Subgroups — C₂×#21425932 #28741916

Order 15104.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837820	C₂	1	2	yes	maximal, complemented
2	C₂×#16837520	C₂	1	1	yes	maximal, characteristic
3	#21425932	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837187	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273749	C₄	1	1	yes	characteristic
8	C₂×#14273733	C₂²	1	2	yes	—
9	#16837820	—	2	4	no	—
10	#16837820	C₂²	1	8	yes	complemented
11	C₂×#14273666	—	2	2	no	—
12	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₉	—	2	2	no	—
15	#16837520	C₄	1	4	yes	—
16	#16837520	C₂²	1	4	yes	—
17	C₂×#14273663	C₄	1	1	yes	characteristic
18	C₂×#14273655	C₄	1	2	yes	—
19	#16837187	C₂²	1	8	yes	complemented
20	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	complemented
21	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#364	C₂²	1	4	yes	complemented
24	C₂×#11745685	C₂×C₄	1	2	yes	—
25	#14273749	C₈	1	4	yes	—
26	#14273749	C₂×C₄	1	4	yes	—
27	C₂×Dic₂₃₆	C₈	1	1	yes	characteristic
28	C₂×#11745660	C₈	1	2	yes	—
29	C₂×D₄₇₂	C₈	1	1	yes	characteristic
30	#14273733	—	2	2	no	—
31	#14273733	—	2	2	no	—
32	#14273733	C₂×C₄	1	4	yes	—
33	#14273733	C₂³	1	4	yes	—
34	C₂×#11745664	C₂×C₄	1	2	yes	—
35	C₂×#11745659	—	2	2	no	—
36	C₂×C₈×D₅₉	—	2	2	no	—
37	#14273666	—	2	8	no	—
38	C₂×C₅₉⋊C₁₆	—	2	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
42	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
43	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
45	C₁₆×D₅₉	—	2	8	no	—
46	#14273663	C₂×C₄	1	4	yes	—
47	#14273655	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₉	—	2	2	no	—
52	M₃₂×C₅₉	C₂³	1	4	yes	—
53	C₂×C₁₆×C₅₉	—	2	2	no	—
54	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₅₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₅₉	—	2	1	no	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₅₉	—	2	2	no	—
59	#11745685	—	2	2	no	—
60	#11745685	—	2	2	no	—
61	#11745685	C₂×C₈	1	4	yes	—
62	#11745685	C₂²×C₄	1	4	yes	—
63	C₂×D₂₃₆	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₅₉	—	2	2	no	—
65	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
74	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
76	C₄²×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
78	#11745664	—	2	2	no	—
79	#11745664	C₂²×C₄	1	4	yes	—
80	C₂×C₅₉⋊C₈	—	2	2	no	—
81	Dic₂₃₆	C₂×C₈	1	4	yes	—
82	D₄₇₂	C₂×C₈	1	4	yes	—
83	#11745660	C₂×C₈	1	8	yes	—
84	#11745659	—	2	4	no	—
85	#11745659	—	2	4	no	—
86	C₈×D₅₉	—	2	4	no	—
87	C₈×D₅₉	—	2	4	no	—
88	C₁₆×C₅₉	—	2	4	no	—
89	C₅₉⋊C₁₆	—	2	2	no	—
90	C₅₉⋊C₁₆	C₂×D₄	1	2	yes	—
91	C₅₉⋊C₁₆	C₂×D₄	1	2	yes	—
92	C₂³×C₅₉	#131	1	1	yes	characteristic
93	C₂²×D₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	4	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
102	C₂²⋊D₅₉	—	2	4	no	—
103	C₂²⋊D₅₉	—	2	4	no	—
104	C₂×Dic₅₉	—	2	2	no	—
105	D₂₃₆	—	2	2	no	—
106	D₂₃₆	C₂²×C₈	1	4	yes	—
107	C₄×D₅₉	—	2	4	no	—
108	C₄×D₅₉	—	2	4	no	—
109	Dic₁₁₈	—	2	2	no	—
110	Dic₁₁₈	C₂²×C₈	1	4	yes	—
111	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
112	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
113	C₈×C₅₉	C₂²×C₈	1	2	yes	—
114	C₈×C₅₉	C₂²×C₈	1	2	yes	—
115	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
117	C₅₉⋊C₈	—	2	4	no	—
118	C₂×#2170	—	59	1	no	maximal, Sylow
119	C₂²×C₅₉	—	2	2	no	—
120	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₅₉	C₂×#131	1	1	yes	characteristic
123	C₂×D₅₉	—	2	4	no	—
124	C₂×D₅₉	—	4	4	no	—
125	C₄×C₅₉	—	2	4	no	—
126	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
127	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
128	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
130	Dic₅₉	—	2	4	no	—
131	C₂×#504	—	59	2	no	—
132	C₂×#443	—	59	1	no	—
133	#2170	—	59	8	no	—
134	C₂×#364	D₅₉	1	1	yes	characteristic, complemented
135	C₂×#350	—	59	2	no	—
136	C₂×C₄×C₁₆	—	59	1	no	—
137	C₂×C₅₉	—	2	1	no	—
138	C₂×C₅₉	—	2	1	no	—
139	C₂×C₅₉	#2170	1	2	yes	complemented
140	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
141	D₅₉	—	4	4	no	—
142	C₂×#135	—	59	1	no	—
143	C₂×#131	—	59	2	no	—
144	#504	—	59	8	no	—
145	#504	—	118	4	no	—
146	C₂×M₃₂	—	118	2	no	—
147	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
148	C₂²×C₁₆	—	59	1	no	—
149	C₂²×C₁₆	—	59	1	no	—
150	C₂²×C₁₆	—	118	2	no	—
151	#443	—	59	4	no	—
152	#443	—	59	4	no	—
153	C₂×#108	—	59	1	no	—
154	C₂×#104	—	59	2	no	—
155	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
156	#364	C₂×D₅₉	1	4	yes	complemented
157	#350	—	59	8	no	—
158	C₄×C₁₆	—	59	4	no	—
159	C₅₉	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	—	59	2	no	—
161	#135	—	59	4	no	—
162	#135	—	59	4	no	—
163	C₂×Q₁₆	—	59	1	no	—
164	C₂×SD₁₆	—	59	2	no	—
165	C₂×D₈	—	59	1	no	—
166	#131	—	59	4	no	—
167	#131	—	59	4	no	—
168	#131	—	118	2	no	—
169	#131	—	118	2	no	—
170	C₂×M₁₆	—	59	2	no	—
171	C₂×M₁₆	—	118	2	no	—
172	C₂²×C₈	—	118	2	no	—
173	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
176	M₃₂	—	2	2	no	—
177	M₃₂	—	118	8	no	—
178	M₃₂	C₂²×D₅₉	1	4	yes	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	59	2	no	—
181	C₂×C₁₆	—	59	1	no	—
182	C₂×C₁₆	—	59	1	no	—
183	C₂×C₁₆	—	59	2	no	—
184	C₂×C₁₆	—	59	2	no	—
185	C₂×C₁₆	—	118	8	no	—
186	C₂×C₁₆	—	118	1	no	—
187	C₂×C₁₆	—	118	1	no	—
188	#108	—	59	4	no	—
189	#104	—	59	8	no	—
190	C₄×C₈	C₄×D₅₉	1	2	yes	—
191	C₄×C₈	C₂²×D₅₉	1	2	yes	—
192	#41	—	59	4	no	—
193	#41	—	59	4	no	—
194	#41	—	118	2	no	—
195	#41	—	118	2	no	—
196	C₂×Q₈	—	59	2	no	—
197	C₂×D₄	—	59	2	no	—
198	C₂×D₄	—	118	2	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	118	2	no	—
201	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
202	Q₁₆	—	59	4	no	—
203	SD₁₆	—	59	8	no	—
204	D₈	—	59	4	no	—
205	M₁₆	—	59	4	no	—
206	M₁₆	—	118	2	no	—
207	M₁₆	—	118	4	no	—
208	M₁₆	—	118	4	no	—
209	C₂×C₈	—	118	2	no	—
210	C₂×C₈	—	118	4	no	—
211	C₂×C₈	—	118	4	no	—
212	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
213	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
214	C₂×C₈	C₈×D₅₉	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
217	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
218	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
219	C₂×C₈	#11745689	1	1	yes	characteristic
220	C₂×C₈	#11745689	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
222	C₄²	—	2	1	no	—
223	C₄²	C₂×C₄×D₅₉	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₆	—	59	2	no	—
226	C₁₆	—	59	2	no	—
227	C₁₆	—	118	2	no	—
228	C₂³	—	118	2	no	—
229	C₂³	#14273735	1	1	yes	characteristic
230	Q₈	—	59	4	no	—
231	Q₈	—	118	2	no	—
232	D₄	—	59	4	no	—
233	D₄	—	118	4	no	—
234	D₄	—	118	4	no	—
235	D₄	—	118	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	118	2	no	—
239	C₂×C₄	—	118	4	no	—
240	C₂×C₄	—	118	4	no	—
241	C₂×C₄	#14273689	1	1	yes	characteristic
242	C₂×C₄	#14273689	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
245	C₂×C₄	#14273735	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
247	C₈	—	118	4	no	—
248	C₈	#14273689	1	1	yes	characteristic
249	C₈	#14273689	1	1	yes	characteristic
250	C₈	C₂×C₈×D₅₉	1	2	yes	—
251	C₈	C₂×C₈×D₅₉	1	2	yes	—
252	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
253	C₈	C₂×#11745689	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	118	4	no	—
256	C₂²	—	236	4	no	—
257	C₂²	#16837467	1	1	yes	characteristic
258	C₂²	#16837467	1	1	yes	characteristic
259	C₂²	C₂×#14273735	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	118	4	no	—
262	C₄	#16837467	1	1	yes	characteristic
263	C₄	#16837467	1	1	yes	characteristic
264	C₄	C₂×#14273689	1	1	yes	characteristic
265	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	236	4	no	—
269	C₂	#21425932	1	2	yes	direct product factor
270	C₂	C₂×#16837467	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂×#14273749	C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	C₂×#2170
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#16837820	C₂	semidirect
3. #21425932	C₂	direct (N × H)
4. C₂×#16837187	C₂	semidirect
5. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
6. C₂×C₅₉×#364	C₂	semidirect
10. #16837820	C₂²	semidirect
19. #16837187	C₂²	semidirect
20. C₄×C₅₉⋊C₁₆	C₂²	semidirect
23. C₅₉×#364	C₂²	semidirect
134. C₂×#364	D₅₉	semidirect
139. C₂×C₅₉	#2170	semidirect
156. #364	C₂×D₅₉	semidirect
159. C₅₉	C₂×#2170	semidirect
269. C₂	#21425932	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837820	2
C₂×#16837520	2
#21425932	2
C₂×#16837187	2
C₂×C₄×C₅₉⋊C₁₆	2
C₂×C₅₉×#364	2
C₂×#2170	59

Subgroups — C2×#21425932 #28741916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425932 #28741916