Subgroups — #28589102

Order 15104.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837440	C₂	1	4	yes	maximal
2	#21426503	C₂	1	2	yes	maximal
3	#21424260	C₂	1	4	yes	maximal, complemented
4	#21424256	C₂	1	2	yes	maximal, complemented
5	#21424253	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273686	C₂²	1	4	yes	—
8	#16837440	—	2	16	no	—
9	C₂×#14273661	C₂²	1	8	yes	—
10	C₂×#14273659	C₂²	1	8	yes	—
11	#16837272	C₂²	1	4	yes	—
12	#16837268	C₂²	1	4	yes	—
13	#16837262	—	2	4	no	—
14	C₄×C₄⋊Dic₅₉	—	2	4	no	—
15	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₉	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
21	#14273686	—	2	16	no	—
22	C₂×C₄⋊Dic₅₉	—	2	8	no	—
23	C₂×C₄⋊Dic₅₉	C₂³	1	8	yes	—
24	C₂×C₂²⋊Dic₅₉	—	2	4	no	—
25	C₂×C₄×Dic₅₉	—	2	4	no	—
26	#14273661	—	2	16	no	—
27	#14273659	—	2	16	no	—
28	C₅₉⋊#95	—	2	8	no	—
29	C₂³×C₄×C₅₉	D₄	1	2	yes	—
30	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₅₉	D₄	1	8	yes	—
33	C₂²×C₈×C₅₉	Q₈	1	8	yes	—
34	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₉	D₄	1	2	yes	—
38	C₂×C₄²×C₅₉	Q₈	1	4	yes	—
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₉	C₂×C₄	1	8	yes	—
44	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	—	2	8	no	—
46	C₂²×C₄×C₅₉	C₄⋊C₄	1	8	yes	—
47	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	2	8	no	—
56	C₂×C₈×C₅₉	C₄⋊C₄	1	32	yes	complemented
57	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
60	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
63	C₄²×C₅₉	C₄⋊C₄	1	8	yes	—
64	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
65	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
66	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
67	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
68	C₂²⋊Dic₅₉	—	4	8	no	—
69	C₄⋊Dic₅₉	—	2	16	no	—
70	C₄⋊Dic₅₉	—	4	16	no	—
71	C₄×Dic₅₉	—	4	8	no	—
72	C₂³×C₅₉	—	2	4	no	—
73	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₅₉	#127	1	2	yes	—
75	C₂³×C₅₉	C₄⋊Q₈	1	2	yes	—
76	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	—	2	4	no	—
78	C₂×C₄×C₅₉	—	2	16	no	—
79	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₅₉	#127	1	2	yes	—
84	C₂×C₄×C₅₉	C₄⋊Q₈	1	8	yes	—
85	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₅₉	#135	1	8	yes	—
87	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
89	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
90	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂×Dic₅₉	—	4	8	no	—
92	C₂×Dic₅₉	—	4	16	no	—
93	C₈×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
94	C₈×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
95	#20797	—	59	1	no	maximal, Sylow
96	C₂²×C₅₉	—	2	4	no	—
97	C₂²×C₅₉	—	2	4	no	—
98	C₂²×C₅₉	—	2	4	no	—
99	C₂²×C₅₉	#384	1	4	yes	—
100	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
102	C₂²×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₂²×C₅₉	C₂×#135	1	4	yes	—
104	C₄×C₅₉	—	2	4	no	—
105	C₄×C₅₉	#384	1	4	yes	—
106	C₄×C₅₉	#427	1	8	yes	complemented
107	C₄×C₅₉	C₂²×C₄⋊C₄	1	2	yes	—
108	C₄×C₅₉	C₂×C₄⋊Q₈	1	2	yes	—
109	Dic₅₉	—	8	8	no	—
110	C₂×#427	—	59	4	no	—
111	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
112	#2285	—	59	2	no	—
113	#1846	—	59	4	no	—
114	#1845	—	59	2	no	—
115	#1844	—	59	2	no	—
116	C₂×C₅₉	—	2	4	no	—
117	C₂×C₅₉	C₂×#384	1	1	yes	characteristic
118	C₂×C₅₉	C₂×#427	1	2	yes	—
119	C₂×C₅₉	#3147	1	2	yes	—
120	C₂×C₅₉	#3160	1	2	yes	—
121	C₂³×C₈	C₂×D₅₉	1	2	yes	—
122	C₂×#117	—	59	4	no	—
123	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
124	#427	—	118	16	no	—
125	C₂×#107	—	59	8	no	—
126	C₂×#106	—	59	8	no	—
127	C₂×C₄×C₈	Dic₅₉	1	8	yes	complemented
128	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
131	#384	—	59	4	no	—
132	#382	—	59	4	no	—
133	#380	—	118	4	no	—
134	C₄×C₄⋊C₄	—	118	4	no	—
135	C₅₉	#20797	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	D₂₃₆	1	2	yes	—
137	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
138	C₂²×C₈	Dic₁₁₈	1	8	yes	—
139	C₂²×C₈	D₂₃₆	1	8	yes	—
140	C₂²×C₈	C₂×Dic₅₉	1	8	yes	—
141	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
142	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
143	#117	—	118	16	no	—
144	C₂×C₄⋊C₄	—	59	8	no	—
145	C₂×C₄⋊C₄	—	118	8	no	—
146	C₂×C₂²⋊C₄	—	118	4	no	—
147	C₂×C₄²	—	118	4	no	—
148	C₂×C₄²	Dic₁₁₈	1	4	yes	—
149	C₂×C₄²	D₂₃₆	1	2	yes	—
150	C₂×C₄²	C₂×Dic₅₉	1	4	yes	—
151	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
153	#107	—	118	16	no	—
154	#106	—	118	16	no	—
155	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
156	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
157	C₄×C₈	C₂×Dic₅₉	1	8	yes	—
158	#95	—	118	8	no	—
159	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	118	8	no	—
162	C₂²×C₄	C₄⋊Dic₅₉	1	8	yes	—
163	C₂²×C₄	C₂×Dic₁₁₈	1	4	yes	—
164	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
165	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
166	C₂²×C₄	C₂×D₂₃₆	1	4	yes	—
167	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
168	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
169	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
170	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
171	C₂×C₈	C₄⋊Dic₅₉	1	32	yes	complemented
172	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
173	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
174	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
175	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
176	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
177	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
178	C₄⋊C₄	—	118	16	no	—
179	C₄⋊C₄	—	236	16	no	—
180	C₂²⋊C₄	—	236	8	no	—
181	C₄²	—	236	8	no	—
182	C₄²	C₄⋊Dic₅₉	1	8	yes	—
183	C₄²	C₂×Dic₁₁₈	1	2	yes	—
184	C₄²	C₂×Dic₁₁₈	1	2	yes	—
185	C₄²	C₂²×Dic₅₉	1	2	yes	—
186	C₄²	C₂²×Dic₅₉	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
189	C₂³	#14273724	1	2	yes	—
190	C₂³	#14273727	1	2	yes	—
191	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	236	16	no	—
195	C₂×C₄	—	236	8	no	—
196	C₂×C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
197	C₂×C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
198	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₅₉	1	8	yes	—
200	C₂×C₄	#14273724	1	2	yes	—
201	C₂×C₄	#14273727	1	8	yes	—
202	C₂×C₄	#14273727	1	2	yes	—
203	C₂×C₄	#14273749	1	8	yes	—
204	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₂₃₆	1	2	yes	—
206	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₁₁₈	1	2	yes	—
208	C₈	C₂×C₄⋊Dic₅₉	1	8	yes	—
209	C₈	C₂×C₄⋊Dic₅₉	1	8	yes	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	#16837272	1	4	yes	—
214	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
215	C₂²	C₂×#14273724	1	1	yes	characteristic
216	C₂²	C₂×#14273727	1	1	yes	characteristic
217	C₂²	C₂×#14273749	1	4	yes	—
218	C₄	—	2	4	no	—
219	C₄	—	472	8	no	—
220	C₄	#16837272	1	4	yes	—
221	C₄	#16837440	1	8	yes	complemented
222	C₄	C₂²×C₄⋊Dic₅₉	1	2	yes	—
223	C₄	C₂×#14273727	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	C₂×#16837272	1	1	yes	characteristic
226	C₂	C₂×#16837440	1	2	yes	—
227	C₂	#21434524	1	2	yes	—
228	C₂	#21434610	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×D₄
prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	#20797
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #21424260	C₂	semidirect
4. #21424256	C₂	semidirect
5. #21424253	C₂	semidirect
17. C₂×C₄×C₈×C₅₉	C₄	semidirect
56. C₂×C₈×C₅₉	C₄⋊C₄	semidirect
106. C₄×C₅₉	#427	semidirect
127. C₂×C₄×C₈	Dic₅₉	semidirect
135. C₅₉	#20797	semidirect
171. C₂×C₈	C₄⋊Dic₅₉	semidirect
221. C₄	#16837440	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837440	2
#21426503	2
#21424260	2
#21424256	2
#21424253	2
C₂²×C₄×C₈×C₅₉	2
#20797	59