Subgroups — C₄×#16837222 #28504008

Order 15104.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837222	C₂	1	2	yes	maximal
2	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	#21422177	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
13	#16837222	C₄	1	8	yes	direct product factor
14	#16837178	—	2	1	no	—
15	C₄×C₅₉⋊C₁₆	—	2	1	no	—
16	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#346	—	2	1	no	—
19	C₄×C₁₆×C₅₉	—	2	1	no	—
20	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₉	—	2	2	no	—
23	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
25	#14273667	—	2	2	no	—
26	#14273667	C₂×C₄	1	4	yes	—
27	C₂×C₅₉⋊C₁₆	—	2	2	no	—
28	C₄×C₅₉⋊C₈	C₈	1	4	yes	—
29	C₈×Dic₅₉	—	2	2	no	—
30	C₈×Dic₅₉	—	2	2	no	—
31	C₈×Dic₅₉	C₈	1	4	yes	—
32	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₅₉	—	2	2	no	—
37	M₃₂×C₅₉	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₉	—	2	1	no	—
46	C₂²×Dic₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	2	yes	—
55	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₅₉	C₂×C₈	1	2	yes	—
58	C₄×Dic₅₉	—	2	8	no	—
59	C₄×Dic₅₉	—	2	1	no	—
60	C₄×Dic₅₉	—	2	1	no	—
61	C₄×Dic₅₉	C₄²	1	2	yes	—
62	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
63	C₂×C₅₉⋊C₈	—	2	1	no	—
64	C₂×C₅₉⋊C₈	—	2	1	no	—
65	C₂×C₅₉⋊C₈	C₄²	1	2	yes	—
66	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
67	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
68	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
69	C₁₆×C₅₉	—	2	4	no	—
70	C₅₉⋊C₁₆	—	2	4	no	—
71	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₉	—	2	4	no	—
73	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₅₉	—	2	4	no	—
82	C₂×Dic₅₉	—	2	4	no	—
83	C₂×Dic₅₉	—	2	2	no	—
84	C₂×Dic₅₉	—	2	2	no	—
85	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
86	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
87	C₈×C₅₉	C₄×D₄	1	2	yes	—
88	C₈×C₅₉	C₄×Q₈	1	2	yes	—
89	C₅₉⋊C₈	—	2	2	no	—
90	C₅₉⋊C₈	C₄×C₈	1	4	yes	—
91	C₄×#364	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	—	2	2	no	—
93	C₂²×C₅₉	C₄×M₁₆	1	1	yes	characteristic
94	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂²×C₅₉	#431	1	1	yes	characteristic
96	C₄×C₅₉	—	2	2	no	—
97	C₄×C₅₉	#364	1	4	yes	complemented
98	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
100	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₅₉	#431	1	1	yes	characteristic
102	Dic₅₉	—	2	4	no	—
103	Dic₅₉	—	2	4	no	—
104	C₂×#364	—	59	2	no	—
105	C₄×M₃₂	—	59	1	no	—
106	C₄×M₃₂	D₅₉	1	1	yes	characteristic
107	C₄²×C₈	—	59	1	no	—
108	#1376	—	59	2	no	—
109	C₂×C₅₉	—	2	1	no	—
110	C₂×C₅₉	—	2	1	no	—
111	C₂×C₅₉	C₄×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₅₉	C₂×#364	1	2	yes	—
113	C₂×M₃₂	—	59	2	no	—
114	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
115	C₂×C₄×C₈	—	59	1	no	—
116	C₂×C₄×C₈	—	59	1	no	—
117	C₂×C₄×C₈	—	59	1	no	—
118	C₂×C₄×C₈	—	59	1	no	—
119	C₂×C₄×C₈	—	59	1	no	—
120	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
121	C₄³	—	59	1	no	—
122	#364	—	59	8	no	—
123	#346	—	2	1	no	—
124	#346	—	118	1	no	—
125	C₄×C₁₆	—	2	1	no	—
126	C₄×C₁₆	—	118	1	no	—
127	C₅₉	C₄×#364	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₈	—	59	1	no	—
129	C₂²×C₈	—	59	1	no	—
130	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
132	C₂×C₄²	—	59	1	no	—
133	C₂×C₄²	—	59	1	no	—
134	C₂×C₄²	—	118	2	no	—
135	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
136	M₃₂	—	2	2	no	—
137	M₃₂	—	59	4	no	—
138	M₃₂	—	118	2	no	—
139	M₃₂	C₄×D₅₉	1	4	yes	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	118	2	no	—
142	C₄×C₈	—	59	4	no	—
143	C₄×C₈	—	59	4	no	—
144	C₄×C₈	—	59	4	no	—
145	C₄×C₈	—	118	2	no	—
146	C₄×C₈	—	118	2	no	—
147	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
148	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂²×C₄	—	118	1	no	—
152	C₂²×C₄	—	118	1	no	—
153	C₂²×C₄	#11745659	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
155	C₂×C₈	—	59	2	no	—
156	C₂×C₈	—	59	2	no	—
157	C₂×C₈	—	59	2	no	—
158	C₂×C₈	—	59	2	no	—
159	C₂×C₈	—	118	1	no	—
160	C₂×C₈	—	118	1	no	—
161	C₂×C₈	C₈×D₅₉	1	2	yes	—
162	C₂×C₈	#11745659	1	2	yes	—
163	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
164	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
167	C₂×C₈	#11745685	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
169	C₄²	—	59	2	no	—
170	C₄²	—	59	2	no	—
171	C₄²	—	118	8	no	—
172	C₄²	—	118	1	no	—
173	C₄²	—	118	1	no	—
174	C₄²	C₈×D₅₉	1	2	yes	—
175	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
176	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
177	C₁₆	—	2	4	no	—
178	C₁₆	—	118	4	no	—
179	C₂³	C₂×#11745659	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	118	4	no	—
182	C₂×C₄	—	118	4	no	—
183	C₂×C₄	—	118	2	no	—
184	C₂×C₄	—	118	2	no	—
185	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
186	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
187	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
189	C₂×C₄	#14273684	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
191	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
192	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
193	C₈	—	59	4	no	—
194	C₈	—	118	2	no	—
195	C₈	Dic₅₉⋊C₈	1	2	yes	—
196	C₈	Dic₅₉⋊C₈	1	2	yes	—
197	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
198	C₈	C₄×Dic₁₁₈	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	C₄×#11745659	1	1	yes	characteristic
201	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
202	C₂²	#16837453	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	118	4	no	—
205	C₄	—	118	4	no	—
206	C₄	#16837222	1	4	yes	direct product factor
207	C₄	C₄×Dic₅₉⋊C₄	1	1	yes	characteristic
208	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
209	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
210	C₄	#16837453	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	C₄×Dic₅₉⋊C₈	1	1	yes	characteristic
214	C₂	C₂×#16837222	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	C₄×#364
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #16837222	C₄	direct (N × H)
97. C₄×C₅₉	#364	semidirect
127. C₅₉	C₄×#364	semidirect
206. C₄	#16837222	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×#16837222	2
C₄×#14273667	2
C₄×C₈×Dic₅₉	2
#21422177	2
C₄×M₃₂×C₅₉	2
C₄×#364	59

Subgroups — C4×#16837222 #28504008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837222 #28504008