Subgroups — #28475280

Order 15104.

292 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424248	C₂	1	1	yes	maximal, characteristic
2	#21424213	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
5	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
8	#16837157	—	2	1	no	—
9	#16837157	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
11	C₄³×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#336	—	2	1	no	—
13	C₅₉×#336	C₄	1	2	yes	—
14	C₂²×C₅₉⋊C₈	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
17	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
18	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
19	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
20	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
21	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
22	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
23	C₂²×C₈×C₅₉	—	2	2	no	—
24	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	4	1	no	—
32	C₄⋊C₈×C₅₉	—	4	1	no	—
33	C₂²×C₄×C₅₉	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	—	4	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	—	4	2	no	—
43	C₄²×C₅₉	—	2	1	no	—
44	C₄²×C₅₉	—	2	1	no	—
45	C₄²×C₅₉	—	2	2	no	—
46	C₄²×C₅₉	—	4	1	no	—
47	C₄²×C₅₉	—	4	1	no	—
48	C₄²×C₅₉	—	4	1	no	—
49	C₄²×C₅₉	D₈	1	1	yes	characteristic
50	C₄²×C₅₉	D₈	1	1	yes	characteristic
51	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
52	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
53	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
54	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
55	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
56	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
57	C₂×C₅₉⋊C₈	—	4	2	no	—
58	C₂×C₅₉⋊C₈	—	4	2	no	—
59	C₂×C₅₉⋊C₈	—	4	2	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂×C₅₉⋊C₈	—	4	2	no	—
63	C₂³×C₅₉	#95	1	1	yes	characteristic
64	C₂×C₄×C₅₉	—	2	1	no	—
65	C₂×C₄×C₅₉	—	2	1	no	—
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	2	1	no	—
69	C₂×C₄×C₅₉	—	2	1	no	—
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	2	1	no	—
73	C₂×C₄×C₅₉	—	2	1	no	—
74	C₂×C₄×C₅₉	—	2	1	no	—
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	1	no	—
77	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#101	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
84	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
85	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#108	1	2	yes	—
92	C₈×C₅₉	—	8	2	no	—
93	C₅₉⋊C₈	—	8	2	no	—
94	C₅₉⋊C₈	—	8	2	no	—
95	#7040	—	59	1	no	maximal, Sylow
96	C₂²×C₅₉	H(3,ℤ/4)	1	1	yes	characteristic
97	C₂²×C₅₉	#340	1	1	yes	characteristic
98	C₂²×C₅₉	#340	1	1	yes	characteristic
99	C₂²×C₅₉	#341	1	1	yes	characteristic
100	C₂²×C₅₉	#343	1	1	yes	characteristic
101	C₂²×C₅₉	#344	1	1	yes	characteristic
102	C₂²×C₅₉	#344	1	1	yes	characteristic
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	—	2	1	no	—
106	C₄×C₅₉	—	2	1	no	—
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	1	no	—
112	C₄×C₅₉	—	4	1	no	—
113	C₄×C₅₉	#331	1	1	yes	characteristic
114	C₄×C₅₉	#332	1	1	yes	characteristic
115	C₄×C₅₉	#332	1	1	yes	characteristic
116	C₄×C₅₉	#333	1	1	yes	characteristic
117	#1842	—	59	1	no	—
118	#1831	—	59	1	no	—
119	#1831	D₅₉	1	1	yes	characteristic
120	C₂×C₅₉	#1286	1	1	yes	characteristic
121	C₂×C₅₉	#1286	1	1	yes	characteristic
122	C₂×C₅₉	#1290	1	1	yes	characteristic
123	C₂×C₅₉	#1296	1	1	yes	characteristic
124	C₂×C₅₉	#1301	1	1	yes	characteristic
125	C₂×C₅₉	#1301	1	1	yes	characteristic
126	C₂×C₅₉	#1307	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	—	59	2	no	—
129	C₂×C₄⋊C₈	—	118	1	no	—
130	C₂×C₄⋊C₈	—	118	1	no	—
131	C₂×C₄⋊C₈	—	118	1	no	—
132	C₄³	C₂×D₅₉	1	1	yes	characteristic
133	#336	—	2	1	no	—
134	#336	—	59	2	no	—
135	#336	—	118	1	no	—
136	#336	Dic₅₉	1	2	yes	—
137	C₅₉	#7040	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	118	2	no	—
140	C₂²×C₈	—	118	2	no	—
141	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
144	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	236	2	no	—
151	C₄⋊C₈	—	236	1	no	—
152	C₄⋊C₈	—	236	1	no	—
153	C₄⋊C₈	—	236	1	no	—
154	C₄⋊C₈	—	236	1	no	—
155	C₄⋊C₈	—	236	1	no	—
156	C₄⋊C₈	—	236	1	no	—
157	C₂²×C₄	C₄×Dic₅₉	1	1	yes	characteristic
158	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
161	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	236	2	no	—
168	C₂×C₈	—	236	2	no	—
169	C₂×C₈	—	236	2	no	—
170	C₂×C₈	—	236	2	no	—
171	C₂×C₈	—	236	2	no	—
172	C₂×C₈	—	236	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	1	no	—
177	C₄²	—	4	1	no	—
178	C₄²	—	4	1	no	—
179	C₄²	#11745660	1	1	yes	characteristic
180	C₄²	#11745660	1	1	yes	characteristic
181	C₄²	D₄₇₂	1	1	yes	characteristic
182	C₄²	Dic₂₃₆	1	1	yes	characteristic
183	C₄²	#11745669	1	1	yes	characteristic
184	C₄²	#11745670	1	1	yes	characteristic
185	C₄²	#11745671	1	1	yes	characteristic
186	C₄²	#11745672	1	1	yes	characteristic
187	C₂³	C₅₉⋊#95	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	#14273643	1	1	yes	characteristic
202	C₂×C₄	#14273645	1	1	yes	characteristic
203	C₂×C₄	#14273647	1	1	yes	characteristic
204	C₂×C₄	#14273648	1	1	yes	characteristic
205	C₂×C₄	#14273649	1	1	yes	characteristic
206	C₂×C₄	#14273649	1	1	yes	characteristic
207	C₂×C₄	#14273650	1	1	yes	characteristic
208	C₂×C₄	#14273651	1	1	yes	characteristic
209	C₂×C₄	#14273652	1	1	yes	characteristic
210	C₂×C₄	#14273652	1	1	yes	characteristic
211	C₂×C₄	#14273658	1	1	yes	characteristic
212	C₂×C₄	#14273659	1	1	yes	characteristic
213	C₂×C₄	#14273660	1	1	yes	characteristic
214	C₂×C₄	#14273661	1	1	yes	characteristic
215	C₂×C₄	#14273662	1	2	yes	—
216	C₈	—	8	2	no	—
217	C₈	—	472	2	no	—
218	C₈	—	472	2	no	—
219	C₂²	#16837158	1	1	yes	characteristic
220	C₂²	#16837164	1	1	yes	characteristic
221	C₂²	#16837165	1	1	yes	characteristic
222	C₂²	#16837166	1	1	yes	characteristic
223	C₂²	#16837170	1	1	yes	characteristic
224	C₂²	#16837172	1	1	yes	characteristic
225	C₂²	#16837173	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	1	no	—
236	C₄	#16837145	1	1	yes	characteristic
237	C₄	#16837147	1	1	yes	characteristic
238	C₄	#16837149	1	1	yes	characteristic
239	C₄	#16837150	1	1	yes	characteristic
240	C₂	#21421954	1	1	yes	characteristic
241	C₂	#21421955	1	1	yes	characteristic
242	C₂	#21421965	1	1	yes	characteristic
243	C₂	#21421980	1	1	yes	characteristic
244	C₂	#21421991	1	1	yes	characteristic
245	C₂	#21421992	1	1	yes	characteristic
246	C₂	#21422006	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₅₉⋊#95
commutator	C₂²×C₄×C₅₉	C₄²
Frattini	C₄³	C₂×D₅₉
Fitting	C₅₉×#1831	C₂
socle, prime radical	C₂³×C₅₉	#95
cosocle	C₄³×C₅₉	C₂²
hypercenter	#1831	D₅₉
hypocenter	C₅₉	#7040
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
137. C₅₉	#7040	semidirect

Maximal subgroups

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

subgroup	index
#21424248	2
#21424213	2
C₅₉×#1831	2
#7040	59