Subgroups — C₂×#10618472 #13006141

Order 1280.

7022 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#543	C₂	1	1	yes	maximal, characteristic
2	#10618472	C₂	1	32	yes	maximal, direct product factor
3	C₂×#65957	C₂	1	12	yes	maximal, complemented
4	C₂×#65956	C₂	1	12	yes	maximal, complemented
5	C₂×#65817	C₂	1	3	yes	maximal
6	C₂²×#4023	C₂	1	3	yes	maximal
7	C₂⁴×Dic₅	C₂²	1	1	yes	characteristic
8	C₅×#543	C₂²	1	16	yes	—
9	C₂×C₅×C₂²⋊Q₈	C₂²	1	12	yes	—
10	C₂²×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
11	C₂²×C₂²⋊Dic₅	C₂²	1	3	yes	—
12	C₂×Q₈×Dic₅	C₂²	1	4	yes	complemented
13	C₂×#4090	C₂²	1	12	yes	complemented
14	C₂×#4070	C₂²	1	12	yes	—
15	C₂²×#683	C₂²	1	6	yes	—
16	C₂²×C₄×Dic₅	C₂²	1	6	yes	—
17	#65957	C₂²	1	192	yes	complemented
18	#65956	C₂²	1	192	yes	complemented
19	C₂×#4035	C₂²	1	24	yes	complemented
20	C₂×#4034	C₂²	1	12	yes	complemented
21	C₂×#4033	C₂²	1	12	yes	complemented
22	#65817	C₂²	1	48	yes	—
23	C₂×#4024	C₂²	1	24	yes	—
24	C₂×#4023	C₂²	1	12	yes	—
25	C₂×#4023	C₂²	1	12	yes	—
26	C₂×#4023	C₂²	1	48	yes	—
27	C₂×#3458	—	5	1	no	maximal, Sylow
28	C₂⁵×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₅×Q₈	C₂³	1	4	yes	—
30	C₂³×C₄×C₅	C₂³	1	6	yes	—
31	C₂³×Dic₅	Q₈	1	16	yes	complemented
32	C₂³×Dic₅	C₂³	1	6	yes	—
33	C₂³×Dic₅	C₂³	1	3	yes	—
34	C₂³×Dic₅	C₂³	1	3	yes	—
35	C₂³×Dic₅	C₂³	1	8	yes	—
36	C₂²×Dic₁₀	C₂³	1	12	yes	—
37	C₅×C₂²⋊Q₈	C₂³	1	96	yes	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂³	1	12	yes	—
40	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	—
41	C₂×C₂²⋊Dic₅	C₂³	1	12	yes	—
42	C₂×C₂²⋊Dic₅	C₂³	1	24	yes	—
43	Q₈×Dic₅	C₂³	1	32	yes	complemented
44	#4090	C₂³	1	96	yes	complemented
45	C₂×C₄⋊Dic₅	C₂³	1	12	yes	—
46	#4070	C₂³	1	96	yes	—
47	C₂×#683	C₂³	1	24	yes	—
48	C₂×#683	C₂³	1	48	yes	—
49	C₂×#683	C₂³	1	24	yes	—
50	C₂×C₄×Dic₅	C₂³	1	12	yes	—
51	C₂×C₄×Dic₅	C₂³	1	12	yes	—
52	C₂×C₄×Dic₅	C₂³	1	48	yes	—
53	#4035	C₂³	1	192	yes	complemented
54	#4034	C₂³	1	96	yes	complemented
55	#4033	C₂³	1	96	yes	complemented
56	#4024	C₂³	1	192	yes	—
57	#4023	C₂³	1	96	yes	—
58	#4023	C₂³	1	96	yes	—
59	#3458	—	5	32	no	—
60	C₂×#544	—	5	12	no	—
61	C₂×#543	—	5	3	no	—
62	C₂×#543	D₅	1	1	yes	characteristic
63	C₂×#533	—	5	12	no	—
64	C₂²×#117	—	5	3	no	—
65	C₂⁴×C₅	—	4	4	no	—
66	C₂⁴×C₅	C₂×Q₈	1	8	yes	—
67	C₂⁴×C₅	C₂⁴	1	3	yes	—
68	C₂⁴×C₅	C₂⁴	1	4	yes	—
69	C₂×C₅×Q₈	—	4	8	no	—
70	C₂×C₅×Q₈	C₂⁴	1	16	yes	—
71	C₂²×C₄×C₅	—	4	12	no	—
72	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
73	C₂²×C₄×C₅	C₂⁴	1	24	yes	—
74	C₂²×Dic₅	—	4	12	no	—
75	C₂²×Dic₅	C₂×Q₈	1	24	yes	—
76	C₂²×Dic₅	C₂×Q₈	1	24	yes	—
77	C₂²×Dic₅	C₂×Q₈	1	64	yes	complemented
78	C₂²×Dic₅	C₂⁴	1	12	yes	—
79	C₂²×Dic₅	C₂⁴	1	24	yes	—
80	C₂²×Dic₅	C₂⁴	1	12	yes	—
81	C₂²×Dic₅	C₂⁴	1	12	yes	—
82	C₂²×Dic₅	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₅	C₂⁴	1	3	yes	—
84	C₂×Dic₁₀	—	4	24	no	—
85	C₂×Dic₁₀	C₂⁴	1	48	yes	—
86	C₅×C₄⋊C₄	C₂⁴	1	48	yes	—
87	C₅×C₂²⋊C₄	C₂⁴	1	48	yes	—
88	C₂²⋊Dic₅	C₂⁴	1	48	yes	—
89	C₄⋊Dic₅	C₂⁴	1	48	yes	—
90	#683	C₂⁴	1	96	yes	—
91	#683	C₂⁴	1	96	yes	—
92	C₄×Dic₅	C₂⁴	1	48	yes	—
93	C₄×Dic₅	C₂⁴	1	48	yes	—
94	C₂⁴×C₄	—	5	1	no	—
95	#544	—	5	192	no	—
96	#543	—	5	48	no	—
97	#543	C₂×D₅	1	16	yes	—
98	#533	—	5	192	no	—
99	C₂×C₄⋊Q₈	—	5	12	no	—
100	C₂×C₄⋊Q₈	—	5	12	no	—
101	C₂×#125	—	5	24	no	—
102	C₂×C₂²⋊Q₈	—	5	12	no	—
103	C₂×C₂²⋊Q₈	—	5	24	no	—
104	C₂×C₂²⋊Q₈	C₂×D₅	1	12	yes	—
105	C₂×C₄×Q₈	—	5	4	no	—
106	C₂×C₄×Q₈	—	5	12	no	—
107	C₂×#117	—	5	12	no	—
108	C₂×#117	—	5	12	no	—
109	C₂×#117	—	5	48	no	—
110	C₂²×C₄⋊C₄	—	5	6	no	—
111	C₂²×C₂²⋊C₄	—	5	3	no	—
112	C₂²×C₂²⋊C₄	C₂×D₅	1	3	yes	—
113	C₂²×C₄²	—	5	6	no	—
114	C₂³×C₅	—	2	12	no	—
115	C₂³×C₅	—	4	16	no	—
116	C₂³×C₅	—	4	8	no	—
117	C₂³×C₅	C₂²×Q₈	1	12	yes	—
118	C₂³×C₅	C₂²×Q₈	1	16	yes	—
119	C₂³×C₅	C₂⁵	1	6	yes	—
120	C₂³×C₅	C₂⁵	1	1	yes	characteristic
121	C₅×Q₈	—	4	16	no	—
122	C₂×C₄×C₅	—	2	24	no	—
123	C₂×C₄×C₅	—	4	24	no	—
124	C₂×C₄×C₅	C₂⁵	1	24	yes	—
125	C₂×Dic₅	—	2	24	no	—
126	C₂×Dic₅	—	4	24	no	—
127	C₂×Dic₅	C₂²×Q₈	1	4	yes	—
128	C₂×Dic₅	C₂²×Q₈	1	12	yes	—
129	C₂×Dic₅	C₂²×Q₈	1	48	yes	—
130	C₂×Dic₅	C₂²×Q₈	1	48	yes	—
131	C₂×Dic₅	C₂⁵	1	24	yes	—
132	C₂×Dic₅	C₂⁵	1	2	yes	—
133	C₂×Dic₅	C₂⁵	1	6	yes	—
134	Dic₁₀	—	4	48	no	—
135	C₂⁵	C₂²×D₅	1	1	yes	characteristic
136	C₂²×Q₈	—	5	12	no	—
137	C₂²×Q₈	C₂²×D₅	1	4	yes	—
138	C₂³×C₄	—	5	16	no	—
139	C₂³×C₄	—	5	3	no	—
140	C₂³×C₄	—	5	8	no	—
141	C₂³×C₄	—	5	3	no	—
142	C₂³×C₄	—	5	6	no	—
143	C₂³×C₄	C₂²×D₅	1	6	yes	—
144	C₄⋊Q₈	—	5	96	no	—
145	C₄⋊Q₈	—	5	96	no	—
146	#125	—	5	192	no	—
147	C₂²⋊Q₈	—	5	96	no	—
148	C₂²⋊Q₈	—	5	192	no	—
149	C₂²⋊Q₈	C₂²×D₅	1	96	yes	—
150	C₄×Q₈	—	5	32	no	—
151	C₄×Q₈	—	5	96	no	—
152	#117	—	5	96	no	—
153	#117	—	5	96	no	—
154	C₂×C₄⋊C₄	—	5	24	no	—
155	C₂×C₄⋊C₄	—	5	48	no	—
156	C₂×C₄⋊C₄	—	5	24	no	—
157	C₂×C₄⋊C₄	—	5	12	no	—
158	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	—
159	C₂×C₂²⋊C₄	—	5	12	no	—
160	C₂×C₂²⋊C₄	—	5	24	no	—
161	C₂×C₂²⋊C₄	C₂²×D₅	1	12	yes	—
162	C₂×C₂²⋊C₄	C₂²×D₅	1	24	yes	—
163	C₂×C₄²	—	5	12	no	—
164	C₂×C₄²	—	5	12	no	—
165	C₂×C₄²	—	5	48	no	—
166	C₂²×C₅	—	2	12	no	—
167	C₂²×C₅	—	2	24	no	—
168	C₂²×C₅	—	4	16	no	—
169	C₂²×C₅	C₂³×Q₈	1	12	yes	—
170	C₂²×C₅	C₂³×Q₈	1	2	yes	—
171	C₂²×C₅	#585	1	4	yes	—
172	C₂²×C₅	C₂⁶	1	1	yes	characteristic
173	C₄×C₅	—	2	24	no	—
174	Dic₅	—	2	24	no	—
175	Dic₅	C₂³×Q₈	1	4	yes	—
176	Dic₅	C₂³×Q₈	1	12	yes	—
177	C₂⁴	—	4	4	no	—
178	C₂⁴	Q₈×D₅	1	8	yes	—
179	C₂⁴	C₂³×D₅	1	3	yes	—
180	C₂⁴	C₂³×D₅	1	4	yes	—
181	C₂×Q₈	—	4	8	no	—
182	C₂×Q₈	—	5	48	no	—
183	C₂×Q₈	—	20	24	no	—
184	C₂×Q₈	C₂³×D₅	1	16	yes	—
185	C₂²×C₄	—	4	12	no	—
186	C₂²×C₄	—	5	24	no	—
187	C₂²×C₄	—	5	24	no	—
188	C₂²×C₄	—	5	64	no	—
189	C₂²×C₄	—	5	12	no	—
190	C₂²×C₄	—	5	12	no	—
191	C₂²×C₄	—	5	1	no	—
192	C₂²×C₄	—	5	3	no	—
193	C₂²×C₄	—	5	12	no	—
194	C₂²×C₄	—	5	24	no	—
195	C₂²×C₄	—	20	12	no	—
196	C₂²×C₄	C₂³×D₅	1	12	yes	—
197	C₂²×C₄	C₂³×D₅	1	24	yes	—
198	C₄⋊C₄	—	5	96	no	—
199	C₄⋊C₄	—	5	96	no	—
200	C₄⋊C₄	—	5	48	no	—
201	C₄⋊C₄	C₂³×D₅	1	48	yes	—
202	C₂²⋊C₄	—	5	48	no	—
203	C₂²⋊C₄	C₂³×D₅	1	48	yes	—
204	C₄²	—	5	48	no	—
205	C₄²	—	5	48	no	—
206	C₂×C₅	—	2	12	no	—
207	C₂×C₅	#3458	1	4	yes	complemented
208	C₂×C₅	C₂⁴×Q₈	1	1	yes	characteristic
209	C₂×C₅	C₂×#585	1	2	yes	—
210	C₂³	—	2	12	no	—
211	C₂³	—	4	16	no	—
212	C₂³	—	4	8	no	—
213	C₂³	C₂×Q₈×D₅	1	12	yes	—
214	C₂³	C₂×Q₈×D₅	1	16	yes	—
215	C₂³	C₂⁴×D₅	1	6	yes	—
216	C₂³	C₂⁴×D₅	1	1	yes	characteristic
217	Q₈	—	4	16	no	—
218	Q₈	—	20	48	no	—
219	C₂×C₄	—	2	24	no	—
220	C₂×C₄	—	4	24	no	—
221	C₂×C₄	—	5	48	no	—
222	C₂×C₄	—	5	48	no	—
223	C₂×C₄	—	5	4	no	—
224	C₂×C₄	—	5	12	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	5	6	no	—
227	C₂×C₄	—	5	24	no	—
228	C₂×C₄	—	10	24	no	—
229	C₂×C₄	—	20	24	no	—
230	C₂×C₄	C₂⁴×D₅	1	24	yes	—
231	C₅	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
232	C₂²	—	2	12	no	—
233	C₂²	—	2	24	no	—
234	C₂²	—	4	16	no	—
235	C₂²	C₂²×Q₈×D₅	1	12	yes	—
236	C₂²	C₂²×Q₈×D₅	1	2	yes	—
237	C₂²	#66285	1	4	yes	—
238	C₂²	C₂⁵×D₅	1	1	yes	characteristic
239	C₄	—	2	24	no	—
240	C₄	—	5	4	no	—
241	C₄	—	5	12	no	—
242	C₄	—	10	24	no	—
243	C₂	—	2	12	no	—
244	C₂	#10618472	1	4	yes	direct product factor
245	C₂	C₂³×Q₈×D₅	1	1	yes	characteristic
246	C₂	C₂×#66285	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂×C₅×#543	C₂
socle	C₂³×C₅	C₂⁵
prime radical	C₂⁵×C₅	C₂³
hypercenter	C₂×#543	D₅
hypocenter	C₅	C₂×#3458
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
2. #10618472	C₂	direct (N × H)
3. C₂×#65957	C₂	semidirect
4. C₂×#65956	C₂	semidirect
12. C₂×Q₈×Dic₅	C₂²	semidirect
13. C₂×#4090	C₂²	semidirect
17. #65957	C₂²	semidirect
18. #65956	C₂²	semidirect
19. C₂×#4035	C₂²	semidirect
20. C₂×#4034	C₂²	semidirect
21. C₂×#4033	C₂²	semidirect
31. C₂³×Dic₅	Q₈	semidirect
43. Q₈×Dic₅	C₂³	semidirect
44. #4090	C₂³	semidirect
53. #4035	C₂³	semidirect
54. #4034	C₂³	semidirect
55. #4033	C₂³	semidirect
77. C₂²×Dic₅	C₂×Q₈	semidirect
207. C₂×C₅	#3458	semidirect
231. C₅	C₂×#3458	semidirect
244. C₂	#10618472	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×C₅×#543	2
#10618472	2
C₂×#65957	2
C₂×#65956	2
C₂×#65817	2
C₂²×#4023	2
C₂×#3458	5

Subgroups — C2×#10618472 #13006141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10618472 #13006141