Subgroups — #28744013

Order 15104.

590 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	#21434610	C₂	1	1	yes	maximal, characteristic, complemented
2	#21434524	C₂	1	1	yes	maximal, characteristic
3	#21426342	C₂	1	2	yes	maximal, complemented
4	#21426338	C₂	1	2	yes	maximal, complemented
5	#21426333	C₂	1	4	yes	maximal, complemented
6	#21425804	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	—	2	1	no	—
9	C₂×#14273749	C₂²	1	2	yes	—
10	#16837959	—	2	1	no	—
11	#16837952	—	2	1	no	—
12	#16837807	C₂²	1	2	yes	—
13	#16837804	—	2	1	no	—
14	#16837804	C₂²	1	2	yes	—
15	#16837800	—	2	1	no	—
16	#16837786	—	2	1	no	—
17	#16837786	C₂²	1	2	yes	—
18	#16837782	C₂²	1	1	yes	characteristic
19	#16837779	C₂²	1	1	yes	characteristic
20	#16837775	—	2	1	no	—
21	#16837440	—	2	1	no	—
22	#16837440	C₂²	1	2	yes	—
23	#16837207	C₂²	1	8	yes	—
24	#16837204	C₂²	1	8	yes	—
25	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#346	C₂²	1	2	yes	—
28	C₅₉×#346	C₂²	1	2	yes	—
29	C₂×#11745685	—	2	2	no	—
30	#14273749	—	2	8	no	—
31	#14273749	—	4	4	no	—
32	C₂×Dic₂₃₆	—	2	1	no	—
33	C₂×Dic₂₃₆	C₂³	1	2	yes	—
34	C₂×#11745660	—	2	2	no	—
35	C₂×#11745660	—	2	2	no	—
36	C₂×D₄₇₂	—	2	1	no	—
37	C₂×D₄₇₂	C₂³	1	2	yes	—
38	#14273727	—	2	2	no	—
39	#14273727	—	2	1	no	—
40	#14273724	—	2	1	no	—
41	#14273719	—	2	2	no	—
42	#14273715	—	2	2	no	—
43	#14273706	—	4	2	no	—
44	#14273701	—	4	2	no	—
45	C₄×Dic₁₁₈	—	4	2	no	—
46	C₄×D₂₃₆	—	4	2	no	—
47	#14273686	—	2	2	no	—
48	#14273661	—	2	2	no	—
49	#14273661	C₂³	1	4	yes	—
50	#14273659	—	2	2	no	—
51	#14273659	—	2	2	no	—
52	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₅₉	—	2	4	no	—
56	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
57	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	D₄	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	D₈	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₉	—	4	2	no	—
64	#11745685	—	4	4	no	—
65	#11745685	—	4	4	no	—
66	C₂×D₂₃₆	—	2	2	no	—
67	C₂×D₂₃₆	—	4	2	no	—
68	C₂×C₄×D₅₉	—	4	2	no	—
69	C₂×Dic₁₁₈	—	2	2	no	—
70	C₂×Dic₁₁₈	—	4	2	no	—
71	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
72	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	#41	1	4	yes	—
76	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
77	C₄²×C₅₉	D₈	1	2	yes	—
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂²⋊Dic₅₉	—	4	2	no	—
81	(C₂×D₅₉)⋊C₄	—	4	4	no	—
82	C₄⋊Dic₅₉	—	2	4	no	—
83	C₄⋊Dic₅₉	—	4	2	no	—
84	C₄⋊Dic₅₉	—	4	2	no	—
85	Dic₅₉⋊C₄	—	4	4	no	—
86	C₄×Dic₅₉	—	4	2	no	—
87	Dic₂₃₆	—	2	4	no	—
88	Dic₂₃₆	—	4	2	no	—
89	D₄₇₂	—	2	4	no	—
90	D₄₇₂	—	4	2	no	—
91	#11745660	—	4	4	no	—
92	#11745660	—	4	4	no	—
93	C₁₆×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂²×D₅₉	—	4	2	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	#124	1	2	yes	—
98	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
103	C₂²⋊D₅₉	—	8	4	no	—
104	C₂×Dic₅₉	—	4	4	no	—
105	C₂×Dic₅₉	—	4	2	no	—
106	D₂₃₆	—	4	2	no	—
107	D₂₃₆	—	4	2	no	—
108	D₂₃₆	—	8	4	no	—
109	C₄×D₅₉	—	8	4	no	—
110	Dic₁₁₈	—	4	2	no	—
111	Dic₁₁₈	—	4	2	no	—
112	Dic₁₁₈	—	8	4	no	—
113	C₈×C₅₉	#124	1	2	yes	—
114	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
115	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
116	C₈×C₅₉	C₂×#41	1	2	yes	—
117	#33701	—	59	1	no	maximal, Sylow
118	C₂²×C₅₉	—	2	2	no	—
119	C₂²×C₅₉	#486	1	2	yes	—
120	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
121	C₂×D₅₉	—	8	2	no	—
122	C₂×D₅₉	—	8	4	no	—
123	C₄×C₅₉	—	2	2	no	—
124	C₄×C₅₉	—	2	2	no	—
125	C₄×C₅₉	#486	1	2	yes	—
126	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
127	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
128	Dic₅₉	—	8	2	no	—
129	Dic₅₉	—	8	4	no	—
130	#3160	—	59	1	no	—
131	#3147	—	59	1	no	—
132	#2245	—	59	2	no	—
133	#2244	—	59	2	no	—
134	#2243	—	59	4	no	—
135	#2138	—	59	4	no	—
136	C₂×#346	D₅₉	1	1	yes	characteristic, complemented
137	C₂×C₅₉	—	2	2	no	—
138	C₂×C₅₉	C₂×#486	1	1	yes	characteristic
139	C₂×C₅₉	#3414	1	2	yes	—
140	D₅₉	—	16	2	no	—
141	C₂×#135	—	59	2	no	—
142	C₂×#135	—	118	1	no	—
143	#533	—	118	1	no	—
144	#532	—	118	1	no	—
145	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
146	#500	—	59	2	no	—
147	#499	—	59	2	no	—
148	#499	—	118	1	no	—
149	#498	—	118	1	no	—
150	#495	—	59	2	no	—
151	#495	—	118	1	no	—
152	#494	—	59	1	no	—
153	#493	—	59	1	no	—
154	#492	—	118	1	no	—
155	#427	—	59	2	no	—
156	#427	—	118	1	no	—
157	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
158	#358	—	59	8	no	—
159	#357	—	59	8	no	—
160	#346	C₂×D₅₉	1	2	yes	—
161	#346	C₂×D₅₉	1	2	yes	—
162	C₅₉	#33701	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	118	2	no	—
164	#135	—	118	8	no	—
165	#135	—	236	4	no	—
166	C₂×Q₁₆	—	59	2	no	—
167	C₂×Q₁₆	—	118	1	no	—
168	C₂×SD₁₆	—	118	2	no	—
169	C₂×SD₁₆	—	118	2	no	—
170	C₂×D₈	—	59	2	no	—
171	C₂×D₈	—	118	1	no	—
172	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
174	C₄⋊Q₈	—	118	2	no	—
175	C₄⋊Q₈	—	118	1	no	—
176	#127	—	118	1	no	—
177	#125	—	118	2	no	—
178	#124	—	118	2	no	—
179	C₂²⋊Q₈	—	236	2	no	—
180	#121	—	236	2	no	—
181	C₄×Q₈	—	236	2	no	—
182	C₄×D₄	—	236	2	no	—
183	#117	—	118	2	no	—
184	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
187	#107	—	59	4	no	—
188	#107	—	118	2	no	—
189	#106	—	118	2	no	—
190	#106	—	118	2	no	—
191	C₄×C₈	D₂₃₆	1	1	yes	characteristic
192	C₄×C₈	D₂₃₆	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
195	#41	—	236	4	no	—
196	#41	—	236	4	no	—
197	C₂×Q₈	—	118	2	no	—
198	C₂×Q₈	—	236	2	no	—
199	C₂×D₄	—	118	2	no	—
200	C₂×D₄	—	236	2	no	—
201	C₂×D₄	—	236	2	no	—
202	C₂²×C₄	—	236	2	no	—
203	C₂²×C₄	D₄₇₂	1	2	yes	—
204	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
205	Q₁₆	—	118	4	no	—
206	Q₁₆	—	236	2	no	—
207	SD₁₆	—	236	4	no	—
208	SD₁₆	—	236	4	no	—
209	D₈	—	118	4	no	—
210	D₈	—	236	2	no	—
211	C₂×C₈	#11745660	1	4	yes	—
212	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
213	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
214	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
215	C₂×C₈	#11745685	1	4	yes	—
216	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
217	C₄⋊C₄	—	118	4	no	—
218	C₄⋊C₄	—	236	2	no	—
219	C₄⋊C₄	—	236	4	no	—
220	C₄⋊C₄	—	236	2	no	—
221	C₂²⋊C₄	—	236	2	no	—
222	C₂²⋊C₄	—	236	4	no	—
223	C₄²	—	236	2	no	—
224	C₄²	D₄₇₂	1	2	yes	—
225	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
226	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
227	C₁₆	—	2	4	no	—
228	C₂³	—	236	2	no	—
229	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
230	Q₈	—	472	4	no	—
231	Q₈	—	236	2	no	—
232	Q₈	—	236	2	no	—
233	D₄	—	472	4	no	—
234	D₄	—	472	4	no	—
235	D₄	—	236	2	no	—
236	D₄	—	236	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	472	4	no	—
239	C₂×C₄	—	236	4	no	—
240	C₂×C₄	—	236	2	no	—
241	C₂×C₄	#14273715	1	2	yes	—
242	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
243	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
244	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
245	C₂×C₄	C₂×#11745660	1	2	yes	—
246	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
247	C₈	#14273715	1	2	yes	—
248	C₈	C₂×#11745660	1	2	yes	—
249	C₈	C₂×#11745660	1	2	yes	—
250	C₈	C₂×#11745685	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	472	2	no	—
253	C₂²	—	472	4	no	—
254	C₂²	#16837749	1	2	yes	—
255	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	472	2	no	—
259	C₄	—	472	4	no	—
260	C₄	#16837749	1	2	yes	—
261	C₄	C₂×#14273715	1	1	yes	characteristic
262	C₄	C₂²×#11745660	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	944	2	no	—
265	C₂	C₂×#16837749	1	1	yes	characteristic
266	C₂	#21437232	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21434524	C₂
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#33701
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. #21434610	C₂	semidirect
3. #21426342	C₂	semidirect
4. #21426338	C₂	semidirect
5. #21426333	C₂	semidirect
6. #21425804	C₂	semidirect
7. C₂×C₅₉×#346	C₂	semidirect
136. C₂×#346	D₅₉	semidirect
162. C₅₉	#33701	semidirect

Maximal subgroups

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

subgroup	index
#21434610	2
#21434524	2
#21426342	2
#21426338	2
#21426333	2
#21425804	2
C₂×C₅₉×#346	2
#33701	59