Subgroups — #28558162

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433176	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21422914	C₂	1	4	yes	maximal, complemented
4	#21422902	C₂	1	4	yes	maximal, complemented
5	#21422490	C₂	1	2	yes	maximal
6	#21422379	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₃₆	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₁₈	C₂²	1	2	yes	—
10	C₈×Dic₁₁₈	—	2	4	no	—
11	#16837526	C₂²	1	8	yes	—
12	C₄×Dic₂₃₆	C₄	1	2	yes	—
13	C₄×Dic₂₃₆	C₄	1	2	yes	—
14	C₄×Dic₂₃₆	C₂²	1	2	yes	—
15	C₄×Dic₂₃₆	C₂²	1	2	yes	—
16	#16837455	C₂²	1	1	yes	characteristic
17	#16837455	C₂²	1	1	yes	characteristic
18	#16837452	—	2	4	no	—
19	C₂×#14273661	C₄	1	1	yes	characteristic
20	C₂×#14273652	C₄	1	2	yes	—
21	#16837132	C₂²	1	8	yes	—
22	#16837123	C₂²	1	2	yes	—
23	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
24	C₅₉×#431	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#335	C₂²	1	2	yes	—
27	C₅₉×#334	C₂²	1	2	yes	—
28	C₂²×Dic₁₁₈	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₃₆	C₈	1	8	yes	complemented
30	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₁₈	—	2	2	no	—
33	C₄×Dic₁₁₈	—	2	2	no	—
34	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₁₈	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
37	C₂×Dic₅₉⋊C₄	—	2	2	no	—
38	C₂×C₄×Dic₅₉	—	2	2	no	—
39	#14273661	C₂×C₄	1	2	yes	—
40	#14273661	C₂×C₄	1	2	yes	—
41	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
42	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
43	Dic₅₉⋊C₈	—	2	8	no	—
44	#14273652	C₂×C₄	1	4	yes	—
45	#14273652	C₂×C₄	1	4	yes	—
46	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
48	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
49	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
50	C₈×Dic₅₉	—	2	8	no	—
51	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
55	C₂²⋊C₈×C₅₉	—	2	2	no	—
56	C₄×C₈×C₅₉	—	2	2	no	—
57	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
64	C₂²×Dic₅₉	—	2	2	no	—
65	C₂²×Dic₅₉	—	2	2	no	—
66	C₂×Dic₁₁₈	—	2	2	no	—
67	C₂×Dic₁₁₈	—	2	4	no	—
68	C₂×Dic₁₁₈	C₂×C₈	1	8	yes	—
69	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	—	2	4	no	—
72	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	#41	1	1	yes	characteristic
80	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
81	C₄⋊Dic₅₉	—	2	2	no	—
82	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
83	Dic₅₉⋊C₄	—	2	4	no	—
84	Dic₅₉⋊C₄	—	2	4	no	—
85	C₄×Dic₅₉	—	2	4	no	—
86	C₄×Dic₅₉	—	2	4	no	—
87	C₂×C₅₉⋊C₈	—	2	2	no	—
88	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
89	Dic₂₃₆	C₂×C₈	1	8	yes	—
90	Dic₂₃₆	C₂×C₈	1	8	yes	—
91	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	4	1	no	—
94	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₅₉	#131	1	2	yes	—
99	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₅₉	—	2	4	no	—
103	C₂×Dic₅₉	—	2	8	no	—
104	C₂×Dic₅₉	—	2	4	no	—
105	C₂×Dic₅₉	—	8	2	no	—
106	Dic₁₁₈	—	2	4	no	—
107	Dic₁₁₈	—	2	8	no	—
108	Dic₁₁₈	C₂²×C₈	1	8	yes	—
109	C₈×C₅₉	—	2	1	no	—
110	C₈×C₅₉	—	2	1	no	—
111	C₈×C₅₉	—	4	4	no	—
112	C₈×C₅₉	C₂²×C₈	1	2	yes	—
113	C₈×C₅₉	C₂²×C₈	1	2	yes	—
114	C₅₉⋊C₈	—	2	4	no	—
115	C₅₉⋊C₈	—	4	2	no	—
116	#17856	—	59	1	no	maximal, Sylow
117	C₂²×C₅₉	—	2	1	no	—
118	C₂²×C₅₉	—	2	1	no	—
119	C₂²×C₅₉	C₈×D₄	1	2	yes	—
120	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₅₉	—	2	1	no	—
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	C₈×D₄	1	2	yes	—
125	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
127	C₄×C₅₉	#578	1	2	yes	—
128	Dic₅₉	—	2	8	no	—
129	Dic₅₉	—	4	4	no	—
130	#2982	—	59	2	no	—
131	C₂×C₄×Q₁₆	—	59	1	no	—
132	#1579	—	59	4	no	—
133	C₈×Q₁₆	—	59	4	no	—
134	#1564	D₅₉	1	1	yes	characteristic
135	#1479	—	59	2	no	—
136	#1450	—	59	1	no	—
137	C₂×C₅₉	—	2	2	no	—
138	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₅₉	#2950	1	1	yes	characteristic
140	C₂×C₅₉	#2957	1	1	yes	characteristic
141	C₂²×Q₁₆	—	59	1	no	—
142	C₂×C₄×Q₈	—	59	2	no	—
143	C₈×Q₈	—	59	8	no	—
144	C₈×Q₈	—	118	4	no	—
145	C₄×Q₁₆	—	59	2	no	—
146	C₄×Q₁₆	—	59	2	no	—
147	C₄×Q₁₆	—	59	2	no	—
148	C₄×Q₁₆	—	59	2	no	—
149	#431	—	59	1	no	—
150	#431	—	59	1	no	—
151	#431	—	118	4	no	—
152	#431	C₂×D₅₉	1	2	yes	—
153	C₂×#107	—	59	1	no	—
154	C₂×#103	—	59	2	no	—
155	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
156	#335	C₂×D₅₉	1	2	yes	—
157	#334	C₂×D₅₉	1	2	yes	—
158	Q₈⋊C₈	—	59	8	no	—
159	#322	—	59	2	no	—
160	C₈²	—	59	2	no	—
161	C₅₉	#17856	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	59	2	no	—
163	C₂×Q₁₆	—	59	8	no	—
164	C₂×Q₁₆	—	59	2	no	—
165	C₂×Q₁₆	—	59	2	no	—
166	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
168	C₄×Q₈	—	59	4	no	—
169	C₄×Q₈	—	59	4	no	—
170	C₄×Q₈	—	118	2	no	—
171	C₄×Q₈	—	118	2	no	—
172	C₂×C₄⋊C₄	—	59	2	no	—
173	C₂×C₄⋊C₄	—	118	2	no	—
174	C₂×C₄²	—	118	2	no	—
175	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	#107	—	59	2	no	—
177	#107	—	59	2	no	—
178	C₄⋊C₈	—	59	2	no	—
179	C₄⋊C₈	—	118	8	no	—
180	C₄⋊C₈	—	118	1	no	—
181	C₄⋊C₈	C₂²×D₅₉	1	4	yes	—
182	#103	—	59	4	no	—
183	#103	—	59	4	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	59	2	no	—
186	C₂²⋊C₈	—	118	1	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	59	2	no	—
189	C₄×C₈	—	59	2	no	—
190	C₄×C₈	—	118	8	no	—
191	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
192	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
195	C₂×Q₈	—	59	8	no	—
196	C₂×Q₈	—	59	4	no	—
197	C₂×Q₈	—	118	2	no	—
198	C₂×Q₈	—	118	4	no	—
199	C₂²×C₄	—	118	2	no	—
200	C₂²×C₄	—	118	2	no	—
201	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
203	C₂²×C₄	#11745689	1	1	yes	characteristic
204	Q₁₆	—	59	8	no	—
205	Q₁₆	—	59	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	59	4	no	—
209	C₂×C₈	—	118	2	no	—
210	C₂×C₈	C₈×D₅₉	1	4	yes	—
211	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
215	C₄⋊C₄	—	59	4	no	—
216	C₄⋊C₄	—	118	2	no	—
217	C₄⋊C₄	—	118	4	no	—
218	C₄⋊C₄	—	118	4	no	—
219	C₄²	—	118	4	no	—
220	C₄²	—	118	4	no	—
221	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
222	C₄²	D₄×D₅₉	1	1	yes	characteristic
223	C₄²	#11745689	1	1	yes	characteristic
224	C₄²	C₂³×D₅₉	1	1	yes	characteristic
225	C₂³	#14273689	1	1	yes	characteristic
226	Q₈	—	59	8	no	—
227	Q₈	—	118	4	no	—
228	Q₈	—	118	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	118	8	no	—
232	C₂×C₄	—	118	4	no	—
233	C₂×C₄	—	118	4	no	—
234	C₂×C₄	—	472	2	no	—
235	C₂×C₄	#14273689	1	1	yes	characteristic
236	C₂×C₄	#14273689	1	1	yes	characteristic
237	C₂×C₄	#14273689	1	1	yes	characteristic
238	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
239	C₂×C₄	#14273735	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
242	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	4	no	—
246	C₈	—	118	4	no	—
247	C₈	—	236	2	no	—
248	C₈	C₂×C₈×D₅₉	1	2	yes	—
249	C₈	C₂×C₈×D₅₉	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#16837467	1	2	yes	—
253	C₂²	C₂×#14273689	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	118	8	no	—
258	C₄	—	236	4	no	—
259	C₄	#16837467	1	2	yes	—
260	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
261	C₄	C₂×#14273735	1	1	yes	characteristic
262	C₄	#16838216	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×#16837467	1	1	yes	characteristic
265	C₂	#21432963	1	1	yes	characteristic
266	C₂	#21433012	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1564	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#17856
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #21422914	C₂	semidirect
4. #21422902	C₂	semidirect
29. C₂×Dic₂₃₆	C₈	semidirect
161. C₅₉	#17856	semidirect

Maximal subgroups

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

subgroup	index
#21433176	2
C₂×C₄×Dic₂₃₆	2
#21422914	2
#21422902	2
#21422490	2
#21422379	2
C₅₉×#1564	2
#17856	59