Subgroups — C₂²×#65482 #13005791

Order 1280.

2334 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#65482	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#3960	C₂	1	2	yes	maximal, complemented
5	C₂²×#3958	C₂	1	2	yes	maximal, complemented
6	C₂²×#3944	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₅×#41	—	2	1	no	—
8	C₂³×D₂₀	—	2	1	no	—
9	C₂×C₅×#135	—	2	6	no	—
10	C₂×C₅×#135	C₂²	1	12	yes	complemented
11	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₅×SD₁₆	—	2	1	no	—
13	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
14	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂²×D₄₀	—	2	1	no	—
16	C₂²×D₄₀	—	2	6	no	—
17	C₂²×D₄₀	C₂²	1	2	yes	—
18	C₂²×D₄₀	C₂²	1	12	yes	complemented
19	#65482	C₂²	1	64	yes	direct product factor
20	C₂×#3960	C₂²	1	24	yes	complemented
21	C₂×#3958	C₂²	1	24	yes	complemented
22	C₂×#3944	C₂²	1	12	yes	complemented
23	C₂²×#672	C₂²	1	2	yes	complemented
24	C₂²×#509	—	5	1	no	maximal, Sylow
25	C₂⁴×D₅	—	4	1	no	—
26	C₂×C₅×#41	—	2	12	no	—
27	C₂×C₅×#41	—	4	6	no	—
28	C₂²×C₅×Q₈	—	2	1	no	—
29	C₂²×C₅×D₄	—	2	1	no	—
30	C₂²×C₅×D₄	—	4	1	no	—
31	C₂³×C₄×C₅	—	4	1	no	—
32	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂²×D₂₀	—	2	2	no	—
34	C₂²×D₂₀	—	2	12	no	—
35	C₂²×D₂₀	—	4	1	no	—
36	C₂²×D₂₀	—	4	6	no	—
37	C₅×#135	—	2	24	no	—
38	C₅×#135	C₂³	1	16	yes	complemented
39	C₂×C₅×Q₁₆	—	2	3	no	—
40	C₂×C₅×Q₁₆	C₂³	1	6	yes	complemented
41	C₂×C₅×SD₁₆	—	2	12	no	—
42	C₂×C₅×D₈	—	2	3	no	—
43	C₂×C₅×D₈	C₂³	1	6	yes	complemented
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	D₄	1	6	yes	—
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂³	1	6	yes	—
48	C₂×D₄₀	—	2	6	no	—
49	C₂×D₄₀	—	2	24	no	—
50	C₂×D₄₀	—	2	12	no	—
51	C₂×D₄₀	C₂³	1	12	yes	—
52	C₂×D₄₀	C₂³	1	16	yes	complemented
53	#3960	C₂³	1	32	yes	complemented
54	#3958	C₂³	1	32	yes	complemented
55	#3944	C₂³	1	16	yes	complemented
56	C₂×#672	C₂³	1	12	yes	complemented
57	C₂²×#135	D₅	1	1	yes	characteristic, complemented
58	C₂³×D₈	—	5	1	no	—
59	C₂×#509	—	5	24	no	—
60	C₂²×SD₃₂	—	5	2	no	—
61	C₂²×D₁₆	—	5	2	no	—
62	C₂²×M₃₂	—	5	1	no	—
63	C₂⁴×C₅	—	4	1	no	—
64	C₂⁴×C₅	D₈	1	1	yes	characteristic
65	C₂³×D₅	—	4	12	no	—
66	C₂³×D₅	—	4	2	no	—
67	C₂³×D₅	—	8	8	no	—
68	C₅×#41	—	2	16	no	—
69	C₅×#41	—	4	24	no	—
70	C₂×C₅×Q₈	—	2	6	no	—
71	C₂×C₅×Q₈	—	4	3	no	—
72	C₂×C₅×D₄	—	2	6	no	—
73	C₂×C₅×D₄	—	4	12	no	—
74	C₂×C₅×D₄	—	4	3	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	—	4	12	no	—
77	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	D₈	1	6	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
81	C₂×D₂₀	—	2	12	no	—
82	C₂×D₂₀	—	2	16	no	—
83	C₂×D₂₀	—	4	12	no	—
84	C₂×D₂₀	—	4	6	no	—
85	C₂×D₂₀	—	4	24	no	—
86	C₅×Q₁₆	—	2	6	no	—
87	C₅×Q₁₆	C₂⁴	1	4	yes	complemented
88	C₅×SD₁₆	—	2	16	no	—
89	C₅×D₈	—	2	6	no	—
90	C₅×D₈	C₂⁴	1	4	yes	complemented
91	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
92	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	12	yes	—
94	C₂×C₅×C₈	C₂⁴	1	3	yes	—
95	C₂×C₅×C₈	C₂⁴	1	4	yes	—
96	D₄₀	—	2	12	no	—
97	D₄₀	—	2	16	no	—
98	D₄₀	C₂⁴	1	8	yes	—
99	#672	C₂⁴	1	8	yes	complemented
100	C₂²×#41	—	2	1	no	—
101	C₂³×D₄	—	10	1	no	—
102	C₂×#135	—	2	6	no	—
103	C₂×#135	C₂×D₅	1	12	yes	complemented
104	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
105	C₂²×SD₁₆	—	2	1	no	—
106	C₂²×D₈	—	5	2	no	—
107	C₂²×D₈	—	5	12	no	—
108	C₂²×D₈	—	10	1	no	—
109	C₂²×D₈	—	10	6	no	—
110	C₂²×D₈	C₂×D₅	1	1	yes	characteristic, complemented
111	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
112	#509	—	5	64	no	—
113	C₂×SD₃₂	—	5	24	no	—
114	C₂×D₁₆	—	5	24	no	—
115	C₂×M₃₂	—	5	12	no	—
116	C₂²×C₁₆	—	5	2	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	—	4	6	no	—
119	C₂³×C₅	—	8	4	no	—
120	C₂³×C₅	C₂×D₈	1	6	yes	—
121	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
122	C₂²×D₅	—	4	16	no	—
123	C₂²×D₅	—	4	12	no	—
124	C₂²×D₅	—	8	48	no	—
125	C₂²×D₅	—	8	8	no	—
126	C₅×Q₈	—	2	4	no	—
127	C₅×Q₈	—	4	6	no	—
128	C₅×D₄	—	2	4	no	—
129	C₅×D₄	—	4	16	no	—
130	C₅×D₄	—	4	6	no	—
131	C₂×C₄×C₅	—	4	6	no	—
132	C₂×C₄×C₅	—	4	16	no	—
133	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
134	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
135	C₂×C₄×C₅	C₂×D₈	1	12	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
137	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
138	D₂₀	—	2	8	no	—
139	D₂₀	—	4	16	no	—
140	D₂₀	—	4	12	no	—
141	C₅×C₈	C₂²×D₄	1	3	yes	—
142	C₅×C₈	C₂²×D₄	1	4	yes	—
143	C₅×C₈	C₂⁵	1	1	yes	characteristic
144	C₂⁵	—	20	1	no	—
145	C₂×#41	—	2	12	no	—
146	C₂×#41	—	4	6	no	—
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×D₄	—	2	1	no	—
149	C₂²×D₄	—	4	1	no	—
150	C₂²×D₄	—	10	12	no	—
151	C₂²×D₄	—	10	2	no	—
152	C₂²×D₄	—	20	6	no	—
153	C₂²×D₄	—	20	1	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
156	#135	—	2	24	no	—
157	#135	C₂²×D₅	1	16	yes	complemented
158	C₂×Q₁₆	—	2	3	no	—
159	C₂×Q₁₆	C₂²×D₅	1	6	yes	complemented
160	C₂×SD₁₆	—	2	12	no	—
161	C₂×D₈	—	2	3	no	—
162	C₂×D₈	—	5	12	no	—
163	C₂×D₈	—	5	16	no	—
164	C₂×D₈	—	10	6	no	—
165	C₂×D₈	—	10	12	no	—
166	C₂×D₈	—	10	24	no	—
167	C₂×D₈	C₂²×D₅	1	6	yes	complemented
168	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
169	C₂²×C₈	C₂²⋊D₅	1	6	yes	—
170	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅	1	6	yes	—
172	SD₃₂	—	5	32	no	—
173	D₁₆	—	5	32	no	—
174	M₃₂	—	5	16	no	—
175	C₂×C₁₆	—	5	12	no	—
176	C₂²×C₅	—	2	12	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	8	12	no	—
179	C₂²×C₅	#509	1	4	yes	complemented
180	C₂²×C₅	C₂²×D₈	1	4	yes	—
181	C₂²×C₅	C₂²×D₈	1	3	yes	—
182	C₂×D₅	—	4	8	no	—
183	C₂×D₅	—	8	32	no	—
184	C₂×D₅	—	8	24	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	C₂²×D₈	1	4	yes	—
187	C₄×C₅	C₂²×D₈	1	3	yes	—
188	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
189	C₂⁴	—	4	1	no	—
190	C₂⁴	—	20	12	no	—
191	C₂⁴	—	20	2	no	—
192	C₂⁴	—	40	8	no	—
193	C₂⁴	#686	1	1	yes	characteristic
194	#41	—	2	16	no	—
195	#41	—	4	24	no	—
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	4	3	no	—
198	C₂×D₄	—	2	6	no	—
199	C₂×D₄	—	4	12	no	—
200	C₂×D₄	—	4	3	no	—
201	C₂×D₄	—	10	12	no	—
202	C₂×D₄	—	10	16	no	—
203	C₂×D₄	—	20	12	no	—
204	C₂×D₄	—	20	6	no	—
205	C₂×D₄	—	20	24	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	12	no	—
208	C₂²×C₄	#686	1	6	yes	—
209	C₂²×C₄	#686	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₅	1	6	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
212	Q₁₆	—	2	6	no	—
213	Q₁₆	C₂³×D₅	1	4	yes	complemented
214	SD₁₆	—	2	16	no	—
215	D₈	—	2	6	no	—
216	D₈	—	5	8	no	—
217	D₈	—	10	12	no	—
218	D₈	—	10	16	no	—
219	D₈	C₂³×D₅	1	4	yes	complemented
220	C₂×C₈	C₂×C₂²⋊D₅	1	3	yes	—
221	C₂×C₈	C₂×C₂²⋊D₅	1	6	yes	—
222	C₂×C₈	C₂×C₂²⋊D₅	1	12	yes	—
223	C₂×C₈	C₂³×D₅	1	3	yes	—
224	C₂×C₈	C₂³×D₅	1	4	yes	—
225	C₁₆	—	5	8	no	—
226	C₂×C₅	—	2	4	no	—
227	C₂×C₅	—	8	4	no	—
228	C₂×C₅	C₂×#509	1	6	yes	complemented
229	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
230	D₅	—	8	8	no	—
231	C₂³	—	2	4	no	—
232	C₂³	—	4	6	no	—
233	C₂³	—	8	4	no	—
234	C₂³	—	20	16	no	—
235	C₂³	—	20	12	no	—
236	C₂³	—	40	48	no	—
237	C₂³	—	40	8	no	—
238	C₂³	C₂×#686	1	6	yes	—
239	C₂³	C₂×#686	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	6	no	—
242	D₄	—	2	4	no	—
243	D₄	—	4	16	no	—
244	D₄	—	4	6	no	—
245	D₄	—	10	8	no	—
246	D₄	—	20	16	no	—
247	D₄	—	20	12	no	—
248	C₂×C₄	—	4	6	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	C₂×#686	1	6	yes	—
251	C₂×C₄	C₂×#686	1	3	yes	—
252	C₂×C₄	C₂×#686	1	12	yes	—
253	C₂×C₄	C₂²×C₂²⋊D₅	1	3	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₅	1	4	yes	—
255	C₈	C₂²×C₂²⋊D₅	1	3	yes	—
256	C₈	C₂²×C₂²⋊D₅	1	4	yes	—
257	C₈	C₂⁴×D₅	1	1	yes	characteristic
258	C₅	C₂²×#509	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	2	12	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	8	12	no	—
262	C₂²	—	20	8	no	—
263	C₂²	—	40	32	no	—
264	C₂²	—	40	24	no	—
265	C₂²	#65482	1	4	yes	direct product factor
266	C₂²	C₂²×#686	1	4	yes	—
267	C₂²	C₂²×#686	1	3	yes	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×#686	1	4	yes	—
270	C₄	C₂²×#686	1	3	yes	—
271	C₄	C₂³×C₂²⋊D₅	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	8	4	no	—
274	C₂	—	40	8	no	—
275	C₂	C₂×#65482	1	6	yes	direct product factor
276	C₂	C₂³×#686	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator, cosocle	C₅×C₈	C₂⁵
Frattini	C₈	C₂⁴×D₅
Fitting	C₂²×C₅×#135	C₂
socle	C₂³×C₅	C₂×D₈
hypercenter	C₂²×#135	D₅
hypocenter	C₅	C₂²×#509
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#135	C₂	semidirect
2. C₂³×D₄₀	C₂	semidirect
3. C₂×#65482	C₂	direct (N × H)
4. C₂²×#3960	C₂	semidirect
5. C₂²×#3958	C₂	semidirect
6. C₂²×#3944	C₂	semidirect
10. C₂×C₅×#135	C₂²	semidirect
11. C₂²×C₅×Q₁₆	C₂²	semidirect
13. C₂²×C₅×D₈	C₂²	semidirect
18. C₂²×D₄₀	C₂²	semidirect
19. #65482	C₂²	direct (N × H)
20. C₂×#3960	C₂²	semidirect
21. C₂×#3958	C₂²	semidirect
22. C₂×#3944	C₂²	semidirect
23. C₂²×#672	C₂²	semidirect
38. C₅×#135	C₂³	semidirect
40. C₂×C₅×Q₁₆	C₂³	semidirect
43. C₂×C₅×D₈	C₂³	semidirect
52. C₂×D₄₀	C₂³	semidirect
53. #3960	C₂³	semidirect
54. #3958	C₂³	semidirect
55. #3944	C₂³	semidirect
56. C₂×#672	C₂³	semidirect
57. C₂²×#135	D₅	semidirect
87. C₅×Q₁₆	C₂⁴	semidirect
90. C₅×D₈	C₂⁴	semidirect
99. #672	C₂⁴	semidirect
103. C₂×#135	C₂×D₅	semidirect
104. C₂²×Q₁₆	C₂×D₅	semidirect
110. C₂²×D₈	C₂×D₅	semidirect
157. #135	C₂²×D₅	semidirect
159. C₂×Q₁₆	C₂²×D₅	semidirect
167. C₂×D₈	C₂²×D₅	semidirect
179. C₂²×C₅	#509	semidirect
213. Q₁₆	C₂³×D₅	semidirect
219. D₈	C₂³×D₅	semidirect
228. C₂×C₅	C₂×#509	semidirect
258. C₅	C₂²×#509	semidirect
265. C₂²	#65482	direct (N × H)
275. C₂	C₂×#65482	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#135	2
C₂³×D₄₀	2
C₂×#65482	2
C₂²×#3960	2
C₂²×#3958	2
C₂²×#3944	2
C₂²×#509	5

Subgroups — C22×#65482 #13005791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65482 #13005791