Subgroups — #28487461

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273727	C₂	1	1	yes	maximal, characteristic
2	#21424248	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837134	C₂	1	4	yes	maximal
4	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273727	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	#16837308	C₄	1	2	yes	—
10	#16837300	—	2	2	no	—
11	#16837272	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₅₉	—	2	2	no	—
13	#16837134	—	2	8	no	—
14	C₂×C₄×C₈×C₅₉	—	2	2	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
16	C₄³×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₁₈	C₂×C₄	1	2	yes	—
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	#14273727	—	2	2	no	—
20	#14273727	C₈	1	4	yes	complemented
21	C₄×Dic₁₁₈	—	2	4	no	—
22	C₄×Dic₁₁₈	—	2	4	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
26	C₂×Dic₅₉⋊C₄	—	2	2	no	—
27	C₂×C₄×Dic₅₉	—	2	2	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
29	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
30	C₅₉⋊#95	—	2	4	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	D₄	1	2	yes	—
35	C₂×C₄²×C₅₉	D₄	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₅₉	—	2	8	no	—
38	C₄×C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	D₄	1	4	yes	—
40	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₉	—	2	2	no	—
46	C₂²×Dic₅₉	—	2	2	no	—
47	C₂×Dic₁₁₈	—	2	4	no	—
48	C₂×Dic₁₁₈	—	2	4	no	—
49	C₂×Dic₁₁₈	—	2	2	no	—
50	C₂×Dic₁₁₈	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₅₉	—	2	8	no	—
52	C₂×C₈×C₅₉	—	2	4	no	—
53	C₄²×C₅₉	—	2	4	no	—
54	C₄²×C₅₉	—	2	2	no	—
55	C₄²×C₅₉	C₂×C₈	1	2	yes	—
56	C₄²×C₅₉	M₁₆	1	2	yes	—
57	C₄²×C₅₉	SD₁₆	1	4	yes	—
58	C₄²×C₅₉	Q₁₆	1	4	yes	—
59	C₄²×C₅₉	C₂×D₄	1	2	yes	—
60	C₄²×C₅₉	#41	1	2	yes	—
61	C₄⋊Dic₅₉	—	2	8	no	—
62	C₄⋊Dic₅₉	—	2	4	no	—
63	C₄⋊Dic₅₉	—	4	4	no	—
64	Dic₅₉⋊C₄	—	4	4	no	—
65	C₄×Dic₅₉	—	4	2	no	—
66	C₄×Dic₅₉	—	4	2	no	—
67	C₂×C₅₉⋊C₈	—	4	2	no	—
68	C₂×C₅₉⋊C₈	—	4	4	no	—
69	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₅₉	#103	1	4	yes	—
75	C₂×C₄×C₅₉	#103	1	4	yes	—
76	C₂×C₄×C₅₉	#104	1	4	yes	—
77	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₅₉	#121	1	2	yes	—
79	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₉	#131	1	2	yes	—
84	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
86	C₂×Dic₅₉	—	2	4	no	—
87	C₂×Dic₅₉	—	4	2	no	—
88	C₂×Dic₅₉	—	4	2	no	—
89	C₂×Dic₅₉	—	4	2	no	—
90	C₂×Dic₅₉	—	4	4	no	—
91	Dic₁₁₈	—	2	8	no	—
92	Dic₁₁₈	—	4	4	no	—
93	C₈×C₅₉	—	2	8	no	—
94	C₅₉⋊C₈	—	8	2	no	—
95	#9637	—	59	1	no	maximal, Sylow
96	C₂²×C₅₉	#390	1	1	yes	characteristic
97	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₅₉	#408	1	1	yes	characteristic
99	C₂²×C₅₉	C₂×#103	1	2	yes	—
100	C₂²×C₅₉	C₂×#104	1	2	yes	—
101	C₄×C₅₉	—	2	2	no	—
102	C₄×C₅₉	—	2	4	no	—
103	C₄×C₅₉	Q₈⋊C₈	1	8	yes	complemented
104	C₄×C₅₉	C₈×D₄	1	2	yes	—
105	C₄×C₅₉	#436	1	2	yes	—
106	C₄×C₅₉	#487	1	2	yes	—
107	C₄×C₅₉	#492	1	1	yes	characteristic
108	C₄×C₅₉	#494	1	1	yes	characteristic
109	Dic₅₉	—	4	4	no	—
110	Dic₅₉	—	8	2	no	—
111	C₄×C₄⋊Q₈	—	59	1	no	—
112	#1842	—	59	1	no	—
113	C₄²×C₈	D₅₉	1	1	yes	characteristic
114	C₂×Q₈⋊C₈	—	59	4	no	—
115	C₂×C₅₉	C₂×Q₈⋊C₈	1	2	yes	—
116	C₂×C₅₉	#1493	1	2	yes	—
117	C₂×C₅₉	#1954	1	1	yes	characteristic
118	C₂×C₅₉	#1956	1	1	yes	characteristic
119	C₂×C₅₉	#1962	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	—	59	1	no	—
121	C₂×C₄×Q₈	—	59	2	no	—
122	C₂×C₄⋊C₈	—	59	2	no	—
123	C₂×C₄⋊C₈	—	118	2	no	—
124	C₂×C₄×C₈	—	2	2	no	—
125	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
126	#391	—	59	2	no	—
127	#389	—	118	2	no	—
128	#384	—	59	1	no	—
129	C₄×C₄⋊C₄	—	118	2	no	—
130	C₄³	C₂×D₅₉	1	1	yes	characteristic
131	Q₈⋊C₈	—	118	8	no	—
132	C₅₉	#9637	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	59	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	118	2	no	—
136	C₄⋊Q₈	—	59	4	no	—
137	C₄⋊Q₈	—	118	2	no	—
138	C₄×Q₈	—	118	4	no	—
139	C₄×Q₈	—	118	4	no	—
140	C₂×C₄⋊C₄	—	59	2	no	—
141	C₂×C₄⋊C₄	—	118	2	no	—
142	C₂×C₄⋊C₄	—	118	2	no	—
143	C₂×C₄⋊C₄	—	118	2	no	—
144	C₂×C₄²	—	118	2	no	—
145	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
147	C₂×C₄²	D₂₃₆	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
149	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
150	C₄⋊C₈	—	118	4	no	—
151	C₄⋊C₈	—	236	4	no	—
152	C₄×C₈	—	2	8	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	D₂₃₆	1	4	yes	—
155	#95	—	118	4	no	—
156	C₂×Q₈	—	59	4	no	—
157	C₂×Q₈	—	118	2	no	—
158	C₂×Q₈	—	118	4	no	—
159	C₂×Q₈	—	118	4	no	—
160	C₂²×C₄	—	118	2	no	—
161	C₂²×C₄	—	118	2	no	—
162	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
163	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
164	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	236	2	no	—
170	C₂×C₈	—	236	4	no	—
171	C₄⋊C₄	—	118	8	no	—
172	C₄⋊C₄	—	118	4	no	—
173	C₄⋊C₄	—	236	4	no	—
174	C₄⋊C₄	—	236	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	236	2	no	—
178	C₄²	—	236	2	no	—
179	C₄²	C₈×D₅₉	1	2	yes	—
180	C₄²	#11745659	1	2	yes	—
181	C₄²	#11745660	1	4	yes	—
182	C₄²	Dic₂₃₆	1	4	yes	—
183	C₄²	C₂×D₂₃₆	1	2	yes	—
184	C₄²	#11745685	1	2	yes	—
185	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
186	Q₈	—	118	8	no	—
187	Q₈	—	236	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	118	4	no	—
192	C₂×C₄	—	236	4	no	—
193	C₂×C₄	—	236	2	no	—
194	C₂×C₄	—	236	2	no	—
195	C₂×C₄	—	236	2	no	—
196	C₂×C₄	(C₂×D₅₉)⋊C₈	1	4	yes	—
197	C₂×C₄	#14273652	1	4	yes	—
198	C₂×C₄	#14273652	1	4	yes	—
199	C₂×C₄	#14273655	1	4	yes	—
200	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
201	C₂×C₄	#14273701	1	2	yes	—
202	C₂×C₄	#14273715	1	1	yes	characteristic
203	C₂×C₄	#14273724	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
205	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
206	C₂×C₄	#14273733	1	2	yes	—
207	C₂×C₄	C₂×#11745660	1	2	yes	—
208	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
209	C₈	—	2	8	no	—
210	C₈	—	472	2	no	—
211	C₂²	#16837304	1	1	yes	characteristic
212	C₂²	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
213	C₂²	#16837368	1	1	yes	characteristic
214	C₂²	C₂×#14273652	1	2	yes	—
215	C₂²	C₂×#14273655	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	4	no	—
218	C₄	—	472	2	no	—
219	C₄	—	236	4	no	—
220	C₄	#16837134	1	8	yes	complemented
221	C₄	C₈×D₂₃₆	1	2	yes	—
222	C₄	#16837481	1	2	yes	—
223	C₄	#16837753	1	2	yes	—
224	C₄	#16837775	1	1	yes	characteristic
225	C₄	#16837782	1	1	yes	characteristic
226	C₂	C₂×#16837134	1	2	yes	—
227	C₂	#21422552	1	2	yes	—
228	C₂	#21424890	1	1	yes	characteristic
229	C₂	#21424898	1	1	yes	characteristic
230	C₂	#21424924	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#9637
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
20. #14273727	C₈	semidirect
103. C₄×C₅₉	Q₈⋊C₈	semidirect
132. C₅₉	#9637	semidirect
220. C₄	#16837134	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273727	2
#21424248	2
C₂×#16837134	2
C₄²×C₈×C₅₉	2
#9637	59