Subgroups — #28478584

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837459	C₂	1	2	yes	maximal
2	#21421966	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₉×#431	C₂	1	1	yes	maximal, characteristic
4	#16837459	—	2	2	no	—
5	#16837459	—	2	2	no	—
6	#16837459	C₄	1	4	yes	—
7	#16837459	C₂²	1	4	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
9	C₂×C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
10	C₂×#14273638	C₄	1	4	yes	—
11	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#431	—	2	2	no	—
13	C₅₉×#431	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₅₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
17	C₂²×C₅₉⋊C₈	—	2	4	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
22	#14273638	—	2	4	no	—
23	#14273638	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₉	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	D₄	1	2	yes	—
33	C₂×C₄²×C₅₉	D₄	1	2	yes	—
34	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₉	—	2	4	no	—
37	C₂²⋊C₈×C₅₉	—	2	4	no	—
38	C₄×C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	—	2	2	no	—
40	C₂⁴×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	—	2	4	no	—
42	C₂²×C₄×C₅₉	—	4	2	no	—
43	C₂²×C₄×C₅₉	—	4	2	no	—
44	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₅₉	—	2	4	no	—
57	C₂×C₈×C₅₉	—	4	4	no	—
58	C₄²×C₅₉	C₄²	1	1	yes	characteristic
59	C₄²×C₅₉	C₄²	1	1	yes	characteristic
60	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₉	C₂×D₄	1	2	yes	—
68	C₂×C₅₉⋊C₈	—	2	8	no	—
69	C₂×C₅₉⋊C₈	—	4	8	no	—
70	C₂³×C₅₉	—	4	1	no	—
71	C₂³×C₅₉	—	4	1	no	—
72	C₂³×C₅₉	#95	1	2	yes	—
73	C₂³×C₅₉	#108	1	4	yes	—
74	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	—	2	4	no	—
76	C₂×C₄×C₅₉	—	2	4	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	4	no	—
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	2	4	no	—
83	C₂×C₄×C₅₉	—	4	2	no	—
84	C₂×C₄×C₅₉	—	4	2	no	—
85	C₂×C₄×C₅₉	—	4	1	no	—
86	C₂×C₄×C₅₉	—	4	1	no	—
87	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#95	1	2	yes	—
91	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₈×C₅₉	—	4	4	no	—
97	C₅₉⋊C₈	—	4	8	no	—
98	#7986	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	—	2	2	no	—
100	C₂²×C₅₉	—	2	1	no	—
101	C₂²×C₅₉	—	2	2	no	—
102	C₂²×C₅₉	—	2	1	no	—
103	C₂²×C₅₉	—	4	4	no	—
104	C₂²×C₅₉	#341	1	4	yes	—
105	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
106	C₂²×C₅₉	C₂×#108	1	2	yes	—
107	C₄×C₅₉	—	2	2	no	—
108	C₄×C₅₉	—	2	1	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	—	2	1	no	—
111	C₄×C₅₉	—	2	2	no	—
112	C₄×C₅₉	—	2	2	no	—
113	C₄×C₅₉	—	2	1	no	—
114	C₄×C₅₉	H(3,ℤ/4)	1	2	yes	—
115	C₄×C₅₉	#413	1	2	yes	—
116	C₂×#432	—	59	2	no	—
117	C₂×#431	D₅₉	1	1	yes	characteristic
118	#1290	—	59	4	no	—
119	C₂×C₅₉	—	2	4	no	—
120	C₂×C₅₉	—	2	2	no	—
121	C₂×C₅₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic
122	C₂×C₅₉	C₂×#341	1	1	yes	characteristic
123	C₂×C₅₉	#1742	1	1	yes	characteristic
124	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
125	#432	—	59	4	no	—
126	#432	—	59	4	no	—
127	#432	—	118	2	no	—
128	#432	—	118	2	no	—
129	#431	—	2	2	no	—
130	#431	—	2	2	no	—
131	C₂×C₄⋊C₈	—	118	2	no	—
132	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
133	C₂×C₂²⋊C₈	—	118	2	no	—
134	C₂×C₂²⋊C₈	Dic₅₉	1	2	yes	—
135	C₂×#97	—	59	4	no	—
136	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
137	C₅₉	#7986	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	Dic₁₁₈	1	1	yes	characteristic
139	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
140	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
141	C₂²×C₈	—	118	4	no	—
142	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
143	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
144	C₂×C₄²	C₄×D₅₉	1	2	yes	—
145	C₂×C₄²	C₄×D₅₉	1	2	yes	—
146	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
149	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
150	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	118	4	no	—
153	C₄⋊C₈	—	118	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	118	4	no	—
156	C₂²⋊C₈	—	118	4	no	—
157	#97	—	59	8	no	—
158	#97	—	118	4	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₂⁴	C₄⋊Dic₅₉	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
166	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
167	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
168	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
171	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
172	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
173	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	118	8	no	—
180	C₂×C₈	—	236	8	no	—
181	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
182	C₄²	C₄×Dic₅₉	1	1	yes	characteristic
183	C₄²	Dic₅₉⋊C₄	1	2	yes	—
184	C₄²	Dic₅₉⋊C₄	1	2	yes	—
185	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
186	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
187	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₅₉	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
191	C₂³	—	4	1	no	—
192	C₂³	—	4	1	no	—
193	C₂³	C₅₉⋊#95	1	2	yes	—
194	C₂³	#14273663	1	4	yes	—
195	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
209	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
210	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
211	C₂×C₄	C₅₉⋊#95	1	2	yes	—
212	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
214	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	236	8	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	4	4	no	—
224	C₂²	#16837167	1	4	yes	—
225	C₂²	C₂×C₅₉⋊#95	1	1	yes	characteristic
226	C₂²	C₂×#14273663	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	#16837159	1	2	yes	—
235	C₄	#16837386	1	2	yes	—
236	C₂	—	2	4	no	—
237	C₂	—	2	2	no	—
238	C₂	C₂×#16837159	1	1	yes	characteristic
239	C₂	C₂×#16837167	1	1	yes	characteristic
240	C₂	#21423835	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₅₉⋊#95
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×#95
prime radical	C₂⁴×C₅₉	C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#431	D₅₉
hypocenter	C₅₉	#7986
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #21421966	C₂	semidirect
137. C₅₉	#7986	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837459	2
#21421966	2
C₂×C₅₉×#431	2
#7986	59