Subgroups — #28742001

Order 15104.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837834	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273749	C₂	1	2	yes	maximal, complemented
3	#21425973	C₂	1	2	yes	maximal, complemented
4	#21425968	C₂	1	2	yes	maximal, complemented
5	#21425962	C₂	1	4	yes	maximal, complemented
6	#21425880	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425804	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	2	yes	—
10	C₄×#11745685	—	2	2	no	—
11	#16837834	C₄	1	8	yes	complemented
12	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
13	C₂×#14273672	C₂²	1	2	yes	—
14	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₃₆	C₂²	1	4	yes	—
16	C₄×#11745660	—	2	4	no	—
17	C₄×D₄₇₂	C₂²	1	4	yes	—
18	#16837440	C₂²	1	2	yes	—
19	#16837399	—	2	2	no	—
20	#16837230	C₂²	1	2	yes	—
21	#16837228	C₂²	1	2	yes	—
22	#16837207	C₂²	1	4	yes	—
23	#16837204	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
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	4	no	—
31	#14273749	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₂₃₆	C₂³	1	2	yes	—
33	C₂×#11745660	—	2	2	no	—
34	C₂×D₄₇₂	C₂³	1	2	yes	—
35	C₄×Dic₁₁₈	—	2	4	no	—
36	C₄×D₂₃₆	—	2	4	no	—
37	C₄×C₂²⋊D₅₉	—	4	4	no	—
38	#14273686	—	2	2	no	—
39	#14273684	—	4	2	no	—
40	C₄²×D₅₉	—	4	2	no	—
41	Dic₄₇₂	C₂×C₄	1	4	yes	complemented
42	#14273672	C₂×C₄	1	8	yes	complemented
43	D₉₄₄	C₂×C₄	1	4	yes	complemented
44	#14273661	C₂³	1	4	yes	—
45	#14273659	—	2	2	no	—
46	#14273652	—	2	4	no	—
47	#14273649	—	2	4	no	—
48	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₉	—	2	2	no	—
52	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₉	D₄	1	2	yes	—
57	C₄×C₈×C₅₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₅₉	D₈	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₉	—	4	2	no	—
61	#11745685	—	2	8	no	—
62	#11745685	—	4	4	no	—
63	C₂×D₂₃₆	—	2	2	no	—
64	C₂×C₄×D₅₉	—	4	2	no	—
65	C₂×C₄×D₅₉	—	4	4	no	—
66	C₂×Dic₁₁₈	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	#41	1	2	yes	—
72	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₄²×C₅₉	D₈	1	2	yes	—
74	C₄²×C₅₉	C₂×D₄	1	2	yes	—
75	C₂²⋊Dic₅₉	—	4	2	no	—
76	(C₂×D₅₉)⋊C₄	—	4	4	no	—
77	C₄⋊Dic₅₉	—	2	4	no	—
78	Dic₅₉⋊C₄	—	4	4	no	—
79	C₄×Dic₅₉	—	4	2	no	—
80	C₄×Dic₅₉	—	4	4	no	—
81	Dic₂₃₆	—	2	2	no	—
82	Dic₂₃₆	C₂²×C₄	1	4	yes	—
83	D₄₇₂	—	2	2	no	—
84	D₄₇₂	C₂²×C₄	1	4	yes	—
85	#11745660	—	2	8	no	—
86	C₁₆×C₅₉	—	2	2	no	—
87	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
90	C₂²×D₅₉	—	4	2	no	—
91	C₂×C₄×C₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₉	#135	1	2	yes	—
96	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₅₉	—	4	8	no	—
98	C₂×Dic₅₉	—	4	2	no	—
99	C₂×Dic₅₉	—	4	4	no	—
100	D₂₃₆	—	2	4	no	—
101	D₂₃₆	—	4	2	no	—
102	C₄×D₅₉	—	4	8	no	—
103	C₄×D₅₉	—	8	8	no	—
104	Dic₁₁₈	—	2	4	no	—
105	Dic₁₁₈	—	4	2	no	—
106	C₈×C₅₉	—	2	2	no	—
107	C₈×C₅₉	C₄×D₄	1	2	yes	—
108	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
110	#33451	—	59	1	no	maximal, Sylow
111	C₂²×C₅₉	—	2	2	no	—
112	C₂²×C₅₉	C₄×D₈	1	2	yes	—
113	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
114	C₂×D₅₉	—	4	4	no	—
115	C₂×D₅₉	—	8	4	no	—
116	C₄×C₅₉	—	2	4	no	—
117	C₄×C₅₉	C₄×D₈	1	2	yes	—
118	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
120	Dic₅₉	—	4	4	no	—
121	Dic₅₉	—	8	4	no	—
122	C₂×#508	—	59	1	no	—
123	C₄×#135	—	59	2	no	—
124	#2177	—	59	2	no	—
125	#2176	—	59	2	no	—
126	#2175	—	59	4	no	—
127	#2158	—	59	1	no	—
128	#2138	—	59	2	no	—
129	C₂×#346	D₅₉	1	1	yes	characteristic, complemented
130	C₂×C₅₉	—	2	2	no	—
131	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
132	C₂×C₅₉	#3414	1	2	yes	—
133	D₅₉	—	8	4	no	—
134	C₂×#135	—	59	2	no	—
135	C₄×#41	—	118	2	no	—
136	#508	—	59	8	no	—
137	C₂×Q₃₂	—	59	1	no	—
138	C₂×SD₃₂	—	59	2	no	—
139	C₂×D₁₆	—	59	1	no	—
140	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
141	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
142	C₄×Q₁₆	—	59	4	no	—
143	C₄×SD₁₆	—	118	4	no	—
144	C₄×D₈	—	59	4	no	—
145	#427	—	59	2	no	—
146	#416	—	118	2	no	—
147	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
148	#367	—	59	2	no	—
149	#366	—	59	2	no	—
150	#358	—	59	4	no	—
151	#357	—	59	4	no	—
152	#346	C₂×D₅₉	1	2	yes	—
153	#346	C₂×D₅₉	1	2	yes	—
154	C₅₉	#33451	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	118	2	no	—
156	#135	—	59	8	no	—
157	#135	—	118	4	no	—
158	C₂×Q₁₆	—	59	2	no	—
159	C₂×SD₁₆	—	118	2	no	—
160	C₂×D₈	—	59	2	no	—
161	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
163	C₄×Q₈	—	118	4	no	—
164	C₄×D₄	—	118	4	no	—
165	C₄×D₄	—	236	4	no	—
166	#117	—	118	2	no	—
167	#117	—	236	2	no	—
168	C₂×C₄²	—	236	2	no	—
169	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
170	Q₃₂	—	59	4	no	—
171	SD₃₂	—	59	8	no	—
172	D₁₆	—	59	4	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₄×D₅₉	1	4	yes	complemented
175	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
176	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
178	#107	—	59	4	no	—
179	#106	—	118	2	no	—
180	#103	—	118	4	no	—
181	#102	—	118	4	no	—
182	C₄×C₈	D₂₃₆	1	2	yes	—
183	C₄×C₈	C₂²×D₅₉	1	2	yes	—
184	#41	—	118	8	no	—
185	#41	—	236	4	no	—
186	C₂×Q₈	—	118	2	no	—
187	C₂×D₄	—	118	2	no	—
188	C₂×D₄	—	236	2	no	—
189	C₂²×C₄	—	236	4	no	—
190	C₂²×C₄	—	236	2	no	—
191	C₂²×C₄	D₄₇₂	1	2	yes	—
192	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
193	Q₁₆	—	59	4	no	—
194	Q₁₆	—	118	2	no	—
195	SD₁₆	—	118	8	no	—
196	D₈	—	59	4	no	—
197	D₈	—	118	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
200	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
201	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
202	C₂×C₈	#11745685	1	2	yes	—
203	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
204	C₄⋊C₄	—	118	4	no	—
205	C₄⋊C₄	—	236	4	no	—
206	C₂²⋊C₄	—	236	2	no	—
207	C₂²⋊C₄	—	236	4	no	—
208	C₄²	—	236	4	no	—
209	C₄²	—	236	2	no	—
210	C₄²	D₄₇₂	1	2	yes	—
211	C₄²	C₂×D₂₃₆	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂×C₄×D₅₉	1	2	yes	—
214	C₁₆	C₂×C₄×D₅₉	1	2	yes	—
215	C₂³	—	236	2	no	—
216	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
217	Q₈	—	118	4	no	—
218	Q₈	—	236	2	no	—
219	D₄	—	118	4	no	—
220	D₄	—	236	8	no	—
221	D₄	—	236	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	472	8	no	—
224	C₂×C₄	—	236	4	no	—
225	C₂×C₄	—	236	2	no	—
226	C₂×C₄	—	236	8	no	—
227	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
228	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
229	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
230	C₂×C₄	#14273749	1	2	yes	—
231	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₂₃₆	1	2	yes	—
234	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
235	C₈	C₂×#11745685	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	472	4	no	—
238	C₂²	—	236	4	no	—
239	C₂²	C₄×D₄₇₂	1	2	yes	—
240	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	472	4	no	—
243	C₄	—	236	4	no	—
244	C₄	C₄×D₄₇₂	1	2	yes	—
245	C₄	C₂×C₄×D₂₃₆	1	1	yes	characteristic
246	C₄	C₂×#14273749	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	472	4	no	—
249	C₂	C₂×C₄×D₄₇₂	1	1	yes	characteristic
250	C₂	#21437232	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×#16837834	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#33451
LCS	G ▷ 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
2. C₄×#14273749	C₂	semidirect
3. #21425973	C₂	semidirect
4. #21425968	C₂	semidirect
5. #21425962	C₂	semidirect
6. #21425880	C₂	semidirect
7. #21425804	C₂	semidirect
8. C₂×C₅₉×#346	C₂	semidirect
11. #16837834	C₄	semidirect
31. #14273749	C₂×C₄	semidirect
41. Dic₄₇₂	C₂×C₄	semidirect
42. #14273672	C₂×C₄	semidirect
43. D₉₄₄	C₂×C₄	semidirect
52. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
129. C₂×#346	D₅₉	semidirect
154. C₅₉	#33451	semidirect
174. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837834	2
C₄×#14273749	2
#21425973	2
#21425968	2
#21425962	2
#21425880	2
#21425804	2
C₂×C₅₉×#346	2
#33451	59