×C₄²	—	236	4	no	—
151	C₂×C₄²	—	236	4	no	—
152	C₂×C₄²	Dic₁₁₈	1	2	yes	—
153	C₂×C₄²	C₄×D₅₉	1	4	yes	—
154	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₅₉	1	4	yes	—
156	C₄×C₈	—	2	8	no	—
157	#95	—	118	8	no	—
158	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	118	8	no	—
161	C₂²×C₄	—	118	16	no	—
162	C₂²×C₄	—	236	4	no	—
163	C₂²×C₄	—	236	4	no	—
164	C₂²×C₄	—	236	8	no	—
165	C₂²×C₄	#11745660	1	4	yes	—
166	C₂²×C₄	D₄₇₂	1	2	yes	—
167	C₂²×C₄	Dic₂₃₆	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
170	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
171	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	4	yes	—
172	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₅₉	1	4	yes	—
175	C₂²×C₄	#11745685	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	16	no	—
181	C₂×C₈	C₄×Dic₅₉	1	8	yes	complemented
182	C₄⋊C₄	—	59	16	no	—
183	C₄⋊C₄	—	118	8	no	—
184	C₄⋊C₄	—	118	16	no	—
185	C₄²	—	236	16	no	—
186	C₄²	C₄×Dic₅₉	1	4	yes	—
187	C₄²	Dic₅₉⋊C₄	1	8	yes	—
188	C₄²	C₂²⋊Dic₅₉	1	4	yes	—
189	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	2	yes	—
190	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
191	C₂³	C₄×D₂₃₆	1	4	yes	—
192	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
193	C₂³	C₂×#11745660	1	2	yes	—
194	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
195	C₂³	#14273749	1	4	yes	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	118	16	no	—
199	C₂×C₄	—	236	8	no	—
200	C₂×C₄	—	236	8	no	—
201	C₂×C₄	—	236	16	no	—
202	C₂×C₄	—	236	16	no	—
203	C₂×C₄	C₅₉⋊#95	1	4	yes	—
204	C₂×C₄	C₅₉⋊#95	1	4	yes	—
205	C₂×C₄	#14273649	1	8	yes	—
206	C₂×C₄	#14273652	1	8	yes	—
207	C₂×C₄	#14273659	1	4	yes	—
208	C₂×C₄	#14273661	1	4	yes	—
209	C₂×C₄	C₄²×D₅₉	1	4	yes	—
210	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
213	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
214	C₂×C₄	#14273684	1	4	yes	—
215	C₂×C₄	#14273686	1	2	yes	—
216	C₂×C₄	C₄×C₂²⋊D₅₉	1	8	yes	—
217	C₂×C₄	C₄×Dic₁₁₈	1	4	yes	—
218	C₈	—	2	8	no	—
219	C₂²	C₂×C₅₉⋊#95	1	1	yes	characteristic
220	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	4	yes	—
221	C₂²	C₄×C₄⋊Dic₅₉	1	2	yes	—
222	C₂²	C₂×#14273649	1	2	yes	—
223	C₂²	C₂×#14273652	1	2	yes	—
224	C₂²	#16837399	1	4	yes	—
225	C₂²	C₂×#14273659	1	1	yes	characteristic
226	C₂²	C₂×#14273661	1	1	yes	characteristic
227	C₂²	#16837440	1	2	yes	—
228	C₂²	C₄×D₄₇₂	1	4	yes	—
229	C₂²	C₄×#11745660	1	8	yes	—
230	C₂²	C₄×Dic₂₃₆	1	4	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	236	16	no	—
233	C₄	#16837165	1	8	yes	direct product factor
234	C₄	C₄²×Dic₅₉	1	1	yes	characteristic
235	C₄	#16837248	1	1	yes	characteristic
236	C₄	C₄×C₂²⋊Dic₅₉	1	2	yes	—
237	C₄	C₄×Dic₅₉⋊C₄	1	4	yes	—
238	C₂	C₄×C₅₉⋊#95	1	1	yes	characteristic
239	C₂	C₂×#16837165	1	1	yes	characteristic
240	C₂	#21423813	1	1	yes	characteristic
241	C₂	C₄×#14273649	1	4	yes	—
242	C₂	C₄×#14273652	1	4	yes	—
243	C₂	C₄×#14273659	1	2	yes	—
244	C₂	C₄×#14273661	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₃₆
commutator	C₄×C₅₉	C₄³
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	C₄×#340
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₅₉	C₄	semidirect
9. #16837165	C₄	direct (N × H)
13. C₂×C₄×C₈×C₅₉	C₄	semidirect
58. C₂×C₈×C₅₉	C₄²	semidirect
64. C₄⋊Dic₅₉	C₄²	semidirect
110. C₄×C₅₉	#340	semidirect
131. C₂×C₄×C₈	Dic₅₉	semidirect
136. C₅₉	C₄×#340	semidirect
181. C₂×C₈	C₄×Dic₅₉	semidirect
233. C₄	#16837165	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₅₉	2
C₂×#16837165	2
C₂²×C₄×C₈×C₅₉	2
C₄×#340	59

Subgroups — C4×#16837165 #28493870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837165 #28493870