Subgroups — C₂×#66125 #10619124

Order 640.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#4099	C₂	1	2	yes	maximal, complemented
4	#66125	C₂	1	16	yes	maximal, direct product factor
5	C₂×#4076	C₂	1	2	yes	maximal, complemented
6	C₂×#4075	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
8	C₂×#4072	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#4070	C₂	1	2	yes	maximal, complemented
10	C₂⁵×C₅	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅	—	2	8	no	—
12	C₂³×C₄×C₅	C₂²	1	2	yes	—
13	C₂³×C₄×C₅	C₂²	1	2	yes	—
14	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₅	C₂²	1	8	yes	complemented
17	C₂²×C₂²⋊D₅	C₂²	1	4	yes	—
18	C₂²×C₂²⋊D₅	C₂²	1	2	yes	—
19	C₂×#709	—	2	8	no	—
20	C₂×#709	C₂²	1	8	yes	complemented
21	C₂²×D₂₀	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₄×D₅	C₂²	1	2	yes	complemented
23	C₂²×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
24	#4099	C₂²	1	16	yes	complemented
25	C₂×C₂²⋊Dic₅	C₂²	1	4	yes	—
26	C₂×C₂²⋊Dic₅	C₂²	1	2	yes	—
27	#4076	C₂²	1	16	yes	complemented
28	#4075	C₂²	1	16	yes	complemented
29	C₄×C₂²⋊D₅	C₂²	1	32	yes	complemented
30	C₂×#685	C₂²	1	4	yes	complemented
31	#4072	C₂²	1	8	yes	complemented
32	C₂×C₄⋊Dic₅	C₂²	1	2	yes	complemented
33	#4070	C₂²	1	16	yes	complemented
34	C₂×#683	C₂²	1	4	yes	complemented
35	C₂×C₄×Dic₅	C₂²	1	2	yes	complemented
36	C₂×#525	—	5	1	no	maximal, Sylow
37	C₂⁴×C₅	—	2	4	no	—
38	C₂⁴×C₅	—	2	8	no	—
39	C₂⁴×C₅	C₂³	1	4	yes	—
40	C₂⁴×C₅	C₂³	1	1	yes	characteristic
41	C₂⁴×C₅	C₂³	1	2	yes	—
42	C₂³×D₅	C₂³	1	2	yes	—
43	C₂²×C₄×C₅	—	2	4	no	—
44	C₂²×C₄×C₅	—	2	8	no	—
45	C₂²×C₄×C₅	—	2	8	no	—
46	C₂²×C₄×C₅	—	2	32	no	—
47	C₂²×C₄×C₅	D₄	1	8	yes	complemented
48	C₂²×C₄×C₅	C₂³	1	2	yes	—
49	C₂²×C₄×C₅	C₂³	1	8	yes	—
50	C₂²×C₄×C₅	C₂³	1	8	yes	—
51	C₂²×C₄×C₅	C₂³	1	4	yes	—
52	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂³	1	4	yes	—
54	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₅	—	2	8	no	—
56	C₂×C₂²⋊D₅	—	2	16	no	—
57	C₂×C₂²⋊D₅	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₅	C₂³	1	16	yes	—
59	C₂²×Dic₅	C₂³	1	2	yes	—
60	C₂²×Dic₅	C₂³	1	4	yes	—
61	#709	—	2	32	no	—
62	C₂×D₂₀	—	2	4	no	—
63	C₂×D₂₀	C₂³	1	4	yes	complemented
64	C₂×C₄×D₅	—	2	8	no	—
65	C₂×C₄×D₅	C₂³	1	8	yes	complemented
66	C₂×Dic₁₀	—	2	4	no	—
67	C₂×Dic₁₀	C₂³	1	4	yes	complemented
68	C₂²⋊Dic₅	C₂³	1	16	yes	—
69	C₂²⋊Dic₅	C₂³	1	8	yes	—
70	#685	C₂³	1	16	yes	complemented
71	C₄⋊Dic₅	C₂³	1	8	yes	complemented
72	#683	C₂³	1	16	yes	complemented
73	C₄×Dic₅	C₂³	1	8	yes	complemented
74	C₂²×#41	—	5	1	no	—
75	C₂⁴×C₄	D₅	1	1	yes	characteristic, complemented
76	#525	—	5	16	no	—
77	C₂×#123	—	5	2	no	—
78	C₂×C₂²⋊Q₈	—	5	2	no	—
79	C₂×#121	—	5	2	no	—
80	C₂×#120	—	5	2	no	—
81	C₂×C₄×D₄	—	5	4	no	—
82	C₂×#117	—	5	1	no	—
83	C₂³×C₅	—	2	32	no	—
84	C₂³×C₅	—	2	8	no	—
85	C₂³×C₅	—	2	4	no	—
86	C₂³×C₅	—	2	4	no	—
87	C₂³×C₅	—	2	4	no	—
88	C₂³×C₅	—	2	16	no	—
89	C₂³×C₅	C₂×D₄	1	4	yes	—
90	C₂³×C₅	#41	1	8	yes	complemented
91	C₂³×C₅	C₂⁴	1	2	yes	—
92	C₂³×C₅	C₂⁴	1	4	yes	—
93	C₂³×C₅	C₂⁴	1	1	yes	characteristic
94	C₂²×D₅	—	2	4	no	—
95	C₂²×D₅	—	2	8	no	—
96	C₂²×D₅	C₂⁴	1	4	yes	—
97	C₂×C₄×C₅	—	2	8	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	16	no	—
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
102	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₅	C₂×D₄	1	16	yes	complemented
104	C₂×C₄×C₅	C₂⁴	1	4	yes	—
105	C₂×C₄×C₅	C₂⁴	1	2	yes	—
106	C₂×C₄×C₅	C₂⁴	1	2	yes	—
107	C₂²⋊D₅	—	2	16	no	—
108	C₂²⋊D₅	—	2	32	no	—
109	C₂×Dic₅	—	2	4	no	—
110	C₂×Dic₅	—	2	8	no	—
111	C₂×Dic₅	C₂⁴	1	8	yes	—
112	C₂×Dic₅	C₂⁴	1	4	yes	—
113	D₂₀	—	2	8	no	—
114	C₄×D₅	—	2	16	no	—
115	Dic₁₀	—	2	8	no	—
116	C₂⁵	C₂×D₅	1	1	yes	characteristic
117	C₂×#41	—	5	8	no	—
118	C₂×#41	—	10	8	no	—
119	C₂²×Q₈	—	5	1	no	—
120	C₂²×D₄	—	5	4	no	—
121	C₂²×D₄	—	5	2	no	—
122	C₂²×D₄	—	5	1	no	—
123	C₂³×C₄	—	2	8	no	—
124	C₂³×C₄	—	5	2	no	—
125	C₂³×C₄	C₂×D₅	1	2	yes	—
126	C₂³×C₄	C₂×D₅	1	1	yes	characteristic
127	C₂³×C₄	C₂×D₅	1	2	yes	—
128	C₂³×C₄	C₂×D₅	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₅	1	8	yes	complemented
130	#123	—	5	16	no	—
131	C₂²⋊Q₈	—	5	16	no	—
132	#121	—	5	16	no	—
133	#120	—	5	16	no	—
134	C₄×D₄	—	5	32	no	—
135	#117	—	5	8	no	—
136	C₂×C₄⋊C₄	—	5	2	no	—
137	C₂×C₄⋊C₄	—	5	4	no	—
138	C₂×C₂²⋊C₄	—	5	2	no	—
139	C₂×C₂²⋊C₄	—	5	4	no	—
140	C₂×C₂²⋊C₄	—	5	4	no	—
141	C₂×C₄²	—	5	2	no	—
142	C₂²×C₅	—	2	32	no	—
143	C₂²×C₅	—	2	8	no	—
144	C₂²×C₅	—	2	16	no	—
145	C₂²×C₅	—	2	4	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	C₂²×D₄	1	4	yes	—
149	C₂²×C₅	C₂²×D₄	1	2	yes	—
150	C₂²×C₅	C₂×#41	1	4	yes	—
151	C₂²×C₅	C₂×#41	1	8	yes	complemented
152	C₂²×C₅	C₂⁵	1	1	yes	characteristic
153	C₂×D₅	—	2	4	no	—
154	C₂×D₅	—	2	8	no	—
155	C₂×D₅	—	4	8	no	—
156	C₄×C₅	—	2	4	no	—
157	C₄×C₅	C₂²×D₄	1	4	yes	—
158	C₄×C₅	C₂²×D₄	1	4	yes	—
159	Dic₅	—	2	4	no	—
160	Dic₅	—	2	8	no	—
161	C₂⁴	—	2	8	no	—
162	C₂⁴	—	2	4	no	—
163	C₂⁴	—	5	2	no	—
164	C₂⁴	C₂²×D₅	1	4	yes	—
165	C₂⁴	C₂²×D₅	1	2	yes	—
166	C₂⁴	C₂²×D₅	1	1	yes	characteristic
167	#41	—	10	32	no	—
168	C₂×Q₈	—	5	4	no	—
169	C₂×Q₈	—	10	4	no	—
170	C₂×D₄	—	5	16	no	—
171	C₂×D₄	—	5	8	no	—
172	C₂×D₄	—	5	4	no	—
173	C₂×D₄	—	10	16	no	—
174	C₂×D₄	—	10	8	no	—
175	C₂×D₄	—	10	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	32	no	—
180	C₂²×C₄	—	5	4	no	—
181	C₂²×C₄	—	5	2	no	—
182	C₂²×C₄	—	5	8	no	—
183	C₂²×C₄	—	10	8	no	—
184	C₂²×C₄	C₂²⋊D₅	1	8	yes	complemented
185	C₂²×C₄	C₂²×D₅	1	2	yes	—
186	C₂²×C₄	C₂²×D₅	1	8	yes	—
187	C₂²×C₄	C₂²×D₅	1	8	yes	—
188	C₂²×C₄	C₂²×D₅	1	4	yes	—
189	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₅	1	4	yes	—
191	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
192	C₄⋊C₄	—	5	8	no	—
193	C₄⋊C₄	—	5	16	no	—
194	C₂²⋊C₄	—	5	8	no	—
195	C₂²⋊C₄	—	5	16	no	—
196	C₂²⋊C₄	—	5	16	no	—
197	C₄²	—	5	8	no	—
198	C₂×C₅	—	2	4	no	—
199	C₂×C₅	—	2	8	no	—
200	C₂×C₅	#525	1	4	yes	complemented
201	C₂×C₅	C₂³×D₄	1	1	yes	characteristic
202	C₂×C₅	C₂²×#41	1	2	yes	—
203	D₅	—	4	4	no	—
204	C₂³	—	2	32	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	5	4	no	—
211	C₂³	—	10	8	no	—
212	C₂³	—	10	4	no	—
213	C₂³	#709	1	8	yes	complemented
214	C₂³	C₂×C₂²⋊D₅	1	4	yes	—
215	C₂³	C₂³×D₅	1	4	yes	—
216	C₂³	C₂³×D₅	1	2	yes	—
217	C₂³	C₂³×D₅	1	1	yes	characteristic
218	Q₈	—	10	8	no	—
219	D₄	—	10	32	no	—
220	D₄	—	10	16	no	—
221	D₄	—	10	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	16	no	—
226	C₂×C₄	—	5	8	no	—
227	C₂×C₄	—	5	4	no	—
228	C₂×C₄	—	10	8	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	—	10	16	no	—
231	C₂×C₄	C₂×C₂²⋊D₅	1	4	yes	—
232	C₂×C₄	C₂×C₂²⋊D₅	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₅	1	16	yes	complemented
234	C₂×C₄	C₂³×D₅	1	4	yes	—
235	C₂×C₄	C₂³×D₅	1	2	yes	—
236	C₂×C₄	C₂³×D₅	1	2	yes	—
237	C₅	C₂×#525	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	32	no	—
239	C₂²	—	2	16	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	10	8	no	—
245	C₂²	—	10	4	no	—
246	C₂²	—	20	8	no	—
247	C₂²	C₂×#709	1	4	yes	—
248	C₂²	C₂×#709	1	8	yes	complemented
249	C₂²	C₂²×C₂²⋊D₅	1	4	yes	—
250	C₂²	C₂²×C₂²⋊D₅	1	2	yes	—
251	C₂²	C₂⁴×D₅	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	10	8	no	—
254	C₄	—	10	4	no	—
255	C₄	C₂²×C₂²⋊D₅	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₅	1	4	yes	—
257	C₂	—	2	8	no	—
258	C₂	—	2	4	no	—
259	C₂	—	20	4	no	—
260	C₂	#66125	1	4	yes	direct product factor
261	C₂	C₂²×#709	1	2	yes	—
262	C₂	C₂³×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₅
Fitting	C₂⁴×C₄×C₅	C₂
socle	C₂³×C₅	C₂⁴
hypercenter	C₂⁴×C₄	D₅
hypocenter	C₅	C₂×#525
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₄×C₅	C₂	semidirect
2. C₂²×#709	C₂	semidirect
3. C₂×#4099	C₂	semidirect
4. #66125	C₂	direct (N × H)
5. C₂×#4076	C₂	semidirect
6. C₂×#4075	C₂	semidirect
7. C₂×C₄×C₂²⋊D₅	C₂	semidirect
8. C₂×#4072	C₂	semidirect
9. C₂×#4070	C₂	semidirect
16. C₂³×C₄×C₅	C₂²	semidirect
20. C₂×#709	C₂²	semidirect
21. C₂²×D₂₀	C₂²	semidirect
22. C₂²×C₄×D₅	C₂²	semidirect
23. C₂²×Dic₁₀	C₂²	semidirect
24. #4099	C₂²	semidirect
27. #4076	C₂²	semidirect
28. #4075	C₂²	semidirect
29. C₄×C₂²⋊D₅	C₂²	semidirect
30. C₂×#685	C₂²	semidirect
31. #4072	C₂²	semidirect
32. C₂×C₄⋊Dic₅	C₂²	semidirect
33. #4070	C₂²	semidirect
34. C₂×#683	C₂²	semidirect
35. C₂×C₄×Dic₅	C₂²	semidirect
47. C₂²×C₄×C₅	D₄	semidirect
63. C₂×D₂₀	C₂³	semidirect
65. C₂×C₄×D₅	C₂³	semidirect
67. C₂×Dic₁₀	C₂³	semidirect
70. #685	C₂³	semidirect
71. C₄⋊Dic₅	C₂³	semidirect
72. #683	C₂³	semidirect
73. C₄×Dic₅	C₂³	semidirect
75. C₂⁴×C₄	D₅	semidirect
90. C₂³×C₅	#41	semidirect
103. C₂×C₄×C₅	C₂×D₄	semidirect
129. C₂³×C₄	C₂×D₅	semidirect
151. C₂²×C₅	C₂×#41	semidirect
184. C₂²×C₄	C₂²⋊D₅	semidirect
200. C₂×C₅	#525	semidirect
213. C₂³	#709	semidirect
233. C₂×C₄	C₂×C₂²⋊D₅	semidirect
237. C₅	C₂×#525	semidirect
248. C₂²	C₂×#709	semidirect
260. C₂	#66125	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₅	2
C₂²×#709	2
C₂×#4099	2
#66125	2
C₂×#4076	2
C₂×#4075	2
C₂×C₄×C₂²⋊D₅	2
C₂×#4072	2
C₂×#4070	2
C₂×#525	5

Subgroups — C2×#66125 #10619124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66125 #10619124